1
QPSKQAM
QPSK,.QAMMATLABQPSKQAMMonte CarloQPSKQAM
QPSK QAM
Title: QPSK, QAM modulation correspondence simulation
Major: Electronic Information of Science and Technology
Name: Signature:
Supervisor: Signature:
ABSTRACT
This article mainly discusses the modulation demodulation
principle of the QPSK ,QAM. And it analyzes their modulation
demodulation realization process. Then it carries on the
communications system simulation with MATLAB to the QPSK, QAM
modulation process, analyzes how each step of the modulation
process specifically realizes. And obtains their error rate diagram
of curves using Monte the Carlo simulation and analyzes. Finally
carries on comparison to the QPSK, QAM two modulation ways, obtains
their good and bad points respectively.
Key words QPSK, QAM, Scatter diagramError rate
Type of ThesisTheories Analysis Software Development
191837F.B.Mores,,;1837();1948(Shannon);2070,
AMFMFSKOOKPWMPPMPAM
//QAMQPSKBPSKQPSKQAM
11
11.1
11.2
11.2.1
21.2.2
21.3
42 MATLAB
42.1MATLAB
73
84 QPSK
84.1QPSK
84.1.1QPSK
94.1.2 QPSK
104.1.3 QPSK
124.2MATLABQPSK
124.2.1
124.2.2
145 QAM
145.1QAM
145.1.1 MQAM
145.1.2MQAM
155.2 QAM
155.2.1
155.2.2
165.3 MQAM
175.4 16QAM
185.5 16QAM
206 QPSKQAM
22
23
24
25
34
1
1.1
1.1.1
1.2
1.2.1
1
1/45351605kHZ187560m,10cm,900MHz
2
4kHz----PCM30
3
10kHz
1.2.2
---m(t),c(t),s(t),m(t)c(t)
1m(t)
1m(t)
2
2c(t)
1
2
3
1PAMASK
2FMFSKPDM
3PMPSKPPM
4
1ASK
2FMPMFSK
1.3
12SNR
2
log(1)
CWSNR
=+
Cbit/s,WHzSNRdB1.3.11.3.2
1.3.1
1.3.2SNR(dB)
h,bit/Symbol,bit/s/Hz
h
hSNR1.3.3
h
1.3.3
2 MATLAB
BasicFortranc, ,Mathwork1967Matrix
LaboratoryMATLAB6.xwindowsMATLABMATLABMATLAB--BasicFortranCMATLAB.
MATLAB
2.1MATLAB
1)randint
(1)OUT=randint 01
(2)OUT=randint(M) MM0101
(3)OUT=randint(M,N) MN0101
(4)OUT=randint(M,N,RANGE) MNRANGE
2) length
Length(M)M
3) mod
Mod(x,y)xy
4)qammod
Y = QAMMOD(X,M) QAMXMMQAMMX0M-1
5scatterplot
Scatterplot(x)xx
6) fft
Fft(x)xFFT
7abs
Abs(x)x
8)plot
(1)ploty
yplot[1 2 3]y
(2)plotx,y
Xy
(3)plotx,y,
9)stem
plot
10)axis
axis[XMIN XMAX YMIN YMAX ]
XY
11) Xlabel, ylabel
xyxlabel
xlabel(text,property1propertyValuel,property1,property
Valye2,)
12)title
xlabel
13)text
Text(x,y,string)(x,y)
14)subplot
H=subplot(m,n,p)mnp
3
1
0
()
2
n
n
S
w
=
w
p
0
n
---W/Hz
(2)
0
n
/2
(
)
(
)
1
0
()
2
nn
n
RFS
twdt
-
==
t
0
30
4 QPSK
4.1QPSK
4.1.1QPSK
M
(
)
(
)
(
)
(
)
2
2(1)/
Re
2
()cos21
22
()cos1cos2()sin(1)sin2
c
jft
jmM
m
c
cc
stgtee
gtftm
M
gtmftgtmft
MM
p
p
p
p
pp
pp
-
=
=+-
=---
m=1,2,M
0
tT
g(t)
2(1)/
m
mM
qp
=-
m=1,2,MM
22
00
11
()()
22
TT
mg
stdtgtdt
ee
===
1
()
ft
2
()
ft
1122
()()()
mmm
stsftsft
=+
1
2
()()cos2
c
g
ftgtft
p
e
=
2
2
()()sin2
c
g
ftgtft
p
e
=-
1,2
mmm
sss
=
22
cos(1),sin(1)
22
gg
m
smm
MM
ee
pp
=--
m=1,2,M
M=44PSKQPSKQPSK0
/2,,3/2
ppp
/4
QPSK
p
-
/4
p
QPSK
2
()cos(1)
2
s
QPSKc
s
E
Stti
T
p
w
=+-
0
S
tT
i=1234
S
E
0
S
tT
c
w
s
T
QPSKQPSK
QPSK4.1.1
4.1.1 QPSK
4.1.2 QPSK
AWGN
(
)
()()
()()cos(2)()sin(2)
m
mccsc
rtutnt
utntftntft
pp
=+
=+-
()
c
nt
()
s
nt
1
()()cos(2)
Tc
tgtft
yp
=
2
()sin(2)
Tc
gtft
yp
=-
1
()
t
y
2
()
t
y
22
