Background noise removal from audio signals using FFT Python

Question

I am currently doing a project at university where I am distinguishing between different instruments playing notes of the same pitch using python.

I have recorded various notes on different instruments using a microphone attached to a computer.

I have also recorded background for the room.

So far I have plots for different notes on different instruments, where on the y-axis I have the amplitude in dB: 20*log10(|FFT(signal)|)

And on the x-axis I have DFT sample frequencies

Some of the harmonic peaks are small enough (or the background is large enough) for noise to be a factor-(can't post images as I'm a noob!) my problem is calculating the level of uncertainty in the height of the peaks when accounting for background noise.

My question is:

Well, how to calculate the level of uncertainty in the height of the peaks (their relative harmonic amplitudes) when accounting for background noise.

Some ideas:

What dB threshold I should use when classifying what is a harmonic peak and what is attributable to noise (should I discount a peak lower than the maximum backgound (~28000dB) or the mean (~15000) or perhaps twice one of these values)?

Also, to take account of the noise introduced by background, is it legitimate to subtract the value in FFT bin n for the background, from FFT bin n for my instrument recording?

Also I have looked at this post how can the noise be removed from a recorded sound,using fft in MATLAB? , there seem to be very differing opinions on there.

If it's relevant I can post segments of my code- wary of putting too much up though in case of classmate plagarism.

Links to literature that would help with the project would be very much appreciated. (Still at the stage where I'm plotting the data every which way I can think of to look for distinguishing attributes for each instrument).

Thanks in advance

Answer 1

You seem to be asking many questions. Let me start by answering your first one:

Well, how to calculate the level of uncertainty in the height of the peaks (their relative harmonic amplitudes) when accounting for background noise.

You would expect the sound to summate linearly (to a first order approximation). The natural thing to do would be to do some recordings of only the background and then measure the mean amplitude and standard deviation of the harmonics in the background.

As an example, say you are looking at 3 harmonics - 20KHz, 11KHz and 33KHz. Do some recordings of only background and you find mean amplitudes of 1.3dB 2.2dB and 2.3dB with standard deviations of say +/-0.1, +/-0.2 and +/-0.4dB. You now have an uncertainty estimate and a mean background harmonic to subtract from.

There are smarter ways to do this but it's a start.

Now then, to get on to your second question

What dB threshold I should use when classifying what is a harmonic peak and what is attributable to noise (should I discount a peak lower than the maximum backgound (~28000dB) or the mean (~15000) or perhaps twice one of these values)?

If a peak is within the mean + the uncertainty (one or two standard deviations, this is arbitrary really and depends on convention) you can say it's significant. Eg if you find that noise level at 3KHz is 1.2dB with an uncertainty of +/- 0.3dB and you measure your harmonic to be 1.3dB with an uncertainty of (measured in the same way) of 0.1dB then it's not significant.

Now for the third part:

Also, to take account of the noise introduced by background, is it legitimate to subtract the value in FFT bin n for the background, from FFT bin n for my instrument recording?

Yes (generally speaking). If you really want to convince yourself of this you can either A)do some simulations with summating waves and doing an FFT of them, B)do an experiment and the same as in A or C)Go through the mathematics of Fourier transforms.

With regard to literature, I think that would depend on what you're doing specifically, if you're a physics student "Mathematical Methods in the Physical Sciences" by Mary Boas treats Fourier transforms well, if you're a computer scientist/engineer you probably want something different.

Let me know if you need more help.