Digital Signal Processing Reference
In-Depth Information
which shows that
a
μ
0
+
a
μ
1
a
μ
1
+
a
μ
0
>
.
So an interchange of
μ
0
and
μ
1
reduces the objective function (12.38) fur-
ther. From Eq. (12.36) we see that such an interchange can be effected by
postmultiplying
U
f
with a permutation matrix (Sec. B.5.1 of Appendix B).
Thus, define the new
U
f
as
U
new
=
U
f
P0
0
P−M
,
(12
.
42)
where
P
is a permutation. This is still unitary, and (
U
new
H
†
HU
new
) is still
of the required form (12.37). Therefore we see that, under the numbering
convention (12.40), the optimal
U
f
should be such that Eq.
(12.41) is
satisfied.
12.4.2 Finding the optimal
U
f
P
matrix
H
†
H
Note that the unitary matrix
U
f
need not diagonalize the
P
×
completely. Only partial diagonalization of the
M
M
leading principal sub-
matrix is called for, as shown in Eq. (12.37). We know however that since
H
†
H
is Hermitian, there exists a unitary
U
f
diagonalizing it fully, that is,
×
⎡
⎣
⎤
⎦
σ
h,
0
0
...
0
σ
h,
1
0
...
0
U
f
H
†
HU
f
=
.
(12
.
43)
.
.
.
.
.
.
0
0
...
σ
h,P−
1
Here
σ
h,
0
≥ σ
h,
1
≥ ...≥ σ
h,P−
1
≥
0
,
(12
.
44)
are the singular values of the channel
H
.
Setting
U
f
=
U
f
certainly satisfies
Eq. (12.37). We will prove the subtle fact that this choice of
U
f
serves as the
optimal choice as well:
Lemma 12.2.
Optimal unitary
U
f
.
The optimal unitary matrix
U
f
,
which
minimizes Eq. (12.34) subject to Eq. (12.35) for fixed
Σ
f
and
H
can be assumed,
without loss of generality, to be the unitary matrix which diagonalizes
H
†
H
.
♠
♦
The idea of the proof is as follows: with
U
f
yielding a partial diagonalization
(12.37), and under the numbering convention (12.41), we will show that
σ
h,
0
≥
μ
0
,σ
h,
1
≥
μ
1
,...,σ
h,M−
1
≥
μ
M−
1
.
(12
.
45)
Thus
M−
1
M−
1
a
k
μ
k
≥
a
k
σ
h,k
E
mse
=
σ
q
σ
q
(12
.
46)
k
=0
k
=0
Search WWH ::
Custom Search