Post on 26-Jan-2023
Page 1
1 ELEC3540-2012-T2-PSB Academy-Chapter 8
ELEC3540 Analog and Digital Communications Chapter 8: Digital Modulation Techniques
1. Linear Modulation Schemes
- M-ary Pulse Amplitude Modulation (M-PAM)
- M-ary Phase Shift Keying (M-PSK) – BPSK, QPSK and Offset QPSK
- M-ary Quadrature Amplitude Modulation (M-QAM)
2. Energy of Signals
3. Power Spectra of Linear Modulation Schemes
4. Multidimensional and Nonlinear Modulation Schemes
5. Non-coherent Detection
- Differential Phase Shift Keying (DPSK), M-ary FSK
2 ELEC3540-2012-T2-PSB Academy-Chapter 8
Linear Modulation Schemes
M-PAM, M-PSK, and M-QAM
Page 2
3 ELEC3540-2012-T2-PSB Academy-Chapter 8 3
Binary Modulation Techniques
3 Common Types
• Amplitude-shift keying
(ASK)
• Frequency-shift keying
(FSK)
• Phase-shift keying
(PSK)
Binary data
ASK (OOK) signal
FSK signal
PSK signal
t
t
t
t
1 0 1 0
T
4 ELEC3540-2012-T2-PSB Academy-Chapter 8 4
Linear Modulation Schemes - PAM, PSK, and QAM
• A number of interesting linear modulation techniques produce an
equivalent lowpass signal of the form
(1)
• The transmit pulse g(t) can be chosen arbitrarily.
• In general, it may be complex, but we will assume here that it is real.
• Moreover, if we do not accept intersymbol interference (ISI) we have to
satisfy the Nyquist condition.
• This can be achieved, e.g., using the Nyquist roll-off pulse,
(2)
with 0 ≤ α ≤ 1, or using an NRZ-rectangular pulse.
• For simplicity, we will often assume that the support of g(t) is limited to the
time-interval [0;T].
Page 3
5 ELEC3540-2012-T2-PSB Academy-Chapter 8 5
Linear Modulation Schemes - PAM, PSK, and QAM
• A block of k bits is taken from the information sequence and mapped into
one data symbol dn according to the signal space constellation of the
employed modulation technique.
• The modulated signal s(t) is described by
(3)
(4)
where
• If g(t) is limited to the time-interval [0;T], we may express s(t) for 0 ≤ t ≤ T
in terms of its amplitude A = | dn | and phase = arg(dn) as
(5)
(6)
6 ELEC3540-2012-T2-PSB Academy-Chapter 8 6
M-ary Pulse Amplitude Modulation (M-PAM)
• M-ary Pulse Amplitude Modulation (M-PAM), sometimes also called
Amplitude Shift Keying (ASK)
• Here the M possible signal points are real with
(7)
where w is the Euclidean distance between adjacent signal points.
• Thus, a PAM modulator transmits one of the signals
(8)
w
Page 4
7 ELEC3540-2012-T2-PSB Academy-Chapter 8 7
M-ary Phase Shift Keying (M-PSK)
• M-ary Phase Shift Keying (M-PSK), with special cases BPSK (M = 2) and
QPSK (M = 4):
• The M possible signal points lie on a unit circle, i.e.,
(9)
8 ELEC3540-2012-T2-PSB Academy-Chapter 8 8
M-ary Phase Shift Keying (M-PSK)
• The Euclidean distance between adjacent signal points is
(10)
• A M-PSK modulator transmits one of the signals
(11)
(assuming )
• For the case of QPSK, M = 4, and
where
w
ic
cm
θtπftg
mmπtπftgts
2cos
4 ,3 ,2 ,14/122cos
23 , ,2 ,0 πππθi
1 nd
Page 5
9 ELEC3540-2012-T2-PSB Academy-Chapter 8 9
M-ary Phase Shift Keying (M-PSK)
• Alternatively, QPSK may also use constellations that are offset in phase by
.