(cossin)
mscss
mm
rsnnn
MM
pp
xx
=+=++
c
n
s
n
1
()()
2
cTc
ngtntdt
-
=
1
()()
2
sTs
ngtntdt
-
=
()
c
nt
()
s
nt
()()0
cs
EnEn
==
()0
cs
Enn
=
c
n
s
n
22
0
()()
2
cs
N
EnEn
==
rM{
m
s
}
(,)
mm
Crsrs
=
m=0,1,M-1
r=(
c
r
,
s
r
)
arctan
s
r
c
r
r
q
=
{
m
s
}
r
q
AWGNPAM
2
0
2
b
PQ
N
x
=
b
x
1M
f
4
M
P
0
0
2
2sin
2
2sin
s
M
b
PQ
NM
k
Q
NM
x
p
x
p
2
log
kM
=
/
4.1.3 QPSK
QPSKPSK4.1.2
4.1.2 QPSK
+1V1-1V02IQ
sin
t
w
cos
t
w
QPSKQPSK4IQ4[0 0][0 1][1 0][1 1]290
2
m
m
M
p
j
=
m=0,1,,M-1,45-45QPSK
sin
t
w
cos
t
w
-QPSK4.1.3
4.1.3 QPSK
QPSKQPSK
,
()()cos()
()cos()
Sck
k
ck
stgtkTt
gtt
wj
wj
=-+
=+
g(t)
k
j
c
w
1
0
00
()cos()cos
11
()cos(2)cos()
22
s
ss
T
ckc
TT
kk
Ugtttdt
gttdtgtdt
wjw
wjj
=+
=++
t=
s
T
I
U
s
T
s
T
s
T
t=
s
T
0t=
s
T
1/2cos
k
j
g(t)I
cos
k
j
1
cos
k
U
j
Q
sin
k
j
sin
kk
U
j
10
4.2MATLABQPSK
4.2.1
1)QPSK
2)MATLAB
3)MATLABM-File
4)
(1)
(2)
(3)
(4)QPSK
(5)
(6)
5)
4.2.2
MATLAB
1) 01
2) IQ
3) BPSKsin(2*pi*Fc*t)cos(2*pi*Fc*t)
4) QPSK
5) Monte Carlo
Monte Carlo
r4201400.25,(0.25,0.5),(0.5,0.75),(0.75,1.0)
00,01,1110,
m
s
c
n
s
n
2
d
2
1
d
=
s
x
SNR
m
rsn
=+
r44.2.1SNR
0
/
b
N
x
/2
bs
xx
=
10000
4.2.1
5 QAM
M
APKMQAM
MQAM4QAM8QAM16QAM64QAM16QAM
5.1QAM
QAMcos2
t
f
c
p
)
2
sin(
t
f
c
p
5.1.1 MQAM
xtytMQAM
t
f
t
y
t
f
t
x
S
c
c
QAM
p
p
2
sin
)
(
2
cos
)
(
+
=
)
(
)
(
b
k
k
kT
t
g
x
t
x
-
=
)
(
)
(
b
k
k
kT
t
g
y
t
y
-
=
b
T
k
x
k
y
3
,
1
MQAM
()Re[()()]Re[]
c
cs
jwt
QAMiiTi
stajagtVe
=+=
i=1,2,3,..,M
0
s
tT
arctan(/)
ii
ab
q
=
,MQAM
5.1.2MQAM
MQAM
1122
()()()
iii
stsftsft
=+
i=1,2,3,..,M
0
s
tT
1
2
()()cos
Tc
g
ftgtwt
E
=
0
s
tT
2
2
()()sin
Tc
g
ftgtwt
E
=-
0
s
tT
1
1
0
()()
2
s
c
T
g
iii
E
sstftdta
==
i=1,2,3,..,M
22
0
()()
2
s
c
T
g
iii
E
sstftdta
==
i=1,2,3,..,M
MQAM
12
,
[,]
22
cs
gg
iiiii
EE
sssaa
==
i=1,2,3,..,M
5.2 QAM
5.2.1
.,MQAM.
MQAM,MQAM
MQAM1.1-abc16QAM(M=4)
5.2.1 MQAM
5.2.2
MQAM,,:MQAM ,dminmin,
(1)MQAM
m QAM MQAM,,
(2)MQAMdmin
dminMQAM,MQAMMQAMdmin,MQAM
(3)MQAMmin
MQAM,MQAM,MQAMmin,
5.2.2
min
q
min
d
g
450
0
0.43
E
1.7
0
30
0
0.59
E
1.3
0
18
0
0.63
E
1.8
5.2.2
,E0,dmin,,,,,,min,-,
, MQAMMQAMMQAMMQAMMQAMMQAM
5.3 MQAM
MQAM5.3.1
5.3 .1 16QAM
{
n
a
}IQ
b
R
/2K/2
M
I(t)Q(t)
M
PAM0I(t)Q(t)ASKQAM
5.3.2 16QAM
5.4 16QAM
MAPKM
1616QAMQAM
16QAM
5.4.1 16QAM
MQAM
t
f
t
y
t
f
t
x
S
c
c
QAM
p
p
2
sin
)
(
2
cos
)
(
+
=
)
(
)
(
b
k
k
kT
t
g
x
t
x
-
=
)
(
)
(
b
k
k
kT
t
g
y
t
y
-
=
b
T
k
x
k
y
3
,
1
16QAM
k
k
y
x
,
3
,
1
)
(
b
kT
t
g
-
a a
5.5 16QAM
16QAM:
1
2
101416QAM4bit Str=[1 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 1 1 0 1]
2IQ1/2IQstr1=[1 0 1 0 0 0 0 1 1 0]str2=[0 1 1 1 1 0 1 1 1 1
]
3BPSKA1Fc1Fs50
4BPSKQAM
Monte Carlo
16
1
b
2
b
3
b
4
b
4
[
]
,
mcms
AA
[
]
,
cs
nn
f
0
[
]
,
mccmss
rAnAn
=++
r5.5.1SNRN=10000
/4
bs
xx
=
5.5.1 16QAM
6 QPSKQAM
h
[bit/(sHz)],
h
=
,,,,,2 bit/(sHz),,.,
1
2
bit/(sHz),,QPSKQAM2 bit/(sHz)QPSK116QAM2
eb
P
BERQPSKQAM
1QPSKMQAMMQAMM4
2MQAMM4MPSKM=16
MPSK:M=4
2sin2
4
p
dAA
p
==
M=16
2sin0.39
16
p
dAA
p
==
MQAM:M=4
2
2
1
Q
A
dA
L
==
-
M=16
22
0.47
13
Q
A
dAA
L
===
-
M4M=16QAM16PSKMMPSK
3 QPSK
2
A
16QAM0.47A,,,QPSK16QAM.
