The relationship between the prior signal-to-noise ratio and the post-test signal-to-noise ratio in noise suppression

reproduced in the original: https://user.qzone.qq.com/314138065/blog/1442843834 thank you very much. in the noise suppression algorithm, the spectral subtraction algorithm uses the post-verification signal-to-noise ratio, the Wiener filter uses the prior signal-to-noise ratio, the MMSE (minimum mean square error) algorithm uses the prior signal-to-noise ratio, but also uses the posterior signal-to-noise ratio, then naturally presents a problem in the process of noise reduction, prior The signal-to-noise ratio is quite large. This conclusion can be obtained by verifying that the prior SNR is the main parameter affecting the noise suppression, and the posterior signal-to-noise ratio is the auxiliary parameter.

Then what is the relationship between the prior SNR and the posterior signal-to-noise ratio, and here is an analysis.
Let's look at one of the following formulas:
Here k is the frame number, M is the frequency, the prior signal-to-noise ratio equals the power of the Pure Speech signal (X) divided by the power of the noise signal (d), assuming that the voice signal is smooth, and the noise is not related to the voice signal, then the noise power (Y) minus the noisy power (d) divided by the noise The posterior signal-to-noise (gamma) ratio minus 1.
In addition, according to the relationship between the prior signal-to-noise ratio and the posterior signal-to-noise ratio in statistics, it is known:

Here, we find that the second formula is very similar to the first one, and if we take the values of the two formulas in one half and the other, we can get the third formula:

Again, if the weight of 1/2 here becomes a variable a, the upper type becomes:

This formula is a well-known decision-guidance formula, many noise reduction algorithms use this formula to estimate the prior Snr.
Reference: https://user.qzone.qq.com/314138065/blog/1442843834