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<article class="md-content__inner md-typeset">
<h1 id="_1">源分离与降噪<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>源分离与降噪从混合音频中恢复单个信号；即计算层面的"鸡尾酒会问题"。本文涵盖ICA、NMF、时频掩蔽、波束成形、深度学习分离网络（Conv-TasNet、SepFormer）、语音增强以及自适应降噪。</em></p>
<ul>
<li>
<p>想象一下你站在一个拥挤的鸡尾酒会上。数十人同时在交谈，音乐在播放，酒杯在碰撞，但你却能专注于一段对话并清晰地跟上它。这种非凡的能力被称为<strong>鸡尾酒会问题</strong>（Cherry, 1953），人类听觉系统可以毫不费力地做到，但机器却觉得异常困难。本文涵盖了尝试解决这一问题的算法：分离混合音频源、消除不必要的噪声以及在不利条件下增强语音。</p>
</li>
<li>
<p>文件01中的信号处理基础（STFT、语谱图、滤波器组）支撑了这里的每一种方法。第02章中的矩阵分解技术（NMF、ICA、SVD）提供了经典工具集。第06章中的深度学习架构（CNN、RNN、注意力机制）以及第04/05章中的概率论则为现代方法提供了理论基础。</p>
</li>
</ul>
<p><img alt="鸡尾酒会问题：多个说话人和声源混合到麦克风阵列中，分离系统必须从混合物中恢复单个源信号" src="../../images/cocktail_party.svg" /></p>
<ul>
<li><strong>问题形式化</strong>：在一个或多个麦克风处观测到混合信号 <span class="arithmatex">\(x(t)\)</span>。在最简单的情况下，混合信号是 <span class="arithmatex">\(C\)</span> 个源信号的和：</li>
</ul>
<div class="arithmatex">\[x(t) = \sum_{c=1}^{C} s_c(t) + n(t)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(s_c(t)\)</span> 是第 <span class="arithmatex">\(c\)</span> 个源信号，<span class="arithmatex">\(n(t)\)</span> 是背景噪声。目标是从 <span class="arithmatex">\(x(t)\)</span> 中恢复出各个 <span class="arithmatex">\(s_c(t)\)</span>。在单麦克风情况下，这是一个严重欠定的问题：一个方程，<span class="arithmatex">\(C\)</span> 个未知数。需要额外的假设（统计独立性、频谱结构、学习先验）才能使问题变得可解。</p>
</li>
<li>
<p>在频域中（通过文件01中的STFT），混合信号变为：</p>
</li>
</ul>
<div class="arithmatex">\[X(t, f) = \sum_{c=1}^{C} S_c(t, f) + N(t, f)\]</div>
<ul>
<li>许多分离方法在时频域中通过为每个源估计一个<strong>掩蔽</strong> <span class="arithmatex">\(M_c(t, f) \in [0, 1]\)</span> 来工作，然后通过 <span class="arithmatex">\(\hat{S}_c(t, f) = M_c(t, f) \cdot X(t, f)\)</span> 恢复源信号。<strong>理想二值掩蔽（IBM）</strong> 设置 <span class="arithmatex">\(M_c(t, f) = 1\)</span> 如果源 <span class="arithmatex">\(c\)</span> 在该时频单元中占主导，否则为0。<strong>理想比率掩蔽（IRM）</strong> 是其软版本：</li>
</ul>
<div class="arithmatex">\[\text{IRM}_c(t, f) = \frac{|S_c(t, f)|^2}{\sum_{j=1}^{C} |S_j(t, f)|^2}\]</div>
<ul>
<li>
<p><strong>独立成分分析（ICA）</strong> 是麦克风数量等于或超过源数量时的经典方法。ICA（第02章）寻找一个线性解混矩阵 <span class="arithmatex">\(W\)</span>，使得 <span class="arithmatex">\(\hat{s} = Wx\)</span>，其中恢复的源 <span class="arithmatex">\(\hat{s}\)</span> 在统计上最大限度地独立。关键假设是源信号是非高斯且独立的，这对于语音和音乐通常是成立的。</p>
</li>
<li>
<p>对于多麦克风瞬时混叠模型 <span class="arithmatex">\(x = As\)</span>（其中 <span class="arithmatex">\(A\)</span> 是混叠矩阵），ICA 通过最大化输出的非高斯性（FastICA 使用负熵）或最小化互信息来恢复 <span class="arithmatex">\(W \approx A^{-1}\)</span>。ICA 在受控环境中表现良好，但当混叠涉及卷积（房间混响）、源数量超过麦克风数量或独立性假设被违反时则会失败。</p>
</li>
<li>
<p><strong>非负矩阵分解（NMF）</strong> 将幅度语谱图 <span class="arithmatex">\(V \in \mathbb{R}_+^{F \times T}\)</span> 分解为两个非负矩阵的乘积（第02章）：</p>
</li>
</ul>
<div class="arithmatex">\[V \approx WH\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(W \in \mathbb{R}_+^{F \times K}\)</span> 是包含 <span class="arithmatex">\(K\)</span> 个频谱基向量的字典，<span class="arithmatex">\(H \in \mathbb{R}_+^{K \times T}\)</span> 包含随时间变化的激活系数。非负约束具有物理动机：幅度是非负的，且声音是加性组合的。</p>
</li>
<li>
<p>对于源分离，NMF 为每个源学习独立的字典：<span class="arithmatex">\(W_{\text{语音}}\)</span> 捕捉语音的频谱模式（共振峰结构），而 <span class="arithmatex">\(W_{\text{噪声}}\)</span> 捕捉噪声模式。混合信号被分解为 <span class="arithmatex">\(V \approx W_{\text{语音}} H_{\text{语音}} + W_{\text{噪声}} H_{\text{噪声}}\)</span>，每个源通过掩蔽来恢复。NMF 使用乘法更新规则进行最小化，代价函数可以是 Frobenius 范数或 KL 散度：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{aligned}
\text{Frobenius:} \quad D_F(V \| WH) &= \|V - WH\|_F^2 \\
\text{KL:} \quad D_{KL}(V \| WH) &= \sum_{f,t} \left[ V_{ft} \log \frac{V_{ft}}{(WH)_{ft}} - V_{ft} + (WH)_{ft} \right]
\end{aligned}
\]</div>
<ul>
<li><strong>波束成形</strong>利用麦克风阵列的空间信息。当一个源信号以不同的延迟到达不同的麦克风（由于空间排列）时，这些延迟可以用来增强来自某个方向的信号，同时抑制其他方向的信号。</li>
</ul>
<p><img alt="波束成形：麦克风阵列从不同方向接收具有不同时间延迟的信号，波束成形器将它们组合以增强目标方向并抑制其他方向" src="../../images/beamforming.svg" /></p>
<ul>
<li><strong>延迟求和波束成形</strong>是最简单的方法。如果目标源相对于阵列的角度为 <span class="arithmatex">\(\theta\)</span>，则在麦克风 <span class="arithmatex">\(m\)</span> 处的时间延迟为 <span class="arithmatex">\(\tau_m(\theta) = d_m \sin \theta / c\)</span>，其中 <span class="arithmatex">\(d_m\)</span> 是麦克风位置，<span class="arithmatex">\(c\)</span> 是声速。波束成形器输出将麦克风信号对齐并求和：</li>
</ul>
<div class="arithmatex">\[y(t) = \frac{1}{M} \sum_{m=1}^{M} x_m(t - \tau_m(\theta))\]</div>
<ul>
<li>
<p>来自目标方向的信号相干相加，而来自其他方向的信号非相干相加，从而实现空间滤波。阵列的几何形状决定了空间分辨率：更大的阵列产生更窄的波束。</p>
</li>
<li>
<p><strong>最小方差无失真响应（MVDR）</strong> 波束成形优化权重，以最小化总输出功率，同时保证目标方向无失真地通过：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{aligned}
\min_{\mathbf{w}} \quad & \mathbf{w}^H \Phi_{nn} \mathbf{w} \\
\text{subject to} \quad & \mathbf{w}^H \mathbf{d}(\theta) = 1
\end{aligned}
\]</div>
<ul>
<li>其中 <span class="arithmatex">\(\Phi_{nn}\)</span> 是噪声空间协方差矩阵，<span class="arithmatex">\(\mathbf{d}(\theta)\)</span> 是方向 <span class="arithmatex">\(\theta\)</span> 的导向向量。闭式解为：</li>
</ul>
<div class="arithmatex">\[\mathbf{w}_{\text{MVDR}} = \frac{\Phi_{nn}^{-1} \mathbf{d}(\theta)}{\mathbf{d}(\theta)^H \Phi_{nn}^{-1} \mathbf{d}(\theta)}\]</div>
<ul>
<li>
<p>MVDR 通过使用估计的噪声协方差自适应地适应噪声环境，比延迟求和提供更好的干扰抑制能力。它广泛用于助听器、智能音箱和远程会议系统。</p>
</li>
<li>
<p><strong>深度学习用于源分离</strong>显著提升了性能，特别是在经典方法难以处理的单麦克风情况下。一般范式是：编码混合信号，通过神经网络估计掩蔽或源表示，然后解码以恢复各个源。</p>
</li>
<li>
<p><strong>深度聚类</strong>（Hershey 等，2016）将每个时频单元嵌入到一个高维空间中，使得属于同一源的单元彼此靠近，而来自不同源的单元则远离。一个双向 LSTM（第06章）将每个时频单元 <span class="arithmatex">\((t, f)\)</span> 映射为一个嵌入向量 <span class="arithmatex">\(v_{t,f} \in \mathbb{R}^D\)</span>。训练目标为：</p>
</li>
</ul>
<div class="arithmatex">\[\mathcal{L} = \|VV^T - YY^T\|_F^2\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(V\)</span> 是嵌入矩阵，<span class="arithmatex">\(Y\)</span> 是源分配的单热矩阵。乘积 <span class="arithmatex">\(VV^T\)</span> 是一个亲和矩阵（两个单元的嵌入有多相似），而 <span class="arithmatex">\(YY^T\)</span> 是理想的亲和度（若属于同一源则为1，否则为0）。推理时，对嵌入进行 K-means 聚类产生二值掩蔽。</p>
</li>
<li>
<p><strong>Conv-TasNet</strong>（Luo 和 Mesgarani，2019）完全在时域中操作，绕过了 STFT。它包含三个组件：</p>
</li>
</ul>
<p><img alt="Conv-TasNet 架构：编码器将混合波形转换为潜在表示，时域卷积网络分离器估计源掩蔽，解码器重建各个源波形" src="../../images/conv_tasnet.svg" /></p>
<ul>
<li>
<p><strong>编码器</strong>：一个一维卷积将混合波形的短片段映射为潜在表示。对于混合信号 <span class="arithmatex">\(x \in \mathbb{R}^T\)</span>，编码器输出为 <span class="arithmatex">\(w = \text{ReLU}(U \ast x) \in \mathbb{R}^{N \times L}\)</span>，其中 <span class="arithmatex">\(U\)</span> 是一个可学习的基（类似于 STFT 基但从数据中学习），<span class="arithmatex">\(N\)</span> 是基函数的数量，<span class="arithmatex">\(L\)</span> 是片段数。编码器核大小和步长（通常为2ms和1ms）决定了时间分辨率。</p>
</li>
<li>
<p><strong>分离器</strong>：一个<strong>时域卷积网络（TCN）</strong>处理编码后的混合信号并输出 <span class="arithmatex">\(C\)</span> 个掩蔽。TCN 堆叠了扩张一维深度可分离卷积（来自第08章的高效卷积），这些卷积以指数增长的扩张因子 <span class="arithmatex">\(1, 2, 4, \ldots, 2^{B-1}\)</span> 排列成块，重复 <span class="arithmatex">\(R\)</span> 次。这提供了非常大的感受野，同时保持计算高效。</p>
</li>
<li>
<p><strong>解码器</strong>：一个转置一维卷积（使用可学习基 <span class="arithmatex">\(V\)</span>）将每个掩蔽后的表示转换回时域：<span class="arithmatex">\(\hat{s}_c = V^T (M_c \odot w)\)</span>。</p>
</li>
<li>
<p>Conv-TasNet 显著优于基于语谱图的方法，因为学习到的编码器-解码器基可以捕捉 STFT 幅度所丢弃的信息（特别是相位）。</p>
</li>
<li>
<p><strong>双路径 RNN（DPRNN）</strong>（Luo 等，2020）解决了分离中的长序列建模问题。DPRNN 不是用单个 RNN 或 TCN 处理整个编码序列，而是将序列分割成重叠的块，并沿着两条路径应用 RNN：<strong>块内</strong>路径（对每个块内的局部模式建模）和<strong>块间</strong>路径（对跨块的全局模式建模）。这使 RNN 序列长度从 <span class="arithmatex">\(L\)</span> 降低到每个维度上的 <span class="arithmatex">\(\sqrt{L}\)</span>：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{aligned}
\text{块内:} \quad & h_{k,n}^{\text{块内}} = \text{BiLSTM}_{\text{块内}}(z_{k,n}) \\
\text{块间:} \quad & h_{k,n}^{\text{块间}} = \text{BiLSTM}_{\text{块间}}(h_{k,n}^{\text{块内}})
\end{aligned}
\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(k\)</span> 索引块，<span class="arithmatex">\(n\)</span> 索引块内的位置。块内 LSTM 对固定 <span class="arithmatex">\(k\)</span> 的各 <span class="arithmatex">\(n\)</span> 处理；块间 LSTM 对固定 <span class="arithmatex">\(n\)</span> 的各 <span class="arithmatex">\(k\)</span> 处理。</p>
</li>
<li>
<p><strong>SepFormer</strong>（Subakan 等，2021）用 Transformer（第07章）替换了双路径框架中的 RNN。块内 Transformer 通过自注意力捕捉局部依赖关系，块间 Transformer 捕捉全局依赖关系。多头注意力能够建模长程依赖关系而不会出现梯度消失问题（第06章），这使得 SepFormer 对于长录音特别有效。SepFormer 在 WSJ0-2mix 基准上达到了最先进的结果。</p>
</li>
<li>
<p><strong>置换不变训练（PIT）</strong> 解决了监督式源分离中的一个基本问题：标签分配歧义。如果网络有两个输出（对应两个说话人），哪个输出应该对应哪个说话人？没有自然的排序。PIT 计算所有可能分配的损失并取最小值：</p>
</li>
</ul>
<div class="arithmatex">\[\mathcal{L}_{\text{PIT}} = \min_{\pi \in \mathcal{P}} \sum_{c=1}^{C} \ell(\hat{s}_{\pi(c)}, s_c)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\mathcal{P}\)</span> 是 <span class="arithmatex">\(\{1, \ldots, C\}\)</span> 的所有排列集合，<span class="arithmatex">\(\ell\)</span> 是每个源的损失（通常是尺度不变信号失真比 SI-SDR）。对于 <span class="arithmatex">\(C = 2\)</span> 个源只有2种排列；对于 <span class="arithmatex">\(C = 3\)</span> 有6种。对于更大的 <span class="arithmatex">\(C\)</span>，可以使用匈牙利算法高效计算。</p>
</li>
<li>
<p><strong>尺度不变信号失真比（SI-SDR）</strong> 是源分离的标准评估指标：</p>
</li>
</ul>
<div class="arithmatex">\[
\begin{aligned}
s_{\text{target}} &= \frac{\langle \hat{s}, s \rangle}{\|s\|^2} s \\
e_{\text{noise}} &= \hat{s} - s_{\text{target}} \\
\text{SI-SDR} &= 10 \log_{10} \frac{\|s_{\text{target}}\|^2}{\|e_{\text{noise}}\|^2}
\end{aligned}
\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\hat{s}\)</span> 是估计的源，<span class="arithmatex">\(s\)</span> 是真实值。SI-SDR 对估计的总体尺度不变，这是期望的特性，因为绝对音量不如分离质量重要。较高的 SI-SDR（以 dB 为单位）更好。最先进的系统在 WSJ0-2mix 上实现了约 20-22 dB 的 SI-SDR 改进。</p>
</li>
<li>
<p><strong>音乐源分离</strong>将音乐录音分离成声部：人声、鼓、贝斯和其他乐器。这实现了卡拉OK（去除人声）、重新混音（调整乐器电平）和转录（一次分析一种乐器）等应用。</p>
</li>
<li>
<p><strong>Open-Unmix</strong>（Stoter 等，2019）是一个参考基线，使用三层双向 LSTM 在幅度 STFT 域中为每个源预测软掩蔽。它使用专用模型独立处理每个源。Open-Unmix 虽简单但有效，在 MUSDB18 上建立了可重复的基准。</p>
</li>
<li>
<p><strong>Demucs</strong>（Defossez 等，2019；2021年更新为 Hybrid Demucs）使用直接在波形上操作的 U-Net 架构（第08章）。编码器通过步长卷积压缩混合信号，解码器通过转置卷积和跳跃连接将其扩展回来，每个源有各自的解码器头。<strong>Hybrid Demucs</strong> 结合了时域和频域处理：编码器具有并行的时域和 STFT 分支，其特征在解码器之前融合。这同时捕捉了精细的时间细节和频谱结构。</p>
</li>
<li>
<p>Demucs 在 MUSDB18 上达到了最先进的分离质量，特别是人声分离方面。其 U-Net 架构让人联想到第08章中的图像分割架构，将分离问题视为一种"音频分割"形式。</p>
</li>
<li>
<p><strong>主动降噪（ANC）</strong> 通过生成一个与噪声相消干涉的反噪声信号来减少不需要的声音。想象一下降噪耳机：麦克风拾取环境噪声，ANC 系统生成一个反相版本，混合信号（噪声 + 反噪声）理想情况下抵消为静音。</p>
</li>
<li>
<p>物理原理很简单：如果噪声是 <span class="arithmatex">\(n(t)\)</span>，在空间同一点生成 <span class="arithmatex">\(-n(t)\)</span> 则产生静音：<span class="arithmatex">\(n(t) + (-n(t)) = 0\)</span>。挑战在于反噪声必须在时间、幅度和相位上精确对齐。即使很小的误差也会产生残留噪声或伪影。</p>
</li>
<li>
<p><strong>前馈式 ANC</strong> 使用一个参考麦克风，在噪声到达听者之前拾取噪声。系统有时间处理噪声并生成反噪声。参考信号通过一个自适应滤波器，其输出在误差麦克风（靠近听者）处从噪声中减去。这适用于可预测的宽带噪声（引擎嗡嗡声、风扇噪声）。</p>
</li>
<li>
<p><strong>反馈式 ANC</strong> 仅使用听者耳边的误差麦克风。系统从残余信号（听者实际听到的）中估计噪声并调整反噪声。反馈式 ANC 更简单（不需要参考麦克风），但带宽有限且可能变得不稳定。</p>
</li>
<li>
<p><strong>自适应滤波</strong>是 ANC 背后的数学引擎。滤波器系数必须不断适应变化的噪声环境。最常用的算法是<strong>最小均方（LMS）</strong>滤波器。</p>
</li>
</ul>
<p><img alt="LMS 自适应滤波器：参考信号通过 FIR 滤波器，输出从期望信号中减去产生误差，误差反馈更新滤波器系数" src="../../images/lms_adaptive_filter.svg" /></p>
<ul>
<li><strong>LMS 算法</strong>：一个具有系数 <span class="arithmatex">\(\mathbf{w} = [w_0, w_1, \ldots, w_{L-1}]^T\)</span> 的 FIR 滤波器处理参考信号 <span class="arithmatex">\(\mathbf{x}(n) = [x(n), x(n-1), \ldots, x(n-L+1)]^T\)</span>。输出为 <span class="arithmatex">\(y(n) = \mathbf{w}^T \mathbf{x}(n)\)</span>，误差为 <span class="arithmatex">\(e(n) = d(n) - y(n)\)</span>（其中 <span class="arithmatex">\(d(n)\)</span> 是期望/主信号），权重更新为：</li>
</ul>
<div class="arithmatex">\[\mathbf{w}(n+1) = \mathbf{w}(n) + \mu \, e(n) \, \mathbf{x}(n)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\mu\)</span> 是步长（学习率）。这是对均方误差 <span class="arithmatex">\(E[e^2(n)]\)</span> 的一个随机梯度下降步骤，使用瞬时梯度估计 <span class="arithmatex">\(-2 e(n) \mathbf{x}(n)\)</span> 代替真实梯度（第03章的梯度下降和第06章的 SGD）。</p>
</li>
<li>
<p>步长 <span class="arithmatex">\(\mu\)</span> 控制收敛速度与稳态误差之间的权衡。过大则滤波器振荡或发散；过小则自适应速度迟缓。稳定条件为 <span class="arithmatex">\(0 &lt; \mu &lt; 2 / (\lambda_{\max})\)</span>，其中 <span class="arithmatex">\(\lambda_{\max}\)</span> 是输入自相关矩阵 <span class="arithmatex">\(R = E[\mathbf{x}\mathbf{x}^T]\)</span> 的最大特征值。</p>
</li>
<li>
<p><strong>归一化 LMS（NLMS）</strong> 通过输入功率对步长进行归一化，使收敛与信号电平无关：</p>
</li>
</ul>
<div class="arithmatex">\[\mathbf{w}(n+1) = \mathbf{w}(n) + \frac{\mu}{\|\mathbf{x}(n)\|^2 + \epsilon} \, e(n) \, \mathbf{x}(n)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\epsilon\)</span> 是一个小的正则化常数，以防止除零。NLMS 比 LMS 更可靠地收敛，因为有效步长自适应地适应输入功率。</p>
</li>
<li>
<p><strong>递归最小二乘（RLS）</strong> 是一种收敛更快的替代方法，它最小化加权最小二乘代价 <span class="arithmatex">\(\sum_{k=1}^{n} \lambda^{n-k} e^2(k)\)</span>，其中 <span class="arithmatex">\(\lambda \in (0, 1]\)</span> 是遗忘因子。RLS 维护逆自相关矩阵的估计并递归更新，以每个样本 <span class="arithmatex">\(O(L^2)\)</span> 的计算成本（相对于 LMS 的 <span class="arithmatex">\(O(L)\)</span>）实现最优收敛。</p>
</li>
<li>
<p><strong>降噪与语音增强</strong>旨在提高嘈杂录音中的语音质量和可懂度。与源分离（分离不同的源）不同，语音增强专门针对语音加噪声的情况，从带噪观测中恢复干净的语音。</p>
</li>
<li>
<p><strong>谱减法</strong>是最简单的方法。在纯噪声帧（由文件03中的 VAD 检测）期间，估计噪声频谱 <span class="arithmatex">\(|\hat{N}(f)|^2\)</span>。然后将其从每个帧中减去：</p>
</li>
</ul>
<div class="arithmatex">\[|\hat{S}(f)|^2 = \max(|X(f)|^2 - \alpha |\hat{N}(f)|^2, \beta |X(f)|^2)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(\alpha\)</span> 是过减因子（通常为1-4，激进的减法去除更多噪声但引入更多伪影），<span class="arithmatex">\(\beta\)</span> 是频谱地板，防止出现负值并减少"音乐噪声"伪影（听起来像随机音符的孤立音调残留）。</p>
</li>
<li>
<p><strong>维纳滤波</strong>提供了干净语音频谱的最小均方误差估计：</p>
</li>
</ul>
<div class="arithmatex">\[\hat{S}(t, f) = \frac{|S(t,f)|^2}{|S(t,f)|^2 + |N(t,f)|^2} \cdot X(t, f) = G(t, f) \cdot X(t, f)\]</div>
<ul>
<li>
<p>维纳增益 <span class="arithmatex">\(G(t, f) = \text{SNR}(t, f) / (1 + \text{SNR}(t, f))\)</span> 的范围从0（纯噪声）到1（纯语音），作为一个软掩蔽。挑战在于估计语音和噪声的功率谱。<strong>先验 SNR</strong> <span class="arithmatex">\(\xi(t, f) = |S(t,f)|^2 / |N(t,f)|^2\)</span> 使用"决策导向"方法估计：当前帧估计与前一帧维纳滤波输出的平滑组合。</p>
</li>
<li>
<p><strong>神经语音增强</strong>使用深度学习来估计掩蔽（如维纳增益）或直接估计干净语谱图。架构从简单的前馈网络到 U-Net（第08章）、CRN（卷积递归网络）和 Transformer。</p>
</li>
<li>
<p><strong>DCCRN</strong>（深度复数卷积递归网络）在复数 STFT（幅度和相位）上操作，使用自然处理实部和虚部的复数值卷积。这避免了仅幅度方法所困扰的相位估计问题。</p>
</li>
<li>
<p><strong>FullSubNet</strong> 使用双路径架构，包含一个全频带模型（捕捉全局频谱模式）和一个子频带模型（捕捉局部谐波细节）。全频带模型处理整个频谱，而子频带模型处理以每个频率单元为中心的窄频带。它们的输出被组合用于最终的掩蔽估计。</p>
</li>
<li>
<p><strong>DNS（深度噪声抑制）挑战赛</strong>由微软每年举办，对语音增强系统进行基准测试。获胜者通常使用大规模训练，包含多种噪声类型、数据增强（以各种 SNR 添加噪声、混响、编解码器伪影）以及支持实时处理的架构。</p>
</li>
<li>
<p><strong>回声消除</strong>在双向通信中去除声学回声。当你在电话通话中时，远端说话人的声音通过你的扬声器播放，在房间内反弹，并被你的麦克风拾取，产生远端说话人听到的回声。<strong>声学回声消除（AEC）</strong> 对从扬声器到麦克风的声学路径进行建模并减去预测的回声。</p>
</li>
<li>
<p>声学路径被建模为一个自适应 FIR 滤波器（使用 LMS 或 NLMS），以远端信号为输入。滤波器对房间脉冲响应进行建模，包括直达路径、早期反射和晚期混响。房间脉冲响应可能长达数百毫秒，需要数千个抽头的滤波器。</p>
</li>
<li>
<p><strong>双讲检测</strong>对 AEC 至关重要：当近端和远端说话人同时说话时，自适应滤波器必须冻结（停止更新），以防止其抵消近端说话人的声音。双讲检测器将误差信号的能量与远端信号能量进行比较；无法用远端信号解释的误差能量突然增加表明存在近端语音。</p>
</li>
<li>
<p>远端信号 <span class="arithmatex">\(x(n)\)</span> 与麦克风信号 <span class="arithmatex">\(d(n)\)</span> 之间的<strong>归一化互相关</strong>提供了一个双讲指示符：</p>
</li>
</ul>
<div class="arithmatex">\[\xi(n) = \frac{|\sum_{k=0}^{L-1} x(n-k) d(n-k)|}{\sqrt{\sum_{k} x^2(n-k)} \sqrt{\sum_{k} d^2(n-k)}}\]</div>
<ul>
<li>
<p>在单讲期间（仅远端），<span class="arithmatex">\(\xi\)</span> 较高，因为 <span class="arithmatex">\(d\)</span> 主要是 <span class="arithmatex">\(x\)</span> 的回声。在双讲期间，<span class="arithmatex">\(\xi\)</span> 下降，因为近端语音与 <span class="arithmatex">\(x\)</span> 不相关。</p>
</li>
<li>
<p>现代 AEC 系统将自适应滤波与神经网络相结合：自适应滤波器提供初始回声估计，神经网络（类似于上述语音增强模型）清理残余回声并处理线性滤波器无法捕捉的非线性（扬声器失真）。</p>
</li>
<li>
<p><strong>分离与增强的评估指标</strong>：</p>
<ul>
<li><strong>SI-SDR</strong>（如上定义）：源分离的标准指标。</li>
<li><strong>SDR</strong>（信号失真比）：来自 BSS Eval，衡量包括伪影和干扰在内的整体分离质量。</li>
<li><strong>PESQ</strong>（语音质量感知评估）：ITU 标准，预测主观质量分数。范围：-0.5 至 4.5。</li>
<li><strong>STOI</strong>（短时客观可懂度）：预测语音可懂度。范围：0 至 1。</li>
<li><strong>DNSMOS</strong>：微软的深度噪声抑制 MOS 预测器，一个训练用于预测人类 MOS 分数的神经网络，无需干净的参考音频。</li>
</ul>
</li>
</ul>
<h2 id="colab-notebook">编程任务（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">&para;</a></h2>