4 MPSKM2
p
M
2
p
/MM
16QPSK8PSK
MATLABQPSKQAMQPSKQAM
QPSKQAM
QPSKQAMMATLABMonte CarloQPSKQAM
MATLABMATLABMATLAB
QPSKQAM
[1]..1994Vol.14(No.1)
[2].m-QAM.1997Vol.2(No.11)
[3].m-QAM.-
[4].16QAMHDTV.
[5].FSKPSKQAM.
[6].M-QAM.1996Vol.24(No.10)
[7] .QAM modulations with q-ary turbo codes.SCIENCE IN
CHINA,1997,Vol.40(No.1)
[8].QAMHFC.2002
[1]...2001
[2].MATLAB..2001
[3].G.....2001
[4]...2000
5.3 Amplitude and Phase Modulation
In amplitude and phase modulation the information bit stream is
encoded in the amplitude and/or phase of the transmitted signal.
Specifically, over a time interval of
s
T
,
M
2
log
=
k
bits are encoded into the amplitude and/or phase of the
transmitted signal
s
T
t
t
s
p/p
p0/p
p),/p
p(/p
. The transmitted signal over this period div class="embedded"
id="_1207225318"/p)/p
p2/p
psin(/p
p)/p
p(/p
p)/p
p2/p
pcos(/p
p)/p
p(/p
p)/p
p(/p
pt/p
pf/p
pt/p
ps/p
pt/p
pf/p
pt/p
ps/p
pt/p
ps/p
pc/p
pQ/p
pc/p
pI/p
pp/p
pp/p
p-/p
p=/p
can be written in terms of its signal space representation as
div class="embedded" id="_1207225542"/p)/p
p(/p
p)/p
p(/p
p)/p
p(/p
p2/p
p2/p
p1/p
p1/p
pt/p
ps/p
pt/p
ps/p
pt/p
ps/p
pi/p
pi/p
pj/p
pj/p
p+/p
p=/p
with basis functions div class="embedded"
id="_1207228743"/p)/p
p2/p
pcos(/p
p)/p
p(/p
p)/p
p(/p
p0/p
p1/p
pj/p
pp/p
pj/p
p+/p
p=/p
pt/p
pf/p
pt/p
pg/p
pt/p
pc/p
and div class="embedded" id="_1207225850"/p)/p
p2/p
psin(/p
p)/p
p(/p
p)/p
p(/p
p0/p
p2/p
pj/p
pp/p
pj/p
p+/p
p-/p
p=/p
pt/p
pf/p
pt/p
pg/p
pt/p
pc/p
,where div class="embedded" id="_1207226012"/p)/p
p(/p
pt/p
pg/p
is a shaping pulse. To send the div class="embedded"
id="_1207226080"/pi/p
th message over the time interval [div class="embedded"
id="_1207226139"/pT/p
pk/p
pkT/p
p)/p
p1/p
p(/p
p,/p
p+/p
], we setdiv class="embedded" id="_1207226217"/p)/p
p(/p
p)/p
p(/p
p1/p
pt/p
pg/p
ps/p
pt/p
ps/p
pi/p
pI/p
p=/p
anddiv class="embedded" id="_1207226315"/p)/p
p(/p
p)/p
p(/p
p2/p
pt/p
pg/p
ps/p
pt/p
ps/p
pi/p
pQ/p
p=/p
. These in-phase and quadrature signal components are baseband
signals with spectral characteristics determined by the pulse
shapediv class="embedded" id="_1207226012"/p)/p
p(/p
pt/p
pg/p
. In particular, their bandwidth div class="embedded"
id="_1207226478"/pB/p
equals the bandwidth of div class="embedded"
id="_1207226012"/p)/p
p(/p
pt/p
pg/p
, and the transmitted signal div class="embedded"
id="_1207226534"/p)/p
p(/p
pt/p
ps/p
is a passband signal with center frequency div class="embedded"
id="_1207226578"/pc/p
pf/p
and passband bandwidthdiv class="embedded"
id="_1207226607"/pB/p
p2/p
. In practice we take div class="embedded"
id="_1207226636"/ps/p
pg/p
pT/p
pK/p
pB/p
p=/p
where div class="embedded" id="_1207226726"/pg/p
pK/p
depends on the pulse shape: for rectangular pulsesdiv
class="embedded" id="_1207226762"/pg/p
pK/p
= .5 and for raised cosine pulses div class="embedded"
id="_1207226821"/p1/p
p5/p
p./p
p/p
p/p
pg/p
pK/p
, as discussed in Section 5.5. Thus, for rectangular pulses the
bandwidth ofdiv class="embedded" id="_1207226012"/p)/p
p(/p
pt/p
pg/p
isdiv class="embedded" id="_1207226986"/pS/p
pT/p
p5/p
p./p
and the bandwidth ofdiv class="embedded"
id="_1207227067"/p)/p
p(/p
pt/p
ps/p
is div class="embedded" id="_1207227162"/pS/p
pT/p
p1/p
. Since the pulse shape div class="embedded"
id="_1207226012"/p)/p
p(/p
pt/p
pg/p
is fixed, the signal constellation for amplitude and phase
modulation is defined based on the constellation point:div
class="embedded" id="_1207227337"/pM/p
pi/p
pR/p
ps/p
ps/p
pi/p
pi/p
p,...,/p
p1/p
p,/p
p)/p
p,/p
p(/p
p2/p
p2/p
p1/p
p=/p
p/p
. The complex baseband representation of div class="embedded"
id="_1207226534"/p)/p
p(/p
pt/p
ps/p
is /pp class=""div class="embedded" id="_1207227755"/p{/p
