Chapter 2 Modulation and · PDF fileChapter 2 Modulation and Demodulation The most fundamental building block of a digital communication system is the modulator-demodulator (MODEM)

Wong & Lok: Theory of Digital Communications 2. Modulation & Demodulation
Chapter 2
Modulation and Demodulation
The most fundamental building block of a digital communication system is the modulator-demodulator
(MODEM) pair. In this chapter, we study the theory that underlies the design of the MODEM in a
digital communication.
We start with the simple baseband binary communication system over the AWGN channel to in-
troduce the concept of a matched lter. In many cases, we have to modulate the transmitted signal
to higher frequency bands, usually known as theradio frequency(RF) bands, to suit the propagation
characteristics of the communication channels. The simplest RF channel is thenon-dispersivechan-
nel which changes the amplitude and phase of the transmitted signal. There are two ways to perform
demodulation over the non-dispersive channel. The rst way is to estimate the phase distortion and
use the matched lter as in a baseband communication system. This method is usually referred to as
coherentdemodulation. The second approach, known asnoncoherentdemodulation, is to avoid using
the phase information in the demodulation process at all. We discuss and compare the two approaches.
Binary modulation is the simplest modulation format. An immediate extension is to send one out of
M signals instead of one out of two signals. This is calledM -ary communications. Beside introducing
M -ary communications, we present a very useful technique, namely theunion bound, to evaluate the
performance of complex communication systems such as aM -ary system.
The discussion of the modulation and demodulation methods above provide us a solid introduction
to the topic. Our next goal is to study the optimal demodulators, both coherent and noncoherent,
in the sense of achieving the minimum symbol error probability over the non-dispersive channel. It
2.1

s ( t )k"0" or "1"
h ( t )
Tz ( t )k
0
?
n( t )AWGN
Figure 2.1: Baseband binary communication system
turns out that the optimal demodulators can be characterized by maximizing some likelihood functions
of the received signal. Hence, the optimal demodulators are also calledmaximum likelihood (ML)
demodulators.
For the developments in this chapter, we restrict ourselves to the simple AWGN and non-dispersive
channels. The effect of other channel distortions and techniques to overcome them will be discussed
in later chapters.
2.1 Baseband Binary Communications
Consider the baseband binary communication system over the AWGN channel as shown in Figure 2.1.
The goal is to send one bit only (one-shot analysis).
Assumptions:
1. The modulator outputs the signals0(t) if the input bit value is 0 and it outputs the signals1(t)
if the input bit value is 1. In other words, we uses0(t) to represent 0 ands1(t) to represent
1.
2. The additive noisen(t) introduced by the channel is an AWGN process with a two-sided noise
spectral density ofN0/2.
3. The demodulator passes the received signalsk(t) + n(t) (k = 0 or 1) to an LTI lter whose
impulse response ish(t) and samples the lter output at timeT0 to provide thedecision statistic
zk(T0).
2.2

4. The receiver checks the decision statistic against the threshold to decide whether the trans-
mitted bit is 0 or 1. We assume thats0 h(T0) > s1 h(T0). Therefore, we decide 0 ifzk(T0) , and 1 otherwise.
Goals:
1. To determine the probability of error for chosen threshold, lterh(t), sampling timeT0, and
signal pairs0(t) ands1(t).
2. To determine the best
(a) threshold,
(b) linear lterh(t) and sampling timeT0,
(c) signal pairs0(t) ands1(t).
2.1.1 Bit error probability
Suppose that 0 is sent. Lets0(t) andn(t) represent the output of the lterh(t) due to the signal and
the noise, respectively, i.e.,
s0(t) = s0 h(t), (2.1)n(t) = n h(t). (2.2)
The overall outputz0(t) is the sums0(t)+ n(t). The decision is made according to the sample ofz0(t)
at timeT0. Therefore, the decision statistic isz0(T0) given by
z0(T0) = s0(T0) + n(T0). (2.3)
The signal contributions0(T0) is deterministic. We consider the noise contributionn(T0). Sincen(t) is
a WSS zero-mean Gaussian random process,n(t) is also a WSS zero-mean Gaussian random process.
Therefore,n(T0) is a zero-mean Gaussian random variable. The variance ofn(T0) is E[n2(T0)] =
Rn(0). Therefore,z0(T0) is a Gaussian random variable with means0(T0) and varianceRn(0). Given
2.3

