Effect of overlapping percentage on STFT output

Question

I know STFT is generally applied to non-stationary signals but I tried to apply it to a stationary signal to get a working knowledge.

I created a stationary signal composed of three frequencies as below:

x = 3*cos(2*pi*30*t + phi1) + 2*cos(2*pi*45*t - phi2) + 1*cos(2*pi*70*t + phi3); 

I then performed STFT (Short-Time Fourier transform) on this signal using hann window of length = 128.

1. I have tried ranging the overlapping percentage from 75% to 0% (no overlap) but cannot see and difference in the spectrogram generated. Why could that be?

2. On varying the length of the window (keeping overlapping percentage same), the bright lines in the spectrogram thickens or thins. Why could that be?

I am using MATLAB's STFT function and documentation can be found here:1

Edit:

I tried a signal with varying frequency over time. As pointed out, decreasing overlap percentage results in coarser time grid.

However, when increasing window size from 64 to 128 keeping overlap percentage same (75% for both), and 128-point FFT, in both cases the STFT is calculated for 65 frequencies (128/2 + 1). And the 64-hann window gives a better result. Does that mean that a smaller window gives better results almost everytime? Obviously, I understand that a smaller window would mean more computation cost.

Another experiment made me realize that keeping everything constant (window size, overlap), and increasing the N in N-DFT gives better results.

Answer 1

My answers will be intuitive, I hope. There are more rigorous mathematical arguments that can be made, but your example is not stochastic or varying in frequency, so they are not necessary here.

1. Incomplete but intuitive answer: The spectrum estimate does not appear to change with changing overlap because the spectrum is constant. But see below for more details on block-size, overlap, and time resolution versus frequency resolution.

2. If you feed a block of $N$ samples to an FFT (the algorithm that outputs a discrete Fourier transform (DFT)), then the output has $N$ "frequency bins". As $N$ grows, you have more frequency bins, and you have finer resolution in frequency a finer grid of frequencies. Your example has just 3 constant frequencies to estimate, so as the frequency resolution grid of frequencies becomes finer, the "weight" will be concentrated in frequency bins closer and closer to the 3 frequencies. As a result, the lines in the spectrogram will be narrower as $N$ increases. You can also change the window; the thickness of the lines might change as you change the window.

Note that as the block-size ($N$) increases, the waiting-time between feeding blocks of samples to the FFT increases. Hence, as you get greater frequency resolution a finer grid of frequencies, you have lesser time resolution a coarser grid of times. Overlapping blocks compensates for that a little bit: as the overlap increases, the waiting-time between FFTs shrinks. The cost is more computation.

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I offer the diagram below to address @Lobster3321's comment on this answer. The diagram has been corrected per @OverLordGoldDragon's comment on the location of the insertion of zeros for zero-padding.

The diagram below gives a high-level view of the order of processing. If we had 0% overlap, then we would have a new column in the spectrogram after $N$ samples. With, 50% overlap, on the other hand, we have a new column of the spectrogram after just $N/2$ samples.

Answer 2

The DFT/FFT is a block transform, taking the inner-product of a signal vector with rows/columns of a DFT matrix. When applied in a block processing way, it makes sense to do this either back-to-back or with e.g. 50% overlap.

I find it rewarding to take this to the extreme. If you see the DFT matrix not as a block matrix multiplication, but rather as a set of convolution kernels, one can do a convolution for every sample shift of the input signal. Inner-products that are shifted one sample at a time.

With the understanding of maximally overlapped DFT processing as a bank of convolution filters, doing non-overlapped or 50% overlapped STFT is just a kind of degeneration. Namely, dropping most of the convolution results in order to reduce complexity.

Answer 3

Short answer but I'm posting because the other answers have correct general descriptions but are missing the point past question 1:

1. Congratulations, you've generated perfect sines and padded them correctly. Ordinarily you'd observe differences near boundaries. What's happening is, STFT of a pure sine is a perfectly horizontal line in time-frequency, since, well, it's "time vs frequency" and if the frequency never changes then it's a straight line. "Overlap" is alias for "hop size", i.e. stride of convolutions - it's identical to subsampling the STFT along time, like STFT_large_hop == STFT_small_hop[:, ::20]. If this is unclear, see top visual here, where the only difference is the window width doesn't change.

2. Changing window length by itself won't do this. Odds are, you're feeding "length" argument to a generating function like scipy.signal.windows.hanning, which necessarily generate a wider window. The thinning is a consequence of greater frequency resolution, or wider in time window: better separability along frequency, worse along time.

I also discourage the "windowed Fourier" interpretation of STFT as the sole or best one: STFT is convolution with complex bandpass filters, windowed DFT is one way to get there.

Does that mean that a smaller window gives better results almost everytime?

Narrower window is always superior for single-component signals (up to a limit that we likely won't encounter in practice) - i.e. one line you can draw in time-frequency, left-to-right, without lifting your hand. Details in this post.