p}/p
p)/p
p2/p
p(/p
p0/p
p)/p
p(/p
p)/p
p(/p
pt/p
pf/p
pj/p
pj/p
pc/p
pe/p
pe/p
pt/p
px/p
pR/p
pt/p
ps/p
pp/p
pj/p
p=/p
(5.51)/pp class=""wherediv class="embedded"
id="_1207227828"/p)/p
p(/p
p)/p
p(/p
p)/p
p(/p
p)/p
p(/p
p)/p
p(/p
p2/p
p1/p
pt/p
pg/p
pjs/p
ps/p
pt/p
pjs/p
pt/p
ps/p
pt/p
px/p
pi/p
pi/p
pQ/p
pI/p
p+/p
p=/p
p+/p
p=/p
.The constellation pointdiv class="embedded"
id="_1207228036"/p)/p
p,/p
p(/p
p2/p
p1/p
pi/p
pi/p
pi/p
ps/p
ps/p
ps/p
p=/p
is called the symbol associated with the div class="embedded"
id="_1207228134"/pM/p
p2/p
plog/p
bits and div class="embedded" id="_1207228189"/pS/p
pT/p
is called the symbol time. The bit rate for this modulation is
div class="embedded" id="_1207228254"/pK/p
bits per symbol or div class="embedded"
id="_1207228282"/ps/p
pT/p
pM/p
pR/p
p2/p
plog/p
p=/p
bits per second./pp class="" There are three main types of
amplitude/phase modulation:/pp class="" Pulse Amplitude Modulation
(MPAM):information encoded in amplitude only./pp class="" Phase
Shift Keying (MPSK): information encoded in phase only./pp class=""
Quadrature Amplitude Modulation (MQAM): information encoded in both
amplitude and phase./pp class=""The number of bits per symboldiv
class="embedded" id="_1207228390"/pM/p
pK/p
p2/p
plog/p
p=/p
, signal constellationdiv class="embedded"
id="_1207227337"/pM/p
pi/p
pR/p
ps/p
ps/p
pi/p
pi/p
p,...,/p
p1/p
p,/p
p)/p
p,/p
p(/p
p2/p
p2/p
p1/p
p=/p
p/p
, and choice of pulse shape div class="embedded"
id="_1207226012"/p)/p
p(/p
pt/p
pg/p
determines the digital modulation design. The pulse shape div
class="embedded" id="_1207226012"/p)/p
p(/p
pt/p
pg/p
is designed to improve spectral efficiency and combat ISI, as
discussed in Section 5.5 below./pp class=""Amplitude and phase
modulation over a given symbol period can be generated using the
modulator structure shown in Figure 5.10. Note that the basis
functions in this figure have an arbitrary phasediv
class="embedded" id="_1207228604"/p0/p
pj/p
associated with the transmit oscillator. Demodulation over each
symbol period is performed using the demodulation structure of
Figure 5.11, which is equivalent to the structure of Figure 5.7 for
div class="embedded" id="_1207228794"/p)/p
p2/p
pcos(/p
p)/p
p(/p
p)/p
p(/p
p0/p
p1/p
pj/p
pp/p
pj/p
p+/p
p=/p
pt/p
pf/p
pt/p
pg/p
pt/p
pc/p
and div class="embedded" id="_1207225850"/p)/p
p2/p
psin(/p
p)/p
p(/p
p)/p
p(/p
p0/p
p2/p
pj/p
pp/p
pj/p
p+/p
p-/p
p=/p
pt/p
pf/p
pt/p
pg/p
pt/p
pc/p
. Typically the receiver includes some additional circuitry for
carrier phase recovery that matches the carrier phase div
class="embedded" id="_1207302856"/pj/p
at the receiver to the carrier phase div class="embedded"
id="_1207228604"/p0/p
pj/p
at the transmitter, which is called coherent detection. If div
class="embedded" id="_1207302892"/p0/p
p0/p
p/p
pD/p
p=/p
p-/p
pj/p
pj/p
pj/p
then the in-phase branch will have an unwanted term associated
with the quadrature branch and vice versa, i.e. div
class="embedded" id="_1207302984"/p1/p
p2/p
p1/p
p1/p
p)/p
psin(/p
p)/p
pcos(/p
pn/p
ps/p
ps/p
pr/p
pi/p
pi/p
p+/p
pD/p
p+/p
pD/p
p=/p
pj/p
pj/p
and div class="embedded" id="_1207303131"/p2/p
p2/p
p1/p
p2/p
p)/p
pcos(/p
p)/p
psin(/p
pn/p
ps/p
ps/p
pr/p
pi/p
pi/p
p+/p
pD/p
p+/p
pD/p
p=/p
pj/p
pj/p
which can result in significant performance degradation. The
receiver structure also assumes that sampling function everydiv
class="embedded" id="_1207303230"/ps/p
pT/p
seconds is synchronized to the start of the symbol period, which
is called synchronization or timing recovery. Receiver
synchronization and carrier phase recovery are complex receiver
operations that can be highly challenging in wireless environments.
