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I am currently working on some noise removal algorithms and have come across what seems to be an oversight in the design of some of these algorithms.

Two of the algorithms I am looking at use the PSD of the noisy input signal, and calculate the gains that should be applied to retrieve the noise-free signal.

However, both of these algorithms specify the gains should be $\in [0,1]$, with one (an autoencoder) specifying a sigmoid activation should be applied to its output to limit the gains between these values, and another mentioning the gains are $\in [0,1]$. In reality, however, the gains can be any positive integer. Consider a signal $SN$ that is the sum of two signals $S$ and $N$, where $S$ is our noise-free signal and $N$ is our noise. Then, to calculate the (relative) PSD of these signals, we take the FFT of each, and square the absolute value. Now, given $\mathcal F\{x\}$ is the real FFT of $x$, and assuming our signals are real valued:

\begin{align} SN &= S + N\\ \mathcal F\{SN\} &= \mathcal F\{S\} + \mathcal F\{N\} \end{align}

However;

$$ \big(\mathcal F\{SN\}\big)^2 = \big(\mathcal F\{S\} + \mathcal F\{N\}\big)^2 \neq \big(\mathcal F\{S\}\big)^2 + \big(\mathcal F\{N\}\big)^2 $$

Therefore, there are some cases where our speech signal, for a given frequency, may have more power than our combined signal, hence our gains should be $> 1$.

  • Why do the algorithms I have read assume gains should be $\in [0,1]$? Is this because we cannot be certain as to the nature of the input, and so letting the gains be $> 1$ could lead to much more disastrous artifacts?
  • And if the noise we are removing is well modelled, does anything stop us from setting these gains to be greater than the combined signal?
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  • $\begingroup$ thanks @Gilles for reformatting $\endgroup$ – Jack_P Jan 19 at 22:54
  • $\begingroup$ You're welcome :) $\endgroup$ – Gilles Jan 20 at 8:38

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