Analysis of trade-offs between magnitude and phase estimation in loss functions for speech denoising and dereverberation

Sixty Years of Frequency-Domain Monaural Speech Enhancement: From Traditional to Deep Learning Methods

“…The second benefit is that speech distortion and noise reduction can be better balanced when compared with the raw magnitude without compression, resulting in improving speech quality. This may be the case because compression reduces the dynamic range of the magnitude values, facilitating the training process (Luo et al, 2022). Compression of the magnitude of the noisy spectrum can be expressed by: where α cp ∈ false( 0 0.25em 1 false] is the compression factor.…”

“…When the complex spectrum-based MSE loss function is used, the phase estimation error is reduced but spectral magnitude distortion increases. The trade-off between spectral magnitude distortion and phase recovery has been called the “compensation effect” (Wang et al, 2021; Luo et al, 2022). To reduce both magnitude and phase distortion, a combined loss function has been proposed, which is formulated as: where α com is a linear combination coefficient.…”

“…Various easily extracted features of noisy speech have been used in deep neural network (DNN) models designed to extract clean speech from noisy speech, including the LOG-AMP, the log-power spectrum (Xu et al, 2014b(Xu et al, , 2015, spectral amplitudes (Tan & Wang, 2018) and the spectral amplitudes raised to a power less than 1 (Zhao et al, 2020), which represents a form of amplitude compression. The cube-root of the spectral amplitudes generally led to the best performance, perhaps because taking the cube-root reduces the dynamic range of the speech, facilitating the training process (Luo et al, 2022). Tan & Wang (2020) extracted the real and imaginary parts of the complex spectrum of noisy speech as input features.…”

Acoustic echo cancellation based on two‐stage BLSTM

“…Loss function: To partially avoid the compensation effect between the magnitude and RI constraints in [28] and [29], we use the linear combination of magnitude and complex loss:…”

“…To partially avoid the compensation effect between the magnitude and RI constraints in [28] and [29], we use the linear combination of magnitude and complex loss: $$$$\begin{equation} {\cal L} =\frac{{{{\cal L}_{\text{mag}}} + {{\cal L}_{ri}}}}{2} \end{equation}$$$$ $$$$\begin{equation} {{\cal L}_{\text{mag}}}={{\mathbb {E}}_{{S_{\text{mag}}},{{\hat{S}}_{\text{mag}}}}}{\left[ {{{{\left\Vert {{S_{\text{mag}}} - {{\hat{S}}_{\text{mag}}}} \right\Vert} }^2}} \right]} \end{equation}$$$$ $$$$\begin{equation} {{\cal L}_{ri}} ={{\mathbb {E}}_{{S_r},{{\hat{S}}_r}}}{\left[ {{{{\left\Vert {{S_r} - {{\hat{S}}_r}} \right\Vert} }^2}} \right]} + {{\mathbb {E}}_{{S_i},{{\hat{S}}_i}}}{\left[ {{{{\left\Vert {{S_i} - {{\hat{S}}_i}} \right\Vert} }^2}} \right]} \end{equation}$$$$…”