先驗信噪比估計

先驗信噪比估計

很多降噪算法如維納濾波、MMSE估計器等都依賴先驗信噪比(priori SNR)信息

定義先驗信噪比(priori SNR)、後驗信噪比(posteriori SNR)如下
$\xi_{k}(n)=\frac{E\left\{A_{k}^{2}(n)\right\}}{\lambda_{d}(k, n)}\tag1$

$\gamma_k(n)=\frac{{Y_{k}^{2}(n)}}{\lambda_{d}(k, n)}\tag2$

上式中的$A_{k}(n)$爲目標信號的幅度譜、$\lambda_d(n)$爲噪聲的功率譜，這裏先假定噪聲平穩且可以在無語音段估計得到。

假設噪聲和語音不相關且爲加性噪聲，則$\xi_k$可以寫成下式
$\begin{aligned}\xi_{k}(n)&=\frac{E\left\{A_{k}^{2}(n)\right\}}{\lambda_{d}(k, n)}\\&=\frac{E\left\{Y_{k}^{2}(n)-\lambda_d(k,n)\right\}}{\lambda_{d}(k, n)}\\&=\frac{E\left\{Y_{k}^{2}(n)\right\}}{\lambda_{d}(k, n)}-1\\&=E\left\{\lambda_{k}(n)-1\right\}\end{aligned}\tag3$

一般所說的SNR就是指先驗信噪比，維納濾波器就是關於先驗信噪比的函數
$W=\frac{\xi}{1+\xi}\tag4$
後驗信噪比也叫瞬時信噪比，如譜減公式中用到的就是後驗信噪比
$\begin{aligned}A^2&=|Y|^2-\lambda_d\\&=\frac{1}{\lambda_d}(\gamma-1)\end{aligned}\tag5$
因爲譜減法是直接使用瞬時信噪比，因此會更容易產生音樂噪聲，當然，也有很多的改進方法減小音樂噪聲的影響。

合併(1)、(3)兩式得到以下形式
$\xi_k(n)=E\begin{Bmatrix}\frac{1}{2}\frac{A_{k}^{2}(n)}{\lambda_{d}(k, n)}+\frac{1}{2}[\gamma_k(n)-1]\end{Bmatrix}\tag6$
這裏還有一個煩人的期望符號$E$，還是老方法，用一階遞歸平滑代替時間平均，則估計得到的$\xi_k(n)$爲下式：
$\hat{\xi_k}(n)=\alpha\frac{\hat{A}_k^2(n-1)}{\lambda_d(k,n-1)}+(1-\alpha)max(\gamma_k(n)-1,0)\tag7$
其中$\hat{A}_k^2(n-1)$表示上一幀的幅度估計值。

(7)式就是著名的判決引導(Dicision-Directed)公式，被廣泛用在了降噪算法的大街小巷裏

例如，webrtc的單通道降噪算法源碼裏可以看到如下模塊

// 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.
}