Mathematical justification for zero padding?

Question

This question asks what's the point of zero padding. The accepted answer is certainly very insightful, but I don't understand a big chunk of it:

Zero padding allows one to use a longer FFT, which will produce a longer FFT result vector.

A longer FFT result has more frequency bins that are more closely spaced in frequency. But they will be essentially providing the same result as a high quality Sinc interpolation of a shorter non-zero-padded FFT of the original data.

This might result in a smoother looking spectrum when plotted without further interpolation.

Although this interpolation won't help with resolving or the resolution of and/or between adjacent or nearby frequencies, it might make it easier to visually resolve the peak of a single isolated frequency that does not have any significant adjacent signals or noise in the spectrum. Statistically, the higher density of FFT result bins will probably make it more likely that the peak magnitude bin is closer to the frequency of a random isolated input frequency sinusoid, and without further interpolation (parabolic, et.al.).

What exactly is the meaning of "resolve" and "resolution" here, and how, mathematically, is it apparent that zero padding does not increase resolution. Dually, how is apparent mathematically that zero padding means interpolation?

Answer 1

Resolution (in the context of spectral analysis) is the ability to distinguish two or more closely spaced sinusoidal components in a spectrum. If you can't resolve them, this means that you only see one maximum instead of two or more. Resolution is determined by the window length, i.e., by the number of time domain samples (given a certain sampling rate). Windowing will smear out or broaden any narrow band components in the spectrum. Remember that windowing, i.e., multiplication in the time domain, corresponds to convolution in the frequency domain, and this convolution with the spectrum of a standard window is a sort of averaging operation in the frequency domain, hence broadening any (frequency domain) impulses.

Resolution can only be improved by adding more data points, i.e., more information, to your analysis window. Zero-padding obviously doesn't add any information. What zero-padding does is increase the sampling density of the discretet-time Fourier transform (DTFT) of the windowed signal. For a length $N$ signal, the DTFT is given by

$$\tilde{X}(\omega)=\sum_{n=0}^{N-1}x[n]e^{-jn\omega}\tag{1}$$

The discrete Fourier transform is

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{2}$$

Comparing $(1)$ and $(2)$, we see that the DFT is just a sampled version of the DTFT:

$$X[k]=\tilde{X}\left(\frac{2\pi k}{N}\right)\tag{3}$$

Zero-padding increases the number of equidistant samples taken from the DTFT $\tilde{X}(\omega)$.

A detailed analysis of the fact that zero-padding corresponds to interpolation in the frequency domain is given in this answer.

Answer 2

Re: the first part of you question:

There are multiple meanings of the term resolve in common use. The one in use may depend on the context.

In optics, the term resolution is commonly used to indicate the minimum spacing of items that can be determined to actually be separate, such as 2 image lines or 2 spectral peaks with at a dip of at least 3 dB dip in magnitude between them (e.g. two peaks, not just one fat hump). For a spectrum, this resolution can not be increased by zero-padding an FFT. Interpolation will not make a real dip between magnitude peaks magically appear. (One can make inflection points appear in an interpolated curve by overfitting, but these inflections have usually have little to do with the actual signal content.)

Getting a separation dip, assuming multiple sinusoids close in frequency are actually present in the signal, requires more information about the signal, e.g. a longer interval of sampling. The longer one samples, the more likely it will be that two sinusoids close in frequency will be orthogonal, or close to orthogonal, to different sets of FFT basis vectors, and thus show up in separate sets of FFT result bins, instead of mushed together in the same result bins.

But often people use the term resolve to indicate the precision of measurement of the location of a single spectral frequency peak (e.g. what is the frequency of the Nth harmonic of this stationary pitch?). If the spectral peak is sufficiently above the surrounding noise floor, this resolution can be improved (beyond just using the frequency of the bin containing a local maxima in magnitude) by either zero-padding the data and using a longer FFT, or by interpolating between the FFT result points to determine if there is a peak in magnitude between these points in the interpolated curve. (For periodic Sinc interpolation, see Matt L.'s demonstration that these are equivalent here: https://dsp.stackexchange.com/a/24426/4298 , other forms of interpolation can often provide close approximations to this result.)

Resolution is also commonly used to indicate the density of points used in a graphic or plot of the data, and again this can be increased by using a longer zero-padded FFT, or by many various common numeric interpolation methods.