Digital Modulation Techniques

20
Digital Modulation
Techniques
20.1
20.2

Introduction
The Challenge of Digital Modulation

20.3

One-Dimensional Modulation: Pulse-Amplitude
Modulation (PAM)
Two-Dimensional Modulations

Bandwidth • Signal-to-Noise Ratio • Error Probability

20.4

Phase-Shift Keying (PSK) • Quadrature Amplitude
Modulation (QAM)

20.5
20.6

Examples of Lattices • The Coding Gain of aLattice • Carving a Signal Constellation out of a Lattice

Ezio Biglieri
Politecnico di Torino

Multidimensional Modulations: Frequency-Shift
Keying (FSK)
Multidimensional Modulations: Lattices

20.7

Modulations with Memory

20.1 Introduction
The goal of a digital communication system is to deliver information represented by a sequence of binary
symbols, through a physical channel,to a user. The mapping of these symbols into signals, selected to
match the features of the physical channel, is called digital modulation.
The digital modulator is the device used to achieve this mapping. The simplest type of modulator
has no memory, that is, the mapping of blocks of binary digits into signals is performed independently
of the blocks transmitted before or after. If themodulator maps the binary digits 0 and 1 into a set of
two different waveforms, then the modulation is called binary. Alternatively, the modulator may map
h
symbols formed by h binary digits at a time onto M = 2 different waveforms. This modulation is called
multilevel.
The general expression of a signal modulated by a modulator with memory is
∞

v(t) =

∑ s ( t – nT ; x , s )
n

n

n =–∞

where {s(t ; i, j )} is a set of waveforms, xn is the symbol emitted by the source at time nT, and sn is the
state of the modulator at time n. If the modulator has no memory, then there is no dependence on sn.
A linear memoryless modulation scheme is one such that s(t - nT; xn ; sn) = xn s(t - nT).
M
Given a signal set { s i ( t )} i = 1 used for modulation, its compact characterization maybe given in terms
of a geometric representation . From the set of M waveforms we first construct a set of N ≤ M orthonormal

©2002 CRC Press LLC

N

waveforms { y i ( t )} i = 1 (the Gram–Schmidt procedure is the appropriate algorithm to do this). Next we
express each signal sk(t) as a linear combination of these waveforms,
N

sk ( t ) =

Âs

ki

yi ( t )

i=1

The coefficientsof this expansion may be interpreted as the components of M vectors that geometrically
represent the original signal set or, equivalently, as the coordinates in a Euclidean N-dimensional space
of a set of points called the signal constellation.

20.2 The Challenge of Digital Modulation
The selection of a digital modulation scheme should be done by making the best possible use of theresources available for transmission, namely, bandwidth, power, and complexity, in order to achieve the
reliability required.

Bandwidth
There is no unique definition of signal bandwidth. Actually, any signal s(t) strictly limited to a time
interval T would have an infinite bandwidth if the latter were defined as the support of the Fourier
transform of s(t). For example, consider the bandpass linearlymodulated signal

v(t) = ¬

•

 x s ( t – nT ) e

j 2 p f0 t

k

n = –•

where ¬ denotes real part, f0 is the carrier frequency, s(t) is a rectangular pulse with duration T and
amplitude 1, and (xk) is a stationary sequence of complex uncorrelated random variables with E(xn) = 0
2
and E(|xn| ) = 1. Then the power density spectrum of v(t) is given by

1
G ( f ) = -- [ G ( – f –f 0 ) + G ( f – f 0 ) ]
4
where

sin p f T
G ( f ) = T -----------------pfT

2

(20.1)

The function G( f ) is shown in Fig. 20.1.
The following are possible definitions of the bandwidth:
• Half-power bandwidth. This is the interval between the two frequencies at which the power
spectrum is 3 dB below its peak value.
• Equivalent noise bandwidth. This is given by
•

1 Ú –•...