• For such cases, the QPSK signal is given by
where
4π
Re dn
Im dn
4π
QPSK
mc
cm
θtπftg
mπmtπftgts
2cos
4 ,3 ,2 ,14/ 122cos
47 ,45 ,43 ,4 ππππθm
10 ELEC3540-2012-T2-PSB Academy-Chapter 8 10
M-ary PSK : Binary Phase Shift Keying
• The BPSK signal is given by
t Carrier signal
t PSK
signal
t Polar NRZ data
+ A
- A
Balanced Modulator
BPSK Signal
Polar NRZ signal
Carrier
-1
+1
tπfA c2cos
tg
BPSK Modulator
BPSK Modulator
Waveforms
(without pulse shaping)
Bandpass
Filter Polar NRZ signal
Carrier tπfA c2cos
πθθtπftg
mMmπtπftgts
iic
cm
,02cos
2 ,1/122cos
Balanced Modulator
BPSK Signal tsm
-1
+1
Page 6
11 ELEC3540-2012-T2-PSB Academy-Chapter 8 11
M-ary PSK : Binary Phase Shift Keying
Pow
er
spectr
al density (
dB
)
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
10
Frequency
f c f c R b + f c 2 R b + f c 3 R b + f c 4 R b +
BPSK Power Spectral Density
(unfiltered)
12 ELEC3540-2012-T2-PSB Academy-Chapter 8 12
M-ary PSK : Binary Phase Shift Keying
• Coherent detection can be accomplished by the correlation receiver.
• Reference carriers are set to match the incoming phase of the BPSK
signal.
• Outputs of product detectors indicate the presence of either carrier phase.
d b
t T
0
Threshold Detector
Recovered data
S Sample
and Hold
+
_ BPSK Signal
Bandpass Filter
d b
t T
0
Product Detector 1 Integrator 1
Integrator 2
Product Detector 2
12cos θtπfA c
02cos θtπfA c
bT
BPSK Demodulator
Page 7
13 ELEC3540-2012-T2-PSB Academy-Chapter 8 13
M-ary PSK : Binary Phase Shift Keying
t Reference
signal 1
Product detector 1
output
Integrator 1 output
t Sample-and- hold output
t NRZ data
0
1
t
t PSK signal
t
Threshold detector output
V = 0 t
v 1
v 2
t
v 3
t Reference
signal 2
Integrator 2 output
t NRZ data
0
1
t
t PSK signal
t
Product detector 2
output
Integrator 1-2 output t
v 4 v 5
v 1
v 2
v 3
v 4
v 5
0
1
14 ELEC3540-2012-T2-PSB Academy-Chapter 8 14
M-ary PSK : Quadrature Phase Shift Keying
• Consider a QPSK carrier signal defined by
where is the transmitted signal energy per symbol, and is the symbol
duration.
• The carrier frequency equals for some fixed integer .
• The QPSK signal may be written in the form
Ttmπ
mtπfT
Ets cm
0;4 ,3 ,2 ,1
4 122cos
2
E T
cf Tnc cn
tfstfs
tπfT
πmEtπf
T
πmE
tπfπ
mT
Etπf
πm
T
Ets
QI
cc
ccm
21
2sin2
4 12sin2cos
2
4 12cos
2sin4
12sin2
2cos4
12cos2
Page 8
15 ELEC3540-2012-T2-PSB Academy-Chapter 8 15
M-ary PSK : Quadrature Phase Shift Keying
• Based on this representation, there are two orthogonal basis functions and
four message points, defined as
• The signal-space characterization of the QPSK signal may be summarized
as follows:
tπfTtf c2cos/21
tπfTtf c2sin/22
4 ,3 ,2 ,1
4
12sin
4
12cos
m
πmE
πmE
s
s
Q
I
224711
224501
224300
22410
EEπ
EEπ
EEπ
EEπ
ssθd QInn
16 ELEC3540-2012-T2-PSB Academy-Chapter 8 16
M-ary PSK : Quadrature Phase Shift Keying
S +
+ 90 o
Serial-to- Parallel
Converter
In-phase Branch
Quadrature Branch
Pulse Shaping
Filter
Local Oscillator
tf1
I
nd
Q
nd
tf2
tπfTtf c2cos/21
tπfTtf c2sin/22
Pulse Shaping
Filter
QPSK Signal
tsm
nd
NRZ Data Bit Stream
bR
10
11
QPSK Modulator
2bs RR
sR
Is
Qs
Page 9
17 ELEC3540-2012-T2-PSB Academy-Chapter 8 17
M-ary PSK : Quadrature Phase Shift Keying
• Uses 4 phase states to represent the modulating data.
