2.2 Short time fourier transform

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.

• blocks are overlapping
• each block is multiplied by a window that is tapered at its endpoints.

• 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

• 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 $=2\pi$ .

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 $$\mathrm{Overlap}=R-L$$

Spectrogram example

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

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 $N$ .)

$L$ 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.

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.