<ul>
<li><strong>任务 1：用于源分离的独立成分分析。</strong> 实现 FastICA 来分离两个混合音频源，演示确定情况（源与麦克风数量相等）下的经典鸡尾酒会解决方案。</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.random</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jr</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="c1"># 生成两个源信号</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">sr</span> <span class="o">=</span> <span class="mi">8000</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">duration</span> <span class="o">=</span> <span class="mf">1.0</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">t</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">duration</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">sr</span> <span class="o">*</span> <span class="n">duration</span><span class="p">))</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="c1"># 源 1：正弦波（类似音调）</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">s1</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">440</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.3</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">880</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="c1"># 源 2：锯齿波（丰富的谐波）</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">s2</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">t</span> <span class="o">*</span> <span class="mi">200</span> <span class="o">%</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># 200 Hz 锯齿波</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="c1"># 归一化源信号</span>
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="n">s1</span> <span class="o">=</span> <span class="n">s1</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s1</span><span class="p">))</span>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="n">s2</span> <span class="o">=</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">s2</span><span class="p">))</span>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="n">sources</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span><span class="n">s1</span><span class="p">,</span> <span class="n">s2</span><span class="p">])</span> <span class="c1"># (2, T)</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="c1"># 混叠矩阵（算法未知）</span>
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">],</span>
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a> <span class="p">[</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">]])</span>
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="n">mixtures</span> <span class="o">=</span> <span class="n">A</span> <span class="o">@</span> <span class="n">sources</span> <span class="c1"># (2, T)</span>
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a><span class="c1"># FastICA 实现</span>
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="k">def</span><span class="w"> </span><span class="nf">whiten</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;数据中心化与白化。&quot;&quot;&quot;</span>
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a> <span class="n">X_centered</span> <span class="o">=</span> <span class="n">X</span> <span class="o">-</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a> <span class="n">cov</span> <span class="o">=</span> <span class="p">(</span><span class="n">X_centered</span> <span class="o">@</span> <span class="n">X_centered</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">/</span> <span class="n">X_centered</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a> <span class="n">eigvals</span><span class="p">,</span> <span class="n">eigvecs</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">cov</span><span class="p">)</span>
<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a> <span class="n">D_inv_sqrt</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mf">1.0</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">eigvals</span> <span class="o">+</span> <span class="mf">1e-8</span><span class="p">))</span>
<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a> <span class="n">whitening</span> <span class="o">=</span> <span class="n">D_inv_sqrt</span> <span class="o">@</span> <span class="n">eigvecs</span><span class="o">.</span><span class="n">T</span>
<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a> <span class="k">return</span> <span class="n">whitening</span> <span class="o">@</span> <span class="n">X_centered</span><span class="p">,</span> <span class="n">whitening</span>
<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>
<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a><span class="k">def</span><span class="w"> </span><span class="nf">fastica</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">):</span>
<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;使用 tanh 非线性的 FastICA（负熵近似）。&quot;&quot;&quot;</span>
<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a> <span class="n">X_white</span><span class="p">,</span> <span class="n">whitening</span> <span class="o">=</span> <span class="n">whiten</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a> <span class="n">n</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="n">X_white</span><span class="o">.</span><span class="n">shape</span>
<a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a>
<a id="__codelineno-0-42" name="__codelineno-0-42" href="#__codelineno-0-42"></a> <span class="n">key</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<a id="__codelineno-0-43" name="__codelineno-0-43" href="#__codelineno-0-43"></a> <span class="n">W</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="p">(</span><span class="n">n_components</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
<a id="__codelineno-0-44" name="__codelineno-0-44" href="#__codelineno-0-44"></a> <span class="c1"># 正交化 W</span>
<a id="__codelineno-0-45" name="__codelineno-0-45" href="#__codelineno-0-45"></a> <span class="n">U</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">Vt</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">W</span><span class="p">,</span> <span class="n">full_matrices</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<a id="__codelineno-0-46" name="__codelineno-0-46" href="#__codelineno-0-46"></a> <span class="n">W</span> <span class="o">=</span> <span class="n">U</span> <span class="o">@</span> <span class="n">Vt</span>
<a id="__codelineno-0-47" name="__codelineno-0-47" href="#__codelineno-0-47"></a>
<a id="__codelineno-0-48" name="__codelineno-0-48" href="#__codelineno-0-48"></a> <span class="k">for</span> <span class="n">iteration</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_iter</span><span class="p">):</span>
<a id="__codelineno-0-49" name="__codelineno-0-49" href="#__codelineno-0-49"></a> <span class="n">W_old</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<a id="__codelineno-0-50" name="__codelineno-0-50" href="#__codelineno-0-50"></a>
<a id="__codelineno-0-51" name="__codelineno-0-51" href="#__codelineno-0-51"></a> <span class="c1"># 对每个分量</span>
<a id="__codelineno-0-52" name="__codelineno-0-52" href="#__codelineno-0-52"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_components</span><span class="p">):</span>
<a id="__codelineno-0-53" name="__codelineno-0-53" href="#__codelineno-0-53"></a> <span class="n">w</span> <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<a id="__codelineno-0-54" name="__codelineno-0-54" href="#__codelineno-0-54"></a> <span class="c1"># w^T X_white: (T,)</span>
<a id="__codelineno-0-55" name="__codelineno-0-55" href="#__codelineno-0-55"></a> <span class="n">wx</span> <span class="o">=</span> <span class="n">w</span> <span class="o">@</span> <span class="n">X_white</span> <span class="c1"># (T,)</span>
<a id="__codelineno-0-56" name="__codelineno-0-56" href="#__codelineno-0-56"></a>
<a id="__codelineno-0-57" name="__codelineno-0-57" href="#__codelineno-0-57"></a> <span class="c1"># g(u) = tanh(u), g&#39;(u) = 1 - tanh^2(u)</span>
<a id="__codelineno-0-58" name="__codelineno-0-58" href="#__codelineno-0-58"></a> <span class="n">g_wx</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="n">wx</span><span class="p">)</span>
<a id="__codelineno-0-59" name="__codelineno-0-59" href="#__codelineno-0-59"></a> <span class="n">g_prime_wx</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">g_wx</span> <span class="o">**</span> <span class="mi">2</span>
<a id="__codelineno-0-60" name="__codelineno-0-60" href="#__codelineno-0-60"></a>
<a id="__codelineno-0-61" name="__codelineno-0-61" href="#__codelineno-0-61"></a> <span class="c1"># Newton 更新: w_new = E[X * g(w^T X)] - E[g&#39;(w^T X)] * w</span>
<a id="__codelineno-0-62" name="__codelineno-0-62" href="#__codelineno-0-62"></a> <span class="n">w_new</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_white</span> <span class="o">*</span> <span class="n">g_wx</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> \
<a id="__codelineno-0-63" name="__codelineno-0-63" href="#__codelineno-0-63"></a> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">g_prime_wx</span><span class="p">)</span> <span class="o">*</span> <span class="n">w</span>
<a id="__codelineno-0-64" name="__codelineno-0-64" href="#__codelineno-0-64"></a>
<a id="__codelineno-0-65" name="__codelineno-0-65" href="#__codelineno-0-65"></a> <span class="c1"># 与之前的分量去相关（消去法）</span>
<a id="__codelineno-0-66" name="__codelineno-0-66" href="#__codelineno-0-66"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
<a id="__codelineno-0-67" name="__codelineno-0-67" href="#__codelineno-0-67"></a> <span class="n">w_new</span> <span class="o">=</span> <span class="n">w_new</span> <span class="o">-</span> <span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">w_new</span><span class="p">,</span> <span class="n">W</span><span class="p">[</span><span class="n">j</span><span class="p">])</span> <span class="o">*</span> <span class="n">W</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<a id="__codelineno-0-68" name="__codelineno-0-68" href="#__codelineno-0-68"></a>
<a id="__codelineno-0-69" name="__codelineno-0-69" href="#__codelineno-0-69"></a> <span class="n">w_new</span> <span class="o">=</span> <span class="n">w_new</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">w_new</span><span class="p">)</span>
<a id="__codelineno-0-70" name="__codelineno-0-70" href="#__codelineno-0-70"></a> <span class="n">W</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">w_new</span><span class="p">)</span>
<a id="__codelineno-0-71" name="__codelineno-0-71" href="#__codelineno-0-71"></a>
<a id="__codelineno-0-72" name="__codelineno-0-72" href="#__codelineno-0-72"></a> <span class="c1"># 检查收敛</span>
<a id="__codelineno-0-73" name="__codelineno-0-73" href="#__codelineno-0-73"></a> <span class="n">convergence</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">W</span> <span class="o">@</span> <span class="n">W_old</span><span class="o">.</span><span class="n">T</span><span class="p">)))</span>
<a id="__codelineno-0-74" name="__codelineno-0-74" href="#__codelineno-0-74"></a> <span class="k">if</span> <span class="n">convergence</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">tol</span><span class="p">:</span>
<a id="__codelineno-0-75" name="__codelineno-0-75" href="#__codelineno-0-75"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;FastICA 在 </span><span class="si">{</span><span class="n">iteration</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="si">}</span><span class="s2"> 次迭代后收敛&quot;</span><span class="p">)</span>
<a id="__codelineno-0-76" name="__codelineno-0-76" href="#__codelineno-0-76"></a> <span class="k">break</span>
<a id="__codelineno-0-77" name="__codelineno-0-77" href="#__codelineno-0-77"></a>
<a id="__codelineno-0-78" name="__codelineno-0-78" href="#__codelineno-0-78"></a> <span class="c1"># 解混矩阵</span>
<a id="__codelineno-0-79" name="__codelineno-0-79" href="#__codelineno-0-79"></a> <span class="n">unmixing</span> <span class="o">=</span> <span class="n">W</span> <span class="o">@</span> <span class="n">whitening</span>
<a id="__codelineno-0-80" name="__codelineno-0-80" href="#__codelineno-0-80"></a> <span class="n">recovered</span> <span class="o">=</span> <span class="n">unmixing</span> <span class="o">@</span> <span class="n">X</span>
<a id="__codelineno-0-81" name="__codelineno-0-81" href="#__codelineno-0-81"></a> <span class="k">return</span> <span class="n">recovered</span><span class="p">,</span> <span class="n">unmixing</span>
<a id="__codelineno-0-82" name="__codelineno-0-82" href="#__codelineno-0-82"></a>
<a id="__codelineno-0-83" name="__codelineno-0-83" href="#__codelineno-0-83"></a><span class="n">recovered</span><span class="p">,</span> <span class="n">W_unmix</span> <span class="o">=</span> <span class="n">fastica</span><span class="p">(</span><span class="n">mixtures</span><span class="p">)</span>
<a id="__codelineno-0-84" name="__codelineno-0-84" href="#__codelineno-0-84"></a>
<a id="__codelineno-0-85" name="__codelineno-0-85" href="#__codelineno-0-85"></a><span class="c1"># 修复符号歧义（ICA 可能翻转符号）</span>
<a id="__codelineno-0-86" name="__codelineno-0-86" href="#__codelineno-0-86"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
<a id="__codelineno-0-87" name="__codelineno-0-87" href="#__codelineno-0-87"></a> <span class="k">if</span> <span class="n">jnp</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">recovered</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">sources</span><span class="p">[</span><span class="n">i</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&lt;</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">:</span>
<a id="__codelineno-0-88" name="__codelineno-0-88" href="#__codelineno-0-88"></a> <span class="n">recovered</span> <span class="o">=</span> <span class="n">recovered</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="o">-</span><span class="n">recovered</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<a id="__codelineno-0-89" name="__codelineno-0-89" href="#__codelineno-0-89"></a>
<a id="__codelineno-0-90" name="__codelineno-0-90" href="#__codelineno-0-90"></a><span class="c1"># 如果源被交换，修复排列</span>
<a id="__codelineno-0-91" name="__codelineno-0-91" href="#__codelineno-0-91"></a><span class="n">corr_00</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">recovered</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<a id="__codelineno-0-92" name="__codelineno-0-92" href="#__codelineno-0-92"></a><span class="n">corr_01</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">recovered</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sources</span><span class="p">[</span><span class="mi">1</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<a id="__codelineno-0-93" name="__codelineno-0-93" href="#__codelineno-0-93"></a><span class="k">if</span> <span class="n">corr_01</span> <span class="o">&gt;</span> <span class="n">corr_00</span><span class="p">:</span>
<a id="__codelineno-0-94" name="__codelineno-0-94" href="#__codelineno-0-94"></a> <span class="n">recovered</span> <span class="o">=</span> <span class="n">recovered</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-0-95" name="__codelineno-0-95" href="#__codelineno-0-95"></a>
<a id="__codelineno-0-96" name="__codelineno-0-96" href="#__codelineno-0-96"></a><span class="c1"># 归一化以便显示</span>
<a id="__codelineno-0-97" name="__codelineno-0-97" href="#__codelineno-0-97"></a><span class="n">recovered</span> <span class="o">=</span> <span class="n">recovered</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">recovered</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<a id="__codelineno-0-98" name="__codelineno-0-98" href="#__codelineno-0-98"></a>