These operations are discussed in more detail in Section 5.6. We
will assume perfect carrier recovery in our discussion of MPAM,
MPSK and MQAM, and therefore set div class="embedded"
id="_1207303278"/
EMBED Equation.3 p0/p
p0/p
p=/p
p=/p
pj/p
pj/p
for their analysis./pp class=""5.3.1 Pulse Amplitude Modulation
(MPAM)/pp class=""We will start by looking at the form of linear
modulation, one-dimensional MPAM, which has no quadrature component
(div class="embedded" id="_1207303388"/p0/p
p2/p
p=/p
pi/p
ps/p
) For MPAM all of the information is encoded into the signal
amplitudediv class="embedded" id="_1207303424"/pi/p
pA/p
. The transmitted signal over one symbol time is given by/pp
class=""div class="embedded" id="_1207303642"/p{/p
p}/p
pc/p
ps/p
pc/p
pi/p
pt/p
pf/p
pj/p
pi/p
pi/p
pf/p
pT/p
pt/p
pt/p
pf/p
pt/p
pg/p
pA/p
pe/p
pt/p
pg/p
pA/p
pR/p
pt/p
ps/p
pc/p
p1/p
p0/p
p),/p
p2/p
pcos(/p
p)/p
p(/p
p)/p
p(/p
p)/p
p(/p
p2/p
p>>
=
=
p
p
5.52
where
M
i
d
M
i
A
i
,...,
2
,
1
,
)
1
2
(
=
-
-
=
defines the signal constellation, parameterized by the distance
d which is typically a function of the signal energy, and
)
(
t
g
is the pulse shape satisfying (5.12) and (5.13).
N-dimensional space, which must be mapped to a sequence of
constellations in 2-dimensional space in order to be generated by
the modulator shown in Figure 5.10. The general conclusion in [18]
is that for uncoded modulation, the increased complexity of
spherical constellations is not worth their energy gains, since
coding can provide much better performance at less complexity cost.
However, if a complex channel code is already being used and little
further improvement can be obtained by a more complex code,
constellation shaping may obtain around 1 dB of additional gain. An
in-depth discussion of constellation shaping, as well as
constellations that allow a noninteger number of bits per symbol,
can be found in [18].
The minimum distance between constellation points is
d
A
A
d
j
i
j
i
2
|
|
min
,
min
=
-
=
.The amplitude of the transmitted signal takes on M different
values, which implies that each pulse conveys
K
M
=
2
log
bits per symbol time
s
T
.
Over each symbol period the MPAM signal associated with the
i
th constellation has energy
2
0
2
2
2
0
2
)
2
(
cos
)
(
)
(
i
T
c
i
T
i
s
A
dt
t
f
t
g
A
dt
t
s
E
s
s
i
=
=
=
p
(5.53)
Since the pulse shape must satisfy(5.12). Note that the energy
is not the same for each signal
)
(
t
s
i
, i=1,,M. Assuming equally likely symbol, the average energy
is
=
=
M
i
i
s
A
M
E
1
2
1
(5.54)
The constellation mapping is usually done by Gray encoding,
where the message associated with signal amplitudes that are
adjacent to each other differ by one bit value, as illustrated in
Figure 5.12. With this encoding method, if noise causes the
demodulation process to mistake one symbol for an adjacent one
(with most likely type of error), this results in only a single bit
error in the sequence of K bits. Gray codes can be designed for
MPSK and square MQAM constellations, but not rectangular MQAM.
The decision regions
i
Z
, i=1,,M associated with the pulse amplitude
d
M
i
A
i
)
1
2
(
-
-
=
for M=4 and M=8 are shown in Figure 5.13. Mathematically, for
any M, these decision regions are defines by
=
-
-
+
-
=
+
-
=
M
i
d
A
M
i
d
A
d
A
i
d
A
Z
i
i
i
i
i
)
,
[
1
2
)
,
[
1
)
,
(
Form (5.52) we see that MPAM has only a single basis
function
)
2
cos(
)
(
)
(
1
t
f
t
g
t
c
p
f
=
. Thus, the coherent demodulator of Figure 5.11 for MPAM reduces
to the demodulator shown in Figure 5.14, where the multithreshold
device maps x to a decision region
i
Z
and outputs the corresponding bit sequence
)
,
,
{
1
K
i
b
b
m
m
K
=
=
.
5.3.2 Phase Shift Keying (MPSK)
For MPSK all of the information is encoded in the phase of the
transmitted signal. Thus, the transmitted signal over one symbol
time is given by
{
}
t
f
M
i
t
Ag
t
f
M
i
t
Ag
M
i
t
f
t
Ag
T
t
e
e
t
Ag
R
t
s
c
c
c
s
t
f
M
i
j
i
c
p
p
p
p
p
p
p
p
2
sin
)
1
(
2
sin
)
(
2
cos
)
1
(
2
cos
)
(
)
1
(
2
2
cos
)
(
0
,
)
(
)
(
2
/
)
1
(
2
-
-
-
=
-
+
=
=
-
(5.55)
Thus, the constellation points or symbol
)
,
(
2
1
i
i
s
s
are given by
-
=
M
i
A
s
i
)
1
(
2
cos
1
p
and
-
=
M
i
A
s
i
)
1
(
2
sin
2
p
for i=1,,M. The pulse shape
)
(
t
g
satisfied (5.12) and (5.13), and
K
i
M
i
M
i
2
,
,
1
,
)
1
(
2
=
=
-
=
K
p
q
are the different phases in the signal constellation points that
convey the information bits. The minimum distance between
constellation points is
)
sin(
2
min
M
A
d
p
=
, where A is typically a function of the signal energy. 2PSK is
often referred to as binary PSK or BPSK, which 4PSK is often called
quadrature phase shift keying (QPSK), and is the same as MQAM with
M=4 which is defined below.