that 0 is sent, the probability of error is given by
Pb,0 = Pr(z0(T0) < )
=
( s0(T0)
Rn(0)
)
= Q
(s0(T0)
Rn(0)
). (2.4)
Similarly, if 1 is sent, then the decision statisticz1(T0) is a Gaussian random variable with mean
s1(T0) and varianceRn(0). Given that 1 is sent, the probability of error is given by
Pb,1 = Pr(z1(T0) )
= Q
( s1(T0)
Rn(0)
). (2.5)
Suppose that we send 0 with probabilityp0 and 1 with probabilityp1 = 1 p0. Then the averagebit error probability is given by
Pb = p0Pb,0 + p1Pb,1. (2.6)
2.1.2 Threshold optimization
The goal of the optimization is to determine the decision threshold which minimizes the average
bit error probabilityPb. Depending on the knowledge available to the receiver, one of the following
approaches is taken.
Minimax approach
Suppose that the receiver has no knowledge ofp0 andp1. To guarantee a certain performance no matter
whether 0 or 1 is sent, we minimize the maximum ofPb,0 andPb,1. The best way to solve this
optimization problem is to look at the plots ofPb,0 andPb,1 versus as those shown in Figure 2.2.
Consider the following diagram. First, we note from (2.4) and (2.5) that as increases,Pb,0 increases
while Pb,1 decreases. It is obvious from Figure 2.2 that the maximum ofPb,0 andPb,1 reaches its
minimum at wherePb,0 = Pb,1. Therefore under this minimax criterion, the best choice of is the one
2.4

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Minimax
Pb,0
Pb,1
Figure 2.2: Typical plots ofPb,0 andPb,1 versus
2.5

that makes
Pb,0 = Pb,1
Q
(s0(T0)
Rn(0)
)= Q
( s1(T0)
Rn(0)
). (2.7)
Therefore, the best choice of is
=s0(T0) + s1(T0)
2. (2.8)
With this choice of, the average bit error probability becomes
Pb = Pb,0 = Pb,1 = Q
(s0(T0) s1(T0)
2
Rn(0)
). (2.9)
Minimum average bit error probability approach
Suppose that the receiver has clear knowledge ofp0 andp1. Then the obvious approach is to choose
that minimizesPb. We recall that
Pb = p0Q
(s0(T0)
Rn(0)
)+ p1Q
( s1(T0)
Rn(0)
). (2.10)
Differentiating it wrt, we have
dPbd
=p0
2Rn(0)exp
[(s0(T0) )22Rn(0)
] p1
2Rn(0)exp
[( s1(T0))22Rn(0)
]. (2.11)
Setting the result to zero and solving for, we have
=s0(T0) + s1(T0)
2+
Rn(0)
s0(T0) s1(T0) lnp1p0
. (2.12)
We notice that whenp0 = p1, reduces to the minimax threshold. Whenp0 is larger, is reduced so
that the average bit error probabilityPb is reduced. Likewise, whenp1 is larger, is increased so that
the average bit error probabilityPb is reduced.
2.1.3 Filter and sampling time optimization
We consider the best lter when thea priori probabilitiesp0 andp1 are the same and the minimax
threshold is used. With the minimax, Pb is given by (2.9). SinceQ() is monotone decreasing, wecan minimizePb by maximizing
s0(T0) s1(T0)2
Rn(0). (2.13)
2.6

First, let us determine the noise varianceRn(0):
Rn(0) =
n(f)df
=N02
|H(f)|2df
=N02
|h(t)|2dt
=N02h2 (2.14)
where the norm of any functionx is dened by
x =
x2(t)dt. (2.15)
Moreover,
s0(T0) s1(T0) =
[s0(T0 t) s1(T0 t)]h(t)dt. (2.16)
Therefore, in order to minimizePe, we can chooseh(t) such that the expression
1
h
[s0(T0 t) s1(T0 t)]h(t)dt (2.17)
is maximized.
We can use the Cauchy-Schwartz inequality to solve the maximization problem in (2.17). The
Cauchy-Schwartz inequality says that for any two functionsf andg such thatf, g < ,
f(t)g(t)dt
||f || ||g|| (2.18)with equality if and only iff(t) = g(t) where is a constant. Applying (2.18) withf(t) = s0(T0 t) s1(T0 t) andg(t) = h(t), we have1
[s0(T0 t) s1(T0 t)]h(t)dt s0(T0 t) s1(T0 t) h (2.19)
with equality if and only ifh(t) = [s0(T0 t) s1(T0 t)] where is a constant. Therefore, theexpression in (2.17) is maximized if we choose
h(t) = s0(T0 t) s1(T0 t). (2.20)1s0(T0) s1(T0) =
[s0(T0 t) s1(T0 t)]h(t)dt > 0.
2.7

This best linear lter is called the

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Chapter 2 Modulation and · PDF fileChapter 2 Modulation and Demodulation The most fundamental building block of a digital communication system is the modulator-demodulator (MODEM)