Effect of sampling rate of a signal on its Fourier Transform

Question

I am running some experiments on MATLAB, and I have noticed that, keeping the period fixed, increasing the sampling rate of a sine signal causes the different shifted waveforms in the Fourier transform to become more distinct. They get further apart, I think this makes sense because as the sampling rate increases, the difference between the Nyquist rate and the sampling rate increases too, which creates an effect opposed to aliasing. I have also noticed that the amplitude of the peaks of the transform also increase as the sampling rate increases. Even the DC component (frequency = 0) changes. It's shown as being 0 at some sampling rate, but when increasing the sampling rate it's not 0 anymore.

All the sampling rates are above the Nyquist rate. It seems odd to me that the Fourier transform changes its shape, since according to the sampling theorem, the original signal can be recovered if the sampling rate is above the Nyquist rate, no matter if it's 2 times the nyquist rate or 20 times. Wouldn't a different Fourier waveform mean a different recovered signal?

I am wondering, formally, what's the impact of the sampling rate

Thank you.

Answer 1

You're conflating conversion between time-discrete and time-continuous forms of a signal with reversibility of a transform.

The only guarantee is: For a given transform of some discrete signal, its inverse transform will yield the "same" discrete signal back. The discrete signal is abstract from any frequencies. All that the transform does is take some vector of complex values, and give the dimensionally matching vector of complex values back. You can then take this vector, run an inverse transform on it, and get the "original" vector. I use quotes since there may be some numerical errors that depend on the implementation. As you can see, nowhere does the word frequency appear because it's irrelevant.

So, your real question is then, how to get an FFT with values that are useful for something besides getting the original discrete signal back through an inverse transform. Say, how to get an FFT that will tell a human something nice about the frequency content of a signal. A transform "tweaked" for human usefulness, or for use in further signal processing such as automated music transcription, can't reproduce the original signal anymore after inversion. We're trading off veracity for usefulness. Detailed discussion of this can't really fit into one answer, and is off topic here anyway.

Another of your real questions is how to go between a continuous signal and a discrete signal - how to sample the continuous signal, and how to reconstruct it from its discrete representation. The reconstruction means a function (or process) that will yield the values the signal had at points in time between the samples. Again, this is a big topic.

Answer 2

You are seeing several things when you increase the sample rate:

• most (forward) FFT implementations have an implicit scaling factor of N (sometimes sqrt(N)) - if you're increasing your FFT size as you increase the sample rate (i.e. keeping the time window constant) then the apparent magnitude of the peaks in the FFT will increase. When calculating absolute magnitude values you would normally need to take this scaling factor into account.

• I'm guessing that you are not currently applying a window function prior to the FFT - this will result in "smearing" of the spectrum, due to spectral leakage, and the exact nature of this will be very dependent on the relationship between sample rate and the frequencies of the various components in your signal. Apply a window function and the spectrum should look a lot more consistent as you vary the sample rate.