All possible transmitted signals
)
(
t
s
i
have equal energy
2
0
2
)
(
A
dt
t
s
E
s
i
T
i
s
=
=
(5.56)
Note that for
s
s
T
t
T
t
g
=
0
,
2
)
(
, i.e. a rectangular pulse, this signal has constant envelope,
unlike the other amplitude modulation techniques MPAM and MQAM.
However, rectangular pulse are spectrally-inefficient, and more
efficient pulse shapes make MPSK nonconstant envelope. As for MPAM,
constellation mapping is usually done by Gray encoding, where the
messages associated with signal phases that are adjacent to each
other differ by one bit value, as illustrated in Figure 5.15. With
this encoding method, mistaking a symbol for an adjacent one causes
only a single bit error.
The decision regions
i
Z
i=1,,M, associated with MPSK for M=8 are shown in Figure 5.16.
If we represent
2
R
re
r
j
=
q
in polar coordinates then decision for any M are defined by
{
}
M
i
M
i
re
Z
j
i
)
5
.
(
2
)
5
.
(
2
:
+
-
=
p
q
p
q
(5.57)
Form (5.55) we see that MPSK has both in-phase and quadrature
components, and coherent demodulator is as shown in Figure 5.11.
For the special case of BPSK, the decision regions as given in
Example 5.2 simplify to
)
0
:
(
1
>
=
r
r
Z
and
)
0
:
(
1
=
r
r
Z
. Moreover BPSK has only a single basis function
)
2
cos(
)
(
)
(
1
t
f
t
g
t
c
p
f
=
and , since there is only a single bit transmitted per symbol
time
s
T
, the bit time
s
b
T
T
=
. Thus, the coherent demodulator of Figure 5.11for BPSK reduces
to the demodulator shown in Figure 5.17, where the threshold device
maps x to the positive or negative half of the real line, and
outputs the corresponding bit value. We have assumed in this figure
that the message corresponding to a bit value if 1,
1
1
=
m
, is mapped to constellation point
A
s
=
1
and the message corresponding to a bit value of 0,
0
2
=
m
, is mapped to the constellation point
A
s
-
=
2
.
5.3.3 Quadrature Amplitude Modulation (MQAM)
For MQAM, the information bits are encoded in both the amplitude
and phase of the transmitted signal. Thus, whereas both MPAM and
MPSK have one degree of freedom in which to encode the information
bits (amplitude or phase), MQAM has two degrees of freedom. As a
result, MQAM is more spectrally-efficient than MPAM and MPSK, in
that it can encode the most number of bits per symbol for a given
average energy.
The transmitted signal is given by
{
}
s
c
i
i
c
i
i
t
f
j
j
i
i
T
t
t
f
t
g
A
t
f
t
g
A
e
t
g
e
A
R
t
s
c
i
-
=
=
0
),
2
sin(
)
(
)
sin(
)
2
cos(
)
(
)
cos(
)
(
)
(
2
p
q
p
q
p
q
(5.58)
Where the pulse shape
)
(
t
g
satisfies (5.12) and (5.13). The energy in
)
(
t
s
i
is
2
0
2
)
(
i
T
i
s
A
t
s
E
s
i
=
=
(5.59)
The same as for MPAM. The distance between any pair of symbols
in the signal constellation is
2
2
2
2
1
1
)
(
)
(
j
i
j
i
j
i
ij
s
s
s
s
s
s
d
-
+
-
=
-
=
(5.60)
For square signal constellations, where
1
i
s
and
2
i
s
take values on
l
L
i
d
L
i
2
,
,
2
,
1
,
)
1
2
(
=
=
-
-
K
, the minimum distance between signal points reduces
d
d
2
min
=
, the same as for MPAM. In fact, MQAM with square constellation
of size
2
L
is equivalent to MPAM modulation with constellations of size L
on each of the in-phase and quadrature signal components. Common
square constellation are 4QAM and 16QAM, which are shown in Figure
5.18 below. These square constellation have
2
2
2
L
M
l
=
=
constellation points, which are used to send 2l bit/symbol, or l
bits per dimension. It can be shown that the average power of a
square signal constellation with l bits per dimension,
l
S
, is proportional to
3
4
l
,and it follows that the average power for one more bit per
dimension
l
l
S
S
4
1
+
. Thus, for square constellation it takes approximately 6 dB
more power to send an additional 1 bit/dimension or 2bits/symbol
while maintaining the same minimum distance between constellation
points.
Good constellation mapping can be hard to find for QAM signals,
especially for irregular constellation shapes. In particular, it is
hard to find a Gray code mapping where all adjacent symbol differ
by a single bit. The decision regions
i
Z
, i=1,,M, associated with MQAM for M=16 are shown in Figure
5.19. From (5.58)we see that MQAM has both in-phase and quadrature
components, and thus the coherent demodulator is as shown in Figure
5.11.
5.3.4 Differential Modulation
The information in MPSK and MQAM signals is carried in the
signal phase. Thus, these modulation techniques require coherent
demodulation, i.e. the phase of the transmitted signal carrier 0
must be matched to the phase of the receiver carrier . Techniques
for phase recovery typically require more complexity and cost in
the receiver and they are also susceptible to phase drift of the
carrier. Moreover, obtaining a coherent phase reference in a
rapidly fading channel can be difficult. Issues associated with
carrier phase recovery are dicussed in more detail in Section 5.6.
Due to the difficulties as well as the cost and complexity
associated with carrier phase recovery, differential modulation
techniques, which do not require a coherent phase reference at the
receiver, are generally preferred to coherent modulation for
wireless applications.
