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I have a recorded signal which contains several frequency components with stochastic phase and amplitude changes and want to extract the true signal with the other frequency components and noise removed. I have performed several filtering techniques to attempt to achieve this. However I need a robust measure of how correct my filtered signal is and I do not have a method of measuring the signal on it's own, without the noise and other frequency components, in order to validate it that way. One method I have applied is minusing the filtered signal from the raw signal in time space and then taking the fft/psd (power spectral density) of the remaining signal and I see that the peak is gone, is this a robust measure of checking the signal I am tracking is correct? Is there any other methods for analysing this without having a way of measuring the true signal?

Some examples of the raw and differences signals for a couple of different filtering methods are shown below: (blue is the PSD of the original raw signal, and green is the PSD of [raw - filtered])

The psd of the difference of the signals with filtering method 1 The psd of the difference of the signals with filtering method 2

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    $\begingroup$ This is an interesting problem and approach you took. Notice in your second result that you are effectively removing everything within the bandwidth of interest which means signal plus noise in band, while the first approach may be more effective in that noise still exists. Although there is no way to tell from what you described what is actually "signal of interest" versus noise, even within a given bandwidth. It is unfortunate that you cannot have a reference waveform for developing metrics for your various approaches, is this really not the case? $\endgroup$ – Dan Boschen Jun 24 '17 at 9:31
  • $\begingroup$ And I am not suggesting a known waveform for ongoing adaptation, but just a known case that you can test your blind approaches with? Let me know if you could actually do this for test purposes and I will suggest a robust approach for "correctness" of any adaptation approach. $\endgroup$ – Dan Boschen Jun 24 '17 at 9:33
  • $\begingroup$ I believe one my colleagues has developed a simulation of our system, this might be used as a known waveform by which I can compare. I could use his simulation to generate a known waveform and pollute it with noise and then apply my filtering techniques. If I do this, what is your suggested robust approach for testing the "correctness" of the filtered signal? $\endgroup$ – SomeRandomPhysicist Jun 24 '17 at 11:49
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Given you can have a "reference waveform" representing the characteristics of your expected signals, to test the "correctness" of the signal extracted from noise use the Normalized Correlation Coefficient. I reference other posts below where I give the details of computing and using the Normalized Correlation Coefficient:

Noise detection

How can I find SNR, PEAQ, and ODG values by comparing two audios?

This will be an interesting approach to compare your different algorithms as the result of removing portions of your signal of interest or keeping noise contributions will both reduce the correlation coefficient as expected; so this will be an excellent approach to accurately indicate which algorithm you use is optimum for your waveform.

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