EE 562a
Homework Set 7
Due Monday 16 April 2007
1
(The following welldefined problems come from different sources, and the notation used may vary.
Don’t let that bother you!)
1.
Hypothesis testing in discrete time.
(Modified Final Exam Problem  Fall92  Hinedi/
Chugg).
Consider the task of deciding between two hypotheses regarding signals defined on
T
=
Z
:
H
0
:
x
(
u, n
) =
s
0
(
n
) +
w
(
u, n
)
H
1
:
x
(
u, n
) =
s
1
(
n
) +
w
(
u, n
)
,
where the deterministic signals are defined by
s
i
(
n
) =
√
P
cos(
π
(
n
+
i
))
n
∈ {
0
,
1
}
0
n
=
. . .

2
,

1
,
2
,
3
. . .
i
= 0
,
1
,
and the (real) Gaussian noise process,
w
(
u, n
), has PSD given by
S
w
(
ν
) =
σ
2
(1

ρ
2
)
1

2
ρ
cos(2
πν
) +
ρ
2
,
where

ρ

<
1.
(a) Based on observing only
x
(
u, n
) for
n
= 0
,
1 design a good rule for deciding which
hypothesis is true.
(b) What is the bound on the probability of error?
(c) Determine an exact expression for the probability of error. Your answer should involve
the
Q
function:
Q
(
z
) =
∞
z
N
1
(
x
; 0; 1)
dx
=
∞
z
exp

x
2
2
√
2
π
dx.
(d) Using the fact that
Q
(3)
≈
0
.
001, and assuming that the noise is white (i.e.
ρ
= 0),
what is the minimum signaltonoise ratio (
SNR
=
P
σ
2
) required to ensure that the error
probability is at most 1/1000?
(e) Discuss the performance of this system as
ρ
varies.
(f) What is the conditional pdf of
w
(
u,
0)
, w
(
u,
1) given
w
(
u, k
), where
k
is an integer other
than 0 or 1? In other words, determine
f
w
(
u,
0)
,w
(
u,
1)

w
(
u,k
)
(
z
0
, z
1

z
k
)
.
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 Spring '07
 ToddBrun
 Stochastic process, Autocorrelation, Stationary process, random process

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