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Introduction to the Short Time Fourier Transform, which includes it's definition and methods for its use.

Short time fourier transform

The Fourier transforms (FT, DTFT, DFT, etc. ) do not clearly indicate how the frequency content of a signal changes over time.

That information is hidden in the phase - it is not revealed by the plot of the magnitude of the spectrum.

To see how the frequency content of a signal changes over time, we can cut the signal into blocks and compute thespectrum of each block.
To improve the result,
  • blocks are overlapping
  • each block is multiplied by a window that is tapered at its endpoints.
Several parameters must be chosen:
  • Block length, R .
  • The type of window.
  • Amount of overlap between blocks. ( )
  • Amount of zero padding, if any.

Stft: overlap parameter

The short-time Fourier transform is defined as

X m STFT x n DTFT x n m w n n x n m w n n n 0 R 1 x n m w n n
where w n is the window function of length R .
  • The STFT of a signal x n is a function of two variables: time and frequency.
  • The block length is determined by the support of the window function w n .
  • A graphical display of the magnitude of the STFT, X m , is called the spectrogram of the signal. It is often used in speech processing.
  • The STFT of a signal is invertible.
  • One can choose the block length. A long block length will provide higher frequency resolution (because the main-lobeof the window function will be narrow). A short block length will provide higher time resolution because lessaveraging across samples is performed for each STFT value.
  • A narrow-band spectrogram is one computed using a relatively long block length R , (long window function).
  • A wide-band spectrogram is one computed using a relatively short block length R , (short window function).

Sampled stft

To numerically evaluate the STFT, we sample the frequency axis in N equally spaced samples from 0 to 2 .

k 0 k N 1 k 2 N k
We then have the discrete STFT,
X d k m X 2 N k m n 0 R 1 x n m w n n n 0 R 1 x n m w n W N k n DFT N n 0 R 1 x n m w n 0,0
where 0,0 is N R .

In this definition, the overlap between adjacent blocks is R 1 . The signal is shifted along the window one sample at a time. That generates more points than is usuallyneeded, so we also sample the STFT along the time direction. That means we usually evaluate X d k L m where L is the time-skip. The relation between the time-skip, the number ofoverlapping samples, and the block length is Overlap R L

Match each signal to its spectrogram in .

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Spectrogram example

The matlab program for producing the figures above ( and ).

% LOAD DATA load mtlb; x = mtlb; figure(1), clf plot(0:4000,x) xlabel('n') ylabel('x(n)') % SET PARAMETERS R = 256; % R: block length window = hamming(R); % window function of length R N = 512; % N: frequency discretization L = 35; % L: time lapse between blocks fs = 7418; % fs: sampling frequency overlap = R - L; % COMPUTE SPECTROGRAM [B,f,t] = specgram(x,N,fs,window,overlap); % MAKE PLOT figure(2), clf imagesc(t,f,log10(abs(B))); colormap('jet') axis xy xlabel('time') ylabel('frequency') title('SPECTROGRAM, R = 256')

Effect of window length r

Narrow-band spectrogram: better frequency resolution

Wide-band spectrogram: better time resolution

Here is another example to illustrate the frequency/time resolution trade-off (See figures - , , and ).

Effect of window length r

Effect of l and n

A spectrogram is computed with different parameters: L 1 10 N 32 256

  • L = time lapse between blocks.
  • N = FFT length (Each block is zero-padded to length N .)
In each case, the block length is 30 samples.

For each of the four spectrograms in can you tell what L and N are?

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L and N do not effect the time resolution or the frequency resolution. They only affect the'pixelation'.

Effect of r and l

Shown below are four spectrograms of the same signal. Each spectrogram is computed using a different set of parameters. R 120 256 1024 L 35 250 where

  • R = block length
  • L = time lapse between blocks.

For each of the four spectrograms in , match the above values of L and R .

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If you like, you may listen to this signal with the soundsc command; the data is in the file: stft_data.m . Here is a figure of the signal.

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Source:  OpenStax, The dft, fft, and practical spectral analysis. OpenStax CNX. Feb 22, 2007 Download for free at http://cnx.org/content/col10281/1.2
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