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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?

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  • $\begingroup$ Everyone answered as Additive White Noise is the issue. When you say noise, do you mean White Noise? As the interference you describe isn't white noise (Hum, interference). $\endgroup$
    – Royi
    Oct 18, 2019 at 22:04
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    $\begingroup$ @Royi it's not true that the answers only apply to white noise. The noise needn't be white – just uncorrelated between observers, not uncorrelated within the same observation over time. In fact, the noise can be as white as it could be, if it's the same for all receivers, combining won't reduce it. $\endgroup$
    – Marcus Müller
    Oct 19, 2019 at 21:40
  • $\begingroup$ You're correct. I meant white in the meaning between observations. So if you stack the samples it needs to be white along the stacking dimension and not time time dimension. $\endgroup$
    – Royi
    Oct 19, 2019 at 21:46

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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".

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    $\begingroup$ One point to think about - His description doesn't match White Additive Noise. $\endgroup$
    – Royi
    Oct 18, 2019 at 22:04
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    $\begingroup$ that's true, the electrical hum for example will probably even get worse! but that's why he needs a signal model to begin with. $\endgroup$
    – Marcus Müller
    Oct 18, 2019 at 23:26
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    $\begingroup$ Yep. Probably with some tricks adaptive filtering will be a better choice to handle this case. $\endgroup$
    – Royi
    Oct 18, 2019 at 23:36
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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

  • The paths from the source to the 2 locations aren't the same, so they will differ to some extent, so the copies may not add coherently, even if you have an exact alignment.
  • Microphones have directional responses so there is another mechanism that introduces differences between your 2 copies.
  • Ideally, the 2 phones would have synchronized sampling, but they probably have some timing differences which makes the alignment more of a challenge.

You can try. No one is likely to say it is not possible,.

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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.

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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

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    $\begingroup$ Welcome to Signal Processing! Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. $\endgroup$
    – Andrew T.
    Oct 20, 2019 at 16:58
  • $\begingroup$ And the link doesn't work for me. You really should make it public... and do as Andrew says. $\endgroup$
    – Peter K. ♦
    Oct 21, 2019 at 15:18

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