Digital Signal Processing Reference
In-Depth Information
where f
o
= 1 Hz, T = 10 s, and the modulation index is e = 0.2. The instanta-
neous frequency (IF) of x is f
x
= (1/2p)d/
x
/dt = f
o
? 2et (which is a linear
function of time), while the instantaneous frequency of y is f
y
= |(1/2p)d/
y
/
dt| = f
o
? 2e(T - t). The magnitude of the FT's of these two signals are identical,
as shown in Fig.
3.39
.
In this case the magnitude spectrum is not sufficient to reveal the underlying
characteristics of the signal. No information is in the FT, since one can reconstruct
the signal using the inverse FT. The time information is hidden in the phase
response, but it is not easy to deduce much information about the time-varying
nature of the signal from the complicated phase-magnitude relations. Using a time-
frequency representation (TFR), however, it is possible to obtain the contour plots
shown in Figs.
3.40
and
3.41
, which reveal the true IF laws of the two Linear FM
signals. (The particular time-frequency distribution is the Choi-Williams distri-
bution, with parameter r = 11 [
8
].)
In addition to the above dilemma of the FT, there is a more important problem
that the FT cannot deal with. If the signal is composed of more than one FM
component, it is even more difficult to observe the underlying structure from the
FT. Many practical sounds do indeed consist of multiple time-varying frequency
components (eg. EEG, ECG, and animal sound signals).
In particular, consider first the two sinusoidal pulses with the same frequency
occurring at different times as shown in Fig.
3.42
. It is evident that the magnitude
Fourier spectrum cannot reveal the time-frequency structure of the signal, while
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
0
2
4
6
8
10
0
2
4
6
8
10
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−10
−5
0
5
10
−10
−5
0
5
10
2
2
0
0
−2
−2
−10
−5
0
5
10
−10
−5
0
5
10
Fig. 3.39
Two different Linear FM signals with identical Fourier magnitude spectra
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