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Old Oct 20th 2017, 08:27 AM   #1
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Extraction of sinusoidal waves based on beating pattern

I would like to ask if someone might have a tip how to approach a problem of sinusoidal signal separation in a musical recording sample. There is a recording at 44100Hz of 1,3sec (56855 samples) length and we have the knowledge that up to approx. smp 17100 there is just one sinusoid (1-st piano harmonic) – let us call it SIN-A. Starting from smp 17100 up to ca. 49600 there is another sinuosoid SIN-B overlaid with half a tone difference that causes amplitude beating. There is also another component apparently mixed up between 23000 and 30000 but that we can leave out for now. I am most interested in ideas how to best separate the two main components that cause cisible amplitude beating between 17100 and 49600. The first graph from the top – let’s call it Time1 - shows the input time points with all the sinusoids combined. What I am also given is the frequency and phase points of the first sinusoid SIN-A throughout the 17100 – 49600 region. This is output from a DSP filterbank sinusoid tracking software that follows SIN-A using frequency and phase continuity information. However this DSP software cannot cope with frequency beating caused by SIN-A + SIN-B. To separate SIN-A and SIN-B a gave it a first simple try that is I linearly approximated SIN-A amplitude linking the region boundary points. The result is shown on the second graph from above – Time2. Then I did a subtraction Time1 – Time2 hoping that magically I get the SIN-B without beating. But it did not happen – the result is shown on the third graph from the top – it has very irregular amplitude envelope which means that my approach failed knowing that SIN-B is another piano key stroke so it should have a more regular Attack-Decay-Sustain-Release and generally monotonic amplitude envelope. It seems that maybe independent modelling of SIN-B based on the beating pattern may bring better results ? How would you approach such a modelling – that is what would be the equation for the amplitude, frequency and starting phase of SIN-B if I know the combined time points and SIN-A frequency and phase values but not the amplitude in the 17100 – 49600 region ?

Regards, Pawel
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Old Oct 20th 2017, 11:24 AM   #2
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The classic way of approaching this problem would probably be via the Fourier Transform.
However this method basically assumes that your signal is constant through the sample, so you would probably have to select a portion of your time history, where the amplitudes are varying least.
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