Differential modulation falls in the more general class of
modulation with memory, where the symbol transmitted over time
]
)
1
(
,
[
s
s
T
k
kT
+
depends on the bits associated with the current message to be
transmitted and the bits transmitted over prior symbol times. The
basic principle of differential modulation is to use the previous
symbol as a phase reference for the current symbol, thus avoiding
the need for a coherent phase reference at the receiver.
Specifically, the information bits are encoded as the differential
phase between the current symbol and the previous symbol. For
example, in differential BPSK, referred to as DPSK, if the symbol
over time
]
,
)
1
[(
s
s
kT
T
k
-
has phase
p
q
q
q
,
0
,
)
1
(
=
=
-
i
j
i
e
k
, then to encode a 0 bit over
]
)
1
(
,
[
s
s
T
k
kT
+
, the symbol would have phase
i
j
e
k
q
q
=
)
(
and to encode a 1 bit the symbol would have phase
p
q
q
+
=
i
j
e
k
)
(
. In other words, a 0 bit is encoded by no change in phase,
whereas a 1 bit is encoded as a phase change of . Similarly, in
4PSK modulation with differential encoding, the symbol phase over
symbol interval
]
)
1
(
,
[
s
s
T
k
kT
+
depends on the current information bits over this time interval
and the symbol phase over the previous symbol interval. The phase
transitions for DQPSK modulation are summarized in Table 5.1.
Specifically, suppose the symbol over time
]
,
)
1
[(
s
s
kT
T
k
-
has phase
i
j
e
k
q
q
=
-
)
1
(
. Then, over symbol time
]
)
1
(
,
[
s
s
T
k
kT
+
, if the information bits are 00, the corresponding symbol would
have phase
i
j
e
k
q
q
=
)
(
, i.e. to encode the bits 00, the symbol from symbol
interval
]
,
)
1
[(
s
s
kT
T
k
-
is repeated over the next interval
]
)
1
(
,
[
s
s
T
k
kT
+
. If the two information bits to be sent at time interval
]
)
1
(
,
[
s
s
T
k
kT
+
are 01, then the corresponding symbol has phase
)
2
/
(
)
(
p
q
q
+
=
i
j
e
k
. For information bits 10 the symbol phase is
)
2
/
(
)
(
p
q
q
-
=
i
j
e
k
, and for information bits 11 the symbol phase is
)
(
)
(
p
q
q
+
=
k
j
e
n
. We see that the symbol phase over symbol interval
]
)
1
(
,
[
s
s
T
k
kT
+
depends on the current information bits over this time interval
and the symbol phase
i
q
over the previous symbol interval. Note that this mapping of bit
sequences to phase transitions ensures that the most likely
detection error, that of mistaking a received symbol for one of its
nearest neighbors, results in a single bit error. For example, if
the bit sequence 00 is encoded in the kth symbol then the kth
symbol has the same phase as the (k1)th symbol. Assume this phase
is
i
q
. The most likely detection error of the kth symbol is to decode
it as one of its nearest neighbor symbols, which have phase
2
/
p
q
i
. But decoding the received symbol with phase
2
/
p
q
i
would result in a decoded information sequence of either 01 or
10, i.e. it would differ by a single bit from the original sequence
00. More generally, we can use Gray encoding for the phase
transitions in differential MPSK for any M, so that a message of
all 0 bits results in no phase change, a message with a single 1
bit and the rest 0 bits results in the minimum phase change of
M
/
2
p
, a message with two 1 bits and the rest 0 bits results in a
phase change of
M
/
4
p
, and so forth. Differential encoding is most common for MPSK
signals, since the differential mapping is relatively simple.
Differential encoding can also be done for MQAM with a more complex
differential mapping. Differential encoding of MPSK is denoted by
D-MPSK, and for BPSK and QPSK this becomes DPSK and D-QPSK,
respectively.
assuming the transmitted symbol at the (k 1)th symbol time was
s(k 1) =
e
j
A
p
. Solution: The first bit, a 1, results in a phase transition of
, so s(k) =
A
. The next bit, a 0, results in no transition, so s(k + 1) =
A
. The next bit, a 1, results in another transition of , so s(k +
1) =
e
j
A
p
, and so on. The fullsymbol sequence corresponding to 101110
is
A
,
A
,
e
j
A
p
,
A
,
e
j
A
p
,
e
j
A
p
.
The demodulator for differential modulation is shown in Figure
5.20. Assume the transmitted constellation at time k is s(k) =
e
k
j
A
0
)
(
f
q
+
The received vector associated with the sampler outputs is
)
(
)
(
)
(
)
(
0
)
(
2
1
k
n
A
k
jr
k
r
k
z
e
k
j
+
=
+
=
+
f
q
, (5.61)
where n(k) is complex white Gaussian noise. The received vector
at the previous time sample k 1 is thus
)
1
(
)
1
(
)
1
(
)
1
(
0
)
1
(
2
1
-
+
=
-
+
-
=
-
+
-
k
n
A
k
jr
k
r
k
z
e
k
j
f
q
. (5.62)
The phase difference between z(k) and z(k 1) dictates which
symbol was transmitted. Consider
)
1
(
)
(
)
(
)
1
(
)
1
(
)
(
0
0
)
1
(
)
1
(
))
1
(
)
(
(
2
-
+
+
-
+
+
=
-
*
+
-
-
*
+
-
-
-
*
k
n
k
n
k
n
A
k
n
A
A
k
z
k
z
e
e
e
k
j
k
j
k
k
j
f
q
f
q
q
q
. (5.63)
In the absence of noise (n(k) = n(k 1) = 0) only the first term
in (5.63) is nonzero, and this term yields the desired phase
difference. The phase comparator in Figure 5.20 extracts this phase
difference and outputs the corresponding symbol.
