This module introduces short-time Fourier transform.
We saw earlier that Fourier analysis is not well suited to
describing local changes in "frequency content" because thefrequency components defined by the Fourier transform have
infinite (
i.e. , global) time support. For
example, if we have a signal with periodic components plus aglitch at time
, we might want accurate knowledge of both the periodic
component frequencies
and the glitch time
(
[link] ).
The Short-Time Fourier Transform (STFT) provides a means of
joint time-frequency analysis. The STFT pair can be written
assuming real-valued
for which
. The STFT can be interpreted as a "sliding window
CTFT": to calculate
, slide the center of window
to time
, window
the input signal, and compute the CTFT of the result (
[link] ).
The idea is to isolate the signal in the vicinity of time
, then perform a CTFT
analysis in order to estimate the "local" frequency content attime
.
Essentially, the STFT uses the basis elements
over the range
and
. This can be understood as time and frequency shifts
of the window function
. The STFT basis is often illustrated by a tiling of
the time-frequency plane, where each tile represents aparticular basis element (
[link] ):
The height and width of a tile represent the spectral and
temporal widths of the basis element, respectively, and theposition of a tile represents the spectral and temporal centers
of the basis element. Note that, while the
tiling
diagram suggests that the STFT uses a discrete set of
time/frequency shifts, the STFT basis is really constructed froma continuum of time/frequency shifts.
Note that we can decrease spectral width
at the cost of increased temporal width
by stretching basis waveforms in time, although the
time-bandwidth product
(
i.e. , the area of each tile) will
remain constant (
[link] ).
Our observations can be summarized as follows:
the time resolutions and frequency resolutions of every STFT
basis element will equal those of the window
. (All STFT tiles have the same shape.)
the use of a wide window will give good frequency resolution
but poor time resolution, while the use of a narrow windowwill give good time resolution but poor frequency
resolution. (When tiles are stretched in one direction theyshrink in the other.)
The combined time-frequency resolution of the basis,
proportional to
, is determined not by window width but by window
shape. Of all shapes, the Gaussian
The
STFT using a Gaussian window is known as the
Gabor
Transform (1946).
gives the highest time-frequency resolution,
although its infinite time-support makes it impossible toimplement. (The Gaussian window results in tiles with
minimum area.)
Finally, it is interesting to note that the STFT implies a
particular definition of
instantaneous
frequency . Consider the linear chirp
. From casual observation, we might expect an
instantaneous frequency of
at time
since
The STFT, however, will indicate a
time-
instantaneous frequency
of
The phase-derivative interpretation of
instantaneous frequency only makes sense for signals containingexactly
one sinusoid, though! In summary,
always remember that the traditional notion of "frequency"applies only to the CTFT; we must be very careful when bending
the notion to include,
e.g. , "instantaneous
frequency", as the results may be unexpected!
Questions & Answers
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Wine
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Abdullahi
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