Digital Signal Processing Reference
In-Depth Information
in the frequency domain. This is the reason why the sinusoidal signal buried
in additive white Gaussian noise with zero-mean and variance
s
r
n
as shown
in Figure 3.7(a) can be easily detected in the frequency domain.
3.3.2 SNR in the Joint Time-Frequency Domain
As mentioned earlier in this chapter, by taking a time-frequency transform,
the noise tends to spread its energy over the time-frequency domain, while
the signal often concentrates its energy into regions within limited time
intervals and frequency bands. Especially for a frequency-modulated signal,
such as exp{
j
2
p
[
f
0
+
(
h
/2)
t
]
t
} embedded in noise, it is much easier to be
recognized in the joint time-frequency domain than in either the time or
the frequency domain alone. However, when the frequency-changing
rate
h
becomes very large, to keep the same signal energy the signal becomes
a short time impulse that can be easier recognized in the time domain; when
h
approaches zero, the signal becomes a sinusoid exp{
j
2
p
f
0
t
} that can be
more easily recognized in the frequency domain. In general, there is a
question about how much the SNR can be improved with the time-frequency
transform? The answer is that it depends on the type of the time-frequency
transform and the waveform of the signal. In [8], a quantitative analysis of
SNR of a multicomponent signal with the STFT is given. The multicompo-
nent signal consists of a number of monocomponent signals
s
(
t
)=
K
k=
1
s
k
(
t
)
(3.8)
Because the time-frequency transform is especially suitable to represent
signals with time-varying spectrum, the monocomponent signal is assumed
to be a chirp-type signal
s
k
(
t
)=
a
k
(
t
) exp
H
j
2
p
S
f
k
t
D
t
J
h
k
2
+
(3.9)
where
a
k
(
t
) is the amplitude function of the
k
th component,
f
k
is the starting
frequency of the
k
th chirp signal, and
h
k
is the chirp rate of the
k
th
component.
It was proved in [8] that for the multicomponent signal in additive
Gaussian white noise with zero-mean and variance
s
r
n
, the SNR improve-
ment using the STFT with a rectangular window over the SNR in the time
domain is equal to or greater than the order of (
N
/
K
):
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