Differential modulation is less sensitive to a random drift in
the carrier phase. However, if the channel has a nonzero Doppler
frequency, the signal phase can decorrelate between symbol times,
making the previous symbol a very noisy phase reference. This
decorrelation gives rise to an irreducible error floor for
differential modulation over wireless channels with Doppler, as we
shall discuss in Chapter 6.
5.3.5 Constellation Shaping
Rectangular and hexagonal constellations have a better power
efficiency than the square or circular constellations associated
with MQAM and MPSK, respectively. These irregular constellations
can save up to 1.3 dB of power at the expense of increased
complexity in the constellation map [18]. The optimal constellation
shape is a sphere in N-dimensional space, which must be mapped to a
sequence of constellations in 2-dimensional space in order to be
generated by the modulator shown in Figure 5.10. The general
conclusion in [18] is that for uncoded modulation, the increased
complexity of spherical constellations is not worth their energy
gains, since coding can provide much better performance at less
complexity cost. However, if a complex channel code is already
being used and little further improvement can be obtained by a more
complex code, constellation shaping may obtain around 1 dB of
additional gain. An in-depth discussion of constellation shaping,
as well as constellations that allow a noninteger number of bits
per symbol, can be found in [18].
5.3.6 Quadrature Offset
A linearly modulated signal with symbol
)
,
(
2
1
i
i
i
s
s
s
=
will lie in one of the four quadrants of the signal space. At
each symbol time kTs the transition to a new symbol value in a
different quadrant can cause a phase transition of up to 180
degrees, which may cause the signal amplitude to transition through
the zero point: these abrupt phase transitions and large amplitude
variations can be distorted by nonlinear amplifiers and filters.
These abrupt transitions are avoided by offsetting the quadrature
branch pulse g(t) by half a symbol period, as shown in Figure 5.21.
This offset makes the signal less sensitive to distortion during
symbol transitions.
Phase modulation with phase offset is usually abbreviated as
O-MPSK, where the O indicates the offset. For example, QPSK
modulation with quadrature offset is referred to as O-QPSK. O-QPSK
has the same spectral properties as QPSK for linear amplification,
but has higher spectral efficiency under nonlinear amplification,
since the maximum phase transition of the signal is 90 degrees,
corresponding to the maximum phase transition in either the
in-phase or quadrature branch, but not both simultaneously. Another
technique to mitigate the amplitude fluctuations of a 180 degree
phase shift used in the IS-54 standard for digital cellular is
/4-QPSK [13]. This technique allows for a maximum phase transition
of 135 degrees, versus 90 degrees for offset QPSK and 180 degrees
for QPSK. Thus, /4-QPSK does not have as good spectral properties
as O-QPSK under nonlinear amplification. However, /4-QPSK can be
differentially encoded, eliminating the need for a coherent phase
reference, which is a significant advantage. Using differential
encoding with /4-QPSK is called /4-DQPSK. The /4-DQPSK modulation
works as follows: the information bits are first differentially
encoded as in DQPSK, which yields one of the four QPSK
constellation points. Then, every other symbol transmission is
shifted in phase by /4. This periodic phase shift has a similar
effect as the time offset in OQPSK: it reduces the amplitude
fluctuations at symbol transitions, which makes the signal more
robust against noise and fading.
5.4 Frequency Modulation
Frequency modulation encodes information bits into the frequency
of the transmitted signal. Specifically, each symbol time K = log2M
bits are encoded into the frequency of the transmitted signal s(t),
0 t < Ts, resulting in a transmitted signal si(t) = Acos(2fit +
i), where i is the index of the ith message corresponding to the
log2M bits and i is the phase associated with the ith carrier. The
signal space representation is si(t) = _j sijj(t) where sij = A(i
j) and j(t) = cos(2fjt + j), so the basis functions correspond to
carriers at different frequencies and only one such basis function
is transmitted in each symbol period. The orthogonality of the
basis functions requires a minimum separation between different
carrier frequencies of f = minij |fjfi| = .5/Ts. Since frequency
modulation encodes information in the signal frequency, the
transmitted signal s(t) has a constant envelope A. Because the
signal is constant envelope, nonlinear amplifiers can be used with
high power efficiency, and the modulated signal is less sensitive
to amplitude distortion introduced by the channel or the hardware.
The price exacted for this robustness is a lower spectral
efficiency: because the modulation technique is nonlinear, it tends
to have a higher bandwidth occupancy than the amplitude and phase
modulation techniques.
53
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00kk(k1)
i
q
k
2
/
p
q
i
2
/
p
q
i
011000,MPSKM01010MPSKMQAMMPSKD- MPSKBPSKQPSKDPSKD- QPSK
5.20ks(k) =
e
k
j
A
0
)
(
f
q
+
)
(
)
(
)
(
)
(
0
)
(
2
1
k
n
A
k
jr
k
r
k
z
e
k
j
+
=
+
=
+
f
q
(5.61)
)
1
(
)
1
(
)
1
(
)
1
(
0
)
1
(
2
1
-
+
=
-
+
-
=
-
+
-
k
n
A
k
jr
k
r
k
z
e
k
j
f
q
(5.62)
z(k)z(k 1)
)
1
(
)
(
)
(
)
1
(
)
1
(
)
(
0
0
)
1
(
)
1
(
))
1
(
)
(
(
2
-
+
+
-
+
+
=
-
*
+
-
-
*
+
-
-
-
*
k
n
k
n
k
n
A
k
n
A
A
k
z
k
z
e
e
e
k
j
k
j
k
k
j
f
q
f
q
q
q
(5.63)
(5.63)5.20
DopplerDoppler6
5.3.5
MQAMMPSK1.3dBN5.102[18]1dB[18]
PAGE
2
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