(and. therefore, CPM) in which the modulation index h =
carrier in the interval n T $; t $; (n -f l)T is
8" + ~nln -T(
and the modulated carrier signal is
cos [ 2Tifct +
Signal space diagram! for M-ary simplex
The waveforms can be represented in vector form as
DIGIT AL COMMU!'1C' A TIONS
Suppose we have a set of M orthogonal
or, equivalently, their vector representation cfw_sm Their mean is
s = -Mm=I :
Now, let us construct another set of M signals by sub
These waveforms are characterized as having equal energy and crosscorrelation coefficients
:;: 2' /T (
= sin rrT(m
The real part of P1cm is
- k) A[eiICT(m - >A/
tcT(m -k) Af
sin lrrT(m - k) At]
Orthogonal signals for M = N
For the case in which
~f = 1/2T, the M FSK
signals are equivalent to the
.r., + 311/ -+-1
r., + 21lj
1;) +!if -+-+-1
of lime and frequency axes into distinct slots.
- - . .
+- - - t -
. . .- - -
Several signal space diagrams for rectangular
. - -.
- - -_.
- - '
cos 2rcf;.c and sin 2rcf.t. The resulting modulation technique is called quadrature
PAM or QAM. and the corresponding signal waveforms may be expressed as
- R [
Signal space diagrams for PSK signali;.
corresponds to one-dimensional signals, which are 4-3-3. We note that
Signal2 space diagrams for M = 2,
Each of the M waveforms has energy 'it. The cross-correlation between any pair
of waveforms depends on how we select the M waveforms from the 2" possible
waveforms. This topic is treated in Chapter 7. Clearly, any adjacent signa
The trellis diagram for the NRZI signal.
illustrated in Fig. 4-3-14. The trellis provides exactly the same information
concerning the signal dependence as the st
TION OF COMll-1l'1'llCA
Hence, the NRZ modulation is memoryless and is equivalent to a binarv PAM
or a binary PSK signal in a carrier-modulated system.
The NRZI signal is different from the NRZ sig
State trellis for binary CPFSK with h
over L symbol intervals (partial response CPM), the number of phase states may
increase up to a maximum of S, where
' <jl/l~- 21"
Pulse shapes for full response CPM (a, b) and partial response CPM (c, d).
If g(1) = 0 fort> T, the CPM signal is c
where cfJ(t; I) represents
carrier in the interval n T ~ t ~ (n
phase of the carrier. which is defined as
d( r) d t
and when a; = l, the transition matrix is
AND SYSl EMS
0 0 1 0
Thus, these two 4 x 4 state transition matrices characterize the state diagram for
the Miller-encoded sig
characteristics of digital modulation
observe in Section 4-4.
with memory, as we shall
4-3-3 Nonlinear Modulation Methods witb Memory
In this section, we consider a class of digital modulation methods in which the
Signal space diagram for digital PAM signals.
where ~it denotes the energy in the pulse g(t). Clearly. t
Hence, these two signals have the same energy
coefficient of -1. Such signals are called antipodal.
Phase-Modulated Signals In digital
The carrier-modulated PAM signal represented by (4-3-1) is a doublesideband (DSB) signal and requires twice the channel bandwidth of the
equivalent lowpass signal for transmission. Alternatively, we may use singlesideband (SSB)
In the special case in which the stationary stochastic process n(t) is gaussian, the
quadrature components x(t) and y(t + r) are jointly gaussian. Moreover, for r =
0, they are statistically independent, and, hence, their joint
By continuing this procedure, we shall construct a set of n1, orthonormal
vectors, where n, ~ n, in general. If m < n then n1 ~ m, and if m ~ n then n1
4-2-2 Signal Space Concepts
positive reaJ scalar. From the triangle inequality there follows the Cauchy
with equality if v1 = av2 The norm square of the sum of two vectors may be
llv1 + v2ll2 = llvd/2 + l!v2
The combination of ( 4-1-36) with (4-1-30) gives the relationship between
the bandpass output signal r(t) and the equivalent lowpass time functions s,(1)
and h1(t). This simple relatio
That these two properties follow from the stationarity of n(t)
demonstrated. The autocorrelation function cl>nnCr) ot n(t) is
+ r)] = Ecfw_[x(t) cos 2nfct - y(t) sin 2n:ft]
+ r) cos 2nf,.(t + r)
-y(t + r
bandpass input signal is simply obtained from the equivalent lowpass input
signal Suppose that s(t) is a narrowband bandpass signal and s (t) is the equivalent
and the equivalent lowpass impulse response of the system.
Assignment-Chapter 9 (100 points)
1. Write down the commands or functions (not results) in MATLAB to apply
each of the following edge-finding techniques in an image X: (20 points)