Digital Signal Processing Reference
In-Depth Information
Thus, the larger the time duration of
s
(
t
), the smaller the frequency
bandwidth of
S
(
v
). Conversely, the larger the frequency bandwidth of
S
(
v
),
the shorter the time duration of
s
(
t
).
When we use (2.1) to estimate the frequency spectrum of a signal, we
assume that the frequency content of the signal is relatively stable during
the observation time interval. If the frequency content changes with time,
it is not possible to monitor clearly how this variation takes place as a
function of time. The reason can be attributed to the nature of the complex
sinusoidal basis, which is of infinite duration in time. While the frequency
spectrum can still be used to uniquely represent the signal, it does not
adequately reflect the actual characteristics of the signal. In the following
three subsections, three linear time-frequency transforms (viz., STFT, the
CWT, and the adaptive time-frequency representation) are presented. They
can be considered as a generalization of the Fourier transform with alternative
basis sets that can better reflect the time-varying nature of the signal frequency
spectrum.
2.1.1 The STFT
The most standard approach to analyze a signal with time-varying frequency
content is to split the time-domain signal into many segments, and then
take the Fourier transform of each segment (see Figure 2.1). This is known
as the STFT operation and is defined as
v
)=
E
s
(
t
¢
)
w
(
t
¢-
t
) exp{
-
j
v
t
¢
}
dt
¢
STFT
(
t
,
(2.4)
This operation (2.4) differs from the Fourier transform only by the
presence of a window function
w
(
t
). As the name implies, the STFT is
generated by taking the Fourier transform of smaller durations of the original
data. Alternatively, we can interpret the STFT as the projection of the
function
s
(
t
¢
) onto a set of bases
w
*(
t
¢-
t
) exp{
j
v
t
¢
} with parameters
t
and
v
. Since the bases are no longer of infinite extent in time, it is possible
to monitor how the signal frequency spectrum varies as a function of time.
This is accomplished by the translation of the window as a function of time
t
, resulting in a 2D joint time-frequency representation
STFT
(
t
,
v
) of the
original time signal. The magnitude display
|
STFT
(
t
,
v
)
|
is called the
spectrogram of the signal. It shows how the frequency spectrum (i.e., one
vertical column of the spectrogram) varies as a function of the horizontal
time axis.
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