Speech Signal Processing - Noise Suppression

Introduction

• Noise suppression technology is used to eliminate background noise, improve the signal-to-noise ratio and intelligibility of speech signals, and make people and machines hear more clearly
• Common types of noise: human voice noise, street noise, car noise
• Classification of noise suppression methods:
  • According to the number of input channels: single-channel noise reduction, multi-channel noise reduction
  • According to the statistical characteristics of noise: stationary noise suppression, non-stationary noise suppression
  • According to the noise reduction method: passive noise reduction, active noise reduction
• The method described below is used for single-channel, passive, stationary noise suppression

Minima Controlled Recursive Averaging（MCRA）

• The noise reduction method in the traditional signal processing method can be divided into two major steps:

  • One is the estimation of noise (Noise Estimation/Tracking)
  • The second is the estimation of the gain factor

• Common methods for noise estimation:

  • Recursive Averaging: As long as the probability of speech in a certain frequency band is low, the noise spectrum can be estimated/updated using this frequency band
  • Minima Controlled/Tracking: Due to the sparsity of speech signals, even if speech exists, within a short time window (0.5s-1.5s), the minimum value of each frequency band will approach the noise power with a high probability , so the noise estimate for each frequency band can be obtained by tracking the minimum in a short time window
  • Histogram-based Method: Histogram statistics are performed in a short time window for each frequency band of the noisy speech signal, and the value with the highest frequency corresponds to the noise level of this frequency band

• MCRA steps:

• Consider a data model with additive noise: $and\left(n\right)=x(n)+d(n)$

• The input signal goes through STFT: $and(k,l)={\sum }_{n=0}^{N-1}and(n)+lM)h(n)e^{-j\frac{2}{N}nk}$ , get time spectrogram

• in,

  • k is the index of the frequency bin
  • l is the index of the time frame
  • h(n) is the window function
  • M is for frame shift

• Given two hypotheses $H_0(k,l)$ and$H_1(k,l)$ , which respectively represent "no speech" and "speech exists":
  $$\begin{array}{rl}{H}_0(k,l)：& and(k,l)=D(k,l)\\ H_1(k,l)：& and(k,l)=X(k,l)+D(k,l)\end{array}$$

• The estimation of noise is defined as: $l_d(k,l)=E[|D(k,l){|}^2]$, that is, the energy of the noise

• Using time recursive smoothing, when "voice does not exist", iteratively update $l_d(k,l)$
  $$\begin{array}{rl}{H}_0^{\prime }(k,l)：& {\hat{l}}_d(k,l+1)=a_d{\hat{l}}_d(k,l)+(1-a_d)|Y(k,l){|}^2\\ H_1^{\prime }(k,l)：& {\hat{l}}_d(k,l+1)={\hat{l}}_d(k,l)\\ Among\; them& a_d（0<a_d<1)\; is\; the\; smoothing\end{array}$$

• Speech Presence Probability: $p^{\prime }(k,l)=P(H_1^{\prime }(k,l)|Y(k,l))$

• 对星是的是以可以作的：
  $$\begin{array}{rl}{\hat{l}}_d(k,l+1)& ={\hat{l}}_d(k,l)p^{\prime }(k,l)+[a_d{\hat{l}}_d(k,l)+(1-a_d)|Y(k,l){|}^2](1-p^{\prime }(k,l))\\ & ={\hat{a}}_d(k,l){\hat{l}}_d(k,l)+[1-{\hat{a}}_d(k,l)]|Y(k,l){|}^2\\ in,{\hat{a}}_d(k,l)& =a_d+(1-a_d)p^{\prime }(k,l)\end{array}$$

• The question now is: how to determine $p^{\prime }(k,l)$

• Calculate $p^{\prime }(k,l)$ idea: in a short time window, calculate the local energy$S(k,l)$ and minimum energy$S_{min}(k,l)$ ratio

• Calculation of local energy:

  • Area smoothing: $S_f(k,l)={\sum }_{i=-w}^{w}b(i)|Y(k-i,l){|}^2$
  • Temporal smoothing: $S(k,l)=a_{s}S(k,l-1)+(1-a_s)S_f(k,l)$
  • $b\left(i\right)$ is the frequency domain window function,$a_s（0<a_s<1)$ is the time-domain smoothing factor of the local energy

• Calculation of minimum energy:

  • Conventional practice: use the local minimum search method, set a time window L (L is usually 1s), and search for the local minimum
  • Simplified approach:
    

• Calculate the ratio: $S_r(k,l)=\frac{S(k,l)}{S_{min}(k,l)}$

• The discriminant for the existence of speech is:
  $$I(k,l)=\{\begin{array}{c}1，S_r(k,l)>d\\ 0，otherwise\end{array}$$

• Iterative estimation of speech presence probability:

  • ${\hat{p}}^{\prime }(k,l)=a_p{\hat{p}}^{\prime }(k,l-1)+(1-a_p)I(k,l)$
  • $a_p（0<a_p<1)$ is the smoothing factor of the speech existence probability

• MCRA noise estimation process:
  

• MCRA Reference Parameters
  

• After estimating the noise, multiply the noisy speech by the gain factor to perform noise suppression
  $$\begin{array}{rl}\hat{X}(k,l)& =G(k,l)Y(k,l)\\ or|\hat{X}(k){|}^2& =G^2(k)|Y(k){|}^2\end{array}$$

• To determine the gain factor:

  • spectral subtraction
  • Wiener filtering
  • MMSE

• Spectral subtraction: Assuming that the noise changes steadily or slowly, from the noisy speech spectrum, the noise spectrum is subtracted

  • Definitely:
    $$\begin{array}{rl}G(k)& =\sqrt{\frac{|\hat{X}(k){|}^2}{|Y(k){|}^2}}\\ & =\sqrt{\frac{|Y(k){|}^2-|\hat{D}\left(k\right){|}^2}{|Y(k){|}^2}}=\sqrt{\frac{|Y(k){|}^2-l_d(k)}{|Y(k){|}^2}}\\ & =\sqrt{1-\frac{l_d\left(k\right)}{|Y(k){|}^2}}=\sqrt{1-\frac{1}{c\left(k\right)}}\end{array}$$
  • For example, $c\left(k\right)=\frac{|Y(k){|}^2}{l_d(k)}$, known as the posterior SNR

• Frequency-domain Wiener filtering: the mean square error between the estimated pure speech amplitude spectrum and the real amplitude spectrum is the smallest

  • Frequency domain estimation error: $E(k)=X(k)-\hat{X}(k)=X\left(k\right)-G(k)Y(k)$
  • Objective function: $J=E[|E(k){|}^2]$
  • By minimizing the objective function, the expression of the gain factor can be obtained:
    $$G(k)=\frac{l_x(k)}{l_x(k)+l_d\left(k\right)}=\frac{\xi (k}{\xi \left(k\right)+1}$$
  • Among them, $l_x\left(k\right)=E[|X(k){|}^2]$，$\xi \left(k\right)=\frac{l_x(k)}{l_d(k)}$SNR prior
  • 后水信颜比：$c\left(k\right)=\frac{|Y(k){|}^2}{l_d\left(k\right)}$
  • Prior SNR: $\xi \left(k\right)=\frac{l_x\left(k\right)}{l_d\left(k\right)}$
  • Estimating the prior SNR from the posterior SNR—Decision Directed (DD)
    $$\xi (k,l)=a_{DD}\frac{|\hat{X}(k,l-1){|}^2}{l_d(k,l-1)}+(1-a_{DD})max\left\{\gamma \right(k,l)-1,0\}$$
  • Among them, $a_{DD}$The empirical value is 0.95~0.98, $l_d(k,l-1)$ Available$l_d(k,l)$ instead of

• Gain factor for MMSE:
  

• The derivation of MMSE can refer to: Ephraim, Y., and D. Malah. "Speech Enhancement Using a Minimum-Mean Square Error Short-Time Spectral Amplitude Estimator."