176
DIGITAL COMMUNICATIONS
The carriermodulated PAM signal represented by (431) is a doublesideband (DSB) signal and requires twice the channel bandwidth of the
equivalent lowpass signal for transmission. Alternatively, we may use singlesideband (SSB)
CHAPTER
4:
CHAllACffRIZAT!ON
Of' COMMUNICAllON
Hence, these two signals have the same energy
coefficient of 1. Such signals are called antipodal.
PhaseModulated Signals In digital
Sl(iNALS
and
ANO SYSTEMS
177
a crosscorrelation
phase
modulation,
=
the
CHAPTER~:
CHARACTERIZATION
OF COMMl!'ilCATION
SIG!\Al.S
A~D
SYSTEMS
175
0
.
(t1)M=l
00
II
()
01
Ch>M=4
000
011
001
010
IOI
FIGURE 431
11()
Ill
I[)()
Signal space diagram for digital PAM signals.
where ~it denotes the energy in the pulse g(t). Clearly. t
78
Dl<ilTAL
("OMMlJNIC'ATIONS
()
011
()10
()(I;
M= 2
00'.
110
01
Ill
()()
100
IOI
M=H
11
10
FIGURE 4J.J
M"'4
Signal space diagrams for PSK signali;.
=
corresponds to onedimensional signals, which are 433. We note that
Signal2 space diagrams for M = 2,
('l!APTFR
1
01'f<A<TERl/.Allfl:"
ClF COMMl
1'1C"A'll0~
SJ(;l'Al.S
AS()
179
SY~rt.MS
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 [
I
.
.,
.,._
  . .
M ":32
'
M
.
lo

'
=
I

.,
)
I
I
I
I
M=J1
I
I
I
I
+   t 
I
I
I
I
I
.'
I
I
FIGURE 435
I
I
. . .  
.
I
'

..
Several signal space diagrams for rectangular
QAM.
 .
.  .
. I
'
I
  _.
',
I
+
  '

