I'm trying to understand STFT overlapping, why segments are concatenated and what are the consequences of this.
My implementation (from other questions and repositories found) of the STFT in Python is:
def stft(x, L, overlap, window): # Linear Spectrum (LS) [V] opoints = int(L*overlap) hopsize = L - opoints X = np.array([(2 * np.abs(np.fft.fft(window*x[i:i+L])[:L//2])) / np.sum(window) for i in range(0,len(x)-L,hopsize)]) return X
where, x is the time-domain signal sampled at 44.1kHz, L is the FFT window size (4096), overlap is the overlapping factor [0, 1), and window is the window funtion used (Hanning). The absolute value of half the complex dft is scaled by
2/sum(win) so that the units of the linear spectrum are the same as for the input signal (V).
I used to understand the frequency resolution is
fs/L and the width of the time slices or time resolution is
L/fs, so that the number of time windows is
N/L. However, if using overlapping
(overlap!=0), the number of time windows increases
- Why are overlapped segments concatenated instead of added and what are the consequences of this regarding the time axis? (See picture below)
- Is compressing the concatenated segments into the original duration of the signal correct? (i.e 3 segments of 1 second into 2 seconds, see picture below)
Now, I do understand:
Overlap-shift and compression of the time axis of the overlapped segment into the original duration is one way to make a smoother spectrograph image
Overlap is needed in order to avoid lossy signals caused by the window
However, from @user31990 comment I still need to clarify:
- What's the difference between the different methods/implementations?
Implementations I've found/understood:
- Overlap and concatenate side-by-side the overlapped blocks (increasing the number of time windows and compressing the x time axis to the original duration)
- Overlap adding the segments of the block that are overlapped together (the number of time windows doesn't increase = N/L)
- Overlap save (zero-pad the beginning of the block and discard the transformed segment, the number of time windows doesn't increase)
Overlap shift with zero-pad
- I don't totally understand the overlap shift with zero-pad, what is it doing?
My implementation is an example of the overlap and concatenate method.
- Are This and Google's pytfd stft implementations examples of the overlap shift with zero-pad method?
I was confused about the methods, question is answered/solved.
Zero-padding is a method that can be used: 1. To obtain an FFT of size N of a window whose size is smaller 2. To increment the size of the FFT, interpolating values in the frequency axis and obtaining smoother data.
Overlap-add and overlap-save are methods for the synthesis of an IFFT output in order to recover the original signal x.