Zero-padding the middle of a signal

Question

I sample a signal at a certain frequency for a finite amount of time to get a sequence $$(x_n)_{n=1}^N = (x_1, x_2, ... , x_N)$$

with the intention of analyzing its power spectral density by calculating its DFT.

I decide that the tail end of the sequence shouldn't be analyzed (that is, I want to discard all $x_n$ for $n\gt a$ for some $a\leq N$), but I want the DFT to have the same number of samples (in this case, $N$ samples).

One approach would be to truncate the sequence at $x_a$, remove the mean, apply a Hamming window to this truncated sequence, pad the truncated sequence with $N-a$ zeroes, and then calculate its DFT.

Question: Let's say that the section I don't want to analyze is an interval in the middle of the sequence. How do I calculate the DFT of this and make sure the DFT has $N$ samples? Should I remove the mean, apply two Hamming windows (one each on each section I don't want to discard), set the elements in the middle that I want to discard to zero, and calculate the DFT?

This is related to my other question here.

Answer 1

If there's a region in the middle of the sequence that you want to discard, you can't just set those samples to zero and compute the DFT, because the DFT would reflect the fact that your sequence suddenly jumps to the value zero, and then suddenly jumps back again to non-zero values. So the zero values would significantly contribute to the result of the DFT, which is almost the opposite of what you want to achieve.

What you should do is consider the two relevant parts at the beginning and at the end as two realizations of the underlying process, and compute two DFTs, which can then be averaged, just like in Bartlett's method. Of course, each sub-sequence can (and should) be windowed and zero-padded before computing the DFT.

Answer 2

The transform of a rect rotated in a DFT window so it wraps around the window edges is still a Sinc in magnitude. So the effect (Sinc convolution) will be the same as zero padding on just one side of a data vector, except the Sinc will be twisted in phase. But the twist will be coherent (the DFT basis vectors won't be broken). Thus, the advantage of rotating a rect around a FFT aperture to zero pad the middle of a vector is that the longer FFT will give you coherent gain over just using 2 shorter FFTs + Welch or Bartlett summing.

This coherent gain may or may not be meaningful, depending on whether the underlying signal portions are together coherent or not. If the middle of your signal is corrupted by a strong interfering signal burst or ionospheric dropout, your signal of interest might be coherent across the gap. But two separately played musical notes are unlikely to be so.

For narrowband signals that are narrower than the bin separation of DFT result of just the short segments, but are coherent together over a longer time interval, using the phase vocoder algorithm is another method of getting coherent gain for improved spectral frequency estimation from multiple short FFTs. Phase vocoding is usually done with overlapping FFT windows, but the theory works the same for separated windows as long as the sampling clock (and data!) stays strongly synchronized over the total longer time interval. So using separated Von Hann (etc.) windows over portions of a signal and then using a longer FFT across all of the windows may be related to an efficient method of phase vocoding across an entire DFT spectrum.