<a id="__codelineno-0-99" name="__codelineno-0-99" href="#__codelineno-0-99"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">9</span><span class="p">))</span>
<a id="__codelineno-0-100" name="__codelineno-0-100" href="#__codelineno-0-100"></a>
<a id="__codelineno-0-101" name="__codelineno-0-101" href="#__codelineno-0-101"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">s1</span><span class="p">[:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#3498db&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
<a id="__codelineno-0-102" name="__codelineno-0-102" href="#__codelineno-0-102"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;源信号 1（原始）&#39;</span><span class="p">)</span>
<a id="__codelineno-0-103" name="__codelineno-0-103" href="#__codelineno-0-103"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;幅度&#39;</span><span class="p">)</span>
<a id="__codelineno-0-104" name="__codelineno-0-104" href="#__codelineno-0-104"></a>
<a id="__codelineno-0-105" name="__codelineno-0-105" href="#__codelineno-0-105"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">s2</span><span class="p">[:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#e74c3c&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
<a id="__codelineno-0-106" name="__codelineno-0-106" href="#__codelineno-0-106"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;源信号 2（原始）&#39;</span><span class="p">)</span>
<a id="__codelineno-0-107" name="__codelineno-0-107" href="#__codelineno-0-107"></a>
<a id="__codelineno-0-108" name="__codelineno-0-108" href="#__codelineno-0-108"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">mixtures</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#9b59b6&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
<a id="__codelineno-0-109" name="__codelineno-0-109" href="#__codelineno-0-109"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;混合信号 1（麦克风 1）&#39;</span><span class="p">)</span>
<a id="__codelineno-0-110" name="__codelineno-0-110" href="#__codelineno-0-110"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;幅度&#39;</span><span class="p">)</span>
<a id="__codelineno-0-111" name="__codelineno-0-111" href="#__codelineno-0-111"></a>
<a id="__codelineno-0-112" name="__codelineno-0-112" href="#__codelineno-0-112"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">mixtures</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#9b59b6&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
<a id="__codelineno-0-113" name="__codelineno-0-113" href="#__codelineno-0-113"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;混合信号 2（麦克风 2）&#39;</span><span class="p">)</span>
<a id="__codelineno-0-114" name="__codelineno-0-114" href="#__codelineno-0-114"></a>
<a id="__codelineno-0-115" name="__codelineno-0-115" href="#__codelineno-0-115"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">recovered</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#27ae60&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
<a id="__codelineno-0-116" name="__codelineno-0-116" href="#__codelineno-0-116"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;恢复的源信号 1（FastICA）&#39;</span><span class="p">)</span>
<a id="__codelineno-0-117" name="__codelineno-0-117" href="#__codelineno-0-117"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;幅度&#39;</span><span class="p">)</span>
<a id="__codelineno-0-118" name="__codelineno-0-118" href="#__codelineno-0-118"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间 (s)&#39;</span><span class="p">)</span>
<a id="__codelineno-0-119" name="__codelineno-0-119" href="#__codelineno-0-119"></a>
<a id="__codelineno-0-120" name="__codelineno-0-120" href="#__codelineno-0-120"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">recovered</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:</span><span class="mi">1000</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#f39c12&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
<a id="__codelineno-0-121" name="__codelineno-0-121" href="#__codelineno-0-121"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;恢复的源信号 2（FastICA）&#39;</span><span class="p">)</span>
<a id="__codelineno-0-122" name="__codelineno-0-122" href="#__codelineno-0-122"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间 (s)&#39;</span><span class="p">)</span>
<a id="__codelineno-0-123" name="__codelineno-0-123" href="#__codelineno-0-123"></a>
<a id="__codelineno-0-124" name="__codelineno-0-124" href="#__codelineno-0-124"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<a id="__codelineno-0-125" name="__codelineno-0-125" href="#__codelineno-0-125"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<a id="__codelineno-0-126" name="__codelineno-0-126" href="#__codelineno-0-126"></a>
<a id="__codelineno-0-127" name="__codelineno-0-127" href="#__codelineno-0-127"></a><span class="c1"># 报告与原始信号的相关性</span>
<a id="__codelineno-0-128" name="__codelineno-0-128" href="#__codelineno-0-128"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
<a id="__codelineno-0-129" name="__codelineno-0-129" href="#__codelineno-0-129"></a> <span class="n">corr</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">recovered</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">sources</span><span class="p">[</span><span class="n">i</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-0-130" name="__codelineno-0-130" href="#__codelineno-0-130"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;源 </span><span class="si">{</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s2"> 恢复相关性: </span><span class="si">{</span><span class="n">corr</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div>
<ul>
<li><strong>任务 2：基于 NMF 的语谱图源分离。</strong> 使用非负矩阵分解（第02章）将语谱图分离为两个分量，演示 NMF 如何为每个源学习频谱字典。</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.random</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jr</span>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="c1"># 生成两个具有不同频谱特征的信号</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">sr</span> <span class="o">=</span> <span class="mi">8000</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">duration</span> <span class="o">=</span> <span class="mf">1.0</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">t</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">duration</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">sr</span> <span class="o">*</span> <span class="n">duration</span><span class="p">))</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="c1"># 源 1：低频谐波（模拟贝斯）</span>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="n">src1</span> <span class="o">=</span> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">100</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span> <span class="o">+</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">200</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span> <span class="o">+</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a> <span class="mf">0.3</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">300</span> <span class="o">*</span> <span class="n">t</span><span class="p">))</span>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a>
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="c1"># 源 2：高频谐波（模拟长笛）</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="n">src2</span> <span class="o">=</span> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">800</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span> <span class="o">+</span>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a> <span class="mf">0.4</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">1600</span> <span class="o">*</span> <span class="n">t</span><span class="p">))</span>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a>
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="c1"># 时变幅度（源在不同时间激活）</span>
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="n">env1</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">t</span> <span class="o">&lt;</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">)</span>
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="n">env2</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">t</span> <span class="o">&gt;</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">)</span>
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="n">src1</span> <span class="o">=</span> <span class="n">src1</span> <span class="o">*</span> <span class="n">env1</span>
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="n">src2</span> <span class="o">=</span> <span class="n">src2</span> <span class="o">*</span> <span class="n">env2</span>
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a>
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="n">mixture</span> <span class="o">=</span> <span class="n">src1</span> <span class="o">+</span> <span class="n">src2</span>
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="c1"># 计算幅度语谱图（STFT）</span>
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">n_fft</span> <span class="o">=</span> <span class="mi">512</span>
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="n">hop</span> <span class="o">=</span> <span class="mi">128</span>
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="n">window</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">hanning</span><span class="p">(</span><span class="n">n_fft</span><span class="p">)</span>
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a>
<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="k">def</span><span class="w"> </span><span class="nf">compute_stft</span><span class="p">(</span><span class="n">signal</span><span class="p">,</span> <span class="n">n_fft</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">):</span>
<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a> <span class="n">n_frames</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">signal</span><span class="p">)</span> <span class="o">-</span> <span class="n">n_fft</span><span class="p">)</span> <span class="o">//</span> <span class="n">hop</span>
<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a> <span class="n">frames</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span>
<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a> <span class="n">signal</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="n">hop</span> <span class="p">:</span> <span class="n">i</span> <span class="o">*</span> <span class="n">hop</span> <span class="o">+</span> <span class="n">n_fft</span><span class="p">]</span> <span class="o">*</span> <span class="n">window</span>
<a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_frames</span><span class="p">)</span>
<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a> <span class="p">])</span>
<a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a> <span class="k">return</span> <span class="n">jnp</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">frames</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n_fft</span><span class="p">)</span>
<a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a>
<a id="__codelineno-1-41" name="__codelineno-1-41" href="#__codelineno-1-41"></a><span class="n">S_mix</span> <span class="o">=</span> <span class="n">compute_stft</span><span class="p">(</span><span class="n">mixture</span><span class="p">,</span> <span class="n">n_fft</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">)</span>
<a id="__codelineno-1-42" name="__codelineno-1-42" href="#__codelineno-1-42"></a><span class="n">V</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">S_mix</span><span class="p">)</span><span class="o">.</span><span class="n">T</span> <span class="c1"># (F, T) - 频率 x 时间</span>
<a id="__codelineno-1-43" name="__codelineno-1-43" href="#__codelineno-1-43"></a><span class="n">phase</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">angle</span><span class="p">(</span><span class="n">S_mix</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<a id="__codelineno-1-44" name="__codelineno-1-44" href="#__codelineno-1-44"></a>
<a id="__codelineno-1-45" name="__codelineno-1-45" href="#__codelineno-1-45"></a><span class="n">F</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">shape</span>
<a id="__codelineno-1-46" name="__codelineno-1-46" href="#__codelineno-1-46"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;语谱图形状: </span><span class="si">{</span><span class="n">F</span><span class="si">}</span><span class="s2"> 个频率 bin x </span><span class="si">{</span><span class="n">T</span><span class="si">}</span><span class="s2"> 个时间帧&quot;</span><span class="p">)</span>
<a id="__codelineno-1-47" name="__codelineno-1-47" href="#__codelineno-1-47"></a>
<a id="__codelineno-1-48" name="__codelineno-1-48" href="#__codelineno-1-48"></a><span class="c1"># NMF: V ≈ WH 使用乘法更新规则</span>
<a id="__codelineno-1-49" name="__codelineno-1-49" href="#__codelineno-1-49"></a><span class="k">def</span><span class="w"> </span><span class="nf">nmf</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">n_iter</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">jr</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">0</span><span class="p">)):</span>
<a id="__codelineno-1-50" name="__codelineno-1-50" href="#__codelineno-1-50"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;使用 Frobenius 范数的非负矩阵分解。&quot;&quot;&quot;</span>
<a id="__codelineno-1-51" name="__codelineno-1-51" href="#__codelineno-1-51"></a> <span class="n">k1</span><span class="p">,</span> <span class="n">k2</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<a id="__codelineno-1-52" name="__codelineno-1-52" href="#__codelineno-1-52"></a> <span class="n">W</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">jr</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">K</span><span class="p">)))</span> <span class="o">*</span> <span class="mf">0.1</span> <span class="o">+</span> <span class="mf">0.01</span> <span class="c1"># (F, K)</span>
<a id="__codelineno-1-53" name="__codelineno-1-53" href="#__codelineno-1-53"></a> <span class="n">H</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">jr</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">k2</span><span class="p">,</span> <span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">T</span><span class="p">)))</span> <span class="o">*</span> <span class="mf">0.1</span> <span class="o">+</span> <span class="mf">0.01</span> <span class="c1"># (K, T)</span>
<a id="__codelineno-1-54" name="__codelineno-1-54" href="#__codelineno-1-54"></a>
<a id="__codelineno-1-55" name="__codelineno-1-55" href="#__codelineno-1-55"></a> <span class="n">costs</span> <span class="o">=</span> <span class="p">[]</span>
<a id="__codelineno-1-56" name="__codelineno-1-56" href="#__codelineno-1-56"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_iter</span><span class="p">):</span>
<a id="__codelineno-1-57" name="__codelineno-1-57" href="#__codelineno-1-57"></a> <span class="c1"># H 的乘法更新</span>
<a id="__codelineno-1-58" name="__codelineno-1-58" href="#__codelineno-1-58"></a> <span class="n">WtV</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">V</span>
<a id="__codelineno-1-59" name="__codelineno-1-59" href="#__codelineno-1-59"></a> <span class="n">WtWH</span> <span class="o">=</span> <span class="n">W</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">W</span> <span class="o">@</span> <span class="n">H</span> <span class="o">+</span> <span class="mf">1e-8</span>
