Digital Signal Processing Reference
In-Depth Information
assuming the presence of noise, we can express the output vector related to
the
p
th subchannel as
H
x
p
˜
(
n
)
=
p
s
(
n
)
+
ν
p
(
n
)
,
(5.30)
with the convolution matrix associated with the
p
th subchannel, given by
⎡
⎣
⎤
⎦
h
p
(
0
)
h
p
(
1
)
···
h
p
(
L
−
1
)
0
···
0
0
h
p
(
0
)
···
h
p
(
L
−
2
)
h
p
(
L
−
1
)
···
0
H
=
.
p
.
.
.
.
.
.
0
0
···
h
p
(
0
)
h
p
(
1
)
···
h
p
(
L
−
1
)
(5.31)
Finally, stacking the
P
vectors corresponding to the subchannels, one can
obtain the following:
⎤
⎤
⎤
⎡
⎣
⎡
⎣
⎡
⎣
H
x
0
(
˜
n
)
ν
0
(
n
)
˜
0
H
1
.
.
.
⎦
⎦
⎦
x
1
(
˜
n
)
ν
1
(
n
)
˜
=
(
n
)
+
s
,
(5.32)
.
.
.
.
.
.
x
P
−
1
(
˜
n
)
H
ν
P
−
1
(
n
)
P
−
1
or, simply,
)
=
H
˜
x
(
n
s
(
n
)
+
˜
ν
(
n
)
.
(5.33)
5.2.3.2 Representation via the Sylvester Matrix
Another possible way to represent the SIMO model is obtained by arranging
the samples in a different manner. Let us define the vector
)
=
h
0
(
)
T
h
(
n
n
)
h
1
(
n
)
···
h
P
−
1
(
n
(5.34)
containing the samples of the impulse response of all subchannels at a given
time instant
n
. In a similar fashion, let us define the vectors associated with
the outputs and the noise as follows:
)
=
x
0
(
)
T
x
(
n
n
)
x
1
(
n
)
···
x
P
−
1
(
n
(5.35)
)
=
ν
0
)
T
n
(
n
)
ν
1
(
n
)
···
ν
P
−
1
(
n
ν
(
(5.36)
