Digital Signal Processing Reference
In-Depth Information
10
8
6
4
2
0
0
1
2
3
4
5
6
Frequency, Hz
10
8
6
4
2
0
0
1
2
3
4
5
6
Frequency, Hz
Fig. 3.40 TFR of the FM signal x
ð
t
Þ¼
sin
½
2p
ð
f
o
t
þ
et
2
Þ
P
T
ð
t
T
=
2
Þ
with f
o
= 1 Hz, T = 10 s,
and e = 0.2. Top contour plot of the TFD. Bottom the TFR, with red color representing highest
amplitude
the time-frequency distribution shows the number of components as well as their
frequencies and durations.
Second, consider the sum of the two Linear FM signals r(t) = x(t) ? y(t), where
x
ð
t
Þ¼
_
sin
½
2p
ð
f
1
t
þ
e
1
t
2
Þ
P
T
ð
t
T
=
2
Þ;
y
ð
t
Þ¼
sin
½
2p
ð
f
2
t
þ
e
2
t
2
Þ
P
T
ð
t
T
=
2
Þ;
f
1
¼
2Hz
;
f
2
¼
3Hz
;
e
1
¼
0
:
1
;
e
2
¼
0
:
2, and T = 10 s. The FT cannot reveal the mul-
ticomponent nature of this signal, while the TFR can, as shown in Fig.
3.43
. Here
the Choi-Williams distribution is used, with r = 41.
Third, consider a bird sound represented in Fig.
3.44
. The TFR can reveal
the IF laws of the signal components, while one cannot get this information
from the FT.
3.7.2 Some Important TFRs
A TFR is a two-dimensional transform of the signal into the time-frequency
(t-f)
plane.
Signal
components
are
typically
observed
in
the
t-f
plane
by
ridges around the IF laws of the components.
Below some of the commonly used TFRs are introduced.
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