<a id="__codelineno-1-60" name="__codelineno-1-60" href="#__codelineno-1-60"></a> <span class="n">H</span> <span class="o">=</span> <span class="n">H</span> <span class="o">*</span> <span class="p">(</span><span class="n">WtV</span> <span class="o">/</span> <span class="n">WtWH</span><span class="p">)</span>
<a id="__codelineno-1-61" name="__codelineno-1-61" href="#__codelineno-1-61"></a>
<a id="__codelineno-1-62" name="__codelineno-1-62" href="#__codelineno-1-62"></a> <span class="c1"># W 的乘法更新</span>
<a id="__codelineno-1-63" name="__codelineno-1-63" href="#__codelineno-1-63"></a> <span class="n">VHt</span> <span class="o">=</span> <span class="n">V</span> <span class="o">@</span> <span class="n">H</span><span class="o">.</span><span class="n">T</span>
<a id="__codelineno-1-64" name="__codelineno-1-64" href="#__codelineno-1-64"></a> <span class="n">WHHt</span> <span class="o">=</span> <span class="n">W</span> <span class="o">@</span> <span class="n">H</span> <span class="o">@</span> <span class="n">H</span><span class="o">.</span><span class="n">T</span> <span class="o">+</span> <span class="mf">1e-8</span>
<a id="__codelineno-1-65" name="__codelineno-1-65" href="#__codelineno-1-65"></a> <span class="n">W</span> <span class="o">=</span> <span class="n">W</span> <span class="o">*</span> <span class="p">(</span><span class="n">VHt</span> <span class="o">/</span> <span class="n">WHHt</span><span class="p">)</span>
<a id="__codelineno-1-66" name="__codelineno-1-66" href="#__codelineno-1-66"></a>
<a id="__codelineno-1-67" name="__codelineno-1-67" href="#__codelineno-1-67"></a> <span class="n">cost</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">V</span> <span class="o">-</span> <span class="n">W</span> <span class="o">@</span> <span class="n">H</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-1-68" name="__codelineno-1-68" href="#__codelineno-1-68"></a> <span class="n">costs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">cost</span><span class="p">))</span>
<a id="__codelineno-1-69" name="__codelineno-1-69" href="#__codelineno-1-69"></a>
<a id="__codelineno-1-70" name="__codelineno-1-70" href="#__codelineno-1-70"></a> <span class="k">return</span> <span class="n">W</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">costs</span>
<a id="__codelineno-1-71" name="__codelineno-1-71" href="#__codelineno-1-71"></a>
<a id="__codelineno-1-72" name="__codelineno-1-72" href="#__codelineno-1-72"></a><span class="c1"># 运行 K=2 个分量的 NMF</span>
<a id="__codelineno-1-73" name="__codelineno-1-73" href="#__codelineno-1-73"></a><span class="n">K</span> <span class="o">=</span> <span class="mi">2</span>
<a id="__codelineno-1-74" name="__codelineno-1-74" href="#__codelineno-1-74"></a><span class="n">W</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">costs</span> <span class="o">=</span> <span class="n">nmf</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">n_iter</span><span class="o">=</span><span class="mi">300</span><span class="p">)</span>
<a id="__codelineno-1-75" name="__codelineno-1-75" href="#__codelineno-1-75"></a>
<a id="__codelineno-1-76" name="__codelineno-1-76" href="#__codelineno-1-76"></a><span class="c1"># 使用软掩蔽重建每个源</span>
<a id="__codelineno-1-77" name="__codelineno-1-77" href="#__codelineno-1-77"></a><span class="n">V_hat</span> <span class="o">=</span> <span class="n">W</span> <span class="o">@</span> <span class="n">H</span>
<a id="__codelineno-1-78" name="__codelineno-1-78" href="#__codelineno-1-78"></a><span class="n">mask1</span> <span class="o">=</span> <span class="p">(</span><span class="n">W</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">1</span><span class="p">]</span> <span class="o">@</span> <span class="n">H</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span> <span class="p">:])</span> <span class="o">/</span> <span class="p">(</span><span class="n">V_hat</span> <span class="o">+</span> <span class="mf">1e-8</span><span class="p">)</span>
<a id="__codelineno-1-79" name="__codelineno-1-79" href="#__codelineno-1-79"></a><span class="n">mask2</span> <span class="o">=</span> <span class="p">(</span><span class="n">W</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">]</span> <span class="o">@</span> <span class="n">H</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">,</span> <span class="p">:])</span> <span class="o">/</span> <span class="p">(</span><span class="n">V_hat</span> <span class="o">+</span> <span class="mf">1e-8</span><span class="p">)</span>
<a id="__codelineno-1-80" name="__codelineno-1-80" href="#__codelineno-1-80"></a>
<a id="__codelineno-1-81" name="__codelineno-1-81" href="#__codelineno-1-81"></a><span class="n">V_src1</span> <span class="o">=</span> <span class="n">mask1</span> <span class="o">*</span> <span class="n">V</span>
<a id="__codelineno-1-82" name="__codelineno-1-82" href="#__codelineno-1-82"></a><span class="n">V_src2</span> <span class="o">=</span> <span class="n">mask2</span> <span class="o">*</span> <span class="n">V</span>
<a id="__codelineno-1-83" name="__codelineno-1-83" href="#__codelineno-1-83"></a>
<a id="__codelineno-1-84" name="__codelineno-1-84" href="#__codelineno-1-84"></a><span class="c1"># 可视化</span>
<a id="__codelineno-1-85" name="__codelineno-1-85" href="#__codelineno-1-85"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<a id="__codelineno-1-86" name="__codelineno-1-86" href="#__codelineno-1-86"></a>
<a id="__codelineno-1-87" name="__codelineno-1-87" href="#__codelineno-1-87"></a><span class="c1"># 混合信号语谱图</span>
<a id="__codelineno-1-88" name="__codelineno-1-88" href="#__codelineno-1-88"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="n">V</span><span class="p">),</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span> <span class="n">origin</span><span class="o">=</span><span class="s1">&#39;lower&#39;</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;magma&#39;</span><span class="p">)</span>
<a id="__codelineno-1-89" name="__codelineno-1-89" href="#__codelineno-1-89"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;混合信号语谱图 |X|&#39;</span><span class="p">)</span>
<a id="__codelineno-1-90" name="__codelineno-1-90" href="#__codelineno-1-90"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;频率 bin&#39;</span><span class="p">)</span>
<a id="__codelineno-1-91" name="__codelineno-1-91" href="#__codelineno-1-91"></a>
<a id="__codelineno-1-92" name="__codelineno-1-92" href="#__codelineno-1-92"></a><span class="c1"># NMF 收敛</span>
<a id="__codelineno-1-93" name="__codelineno-1-93" href="#__codelineno-1-93"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">costs</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#3498db&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.5</span><span class="p">)</span>
<a id="__codelineno-1-94" name="__codelineno-1-94" href="#__codelineno-1-94"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;NMF 收敛曲线&#39;</span><span class="p">)</span>
<a id="__codelineno-1-95" name="__codelineno-1-95" href="#__codelineno-1-95"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;迭代次数&#39;</span><span class="p">)</span>
<a id="__codelineno-1-96" name="__codelineno-1-96" href="#__codelineno-1-96"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Frobenius 代价&#39;</span><span class="p">)</span>
<a id="__codelineno-1-97" name="__codelineno-1-97" href="#__codelineno-1-97"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_yscale</span><span class="p">(</span><span class="s1">&#39;log&#39;</span><span class="p">)</span>
<a id="__codelineno-1-98" name="__codelineno-1-98" href="#__codelineno-1-98"></a>
<a id="__codelineno-1-99" name="__codelineno-1-99" href="#__codelineno-1-99"></a><span class="c1"># 频谱基向量 W</span>
<a id="__codelineno-1-100" name="__codelineno-1-100" href="#__codelineno-1-100"></a><span class="n">freq_hz</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">F</span><span class="p">)</span> <span class="o">*</span> <span class="n">sr</span> <span class="o">/</span> <span class="n">n_fft</span>
<a id="__codelineno-1-101" name="__codelineno-1-101" href="#__codelineno-1-101"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">freq_hz</span><span class="p">,</span> <span class="n">W</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#27ae60&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span>
<a id="__codelineno-1-102" name="__codelineno-1-102" href="#__codelineno-1-102"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;基 1（低频）&#39;</span><span class="p">)</span>
<a id="__codelineno-1-103" name="__codelineno-1-103" href="#__codelineno-1-103"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">freq_hz</span><span class="p">,</span> <span class="n">W</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#e74c3c&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span>
<a id="__codelineno-1-104" name="__codelineno-1-104" href="#__codelineno-1-104"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;基 2（高频）&#39;</span><span class="p">)</span>
<a id="__codelineno-1-105" name="__codelineno-1-105" href="#__codelineno-1-105"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;学习到的频谱基 W&#39;</span><span class="p">)</span>
<a id="__codelineno-1-106" name="__codelineno-1-106" href="#__codelineno-1-106"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;频率 (Hz)&#39;</span><span class="p">)</span>
<a id="__codelineno-1-107" name="__codelineno-1-107" href="#__codelineno-1-107"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;幅度&#39;</span><span class="p">)</span>
<a id="__codelineno-1-108" name="__codelineno-1-108" href="#__codelineno-1-108"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<a id="__codelineno-1-109" name="__codelineno-1-109" href="#__codelineno-1-109"></a>
<a id="__codelineno-1-110" name="__codelineno-1-110" href="#__codelineno-1-110"></a><span class="c1"># 时域激活 H</span>
<a id="__codelineno-1-111" name="__codelineno-1-111" href="#__codelineno-1-111"></a><span class="n">time_s</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">T</span><span class="p">)</span> <span class="o">*</span> <span class="n">hop</span> <span class="o">/</span> <span class="n">sr</span>
<a id="__codelineno-1-112" name="__codelineno-1-112" href="#__codelineno-1-112"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">time_s</span><span class="p">,</span> <span class="n">H</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#27ae60&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span>
<a id="__codelineno-1-113" name="__codelineno-1-113" href="#__codelineno-1-113"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;激活 1&#39;</span><span class="p">)</span>
<a id="__codelineno-1-114" name="__codelineno-1-114" href="#__codelineno-1-114"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">time_s</span><span class="p">,</span> <span class="n">H</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#e74c3c&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.5</span><span class="p">,</span>
<a id="__codelineno-1-115" name="__codelineno-1-115" href="#__codelineno-1-115"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;激活 2&#39;</span><span class="p">)</span>
<a id="__codelineno-1-116" name="__codelineno-1-116" href="#__codelineno-1-116"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;时域激活 H&#39;</span><span class="p">)</span>
<a id="__codelineno-1-117" name="__codelineno-1-117" href="#__codelineno-1-117"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间 (s)&#39;</span><span class="p">)</span>
<a id="__codelineno-1-118" name="__codelineno-1-118" href="#__codelineno-1-118"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;激活值&#39;</span><span class="p">)</span>
<a id="__codelineno-1-119" name="__codelineno-1-119" href="#__codelineno-1-119"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<a id="__codelineno-1-120" name="__codelineno-1-120" href="#__codelineno-1-120"></a>
<a id="__codelineno-1-121" name="__codelineno-1-121" href="#__codelineno-1-121"></a><span class="c1"># 分离后的语谱图</span>
<a id="__codelineno-1-122" name="__codelineno-1-122" href="#__codelineno-1-122"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="n">V_src1</span><span class="p">),</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span> <span class="n">origin</span><span class="o">=</span><span class="s1">&#39;lower&#39;</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;magma&#39;</span><span class="p">)</span>
<a id="__codelineno-1-123" name="__codelineno-1-123" href="#__codelineno-1-123"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;分离后的源信号 1（低频）&#39;</span><span class="p">)</span>
<a id="__codelineno-1-124" name="__codelineno-1-124" href="#__codelineno-1-124"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;频率 bin&#39;</span><span class="p">)</span>
<a id="__codelineno-1-125" name="__codelineno-1-125" href="#__codelineno-1-125"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间帧&#39;</span><span class="p">)</span>
<a id="__codelineno-1-126" name="__codelineno-1-126" href="#__codelineno-1-126"></a>
<a id="__codelineno-1-127" name="__codelineno-1-127" href="#__codelineno-1-127"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="n">V_src2</span><span class="p">),</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span> <span class="n">origin</span><span class="o">=</span><span class="s1">&#39;lower&#39;</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;magma&#39;</span><span class="p">)</span>
<a id="__codelineno-1-128" name="__codelineno-1-128" href="#__codelineno-1-128"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;分离后的源信号 2（高频）&#39;</span><span class="p">)</span>
<a id="__codelineno-1-129" name="__codelineno-1-129" href="#__codelineno-1-129"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间帧&#39;</span><span class="p">)</span>
<a id="__codelineno-1-130" name="__codelineno-1-130" href="#__codelineno-1-130"></a>
<a id="__codelineno-1-131" name="__codelineno-1-131" href="#__codelineno-1-131"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<a id="__codelineno-1-132" name="__codelineno-1-132" href="#__codelineno-1-132"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<a id="__codelineno-1-133" name="__codelineno-1-133" href="#__codelineno-1-133"></a>
<a id="__codelineno-1-134" name="__codelineno-1-134" href="#__codelineno-1-134"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;重建误差: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">V</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">W</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">H</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-135" name="__codelineno-1-135" href="#__codelineno-1-135"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;NMF 学习到的频谱基能够捕捉每个源的频率特征。&quot;</span><span class="p">)</span>
</code></pre></div>
<ul>
<li><strong>任务 3：用于降噪的 LMS 自适应滤波器。</strong> 实现 LMS 和 NLMS 算法用于回声/降噪，展示收敛行为及步长的影响。</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.random</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jr</span>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="c1"># 模拟回声消除场景</span>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="c1"># 远端信号 -&gt; 房间脉冲响应 -&gt; 麦克风处的回声</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="c1"># 近端语音是我们希望保留的目标信号</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">sr</span> <span class="o">=</span> <span class="mi">8000</span>
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="n">duration</span> <span class="o">=</span> <span class="mf">2.0</span>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="n">n_samples</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">sr</span> <span class="o">*</span> <span class="n">duration</span><span class="p">)</span>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="n">keys</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="c1"># 远端信号（参考）：随机的类语音信号</span>
