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Multiplying a kernel and signal spectrum in Fourier domain lead to a circular convolution and not a linear convolution, so in order to it become linear convolution you must zero pad your signal and kernel before taking the Fourier transform (up to M+N-1 where M is the signal's length and N is the kernels's length). (if you compare the blue and orange signals,...

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Zero padding does not increase the frequency resolution— it just interpolates more samples in frequency but does not give you any more information. The frequency resolution in Hz of an non-windowed (rectangular window) Fourier Transform is 1/T where T is the length of your time domain waveform in seconds. You must have a longer data sample in order to get ...

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The best fit time domain solution can be found by constructing two two basis vectors with your known frequency and calculate the coefficients directly. The magnitude and phase can then be directly determined from these values. Let C be a vector of cosine values over your frame and S be a vector of sine values. You then want to find $(a,b)$ so that \$aC + ...

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Carry the units and it becomes understandable. T = 0.03125 seconds per sample Thus your sampling rate is 1/T = 32 samples per second Each bin corresponds to the cycles per frame. Your frame has four samples. Suppose the bin index is k. k (cycles per frame) * 32 (samples per second) / 4 (samples per frame) = 8k cycles per second = 8k Hz So the ...

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