先验信噪比估计

很多降噪算法如维纳滤波、MMSE估计器等都依赖先验信噪比(priori SNR)信息

定义先验信噪比(priori SNR)、后验信噪比(posteriori SNR)如下
$${\xi }_k(n)=\frac{E\left\{A_k^2(n)\right\}}{{\lambda }_d(k,n)}\text{\hspace{1em}(1)}$$

$${\gamma }_k(n)=\frac{Y_k^2(n)}{{\lambda }_d(k,n)}\text{\hspace{1em}(2)}$$

上式中的$A_k(n)$为目标信号的幅度谱、${\lambda }_d(n)$为噪声的功率谱，这里先假定噪声平稳且可以在无语音段估计得到。

假设噪声和语音不相关且为加性噪声，则${\xi }_k$可以写成下式
ξ k ( n ) = E { A k 2 ( n ) } λ d ( k , n ) = E { Y k 2 ( n ) − λ d ( k , n ) } λ d ( k , n ) = E { Y k 2 ( n ) } λ d ( k , n ) − 1 = E { λ k ( n ) − 1 } (3) \begin{aligned}\xi_{k}(n)&=\frac{E\left\{A_{k}^{2}(n)\right\}}{\lambda_{d}(k, n)}\\&=\frac{E\left\{Y_{k}^{2}(n)-\lambda_d(k,n)\right\}}{\lambda_{d}(k, n)}\\&=\frac{E\left\{Y_{k}^{2}(n)\right\}}{\lambda_{d}(k, n)}-1\\&=E\left\{\lambda_{k}(n)-1\right\}\end{aligned}\tag3 ξk​(n)​=λd​(k,n)E{Ak2​(n)}​=λd​(k,n)E{Yk2​(n)−λd​(k,n)}​=λd​(k,n)E{Yk2​(n)}​−1=E{λk​(n)−1}​(3)

一般所说的SNR就是指先验信噪比，维纳滤波器就是关于先验信噪比的函数
$$W=\frac{\xi }{1+\xi }\text{\hspace{1em}(4)}$$
后验信噪比也叫瞬时信噪比，如谱减公式中用到的就是后验信噪比
$$\begin{array}{rl}{A}^2& =|Y{|}^2-{\lambda }_d\\ & =\frac{1}{{\lambda }_d}(\gamma -1)\end{array}\text{\hspace{1em}(5)}$$
因为谱减法是直接使用瞬时信噪比，因此会更容易产生音乐噪声，当然，也有很多的改进方法减小音乐噪声的影响。

合并(1)、(3)两式得到以下形式
$${\xi }_k(n)=E\left\{\begin{array}{c}\frac{1}{2}\frac{A_k^2(n)}{{\lambda }_d(k,n)}+\frac{1}{2}[{\gamma }_k(n)-1]\end{array}\right\}\text{\hspace{1em}(6)}$$
这里还有一个烦人的期望符号$E$，还是老方法，用一阶递归平滑代替时间平均，则估计得到的${\xi }_k(n)$为下式：
$$\widehat{{\xi }_k}(n)=\alpha \frac{{\hat{A}}_k^2(n-1)}{{\lambda }_d(k,n-1)}+(1-\alpha )max({\gamma }_k(n)-1,0)\text{\hspace{1em}(7)}$$
其中${\hat{A}}_k^2(n-1)$表示上一帧的幅度估计值。

(7)式就是著名的判决引导(Dicision-Directed)公式，被广泛用在了降噪算法的大街小巷里

例如，webrtc的单通道降噪算法源码里可以看到如下模块

// Compute prior and post SNR based on quantile noise estimation.
// Compute DD estimate of prior SNR.
// Inputs:
//   * |magn| is the signal magnitude spectrum estimate.
//   * |noise| is the magnitude noise spectrum estimate.
// Outputs:
//   * |snrLocPrior| is the computed prior SNR.
//   * |snrLocPost| is the computed post SNR.
static void ComputeSnr(const NoiseSuppressionC *self,
                       const float *magn,
                       const float *noise,
                       float *snrLocPrior, float *logSnrLocPrior,
                       float *snrLocPost) {
    size_t i;

    for (i = 0; i < self->magnLen; i++) {
        // Previous post SNR.
        // Previous estimate: based on previous frame with gain filter.
        float previousEstimateStsa = self->magnPrevAnalyze[i] /
                                     (self->noisePrev[i] + 0.0001f) * self->smooth[i];
        // Post SNR.
        snrLocPost[i] = 0.f;
        if (magn[i] > noise[i]) {
            snrLocPost[i] = magn[i] / (noise[i] + 0.0001f) - 1.f;
        }
        // DD estimate is sum of two terms: current estimate and previous estimate.
        // Directed decision update of snrPrior.
        snrLocPrior[i] = 2.f * (
                DD_PR_SNR * previousEstimateStsa + (1.f - DD_PR_SNR) * snrLocPost[i]);
        logSnrLocPrior[i] = logf(snrLocPrior[i] + 1.0f);
    }  // End of loop over frequencies.
}