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="n">far_end</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">keys</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">(</span><span class="n">n_samples</span><span class="p">,))</span> <span class="o">*</span> <span class="mf">0.5</span>
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="c1"># 房间脉冲响应（算法未知）</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="n">rir_length</span> <span class="o">=</span> <span class="mi">64</span>
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="n">rir</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">rir_length</span><span class="p">)</span>
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="n">rir</span> <span class="o">=</span> <span class="n">rir</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mf">0.8</span><span class="p">)</span> <span class="c1"># 直达路径</span>
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="n">rir</span> <span class="o">=</span> <span class="n">rir</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mf">0.3</span><span class="p">)</span> <span class="c1"># 早期反射</span>
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="n">rir</span> <span class="o">=</span> <span class="n">rir</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">12</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="o">-</span><span class="mf">0.2</span><span class="p">)</span> <span class="c1"># 反射</span>
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="n">rir</span> <span class="o">=</span> <span class="n">rir</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">25</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span> <span class="c1"># 晚期反射</span>
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="n">rir</span> <span class="o">=</span> <span class="n">rir</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">40</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="o">-</span><span class="mf">0.05</span><span class="p">)</span>
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a>
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="c1"># 回声：远端信号与 RIR 的卷积</span>
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="n">echo</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">convolve</span><span class="p">(</span><span class="n">far_end</span><span class="p">,</span> <span class="n">rir</span><span class="p">)[:</span><span class="n">n_samples</span><span class="p">]</span>
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a>
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="c1"># 近端语音（在信号的一部分中活跃）</span>
<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="n">near_end</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span>
<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">=</span> <span class="n">n_samples</span> <span class="o">//</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">n_samples</span> <span class="o">//</span> <span class="mi">3</span>
<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="n">near_speech</span> <span class="o">=</span> <span class="mf">0.3</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span>
<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a> <span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mi">300</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">end</span> <span class="o">-</span> <span class="n">start</span><span class="p">)</span> <span class="o">/</span> <span class="n">sr</span><span class="p">,</span> <span class="n">end</span> <span class="o">-</span> <span class="n">start</span><span class="p">)</span>
<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="p">)</span>
<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="n">near_end</span> <span class="o">=</span> <span class="n">near_end</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">near_speech</span><span class="p">)</span>
<a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a>
<a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a><span class="c1"># 麦克风信号：回声 + 近端 + 噪声</span>
<a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a><span class="n">noise</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">keys</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">(</span><span class="n">n_samples</span><span class="p">,))</span> <span class="o">*</span> <span class="mf">0.01</span>
<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="n">mic_signal</span> <span class="o">=</span> <span class="n">echo</span> <span class="o">+</span> <span class="n">near_end</span> <span class="o">+</span> <span class="n">noise</span>
<a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a>
<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="c1"># LMS 自适应滤波器</span>
<a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a><span class="k">def</span><span class="w"> </span><span class="nf">lms_filter</span><span class="p">(</span><span class="n">reference</span><span class="p">,</span> <span class="n">desired</span><span class="p">,</span> <span class="n">filter_length</span><span class="p">,</span> <span class="n">mu</span><span class="p">):</span>
<a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;标准 LMS 自适应滤波器。&quot;&quot;&quot;</span>
<a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a> <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">reference</span><span class="p">)</span>
<a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a> <span class="n">w</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">filter_length</span><span class="p">)</span>
<a id="__codelineno-2-48" name="__codelineno-2-48" href="#__codelineno-2-48"></a> <span class="n">output</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-2-49" name="__codelineno-2-49" href="#__codelineno-2-49"></a> <span class="n">error</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-2-50" name="__codelineno-2-50" href="#__codelineno-2-50"></a> <span class="n">w_history</span> <span class="o">=</span> <span class="p">[]</span>
<a id="__codelineno-2-51" name="__codelineno-2-51" href="#__codelineno-2-51"></a>
<a id="__codelineno-2-52" name="__codelineno-2-52" href="#__codelineno-2-52"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">filter_length</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<a id="__codelineno-2-53" name="__codelineno-2-53" href="#__codelineno-2-53"></a> <span class="n">x</span> <span class="o">=</span> <span class="n">reference</span><span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="o">-</span><span class="n">filter_length</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-2-54" name="__codelineno-2-54" href="#__codelineno-2-54"></a>
<a id="__codelineno-2-55" name="__codelineno-2-55" href="#__codelineno-2-55"></a> <span class="n">y</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<a id="__codelineno-2-56" name="__codelineno-2-56" href="#__codelineno-2-56"></a> <span class="n">e</span> <span class="o">=</span> <span class="n">desired</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">y</span>
<a id="__codelineno-2-57" name="__codelineno-2-57" href="#__codelineno-2-57"></a> <span class="n">w</span> <span class="o">=</span> <span class="n">w</span> <span class="o">+</span> <span class="n">mu</span> <span class="o">*</span> <span class="n">e</span> <span class="o">*</span> <span class="n">x</span>
<a id="__codelineno-2-58" name="__codelineno-2-58" href="#__codelineno-2-58"></a>
<a id="__codelineno-2-59" name="__codelineno-2-59" href="#__codelineno-2-59"></a> <span class="n">output</span> <span class="o">=</span> <span class="n">output</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<a id="__codelineno-2-60" name="__codelineno-2-60" href="#__codelineno-2-60"></a> <span class="n">error</span> <span class="o">=</span> <span class="n">error</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
<a id="__codelineno-2-61" name="__codelineno-2-61" href="#__codelineno-2-61"></a>
<a id="__codelineno-2-62" name="__codelineno-2-62" href="#__codelineno-2-62"></a> <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">500</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<a id="__codelineno-2-63" name="__codelineno-2-63" href="#__codelineno-2-63"></a> <span class="n">w_history</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">w</span><span class="o">.</span><span class="n">copy</span><span class="p">())</span>
<a id="__codelineno-2-64" name="__codelineno-2-64" href="#__codelineno-2-64"></a>
<a id="__codelineno-2-65" name="__codelineno-2-65" href="#__codelineno-2-65"></a> <span class="k">return</span> <span class="n">output</span><span class="p">,</span> <span class="n">error</span><span class="p">,</span> <span class="n">w_history</span>
<a id="__codelineno-2-66" name="__codelineno-2-66" href="#__codelineno-2-66"></a>
<a id="__codelineno-2-67" name="__codelineno-2-67" href="#__codelineno-2-67"></a><span class="c1"># NLMS 自适应滤波器</span>
<a id="__codelineno-2-68" name="__codelineno-2-68" href="#__codelineno-2-68"></a><span class="k">def</span><span class="w"> </span><span class="nf">nlms_filter</span><span class="p">(</span><span class="n">reference</span><span class="p">,</span> <span class="n">desired</span><span class="p">,</span> <span class="n">filter_length</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">):</span>
<a id="__codelineno-2-69" name="__codelineno-2-69" href="#__codelineno-2-69"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;归一化 LMS 自适应滤波器。&quot;&quot;&quot;</span>
<a id="__codelineno-2-70" name="__codelineno-2-70" href="#__codelineno-2-70"></a> <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">reference</span><span class="p">)</span>
<a id="__codelineno-2-71" name="__codelineno-2-71" href="#__codelineno-2-71"></a> <span class="n">w</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">filter_length</span><span class="p">)</span>
<a id="__codelineno-2-72" name="__codelineno-2-72" href="#__codelineno-2-72"></a> <span class="n">output</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-2-73" name="__codelineno-2-73" href="#__codelineno-2-73"></a> <span class="n">error</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<a id="__codelineno-2-74" name="__codelineno-2-74" href="#__codelineno-2-74"></a>
<a id="__codelineno-2-75" name="__codelineno-2-75" href="#__codelineno-2-75"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">filter_length</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<a id="__codelineno-2-76" name="__codelineno-2-76" href="#__codelineno-2-76"></a> <span class="n">x</span> <span class="o">=</span> <span class="n">reference</span><span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="o">-</span><span class="n">filter_length</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<a id="__codelineno-2-77" name="__codelineno-2-77" href="#__codelineno-2-77"></a>
<a id="__codelineno-2-78" name="__codelineno-2-78" href="#__codelineno-2-78"></a> <span class="n">y</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<a id="__codelineno-2-79" name="__codelineno-2-79" href="#__codelineno-2-79"></a> <span class="n">e</span> <span class="o">=</span> <span class="n">desired</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">y</span>
<a id="__codelineno-2-80" name="__codelineno-2-80" href="#__codelineno-2-80"></a> <span class="n">norm_factor</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">eps</span>
<a id="__codelineno-2-81" name="__codelineno-2-81" href="#__codelineno-2-81"></a> <span class="n">w</span> <span class="o">=</span> <span class="n">w</span> <span class="o">+</span> <span class="p">(</span><span class="n">mu</span> <span class="o">/</span> <span class="n">norm_factor</span><span class="p">)</span> <span class="o">*</span> <span class="n">e</span> <span class="o">*</span> <span class="n">x</span>
<a id="__codelineno-2-82" name="__codelineno-2-82" href="#__codelineno-2-82"></a>
<a id="__codelineno-2-83" name="__codelineno-2-83" href="#__codelineno-2-83"></a> <span class="n">output</span> <span class="o">=</span> <span class="n">output</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<a id="__codelineno-2-84" name="__codelineno-2-84" href="#__codelineno-2-84"></a> <span class="n">error</span> <span class="o">=</span> <span class="n">error</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
<a id="__codelineno-2-85" name="__codelineno-2-85" href="#__codelineno-2-85"></a>
<a id="__codelineno-2-86" name="__codelineno-2-86" href="#__codelineno-2-86"></a> <span class="k">return</span> <span class="n">output</span><span class="p">,</span> <span class="n">error</span>
<a id="__codelineno-2-87" name="__codelineno-2-87" href="#__codelineno-2-87"></a>
<a id="__codelineno-2-88" name="__codelineno-2-88" href="#__codelineno-2-88"></a><span class="c1"># 使用不同步长运行 LMS</span>
<a id="__codelineno-2-89" name="__codelineno-2-89" href="#__codelineno-2-89"></a><span class="n">filter_len</span> <span class="o">=</span> <span class="mi">64</span>
<a id="__codelineno-2-90" name="__codelineno-2-90" href="#__codelineno-2-90"></a><span class="n">mu_values</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.001</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">]</span>
<a id="__codelineno-2-91" name="__codelineno-2-91" href="#__codelineno-2-91"></a><span class="n">colors_mu</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;#3498db&#39;</span><span class="p">,</span> <span class="s1">&#39;#e74c3c&#39;</span><span class="p">,</span> <span class="s1">&#39;#27ae60&#39;</span><span class="p">]</span>
<a id="__codelineno-2-92" name="__codelineno-2-92" href="#__codelineno-2-92"></a>
<a id="__codelineno-2-93" name="__codelineno-2-93" href="#__codelineno-2-93"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<a id="__codelineno-2-94" name="__codelineno-2-94" href="#__codelineno-2-94"></a>
<a id="__codelineno-2-95" name="__codelineno-2-95" href="#__codelineno-2-95"></a><span class="c1"># 原始信号</span>
<a id="__codelineno-2-96" name="__codelineno-2-96" href="#__codelineno-2-96"></a><span class="n">t</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="o">/</span> <span class="n">sr</span>
<a id="__codelineno-2-97" name="__codelineno-2-97" href="#__codelineno-2-97"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">mic_signal</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#9b59b6&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span>
<a id="__codelineno-2-98" name="__codelineno-2-98" href="#__codelineno-2-98"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;麦克风（回声 + 近端）&#39;</span><span class="p">)</span>
<a id="__codelineno-2-99" name="__codelineno-2-99" href="#__codelineno-2-99"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">echo</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#e74c3c&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span>
<a id="__codelineno-2-100" name="__codelineno-2-100" href="#__codelineno-2-100"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;回声（待消除）&#39;</span><span class="p">)</span>
<a id="__codelineno-2-101" name="__codelineno-2-101" href="#__codelineno-2-101"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">near_end</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#27ae60&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span>
<a id="__codelineno-2-102" name="__codelineno-2-102" href="#__codelineno-2-102"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;近端语音（需保留）&#39;</span><span class="p">)</span>
