# Noise removal where gains are >1

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?
• thanks @Gilles for reformatting Commented Jan 19, 2021 at 22:54
• You're welcome :) Commented Jan 20, 2021 at 8:38