STFT: why overlapping the window?

Question

For STFT, we impose window of certain size onto the original signal, then we perform fft on each window. The uncertanty about frequency and time is determined by the width of the window, however, I can't understand what is the point of having overlap windows...

If we have a signal, for instance, why can't we just divide the signal into 6 trunks (non-overlapping window), and then we perform fft on each of those trunks?

Maybe let me make it more clearly in my application. I am going to mostly dealing with 60Hz power line wave, and occationally, we want to monitor the 180Hz transient effect at the power line. Since the signal will be mostly periodic, should I use window then?

Answer 1

1. We always want to apply some kind of a window function in order to minimize the effect of leakage. This makes rectangular window (lack of any windowing) case never used, this is why:

2. Any tapering function used is almost always decreasing to zero at boundaries.

This is why we are losing some data. In order to retrieve that somehow you will usually do 50% of overlap when processing. This will retrieve whatever was in between.

3. Another thing is that if you apply the Inverse STFT, you should use complementary window, that is summing to 1, i.e. Hanning with 50%.

Finalising - yes, you should pretty much always use windowing in your applications.

For more comprehensive informations please refer to great white-paper:

Heinzel G. - Spectrum and spectral density estimation by the DFT, including a comprehensive list of window functions and some new flat-top windows

Answer 2

why overlapping the window?

Because otherwise loses information, a ton of it. STFT is equivalently convolutions (rather, cross-correlations) with windowed complex sinusoids, i.e. bandpass filtering. For:

• Spectrograms (modulus): the loss is greatest and in every sense. Otherwise, with maximum overlap, the STFT is invertible within a global phase shift, which is a strong inversion, unlike DFT/FFT modulus. This strong inversion, and STFT's robustness properties, are exclusively enabled by overlapping in both domains.
• Extracting phase/amplitude/frequency vs time: there's tremendous aliasing, worst for phase.
• Non-time-frequency uses (phase vocoding, analysis/synthesis): without modulus, the STFT is perfectly invertible as long as hop_size <= window_size (NOLA), which makes a lot of algorithms possible, but said algorithms may still require analytic information (phase, amplitude, etc), which is aliased or not mapped.

hop_size or window_size - noverlap - gap between windows - is the subsampling factor along time. n_fft - size of frequency dimension - is inversely hop_size along frequency. Simple example:

The spectrogram hence wrongly suggests a pure sine where we have strong F.M., despite hop_len=64 STFT being perfectly invertible.

Further reading

• Equivalence between "windowed Fourier transform" and STFT as convolutions/filtering
• Role of window length and overlap in uncertainty principle? (includes a measure of loss)

Answer 3

You could think of a N-point windowed block DFT/FFT (STFT) as a set of N complex FIR filters (running convolution) where we keep only every Mth output. The question then becomes, not «why do we have overlap», but rather «why do we decimate filter outputs».

Because we can. Because we have sufficient information. And because even with the complexity reduction of using using FFTs, running a new one for each input sample would often add too much compute cost.