<a id="__codelineno-2-103" name="__codelineno-2-103" href="#__codelineno-2-103"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;信号分量&#39;</span><span class="p">)</span>
<a id="__codelineno-2-104" name="__codelineno-2-104" href="#__codelineno-2-104"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间 (s)&#39;</span><span class="p">)</span>
<a id="__codelineno-2-105" name="__codelineno-2-105" href="#__codelineno-2-105"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;幅度&#39;</span><span class="p">)</span>
<a id="__codelineno-2-106" name="__codelineno-2-106" href="#__codelineno-2-106"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">8</span><span class="p">)</span>
<a id="__codelineno-2-107" name="__codelineno-2-107" href="#__codelineno-2-107"></a>
<a id="__codelineno-2-108" name="__codelineno-2-108" href="#__codelineno-2-108"></a><span class="c1"># 不同步长下的 LMS 收敛</span>
<a id="__codelineno-2-109" name="__codelineno-2-109" href="#__codelineno-2-109"></a><span class="k">for</span> <span class="n">mu</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">mu_values</span><span class="p">,</span> <span class="n">colors_mu</span><span class="p">):</span>
<a id="__codelineno-2-110" name="__codelineno-2-110" href="#__codelineno-2-110"></a> <span class="n">_</span><span class="p">,</span> <span class="n">err</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">lms_filter</span><span class="p">(</span><span class="n">far_end</span><span class="p">,</span> <span class="n">mic_signal</span><span class="p">,</span> <span class="n">filter_len</span><span class="p">,</span> <span class="n">mu</span><span class="p">)</span>
<a id="__codelineno-2-111" name="__codelineno-2-111" href="#__codelineno-2-111"></a> <span class="c1"># 平滑后的平方误差</span>
<a id="__codelineno-2-112" name="__codelineno-2-112" href="#__codelineno-2-112"></a> <span class="n">sq_err</span> <span class="o">=</span> <span class="n">err</span> <span class="o">**</span> <span class="mi">2</span>
<a id="__codelineno-2-113" name="__codelineno-2-113" href="#__codelineno-2-113"></a> <span class="n">window_size</span> <span class="o">=</span> <span class="mi">200</span>
<a id="__codelineno-2-114" name="__codelineno-2-114" href="#__codelineno-2-114"></a> <span class="n">smoothed</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">convolve</span><span class="p">(</span><span class="n">sq_err</span><span class="p">,</span> <span class="n">jnp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">window_size</span><span class="p">)</span><span class="o">/</span><span class="n">window_size</span><span class="p">,</span>
<a id="__codelineno-2-115" name="__codelineno-2-115" href="#__codelineno-2-115"></a> <span class="n">mode</span><span class="o">=</span><span class="s1">&#39;valid&#39;</span><span class="p">)</span>
<a id="__codelineno-2-116" name="__codelineno-2-116" href="#__codelineno-2-116"></a> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">smoothed</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.2</span><span class="p">,</span>
<a id="__codelineno-2-117" name="__codelineno-2-117" href="#__codelineno-2-117"></a> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s1">&#39;mu=</span><span class="si">{</span><span class="n">mu</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
<a id="__codelineno-2-118" name="__codelineno-2-118" href="#__codelineno-2-118"></a>
<a id="__codelineno-2-119" name="__codelineno-2-119" href="#__codelineno-2-119"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;LMS 收敛曲线（平滑 MSE）&#39;</span><span class="p">)</span>
<a id="__codelineno-2-120" name="__codelineno-2-120" href="#__codelineno-2-120"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;样本&#39;</span><span class="p">)</span>
<a id="__codelineno-2-121" name="__codelineno-2-121" href="#__codelineno-2-121"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;平方误差&#39;</span><span class="p">)</span>
<a id="__codelineno-2-122" name="__codelineno-2-122" href="#__codelineno-2-122"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_yscale</span><span class="p">(</span><span class="s1">&#39;log&#39;</span><span class="p">)</span>
<a id="__codelineno-2-123" name="__codelineno-2-123" href="#__codelineno-2-123"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<a id="__codelineno-2-124" name="__codelineno-2-124" href="#__codelineno-2-124"></a>
<a id="__codelineno-2-125" name="__codelineno-2-125" href="#__codelineno-2-125"></a><span class="c1"># 最佳 LMS 结果</span>
<a id="__codelineno-2-126" name="__codelineno-2-126" href="#__codelineno-2-126"></a><span class="n">_</span><span class="p">,</span> <span class="n">err_lms</span><span class="p">,</span> <span class="n">w_hist</span> <span class="o">=</span> <span class="n">lms_filter</span><span class="p">(</span><span class="n">far_end</span><span class="p">,</span> <span class="n">mic_signal</span><span class="p">,</span> <span class="n">filter_len</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">)</span>
<a id="__codelineno-2-127" name="__codelineno-2-127" href="#__codelineno-2-127"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">mic_signal</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#9b59b6&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span>
<a id="__codelineno-2-128" name="__codelineno-2-128" href="#__codelineno-2-128"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;消除前&#39;</span><span class="p">)</span>
<a id="__codelineno-2-129" name="__codelineno-2-129" href="#__codelineno-2-129"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">err_lms</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#3498db&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span>
<a id="__codelineno-2-130" name="__codelineno-2-130" href="#__codelineno-2-130"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;LMS 消除后&#39;</span><span class="p">)</span>
<a id="__codelineno-2-131" name="__codelineno-2-131" href="#__codelineno-2-131"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">near_end</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#27ae60&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
<a id="__codelineno-2-132" name="__codelineno-2-132" href="#__codelineno-2-132"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;真实近端&#39;</span><span class="p">)</span>
<a id="__codelineno-2-133" name="__codelineno-2-133" href="#__codelineno-2-133"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;LMS 回声消除结果 (mu=0.01)&#39;</span><span class="p">)</span>
<a id="__codelineno-2-134" name="__codelineno-2-134" href="#__codelineno-2-134"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间 (s)&#39;</span><span class="p">)</span>
<a id="__codelineno-2-135" name="__codelineno-2-135" href="#__codelineno-2-135"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;幅度&#39;</span><span class="p">)</span>
<a id="__codelineno-2-136" name="__codelineno-2-136" href="#__codelineno-2-136"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">8</span><span class="p">)</span>
<a id="__codelineno-2-137" name="__codelineno-2-137" href="#__codelineno-2-137"></a>
<a id="__codelineno-2-138" name="__codelineno-2-138" href="#__codelineno-2-138"></a><span class="c1"># NLMS 结果</span>
<a id="__codelineno-2-139" name="__codelineno-2-139" href="#__codelineno-2-139"></a><span class="n">_</span><span class="p">,</span> <span class="n">err_nlms</span> <span class="o">=</span> <span class="n">nlms_filter</span><span class="p">(</span><span class="n">far_end</span><span class="p">,</span> <span class="n">mic_signal</span><span class="p">,</span> <span class="n">filter_len</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<a id="__codelineno-2-140" name="__codelineno-2-140" href="#__codelineno-2-140"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">mic_signal</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#9b59b6&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span>
<a id="__codelineno-2-141" name="__codelineno-2-141" href="#__codelineno-2-141"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;消除前&#39;</span><span class="p">)</span>
<a id="__codelineno-2-142" name="__codelineno-2-142" href="#__codelineno-2-142"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">err_nlms</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#f39c12&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span>
<a id="__codelineno-2-143" name="__codelineno-2-143" href="#__codelineno-2-143"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;NLMS 消除后&#39;</span><span class="p">)</span>
<a id="__codelineno-2-144" name="__codelineno-2-144" href="#__codelineno-2-144"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">near_end</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#27ae60&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
<a id="__codelineno-2-145" name="__codelineno-2-145" href="#__codelineno-2-145"></a> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;真实近端&#39;</span><span class="p">)</span>
<a id="__codelineno-2-146" name="__codelineno-2-146" href="#__codelineno-2-146"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;NLMS 回声消除结果 (mu=0.5)&#39;</span><span class="p">)</span>
<a id="__codelineno-2-147" name="__codelineno-2-147" href="#__codelineno-2-147"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间 (s)&#39;</span><span class="p">)</span>
<a id="__codelineno-2-148" name="__codelineno-2-148" href="#__codelineno-2-148"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;幅度&#39;</span><span class="p">)</span>
<a id="__codelineno-2-149" name="__codelineno-2-149" href="#__codelineno-2-149"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">8</span><span class="p">)</span>
<a id="__codelineno-2-150" name="__codelineno-2-150" href="#__codelineno-2-150"></a>
<a id="__codelineno-2-151" name="__codelineno-2-151" href="#__codelineno-2-151"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<a id="__codelineno-2-152" name="__codelineno-2-152" href="#__codelineno-2-152"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<a id="__codelineno-2-153" name="__codelineno-2-153" href="#__codelineno-2-153"></a>
<a id="__codelineno-2-154" name="__codelineno-2-154" href="#__codelineno-2-154"></a><span class="c1"># 测量回声衰减</span>
<a id="__codelineno-2-155" name="__codelineno-2-155" href="#__codelineno-2-155"></a><span class="n">echo_power</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">echo</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-2-156" name="__codelineno-2-156" href="#__codelineno-2-156"></a><span class="n">lms_residual</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">err_lms</span><span class="p">[</span><span class="n">n_samples</span><span class="o">//</span><span class="mi">2</span><span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="c1"># 收敛后</span>
<a id="__codelineno-2-157" name="__codelineno-2-157" href="#__codelineno-2-157"></a><span class="n">nlms_residual</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">err_nlms</span><span class="p">[</span><span class="n">n_samples</span><span class="o">//</span><span class="mi">2</span><span class="p">:]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-2-158" name="__codelineno-2-158" href="#__codelineno-2-158"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;回声功率: </span><span class="si">{</span><span class="mi">10</span><span class="o">*</span><span class="n">jnp</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">echo_power</span><span class="p">)</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2"> dB&quot;</span><span class="p">)</span>
<a id="__codelineno-2-159" name="__codelineno-2-159" href="#__codelineno-2-159"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;LMS 残差: </span><span class="si">{</span><span class="mi">10</span><span class="o">*</span><span class="n">jnp</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lms_residual</span><span class="p">)</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2"> dB &quot;</span>
<a id="__codelineno-2-160" name="__codelineno-2-160" href="#__codelineno-2-160"></a> <span class="sa">f</span><span class="s2">&quot;(ERLE: </span><span class="si">{</span><span class="mi">10</span><span class="o">*</span><span class="n">jnp</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">echo_power</span><span class="o">/</span><span class="n">lms_residual</span><span class="p">)</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2"> dB)&quot;</span><span class="p">)</span>
<a id="__codelineno-2-161" name="__codelineno-2-161" href="#__codelineno-2-161"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;NLMS 残差: </span><span class="si">{</span><span class="mi">10</span><span class="o">*</span><span class="n">jnp</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">nlms_residual</span><span class="p">)</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2"> dB &quot;</span>
<a id="__codelineno-2-162" name="__codelineno-2-162" href="#__codelineno-2-162"></a> <span class="sa">f</span><span class="s2">&quot;(ERLE: </span><span class="si">{</span><span class="mi">10</span><span class="o">*</span><span class="n">jnp</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">echo_power</span><span class="o">/</span><span class="n">nlms_residual</span><span class="p">)</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2"> dB)&quot;</span><span class="p">)</span>
</code></pre></div>
<ul>
<li><strong>任务 4：用于语音增强的时频掩蔽。</strong> 实现一个简单的频谱掩蔽方法（理想比率掩蔽），并将其与谱减法进行比较，在合成的带噪语音信号上可视化分离质量。</li>
</ul>
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.random</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jr</span>
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a>
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="c1"># 创建合成的&quot;语音&quot;和&quot;噪声&quot;信号</span>
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="n">sr</span> <span class="o">=</span> <span class="mi">8000</span>
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="n">duration</span> <span class="o">=</span> <span class="mf">2.0</span>
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">t</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">duration</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">sr</span> <span class="o">*</span> <span class="n">duration</span><span class="p">))</span>
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a>
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="c1"># 语音：具有时变幅度的谐波序列（模拟语音）</span>
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="n">speech</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="k">for</span> <span class="n">f0</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">150</span><span class="p">,</span> <span class="mi">300</span><span class="p">,</span> <span class="mi">450</span><span class="p">,</span> <span class="mi">600</span><span class="p">,</span> <span class="mi">900</span><span class="p">]:</span>
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a> <span class="n">amp_env</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span> <span class="c1"># 2 Hz 调制</span>
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a> <span class="n">speech</span> <span class="o">=</span> <span class="n">speech</span> <span class="o">+</span> <span class="p">(</span><span class="mf">0.5</span> <span class="o">/</span> <span class="p">(</span><span class="n">f0</span><span class="o">/</span><span class="mi">150</span><span class="p">))</span> <span class="o">*</span> <span class="n">amp_env</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">f0</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span>
