Digital Signal Processing Reference
In-Depth Information
≈
Py E
N
3
E
N
2
2
s
02
.exp
−
s
y
,,
(3.6)
( )
M
4
M
1
0
0
which will be employed for all M for much of the design work for uncoded systems. If
errors in channel estimation at the receiver are considered, the right side of equation
(3.6) will increase, of course, but it will often it into the same functional form [7], which
is convenient, since the same optimization will apply. Using equation (3.6) yields
=
E
N
EPY E
N
ˆ (
P
2
s
,,
h
ρ
s
XkT
− =
τ
)
h
M
M
s
1
0
0
22
h
ρ
ρ
1
02
.exp
−
1
−

2
2
21
(
−
)
3
2
E
N
(
1
−
ρ
)
1
+
s
(3.7)
(
M
−
1
)
0
ρ
<
1
3
2
E
N
(
1
−
ρ
2
)
≈
,
1
+
s
(
M
−
1
)
0
2
3
4
E
N
h
M
02
.exp
−
s
ρ
=
1
(
−
1
)
0
where the second line is obtained by substituting equation (3.2) and equation (3.6) into
the first line and evaluating the expectation over Y using [34, 6.614.3].
From equation (3.5), equation (3.7) must be evaluated at its supremum on ρ ∈ [ρ min , 1].
Since the right side of equation (3.7) is a continuous function on this closed interval, it
achieves its maximum on this interval at a point that will be denoted ρ ∗ . The following
solution is found by standard calculus techniques. Let
≥
0
h
2
ρ=
.
2
2
(
MN
E
−
1
)
−
(
2
h
)

0
≤≤
h
2
1
+
0
3
2
s
The worst-case autocorrelation is then given by
ρ
ρ ρ
≤
min
min
ρ
*
=
ρ
ρ ρ
<<
1
.
(3.8)
min
1
1
≤
ρ
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