UCSB
Spring 2010
ECE146B:
Laboratory Assignment 4
Lab Objectives:
Introduction to modeling and performance evaluation for signaling on wireless fading
channels.
Let us consider the following simple model of a wireless channel (obtained after filtering and sampling
at the symbol rate, and assuming that there is no ISI). If
{
b
[
n
]
}
is the transmitted symbol sequence,
then the complexvalued received sequence is given by
y
[
n
] =
h
[
n
]
b
[
n
] +
w
[
n
]
(1)
where
{
w
[
n
] =
w
c
[
n
] +
jw
s
[
n
]
}
is an iid complex Gaussian noise sequence with
w
c
[
n
],
w
s
[
n
] i.i.d.
N
(0
, σ
2
=
N
0
2
) random variables.
We say that
w
[
n
] has variance
σ
2
per dimension.
The channel
sequence
{
h
[
n
]
}
is a timevarying sequence of complex gains.
Rayleigh fading:
The channel gain sequence
{
h
[
n
] =
h
c
[
n
] +
jh
s
[
n
]
}
, where
{
h
c
[
n
]
}
and
{
h
s
[
n
]
}
are
zero mean, independent and identically distributed colored Gaussian random processes. The reason this
is called Rayleigh fading is that

h
[
n
]

=
radicalbig
h
2
c
[
n
] +
h
2
s
[
n
] is a Rayleigh random variable.
Remark:
The Gaussianity arises because the overall channel gain results from a superposition of gains
from multiple reflections off scatterers.
Simulation of Rayleigh fading:
We will use a simple firstorder autoregressive (AR(1)) model,
wherein the colored channel gain sequence
{
h
[
n
]
}
is obtained by passing white Gaussian noise through
a firstorder recursive filter, as follows:
h
c
[
n
] =
ρh
c
[
n

1] +
u
[
n
]
h
s
[
n
] =
ρh
s
[
n

1] +
v
[
n
]
(2)
where
{
u
[
n
]
}
and
{
v
[
n
]
}
are independent realvalued white Gaussian sequences, with i.i.d.
N
(0
, β
2
)
elements. The parameter
ρ
(0
< ρ <
1) determines how rapidly the channel varies.
Setting up fading simulator
(a) Set up the AR(1) Rayleigh fading model in matlab, with
ρ
and
β
2
as programmable parameters.
(b) Calculate
E
[

h
[
n
]

2
] = 2
E
bracketleftbig
h
2
c
[
n
]
bracketrightbig
= 2
v
2
analytically as a function of
ρ
and
β
2
. Use simulation to
verify your results, setting
ρ
=
.
99 and
β
=
.
01.
You may choose to initialize
h
c
[0] and
h
s
[0] as iid
N
(0
, v
2
) in your simulation. Use at least 10,000 samples.
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 Fall '08
 Staff
 Additive white Gaussian noise, Diversity scheme, bit error probability, Rayleigh fading

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