Lecture 20: Sequential Communication over a Slow Fading Wireless
Channel
1
4/5/2011
Instructor: Salman Avestimehr
Introduction
Consider the simple
slow
fading wireless channel we arrived at in the previous lecture:
y
[
m
] =
hx
[
m
] +
w
[
m
]
,
m
= 1
,...,N.
(1)
Here
NT
is the time scale (the product of
N
, the number of time samples communicated
over, and
T
, the sampling period) of communication involved. The channel coe±cient
h
is well modeled as independent of
m
if
NT
is much smaller than the coherence time
T
c
of
the channel. This is a very common occurrence in many practical wireless communication
systems and we begin our study of reliable communication over the wireless channel. We
will start with sequential communication, much as we started out with the AWGN channel
(cf. Lecture 3). The goal of this lecture is to be able to compare and contrast the simple
slow fading channel in Equation (1) with the familiar AWGN channel. In particular, we will
calculate the unreliability of communicating a single bit as a function of the SNR. The main
conclusion is the observation of how poorly the unreliability decays with increasing SNR.
This is especially stark when compared to the performance over the AWGN channel.
Comparison with AWGN Channel Model
The channel model in Equation (1) is quite similar to that of the AWGN channel model.
But there are di²erences as well, with some aspects being more important than others.
1. The transmit and receive symbols are complex numbers (pair of voltages) as opposed
to the real numbers in the AWGN channel. This is a relatively minor point and poses
hardly any trouble to our calculations and analysis (as seen further in this lecture).
2. The channel “quality”
h
can be learnt by the receiver (through the transmission of
pilot symbols, cf. Lecture 9), but is not known a priori to the communication engineer.
This is a very important di²erence, as we will see further in this lecture. The important
point is that the calculation of the unreliability level is now to be calculated in terms of
the knowledge the communication engineer has about
h
: the
statistical
characterization
of
h
.
1
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 Spring '11
 SCAGLIONE
 Rayleigh fading, unreliability level

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