Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.
Sign up to join this community
I have two (audio) recordings of one analog signal, I used two separate devices (smart-phones) recording from two different locations.
Each recording device is pretty crappy, lot's of electrical hum and interference etc... My assumption is that each recording contains a unique and distinct noise sample.
Is it possible to reconstruct the original pure signal? one which doesn't include the noise from either of the two recordings?
Is it possible to reconstruct the original pure signal?
No, that is information-theoretical impossible. Also, that signal doesn't exist, probably, to begin with ;)
However, you can definitely increase the the SNR simply by averaging; that becomes pretty obvious when you consider the signal of interest to be correlated within your recording, whereas your noise is claimed to be independent.
If you sum up multiple recordings, the correlated signal's amplitudes will increase linearly with the number of observations, whereas your noises' standard variance only with the square number of that.
I have two (audio) recordings of one analog signal, I used two separate devices (smart-phones) recording from two different locations.
Find a signal model for your analog signal – it's not just "any signal" (that would be completely indistinguishable from noise), but it's probably very specific. Ask a new question – "I have these and that signal, they follow my signal model (maybe you need help formulating one, another good question), how do find a low-variance estimator for the parameters for that signal model".
A "pure" signal, No.
A less noisy signal?
Possible? Yes but there are several complications that may make this impractical.
You basically need to align the 2 recordings and then add them. You might gain 3dB in SNR.
but
You can try. No one is likely to say it is not possible,.
As Stanley Pawlukiewicz said: even under ideal circumstance, you can gain 3 dB of SNR per doubling of recordings. I.e., to increase SNR by, say, 15 dB, you'd need to average $$ 2^{\frac{15}{3}} = 2^{5} = 32$$ recordings. That alone shows that the whole thing isn't really practical: it just doesn't do much unless you use a crazy-high number of recordings.
“Ideal circumstances” are: you know exactly all the phase relations of the desired signals, so they'll properly add up 6 dB. If you don't know the phase relations, then the average gain of both signal and noise will be only 3 dB, i.e. you'd gain no SNR at all. Worse, the phases of all frequencies won't add up in the same way: some may indeed add up for 6 dB gain, but other parts may actually cancel, thus giving an uneven frequency response.
Provided that either you've recorded only a single source or both microphones were at the exact same spot, it's technically speaking possible to infer the exact phase relations from the recordings: first find the highest peak in the cross-correlation of the signals† to get the time alignment. Then divide the (complex) Fourier transforms of the aligned signals to get an IR that fine-adjusts the frequency and phase response of one signal to the other. It's reasonably simple to do that with a small e.g. Python script, but again: just not worth it. Maybe there's software available that has this feature, but I wouldn't bet on it.
†If you're unlucky (likely!) and the signals drift or have very different phase response even in the low frequency range, then this may not be enough, you may need to cross-correlate something like a windowed RMS in chunks to get a time-dependent alignment reconstruction.
This is a beautiful way of noise/interference cancellation technique, please go through the link that I am sharing here. Certainly, you may find a way to implement the concept.
https://drive.google.com/file/d/0B2bUtLEhrWp8Wi1JZzdub0U2Wm9JWlZEX290cHByZi1ES3FZ/view?usp=sharing
