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. 3.41 Time-frequency distribution of the linear FM signal y
ð
t
Þ¼
sin
½
2p
f
f
o
ð
T
t
Þþ
e
ð
T
t
Þ
2
g
P
T
ð
t
T
=
2
Þ
with f
o
= 1 Hz, T = 10 s, e = 0.2. Top contour plot of the TFR. Bottom
the TFR, with red color representing highest amplitude
3.7.2.1 The Short-Time Fourier Transform
The short-time FT, or windowed FT, is the simplest TFR. Instead of transforming the
whole signal s(t) all at once using the FT, it is transformed on a block-by-block basis
using a moving time-window, centered at the time instant t, The STFT is defined as:
q
ð
t
;
f
Þ¼
Z
1
s
ð
k
Þ
h
ð
k
t
Þ
e
j2pf k
dk
¼
FT
k
!
f
f
s
ð
k
Þ
h
ð
k
t
Þg
1
where h(t) is a suitable time-window such as a hamming window. Usually one
refers to the above TFR as the STFT(t, f).
The spectrogram is obtained from the STFT by simply taking the squared
magnitude of the STFT.
3.7.2.2 Cohen's Class of TFRs
Many of the TFRs used in common practice can be shown to be members of a
generalized class known as Cohen's class, or the quadratic class. The general
formula for this class is:
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