EE 5505
Prof. Jindal
Wireless Communication
February 13, 2010
Homework 3
Due: Friday, February 19, 4:00 PM
Please turn in your MATLAB scripts in addition to your solutions and plots.
1. Consider the general tapdelay model
h
(
τ
;
t
) =
L
∑
i
=1
g
i
(
t
)
δ
(
τ

τ
i
)
,
(1)
where the coeﬃcients
g
1
(
t
)
, . . . , g
L
(
t
) are independent and each is complex Gaussian
with zero mean and
E
[

g
i
(
t
)

2
] =
α
i
for
i
= 1
, . . . , L
.
The frequency response with respect to
τ
is given by (we derived this in lecture):
C
(
f
;
t
)
=
∫
∞
∞
c
(
τ, t
)
e

j
2
πfτ
dτ
(2)
=
L
∑
i
=1
g
i
(
t
)
e

j
2
πfτ
i
(3)
(a) Show that
C
(
f
;
t
) is complex Gaussian.
(b) Compute the mean and variance of
C
(
f
;
t
)
Hint: Multiplying a complex Gaussian random variable (with iid real and imaginary
parts, as we consider in this model) by a phase term (e.g.,
e
jθ
) does not change the
distribution of the complex Gaussian.
2. In this problem we will simulate a wideband channel according to a 3 tap model:
c
(
τ, t
) =
3
∑
i
=1
g
i
(
t
)
δ
(
τ

τ
i
)
.
(4)
Let us assume the typical model, as discussed in class (Rayleigh tap coeﬃcients, with
temporal correlation given by the Jakes model):
•
Delays:
τ
1
= 0,
τ
1
= 1
μ
sec,
τ
2
= 3
μ
sec
•
Stationary distribution:
Each of the tap coeﬃcients
g
i
(
t
) are complex Gaus
sian (i.e., Rayleigh fading) and are independent, with powers given by:
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 Spring '08
 Jindal,N
 Normal Distribution, SNR, Rayleigh, block error probability, Pbit error

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