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="n">speech</span> <span class="o">=</span> <span class="n">speech</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">speech</span><span class="p">))</span>
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a>
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="c1"># 噪声：限带噪声</span>
<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="n">noise_raw</span> <span class="o">=</span> <span class="n">jr</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">t</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.4</span>
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a>
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="c1"># 在给定 SNR 下混合</span>
<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="n">snr_db</span> <span class="o">=</span> <span class="mf">5.0</span>
<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="n">speech_power</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">speech</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="n">noise_power</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">noise_raw</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="n">noise_scale</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">speech_power</span> <span class="o">/</span> <span class="p">(</span><span class="n">noise_power</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">**</span> <span class="p">(</span><span class="n">snr_db</span> <span class="o">/</span> <span class="mi">10</span><span class="p">)))</span>
<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="n">noise</span> <span class="o">=</span> <span class="n">noise_raw</span> <span class="o">*</span> <span class="n">noise_scale</span>
<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="n">mixture</span> <span class="o">=</span> <span class="n">speech</span> <span class="o">+</span> <span class="n">noise</span>
<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a>
<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="c1"># STFT</span>
<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="n">n_fft</span> <span class="o">=</span> <span class="mi">512</span>
<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="n">hop</span> <span class="o">=</span> <span class="mi">128</span>
<a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="n">window</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">hanning</span><span class="p">(</span><span class="n">n_fft</span><span class="p">)</span>
<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a>
<a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a><span class="k">def</span><span class="w"> </span><span class="nf">stft</span><span class="p">(</span><span class="n">signal</span><span class="p">,</span> <span class="n">n_fft</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">):</span>
<a id="__codelineno-3-36" name="__codelineno-3-36" href="#__codelineno-3-36"></a> <span class="n">n_frames</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">signal</span><span class="p">)</span> <span class="o">-</span> <span class="n">n_fft</span><span class="p">)</span> <span class="o">//</span> <span class="n">hop</span>
<a id="__codelineno-3-37" name="__codelineno-3-37" href="#__codelineno-3-37"></a> <span class="n">frames</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">stack</span><span class="p">([</span>
<a id="__codelineno-3-38" name="__codelineno-3-38" href="#__codelineno-3-38"></a> <span class="n">signal</span><span class="p">[</span><span class="n">i</span> <span class="o">*</span> <span class="n">hop</span> <span class="p">:</span> <span class="n">i</span> <span class="o">*</span> <span class="n">hop</span> <span class="o">+</span> <span class="n">n_fft</span><span class="p">]</span> <span class="o">*</span> <span class="n">window</span>
<a id="__codelineno-3-39" name="__codelineno-3-39" href="#__codelineno-3-39"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_frames</span><span class="p">)</span>
<a id="__codelineno-3-40" name="__codelineno-3-40" href="#__codelineno-3-40"></a> <span class="p">])</span>
<a id="__codelineno-3-41" name="__codelineno-3-41" href="#__codelineno-3-41"></a> <span class="k">return</span> <span class="n">jnp</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">frames</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n_fft</span><span class="p">)</span>
<a id="__codelineno-3-42" name="__codelineno-3-42" href="#__codelineno-3-42"></a>
<a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a><span class="k">def</span><span class="w"> </span><span class="nf">istft</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">,</span> <span class="n">length</span><span class="p">):</span>
<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a> <span class="n">n_fft</span> <span class="o">=</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="mi">2</span>
<a id="__codelineno-3-45" name="__codelineno-3-45" href="#__codelineno-3-45"></a> <span class="n">n_frames</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></a> <span class="n">frames</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfft</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n_fft</span><span class="p">)</span> <span class="o">*</span> <span class="n">window</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:]</span>
<a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a> <span class="n">output</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">length</span><span class="p">)</span>
<a id="__codelineno-3-48" name="__codelineno-3-48" href="#__codelineno-3-48"></a> <span class="n">window_sum</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">length</span><span class="p">)</span>
<a id="__codelineno-3-49" name="__codelineno-3-49" href="#__codelineno-3-49"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_frames</span><span class="p">):</span>
<a id="__codelineno-3-50" name="__codelineno-3-50" href="#__codelineno-3-50"></a> <span class="n">start</span> <span class="o">=</span> <span class="n">i</span> <span class="o">*</span> <span class="n">hop</span>
<a id="__codelineno-3-51" name="__codelineno-3-51" href="#__codelineno-3-51"></a> <span class="n">end</span> <span class="o">=</span> <span class="n">start</span> <span class="o">+</span> <span class="n">n_fft</span>
<a id="__codelineno-3-52" name="__codelineno-3-52" href="#__codelineno-3-52"></a> <span class="k">if</span> <span class="n">end</span> <span class="o">&lt;=</span> <span class="n">length</span><span class="p">:</span>
<a id="__codelineno-3-53" name="__codelineno-3-53" href="#__codelineno-3-53"></a> <span class="n">output</span> <span class="o">=</span> <span class="n">output</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">frames</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<a id="__codelineno-3-54" name="__codelineno-3-54" href="#__codelineno-3-54"></a> <span class="n">window_sum</span> <span class="o">=</span> <span class="n">window_sum</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">window</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-3-55" name="__codelineno-3-55" href="#__codelineno-3-55"></a> <span class="n">window_sum</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">window_sum</span><span class="p">,</span> <span class="mf">1e-8</span><span class="p">)</span>
<a id="__codelineno-3-56" name="__codelineno-3-56" href="#__codelineno-3-56"></a> <span class="k">return</span> <span class="n">output</span> <span class="o">/</span> <span class="n">window_sum</span>
<a id="__codelineno-3-57" name="__codelineno-3-57" href="#__codelineno-3-57"></a>
<a id="__codelineno-3-58" name="__codelineno-3-58" href="#__codelineno-3-58"></a><span class="n">S_speech</span> <span class="o">=</span> <span class="n">stft</span><span class="p">(</span><span class="n">speech</span><span class="p">,</span> <span class="n">n_fft</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">)</span>
<a id="__codelineno-3-59" name="__codelineno-3-59" href="#__codelineno-3-59"></a><span class="n">S_noise</span> <span class="o">=</span> <span class="n">stft</span><span class="p">(</span><span class="n">noise</span><span class="p">,</span> <span class="n">n_fft</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">)</span>
<a id="__codelineno-3-60" name="__codelineno-3-60" href="#__codelineno-3-60"></a><span class="n">S_mix</span> <span class="o">=</span> <span class="n">stft</span><span class="p">(</span><span class="n">mixture</span><span class="p">,</span> <span class="n">n_fft</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">)</span>
<a id="__codelineno-3-61" name="__codelineno-3-61" href="#__codelineno-3-61"></a>
<a id="__codelineno-3-62" name="__codelineno-3-62" href="#__codelineno-3-62"></a><span class="n">mag_speech</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">S_speech</span><span class="p">)</span>
<a id="__codelineno-3-63" name="__codelineno-3-63" href="#__codelineno-3-63"></a><span class="n">mag_noise</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">S_noise</span><span class="p">)</span>
<a id="__codelineno-3-64" name="__codelineno-3-64" href="#__codelineno-3-64"></a><span class="n">mag_mix</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">S_mix</span><span class="p">)</span>
<a id="__codelineno-3-65" name="__codelineno-3-65" href="#__codelineno-3-65"></a><span class="n">phase_mix</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">angle</span><span class="p">(</span><span class="n">S_mix</span><span class="p">)</span>
<a id="__codelineno-3-66" name="__codelineno-3-66" href="#__codelineno-3-66"></a>
<a id="__codelineno-3-67" name="__codelineno-3-67" href="#__codelineno-3-67"></a><span class="c1"># 方法 1：理想比率掩蔽（oracle - 理论上限）</span>
<a id="__codelineno-3-68" name="__codelineno-3-68" href="#__codelineno-3-68"></a><span class="n">irm</span> <span class="o">=</span> <span class="n">mag_speech</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="n">mag_speech</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">mag_noise</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="mf">1e-8</span><span class="p">)</span>
<a id="__codelineno-3-69" name="__codelineno-3-69" href="#__codelineno-3-69"></a><span class="n">S_irm</span> <span class="o">=</span> <span class="p">(</span><span class="n">irm</span> <span class="o">*</span> <span class="n">mag_mix</span><span class="p">)</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">phase_mix</span><span class="p">)</span>
<a id="__codelineno-3-70" name="__codelineno-3-70" href="#__codelineno-3-70"></a><span class="n">enhanced_irm</span> <span class="o">=</span> <span class="n">istft</span><span class="p">(</span><span class="n">S_irm</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">mixture</span><span class="p">))</span>
<a id="__codelineno-3-71" name="__codelineno-3-71" href="#__codelineno-3-71"></a>
<a id="__codelineno-3-72" name="__codelineno-3-72" href="#__codelineno-3-72"></a><span class="c1"># 方法 2：谱减法</span>
<a id="__codelineno-3-73" name="__codelineno-3-73" href="#__codelineno-3-73"></a><span class="c1"># 从前 0.2s 估计噪声（假设为静音段）</span>
<a id="__codelineno-3-74" name="__codelineno-3-74" href="#__codelineno-3-74"></a><span class="n">noise_frames</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">0.2</span> <span class="o">*</span> <span class="n">sr</span> <span class="o">/</span> <span class="n">hop</span><span class="p">)</span>
<a id="__codelineno-3-75" name="__codelineno-3-75" href="#__codelineno-3-75"></a><span class="n">noise_est</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">mag_mix</span><span class="p">[:</span><span class="n">noise_frames</span><span class="p">]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<a id="__codelineno-3-76" name="__codelineno-3-76" href="#__codelineno-3-76"></a><span class="n">alpha</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="c1"># 过减因子</span>
<a id="__codelineno-3-77" name="__codelineno-3-77" href="#__codelineno-3-77"></a><span class="n">beta</span> <span class="o">=</span> <span class="mf">0.02</span> <span class="c1"># 频谱地板</span>
<a id="__codelineno-3-78" name="__codelineno-3-78" href="#__codelineno-3-78"></a><span class="n">mag_sub</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="n">mag_mix</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">-</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">noise_est</span><span class="p">,</span> <span class="n">beta</span> <span class="o">*</span> <span class="n">mag_mix</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-3-79" name="__codelineno-3-79" href="#__codelineno-3-79"></a><span class="n">mag_sub</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">mag_sub</span><span class="p">)</span>
<a id="__codelineno-3-80" name="__codelineno-3-80" href="#__codelineno-3-80"></a><span class="n">S_sub</span> <span class="o">=</span> <span class="n">mag_sub</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">phase_mix</span><span class="p">)</span>
<a id="__codelineno-3-81" name="__codelineno-3-81" href="#__codelineno-3-81"></a><span class="n">enhanced_sub</span> <span class="o">=</span> <span class="n">istft</span><span class="p">(</span><span class="n">S_sub</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">mixture</span><span class="p">))</span>
<a id="__codelineno-3-82" name="__codelineno-3-82" href="#__codelineno-3-82"></a>
<a id="__codelineno-3-83" name="__codelineno-3-83" href="#__codelineno-3-83"></a><span class="c1"># 方法 3：维纳滤波器</span>
<a id="__codelineno-3-84" name="__codelineno-3-84" href="#__codelineno-3-84"></a><span class="n">snr_est</span> <span class="o">=</span> <span class="n">mag_mix</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="n">noise_est</span> <span class="o">+</span> <span class="mf">1e-8</span><span class="p">)</span>
<a id="__codelineno-3-85" name="__codelineno-3-85" href="#__codelineno-3-85"></a><span class="n">wiener_gain</span> <span class="o">=</span> <span class="n">snr_est</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">snr_est</span><span class="p">)</span>
<a id="__codelineno-3-86" name="__codelineno-3-86" href="#__codelineno-3-86"></a><span class="n">S_wiener</span> <span class="o">=</span> <span class="p">(</span><span class="n">wiener_gain</span> <span class="o">*</span> <span class="n">mag_mix</span><span class="p">)</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">phase_mix</span><span class="p">)</span>
<a id="__codelineno-3-87" name="__codelineno-3-87" href="#__codelineno-3-87"></a><span class="n">enhanced_wiener</span> <span class="o">=</span> <span class="n">istft</span><span class="p">(</span><span class="n">S_wiener</span><span class="p">,</span> <span class="n">hop</span><span class="p">,</span> <span class="n">window</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">mixture</span><span class="p">))</span>
<a id="__codelineno-3-88" name="__codelineno-3-88" href="#__codelineno-3-88"></a>
<a id="__codelineno-3-89" name="__codelineno-3-89" href="#__codelineno-3-89"></a><span class="c1"># 计算每种方法的 SI-SDR</span>
<a id="__codelineno-3-90" name="__codelineno-3-90" href="#__codelineno-3-90"></a><span class="k">def</span><span class="w"> </span><span class="nf">si_sdr</span><span class="p">(</span><span class="n">estimate</span><span class="p">,</span> <span class="n">reference</span><span class="p">):</span>
<a id="__codelineno-3-91" name="__codelineno-3-91" href="#__codelineno-3-91"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;尺度不变信号失真比。&quot;&quot;&quot;</span>
