Digital Signal Processing Reference
In-Depth Information
8.3.4
Second-Order Cyclostationary Statistics
Cyclostationary signals have the property that statistics, such as mean or auto-
correlation function, are periodic. Many anthropogenic signals encountered in
communications contain such periodicities. Conventional models for random
processes such as WSS ignore this valuable information. Gardner
33
discovered
the fact that non-minimum phase channel equalization/identification may be
obtained from the second-order cyclostationary (CS) statistics of the received
signal because cyclic autocorrelation function preserves the phase informa-
tion. Cyclostationary statistics may be obtained by taking several samples per
symbol (sampling rate
P
T
, where
P
is the oversampling factor) or by em-
ploying multiple antennas at the receiver. This leads to vector valued signals
and multiple-output signal models.
Blind equalizers typically use second-order cyclostationary statistics. Hence,
smaller sample sizes are required than in HOS-based receivers, and the algo-
rithms converge faster. The main drawback is that some channel types may not
be identified; see Reference 34. In particular, the channel cannot be identified
if the subchannels resulting from oversampling share common zeros. If the
multiple-output model is obtained by using an antenna array at the receiver,
this limitation is less severe. There is also a phase ambiguity involved in blind
channel estimation that may be resolved, for example, by using differential
coding. However, some loss in SNR results from using such a coding scheme.
The continuous-time received signal is
/
∞
(
)
=
(
)
(
−
)
+
w(
)
y
t
s
k
h
t
kT
t
,
(8.7)
k
=−∞
where
h
(
t
)
is the channel impulse response,
s
(
k
)
the sequence of information
symbols and
)
is assumed to be of finite length. By taking several samples per symbol interval
from
y
w(
t
)
additive white Gaussian noise (AWGN). The response
h
(
t
, the received sequence becomes cyclostationary. Fractionally spaced
channel output resulting from oversampling by factor
P
may be written as
y
k
T
P
(
t
)
h
k
T
P
lT
k
T
P
=
s
(
l
)
−
+
w
.
(8.8)
l
The output of the
i
th subchannel
h
i
(
k
)
can be written as
∞
y
i
(
k
)
=
s
(
l
)
h
i
(
k
−
l
)
+
w
(
k
)
,
for
i
=
0
,
...
,P
−
1
.
(8.9)
i
l
=−∞
An equalizer
d
i
.
An alternate vector representation is obtained by stacking
P
samples taken
in each symbol interval to a
P
-dimensional vector
y
(
k
)
is employed for each subchannel
h
i
(
k
)
]
T
.
(
k
)
=
[
y
0
(
k
)
,
...
,y
P
−
1
(
k
)
]
T
and
Similarly, the noise vector may be written as
w
(
k
)
=
[
w
(
k
)
,
...
,
w
(
k
)
0
P
−
1
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