# 2.2 Short time fourier transform

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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.

The short-time Fourier transform is defined as

$X(, m)=(\mathrm{STFT}(x(n)), \mathrm{DTFT}(x(n-m)w(n)))=\sum_{n=()}$ 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, $\left|X(, m)\right|$ , 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 $=2\pi$ .

$\forall k, 0\le k\le N-1\colon {}_{k}=\frac{2\pi }{N}k$
We then have the discrete STFT,
$({X}^{d}(k, m), X(\frac{2\pi }{N}k, m))=\sum_{n=0}^{R-1} x(n-m)w(n)e^{-(in)}=\sum_{n=0}^{R-1} x(n-m)w(n){W}_{N}^{-(kn)}={\mathrm{DFT}}_{N}((n, \left[0 , R-1\right], x(n-m)w(n)), \text{0,0})$
where $\text{0,0}$ 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 ${X}^{d}(k, Lm)$ where $L$ is the time-skip. The relation between the time-skip, the number ofoverlapping samples, and the block length is $\mathrm{Overlap}=R-L$

Match each signal to its spectrogram in .

## 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

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

## Effect of l and n

A spectrogram is computed with different parameters: $L\in \{1, 10\}$ $N\in \{32, 256\}$

• $L$ = 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?

$L$ 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. $R\in \{120, 256, 1024\}$ $L\in \{35, 250\}$ where

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

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

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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