<a id="__codelineno-3-92" name="__codelineno-3-92" href="#__codelineno-3-92"></a> <span class="n">ref</span> <span class="o">=</span> <span class="n">reference</span><span class="p">[:</span><span class="nb">len</span><span class="p">(</span><span class="n">estimate</span><span class="p">)]</span>
<a id="__codelineno-3-93" name="__codelineno-3-93" href="#__codelineno-3-93"></a> <span class="n">est</span> <span class="o">=</span> <span class="n">estimate</span><span class="p">[:</span><span class="nb">len</span><span class="p">(</span><span class="n">reference</span><span class="p">)]</span>
<a id="__codelineno-3-94" name="__codelineno-3-94" href="#__codelineno-3-94"></a> <span class="n">s_target</span> <span class="o">=</span> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">est</span><span class="p">,</span> <span class="n">ref</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">ref</span><span class="p">,</span> <span class="n">ref</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1e-8</span><span class="p">))</span> <span class="o">*</span> <span class="n">ref</span>
<a id="__codelineno-3-95" name="__codelineno-3-95" href="#__codelineno-3-95"></a> <span class="n">e_noise</span> <span class="o">=</span> <span class="n">est</span> <span class="o">-</span> <span class="n">s_target</span>
<a id="__codelineno-3-96" name="__codelineno-3-96" href="#__codelineno-3-96"></a> <span class="k">return</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">s_target</span><span class="p">,</span> <span class="n">s_target</span><span class="p">)</span> <span class="o">/</span>
<a id="__codelineno-3-97" name="__codelineno-3-97" href="#__codelineno-3-97"></a> <span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">e_noise</span><span class="p">,</span> <span class="n">e_noise</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1e-8</span><span class="p">))</span>
<a id="__codelineno-3-98" name="__codelineno-3-98" href="#__codelineno-3-98"></a>
<a id="__codelineno-3-99" name="__codelineno-3-99" href="#__codelineno-3-99"></a><span class="n">si_sdr_mix</span> <span class="o">=</span> <span class="n">si_sdr</span><span class="p">(</span><span class="n">mixture</span><span class="p">,</span> <span class="n">speech</span><span class="p">)</span>
<a id="__codelineno-3-100" name="__codelineno-3-100" href="#__codelineno-3-100"></a><span class="n">si_sdr_irm_val</span> <span class="o">=</span> <span class="n">si_sdr</span><span class="p">(</span><span class="n">enhanced_irm</span><span class="p">,</span> <span class="n">speech</span><span class="p">)</span>
<a id="__codelineno-3-101" name="__codelineno-3-101" href="#__codelineno-3-101"></a><span class="n">si_sdr_sub_val</span> <span class="o">=</span> <span class="n">si_sdr</span><span class="p">(</span><span class="n">enhanced_sub</span><span class="p">,</span> <span class="n">speech</span><span class="p">)</span>
<a id="__codelineno-3-102" name="__codelineno-3-102" href="#__codelineno-3-102"></a><span class="n">si_sdr_wiener_val</span> <span class="o">=</span> <span class="n">si_sdr</span><span class="p">(</span><span class="n">enhanced_wiener</span><span class="p">,</span> <span class="n">speech</span><span class="p">)</span>
<a id="__codelineno-3-103" name="__codelineno-3-103" href="#__codelineno-3-103"></a>
<a id="__codelineno-3-104" name="__codelineno-3-104" href="#__codelineno-3-104"></a><span class="c1"># 可视化</span>
<a id="__codelineno-3-105" name="__codelineno-3-105" href="#__codelineno-3-105"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<a id="__codelineno-3-106" name="__codelineno-3-106" href="#__codelineno-3-106"></a>
<a id="__codelineno-3-107" name="__codelineno-3-107" href="#__codelineno-3-107"></a><span class="c1"># 语谱图</span>
<a id="__codelineno-3-108" name="__codelineno-3-108" href="#__codelineno-3-108"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="n">mag_speech</span><span class="o">.</span><span class="n">T</span><span class="p">),</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span> <span class="n">origin</span><span class="o">=</span><span class="s1">&#39;lower&#39;</span><span class="p">,</span>
<a id="__codelineno-3-109" name="__codelineno-3-109" href="#__codelineno-3-109"></a> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;magma&#39;</span><span class="p">)</span>
<a id="__codelineno-3-110" name="__codelineno-3-110" href="#__codelineno-3-110"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;干净语音语谱图&#39;</span><span class="p">)</span>
<a id="__codelineno-3-111" name="__codelineno-3-111" href="#__codelineno-3-111"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;频率 bin&#39;</span><span class="p">)</span>
<a id="__codelineno-3-112" name="__codelineno-3-112" href="#__codelineno-3-112"></a>
<a id="__codelineno-3-113" name="__codelineno-3-113" href="#__codelineno-3-113"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="n">mag_mix</span><span class="o">.</span><span class="n">T</span><span class="p">),</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span> <span class="n">origin</span><span class="o">=</span><span class="s1">&#39;lower&#39;</span><span class="p">,</span>
<a id="__codelineno-3-114" name="__codelineno-3-114" href="#__codelineno-3-114"></a> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;magma&#39;</span><span class="p">)</span>
<a id="__codelineno-3-115" name="__codelineno-3-115" href="#__codelineno-3-115"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;带噪混合 (</span><span class="si">{</span><span class="n">snr_db</span><span class="si">:</span><span class="s1">.0f</span><span class="si">}</span><span class="s1"> dB SNR)&#39;</span><span class="p">)</span>
<a id="__codelineno-3-116" name="__codelineno-3-116" href="#__codelineno-3-116"></a>
<a id="__codelineno-3-117" name="__codelineno-3-117" href="#__codelineno-3-117"></a><span class="c1"># 掩蔽</span>
<a id="__codelineno-3-118" name="__codelineno-3-118" href="#__codelineno-3-118"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">irm</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span> <span class="n">origin</span><span class="o">=</span><span class="s1">&#39;lower&#39;</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;RdYlGn&#39;</span><span class="p">)</span>
<a id="__codelineno-3-119" name="__codelineno-3-119" href="#__codelineno-3-119"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;理想比率掩蔽（Oracle）&#39;</span><span class="p">)</span>
<a id="__codelineno-3-120" name="__codelineno-3-120" href="#__codelineno-3-120"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;频率 bin&#39;</span><span class="p">)</span>
<a id="__codelineno-3-121" name="__codelineno-3-121" href="#__codelineno-3-121"></a>
<a id="__codelineno-3-122" name="__codelineno-3-122" href="#__codelineno-3-122"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">wiener_gain</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">aspect</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span> <span class="n">origin</span><span class="o">=</span><span class="s1">&#39;lower&#39;</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;RdYlGn&#39;</span><span class="p">,</span>
<a id="__codelineno-3-123" name="__codelineno-3-123" href="#__codelineno-3-123"></a> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-3-124" name="__codelineno-3-124" href="#__codelineno-3-124"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;估计的维纳增益&#39;</span><span class="p">)</span>
<a id="__codelineno-3-125" name="__codelineno-3-125" href="#__codelineno-3-125"></a>
<a id="__codelineno-3-126" name="__codelineno-3-126" href="#__codelineno-3-126"></a><span class="c1"># 增强后的波形对比</span>
<a id="__codelineno-3-127" name="__codelineno-3-127" href="#__codelineno-3-127"></a><span class="n">n_show</span> <span class="o">=</span> <span class="mi">3000</span>
<a id="__codelineno-3-128" name="__codelineno-3-128" href="#__codelineno-3-128"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">n_show</span><span class="p">],</span> <span class="n">speech</span><span class="p">[:</span><span class="n">n_show</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#27ae60&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span>
<a id="__codelineno-3-129" name="__codelineno-3-129" href="#__codelineno-3-129"></a> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;干净&#39;</span><span class="p">)</span>
<a id="__codelineno-3-130" name="__codelineno-3-130" href="#__codelineno-3-130"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">n_show</span><span class="p">],</span> <span class="n">mixture</span><span class="p">[:</span><span class="n">n_show</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#e74c3c&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
<a id="__codelineno-3-131" name="__codelineno-3-131" href="#__codelineno-3-131"></a> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;带噪&#39;</span><span class="p">)</span>
<a id="__codelineno-3-132" name="__codelineno-3-132" href="#__codelineno-3-132"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">[:</span><span class="n">n_show</span><span class="p">],</span> <span class="n">enhanced_irm</span><span class="p">[:</span><span class="n">n_show</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;#3498db&#39;</span><span class="p">,</span>
<a id="__codelineno-3-133" name="__codelineno-3-133" href="#__codelineno-3-133"></a> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;IRM 增强&#39;</span><span class="p">)</span>
<a id="__codelineno-3-134" name="__codelineno-3-134" href="#__codelineno-3-134"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;波形对比（IRM）&#39;</span><span class="p">)</span>
<a id="__codelineno-3-135" name="__codelineno-3-135" href="#__codelineno-3-135"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">&#39;时间 (s)&#39;</span><span class="p">)</span>
<a id="__codelineno-3-136" name="__codelineno-3-136" href="#__codelineno-3-136"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;幅度&#39;</span><span class="p">)</span>
<a id="__codelineno-3-137" name="__codelineno-3-137" href="#__codelineno-3-137"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">8</span><span class="p">)</span>
<a id="__codelineno-3-138" name="__codelineno-3-138" href="#__codelineno-3-138"></a>
<a id="__codelineno-3-139" name="__codelineno-3-139" href="#__codelineno-3-139"></a><span class="c1"># SI-SDR 柱状图</span>
<a id="__codelineno-3-140" name="__codelineno-3-140" href="#__codelineno-3-140"></a><span class="n">methods</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;混合信号&#39;</span><span class="p">,</span> <span class="s1">&#39;谱减法&#39;</span><span class="p">,</span> <span class="s1">&#39;维纳滤波器&#39;</span><span class="p">,</span> <span class="s1">&#39;理想比率掩蔽&#39;</span><span class="p">]</span>
<a id="__codelineno-3-141" name="__codelineno-3-141" href="#__codelineno-3-141"></a><span class="n">sdr_values</span> <span class="o">=</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="n">si_sdr_mix</span><span class="p">),</span> <span class="nb">float</span><span class="p">(</span><span class="n">si_sdr_sub_val</span><span class="p">),</span>
<a id="__codelineno-3-142" name="__codelineno-3-142" href="#__codelineno-3-142"></a> <span class="nb">float</span><span class="p">(</span><span class="n">si_sdr_wiener_val</span><span class="p">),</span> <span class="nb">float</span><span class="p">(</span><span class="n">si_sdr_irm_val</span><span class="p">)]</span>
<a id="__codelineno-3-143" name="__codelineno-3-143" href="#__codelineno-3-143"></a><span class="n">bar_colors</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;#e74c3c&#39;</span><span class="p">,</span> <span class="s1">&#39;#f39c12&#39;</span><span class="p">,</span> <span class="s1">&#39;#9b59b6&#39;</span><span class="p">,</span> <span class="s1">&#39;#27ae60&#39;</span><span class="p">]</span>
<a id="__codelineno-3-144" name="__codelineno-3-144" href="#__codelineno-3-144"></a><span class="n">bars</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">methods</span><span class="p">,</span> <span class="n">sdr_values</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">bar_colors</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
<a id="__codelineno-3-145" name="__codelineno-3-145" href="#__codelineno-3-145"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;SI-SDR (dB)&#39;</span><span class="p">)</span>
<a id="__codelineno-3-146" name="__codelineno-3-146" href="#__codelineno-3-146"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;增强质量对比&#39;</span><span class="p">)</span>
<a id="__codelineno-3-147" name="__codelineno-3-147" href="#__codelineno-3-147"></a><span class="k">for</span> <span class="n">bar</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">bars</span><span class="p">,</span> <span class="n">sdr_values</span><span class="p">):</span>
<a id="__codelineno-3-148" name="__codelineno-3-148" href="#__codelineno-3-148"></a> <span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">bar</span><span class="o">.</span><span class="n">get_x</span><span class="p">()</span> <span class="o">+</span> <span class="n">bar</span><span class="o">.</span><span class="n">get_width</span><span class="p">()</span><span class="o">/</span><span class="mf">2.</span><span class="p">,</span> <span class="n">bar</span><span class="o">.</span><span class="n">get_height</span><span class="p">()</span> <span class="o">+</span> <span class="mf">0.3</span><span class="p">,</span>
<a id="__codelineno-3-149" name="__codelineno-3-149" href="#__codelineno-3-149"></a> <span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">val</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">&#39;center&#39;</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
<a id="__codelineno-3-150" name="__codelineno-3-150" href="#__codelineno-3-150"></a><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;gray&#39;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
<a id="__codelineno-3-151" name="__codelineno-3-151" href="#__codelineno-3-151"></a>
<a id="__codelineno-3-152" name="__codelineno-3-152" href="#__codelineno-3-152"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<a id="__codelineno-3-153" name="__codelineno-3-153" href="#__codelineno-3-153"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<a id="__codelineno-3-154" name="__codelineno-3-154" href="#__codelineno-3-154"></a>
<a id="__codelineno-3-155" name="__codelineno-3-155" href="#__codelineno-3-155"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;SI-SDR（带噪混合）: </span><span class="si">{</span><span class="n">si_sdr_mix</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2"> dB&quot;</span><span class="p">)</span>
<a id="__codelineno-3-156" name="__codelineno-3-156" href="#__codelineno-3-156"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;SI-SDR（谱减法）: </span><span class="si">{</span><span class="n">si_sdr_sub_val</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2"> dB&quot;</span><span class="p">)</span>
<a id="__codelineno-3-157" name="__codelineno-3-157" href="#__codelineno-3-157"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;SI-SDR（维纳滤波器）: </span><span class="si">{</span><span class="n">si_sdr_wiener_val</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2"> dB&quot;</span><span class="p">)</span>
<a id="__codelineno-3-158" name="__codelineno-3-158" href="#__codelineno-3-158"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;SI-SDR（理想比率掩蔽）: </span><span class="si">{</span><span class="n">si_sdr_irm_val</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2"> dB（oracle 理论上限）&quot;</span><span class="p">)</span>
</code></pre></div>
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