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Subsections

2.1 The Short Time Fourier Transform

In many applications such as speech processing, we are interested in the frequency content of a signal
locally in time. That is, the signal parameters (frequency content etc.) evolve over time. Such signals are
called non-stationary. For a non-stationary signal, , the standard Fourier Transform is not useful for
analyzing the signal. Information which is localized in time such as spikes and high frequency bursts
cannot be easily detected from the Fourier Transform.

Time-localization can be achieved by first windowing the signal so as to cut off only a well-localized slice
of and then taking its Fourier Transform. This gives rise to the Short Time Fourier Transform,
(STFT) or Windowed Fourier Transform. The magnitude of the STFT is called the spectrogram. By
restricting to a discrete range of frequencies and times we can obtain an orthogonal basis of functions.

The Short Time Fourier Transform of a signal using a window function is defined as follows.

Think of the window as sliding along the signal and for each shift we compute the
usual Fourier Transform of the product function . For example, if is the box of width
1/2 then we have (see the Matlab m-file fig1.m):

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In the frequency domain we can use the convolution theorem to recognize as the

convolution of with the Fourier transform of (which is ).

Recall that we have the Fourier Transform pair:

In the case where is a box of width , that is, then . That is,
the nulls of are at multiple of . See the figure below where the box has width .

In the case where the signal is a pure sinusoid of frequency the windowed transform will be the sinc
function shifted by . In the figure below the box has width and the first sinusoid has
frequency Hz.

In the case where the signal consists of two sinusoids of frequencies and the windowed transform
will be the superposition of two shifted sinc functions. The individual frequencies cannot be resolved
unless . In fact, for adequate separation we should have .

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That is, the ``frequency resolution'' of this analysis is .

In the following figure a signal is the sum of two sinusoids with frequencies Hz and
Hz. The window size is . We get two distinct peaks in the frequency response (see fig2.m).

In the case where the signal consists of two spikes close together in time we can resolve the spikes if the
window size is smaller that the time difference between the spikes.

This analysis shows the ``trade-off'' between time resolution and frequency resolution: if we use a window
of length then we have a ``time-resolution'' of but our frequency resolution is .

The magnitude of the Short Time Fourier Transform is called the spectrogram. We can make
2 dimensional plots of the spectrogram with time on the horizontal axis, frequency on the vertical axis and
amplitude given by a gray-scale colour. Alternately we can make 3 dimensional plots where we plot
amplitude on the third axis. The Matlab command specgram can be used to generate these plots.
In the following example, (see fig3.m) a signal is the sum of two sinusoids of frequencies
and and two impulses at times ms and ms. We use a
window width of ms ( Hz).

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The resolution in frequency is Hz. The time resolution is ms. As the plots show, we
can can resolve both the sinusoids and the impulses.

Now suppose that we move the two frequencies closer together. Let's use a signal which is the sum
of two sinusoids of frequencies and and two impulses at times ms
and ms with a window width of ms (see fig4.m).

As the spectrograms now show we cannot resolve the frequencies but we can still resolve the spikes.

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Now suppose that we change the window size to ms. As the spectrograms below show, we can
resolve the frequencies but not the spikes (see fig4cd.m).

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We can obtain an orthogonal basis of functions related to the Short Time Fourier Transform when using
the window function = the box of width as follows. Instead of computing for all
frequencies and all time shifts we restrict the calculation to and . To see that
this corresponds to orthonormal functions define:

Then we have:

Since is non-zero only for it is clear that these are orthogonal functions.

Because we have analysis and synthesis on each interval to it follows that we have
analysis and synthesis in general. That is:

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In summary, if we restrict the STFT calculation to a discrete set of frequencies and times we can regard
the STFT values as the coordinates of our signal with respect to an orthogonal basis. Hence we can
recover our signal from these STFT values.

Next: 3. Time-Scale Analysis Up: Wavelets and Filter Banks Previous: 1. Analysis and Synthesis
Contents
Dr. W. J. Phillips
2003-04-03

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