I

and
=[A,.~
C'HAPTEK
1:
CHARACTERIZ,\TIO!'
OF COMMU./!CATION
Sl(iNALS
AND
181
SYSTEMS
f
.r.,+
411[ ~~..
.r., + 311/ +1
r., + 21lj
++1
1;) +!if ++1
r,.
FIGURE 436
Subdivision
.
0
T
of lime and frequency axes into distinct slots.
.
:!.T
._.
3T
vector. Thus
86
DIGITAL
COMMl:~ICATIONS
Each of the M waveforms has energy 'it. The crosscorrelation 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
188
DIGITAL
C'OMMUNICA TIONS
51=1
flGURE 4314
~:i~.:tr~._.
O/s(r)
The trellis diagram for the NRZI signal.
(1'.<(r)
O/slll
01.(I)
illustrated in Fig. 4314. The trellis provides exactly the same information
concerning the signal dependence as the st
CHAf'llR
4:
CHARACTER17.A
TION OF COMll1l'1'llCA
no~
SlGNAl.S
AND,.,.,
!T\1S
187
Hence, the NRZ modulation is memoryless and is equivalent to a binarv PAM
or a binary PSK signal in a carriermodulated system.
The NRZI signal is different from the NRZ sig
190
DIGITAL COMMUNICATIONS
characteristics of digital modulation
observe in Section 44.
techniques
with memory, as we shall
433 Nonlinear Modulation Methods witb Memory
In this section, we consider a class of digital modulation methods in which the
pha
CHAPTf.R
4:
CHARA(TERIZAT10N
OF COMML~ICATIOl'o
and when a; = l, the transition matrix is
o
SIGNALS
\89
o]
1 0
0 0
T2 =
AND SYSl EMS
(4345)
0 0 1 0
Thus, these two 4 x 4 state transition matrices characterize the state diagram for
the Millerencoded sig
CHAPTER
~:
CHARACfER12.ATION
where cfJ(t; I) represents
the timevarying
efJ(t; I)
= 41CTf,1
is
although
continuous.
d(t)
f"
ANLl SYSTEMS
contains
carrier in the interval n T ~ t ~ (n
191
phase of the carrier. which is defined as
d( r) d t
discontinuities
192
DIGITAi.
COMMUNICATIONS
(jlli
~(/)
!
J_,
~
2T
0
2
T
T
(lll
I
' <jl/l~ 21"
.
I
1C(h~
2itr)
7
qcfw_f)
.!.
T
(I
T
T
(I>)
FIGl.iRE 4J.16
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
96
~:=
DIGITAL COMMl!NKATlONS
Jt
0
FIGURE 4321
2T
T
State trellis for binary CPFSK with h
3T
4T
= ~
over L symbol intervals (partial response CPM), the number of phase states may
increase up to a maximum of S, where
cfw_pMLi
S, 
2pMLI
(evenm)
(436
174
DIGITAL COMMUNICATIONS
from the sequence cfw_a, to the waveforms cfw_sm(t) is performed without any
constraint on previously transmitted waveforms, the modulator is called
memoryless.
In addition to classifying the modulator as either memoryless or ha
CHAPTER
4:
CHARACTERIZATION
OF COMMUNICATION
SIGNALS
AND
73
S~'STEMS
of a set of signals. Another related parameter is the Euclidean distance
between a pair of signals, defined as
d~ = II
=
Sm 
d~'.!,
I!
S~
cfw_J~, cfw_srn(t)  sk(t)2dt1.2
= cfw_im + ~k
EE 5362  Digital Communications, Fall 2014
Homework 1
Problem 1. Consider the signals in Fig. 22.1 (a) (as in the example that was discussed in class). Now,
change the fourth signal to the signal s4 (t) = +1 for 0 t 1 and zero everywhere else
(basically
AssignmentChapter 9 (100 points)
Student Name:
ID:
1. Write down the commands or functions (not results) in MATLAB to apply
each of the following edgefinding techniques in an image X: (20 points)
a. Roberts
edge=edge(img,roberts);
b. Sobel
edge=edge(img
158
DIGIT~l COMMUNICATIONS
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.
1
lowpass
160
DIGITAL COMM(JNICATIONS
That these two properties follow from the stationarity of n(t)
demonstrated. The autocorrelation function cl>nnCr) ot n(t) is
E[n(t)n(t
is now
+ r)] = Ecfw_[x(t) cos 2nfct  y(t) sin 2n:ft]
x [x(t
+ r) cos 2nf,.(t + r)
y(t + r
CHAPTER .i:
CHARACTERIZATION
OF COMMUNICATION.
SIGNALS
AND SYSTEMS
159
The combination of ( 4136) with (4130) gives the relationship between
the bandpass output signal r(t) and the equivalent lowpass time functions s,(1)
and h1(t). This simple relatio
164
DIGITAL COMMUNICATIONS
positive reaJ scalar. From the triangle inequality there follows the Cauchy
Schwartz inequality
(426)
with equality if v1 = av2 The norm square of the sum of two vectors may be
expressed as
(427)
llv1 + v2ll2 = llvd/2 + l!v2
CHAPTER:
CHARACTERIZATION
OF COMMUNICATION
SIGl'IALS
AND SYSTEMS.
165
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
=S;n.
422 Signal Space Concepts
162
DIGITAL COMMUNICATIONS
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
l"HAPrloR
Substituting
obtain
4:
CHARACTERIZATION
Of
COMMUNICATION
(4146) into (4147) and performing
SluNAL.S
ANO SYSTEMS
161
the expectation operation. we
(4148)
Now if the symmetry
properties given in (4140) and (4141)
are used in
(4148), we
166
DIGITAi. COMMUNICATIONS
We may approximate the signal s(t) by a weighted linear combination of
these functions, i.e.,
(4222)
s(t) = l( sd1c(t)
2:
k=I
where cfw_sk, I~ k ~ K are the coefficients in the approximation of s(t). The
approximation error i