Digital Signal Processing Reference
In-Depth Information
Figure 2.1
Illustration of the STFT.
The definition of the STFT can also be expressed in the frequency
domain by manipulating (2.4), with the result
2
p
exp{
-
j
v
t
}
E
S
(
v¢
)
W
(
v-
1
STFT
(
t
,
v
)=
v¢
) exp{
j
v¢
t
}
d
v¢
(2.5)
Here
W
(
v
) is the Fourier transform of
w
(
t
). The dual relationship
between (2.4) and (2.5)
1
is apparent (i.e., the time-frequency representation
can be generated via a moving window in time or a moving window in
frequency). In addition, we make the following observations: (1) Signal
components with durations shorter than the duration of the window will
tend to get smeared out [i.e., the resolution in the time domain is limited
by the width of the window
w
(
t
)]. Similarly, the resolution in the frequency
domain is limited by the width of the frequency window
W
(
v
). (2) The
window width in time and the window width in frequency are inversely
proportional to each other by the uncertainty principle. Therefore, good
resolution in time (small time window) necessarily implies poor resolution
in frequency (large frequency window). Conversely, good resolution in fre-
1.
Equation (2.5) has also been referred to as the running-window Fourier transform [4].
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