ECE 5670 : Digital Communications
Lecture 21: Typical Error Event in a Slow Fading Wireless Channel
1
4/5/2011 and 4/7/2011
Instructor: Salman Avestimehr
Introduction
We studied sequential communication over a slowly varying wireless channel in the previous
lecture.
The key feature of our channel was that it so slowly changing in time that it is
practically timeinvariant over the course of communication. Mathematically, our model is:
y
[
m
] =
hx
[
m
] +
w
[
m
]
,
m
= 1
, . . . , N.
(1)
We defined the probability of error with sequential communication over this channel as the
average of the dynamic probability of error. With Rayleigh fading (i.e.,

h

2
has exponential
density with mean
A
) we had the following relationship between SNR and unreliability of a
single bit:
P
e
=
1
2
parenleftBigg
1
−
radicalbigg
A
SNR
1 +
A
SNR
parenrightBigg
(2)
≈
1
4
A
SNR
,
A
SNR
≫
1
.
(3)
The main conclusion is that the probability of error decays only linearly with increase in
SNR. This is in stark contrast to the situation in the AWGN channel (or a wireline channel)
where we had an exponential decay. This, of course, requires huge improvements in the
P
e
vs SNR interdependence, for any communication over the wireless channel to be acceptable.
As is clear from the Equation (3), there are two factors that cause errors over the wireless
channel: the first is the additive noise w[m] (just as it was for the wireline channel) and the
second (novel) factor
h
is multiplicative in nature. In this lecture we focus on isolating the
effects of each of these factors in an attempt to figure out which one is more crucial. This
insight will be used in later lectures to improve the poor relation between unreliability level
and SNR that is exhibitited in Equation (3).
1
Based on lecture notes of Professor Pramod Viswanath at UIUC.
1
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Sequential Communication
Consider sequential communication of a single bit
x
from the set
±
√
E
. We use the nearest
neighbor rule on a real sufficient statistic
˜
y
[
m
]
=
ℜ
bracketleftbigg
h
*

h

y
[
m
]
bracketrightbigg
(4)
=

h

x
[
m
] + ˜
w
[
m
]
.
(5)
Here ˜
w
[
m
] is real and Gaussian, with zero mean and variance
σ
2
2
. For example, the nearest
neighbor rule at a time
m
(index is suppressed here) could be
decide
x
= +
√
E
if ˜
y >
0
.
(6)
Now suppose we send
x
=
−
√
E
. Then an error occurs exactly when
˜
w >

h

√
E.
(7)
Observe that the error occurs due to the combined effect of

h

and ˜
w
. Thus there are two
factors at play here when an error occurs. At a rough level we see that the smaller the
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 Spring '11
 SCAGLIONE
 Rayleigh fading, Block Communication, ATH, fading wireless channel

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