Digital Signal Processing Reference
In-Depth Information
where
A
is the full banded Toeplitz matrix of channel coe
cients given in Eq.
(8.2). This is a
P
M
matrix, where
P
=
M
+
L
.
We now define certain
submatrices
of
A
which are important for further
discussions. Let
A
K
denote the submatrix obtained by keeping the top
K
rows
of
A
.
For
K
×
≥
M
the matrix
A
K
has full rank
M
(assuming
c
(0)
=0). For
example, with
M
= 3 and
L
=2wehave
⎡
⎤
⎡
⎤
c
(0)
0
0
c
(0)
,
A
4
=
c
(0)
0
0
0
0
⎣
c
(1)
c
(0)
0
⎦
⎣
c
(1)
c
(0)
0
⎦
A
3
=
c
(1)
c
(0)
0
,
A
5
=
c
(2)
c
(1)
c
(0)
.
c
(2)
c
(1)
c
(0)
c
(2)
c
(1)
c
(0)
0
c
(2)
c
(1)
0
c
(2)
c
(1)
0
0
c
(2)
(8
.
19)
Note the following properties of these matrices:
1.
A
K
is lower triangular and Toeplitz for all
K.
2. For
K
=
M
the matrix
A
K
is also a
square
matrix.
3. For
K
=
P
the matrix
A
K
is
full banded
Toeplitz.
Note from Eq. (8.18) that we also have
y
K
(
n
)=
A
K
s
(
n
)+
q
K
(
n
)
,
(8
.
20)
where
y
K
(
n
) contains the first
K
components of
y
(
n
)
,
and
q
K
(
n
)containsthe
first
K
components of
q
(
n
)
.
For
K
≥
M,
since
A
K
has full rank, it has a left
inverse. Letting
A
K
denote the minimum-norm left inverse as usual, we have
A
K
y
K
(
n
)=
s
(
n
)+
A
K
q
K
(
n
)
.
(8
.
21)
Proceeding as in the derivation of Eq. (8.6) we conclude again that
E
reco,K
=
σ
q
A
K
2
,
(8
.
22)
where the subscript
K
is a reminder that we have retained
K
components of the
output vector
y
(
n
) in the reconstruction of
s
(
n
). Since we can write
A
K
+1
=
A
K
×
it follows that the minimum-norm left inverse of
A
K
+1
has a smaller Frobenius
norm than
A
K
(Lemma 8.1). This shows that
E
reco,K
+1
≤E
reco,K
for fixed
M.
That is, the reconstruction error (8.22) can only improve as
K
increases. As we make
A
K
taller and taller, that is,
as we use more and more
output samples from the block
y
(
n
), the effect of channel noise becomes smaller.
This is intuitively obvious as well.
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