• May be viewed as 2 orthogonal channels of BPSK summed together.
S +
+
tπfTtf c2cos/21
tπfTtf c2sin/22
tsm tf1
tf2
Is
Qs
(1,-1) (-1,-1)
(-1,1) (1,1)
I
Q
tf1
tf2
2/E
2/E
Q
n
I
n dd ,
E
18 ELEC3540-2012-T2-PSB Academy-Chapter 8 18
QPSK Modulator Waveforms
d 0 d 1 d 2 d 3 d 4 d 5 d 6 d 7
d 0
d 2 d 4
d 6
d 1
d 3
d 5 d 7
T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 0
nd
I
nd
Q
nd
tsm
t
t
t
t
1fsI
t
2fsQ
t tπfTtf c2cos/21
tπfTtf c2sin/22
S +
+
tsm tf1
tf2
Is
Qs
Page 10
19 ELEC3540-2012-T2-PSB Academy-Chapter 8 19
Unfiltered QPSK Power Spectral Density
Pow
er
spectr
al density (
dB
) 0
-10
-20
-30
-40
-50
-60
-70
-80
-90
10
Frequency
f c f c R b + f c 2 R b + f c 3 R b + f c 4 R b +
Unfiltered QPSK
Unfiltered BPSK
20 ELEC3540-2012-T2-PSB Academy-Chapter 8 20
Unfiltered and Baseband-filtered QPSK Power Spectrum
Po
we
r sp
ectr
al d
en
sity (
dB
)
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
10
f c f c R b + f c 3 R b + f c R b - f c 3 R b -
Frequency
Unfiltered QPSK
QPSK with R /4 Gaussian lowpass
filtering
b
Page 11
21 ELEC3540-2012-T2-PSB Academy-Chapter 8 21
Time Domain Waveform of Filtered QPSK
RF
am
plit
ude
Time
180 phase shift o
90 phase shift o
22 ELEC3540-2012-T2-PSB Academy-Chapter 8 22
QPSK Modulator - Gray Coding
• The constellation shows the four
values of .
• Gray coding is used for the four carrier
phase states (symbols) in the sense
that adjacent states differ by only one
bit (Hamming distance = 1).
• Assuming that the demodulator only
makes errors between adjacent states
(symbols), the QPSK bit error
probability is then approximately
half the symbol error probability .
• In general, for M-ary PSK (M symbols)
and Gray coding,
bP
sP
Q
n
I
n dd ,
M
PP s
b
2log
(1,-1) (-1,-1)
(-1,1) (1,1)
I
Q
tf1
tf2
2/E
2/E
Q
n
I
n dd ,
E
Page 12
23 ELEC3540-2012-T2-PSB Academy-Chapter 8 23
QPSK Demodulator
Threshold
Detector
Threshold Detector
Recovered
Bit Stream
Lowpass
Filter
Carrier Recovery
Circuit
Bandpass Filter
Lowpass
Filter
Symbol Timing
Recovery
Parallel to
Serial Converter
Received
QPSK Signal
90 o
tntstr m
tf1
tf2
bR
24 ELEC3540-2012-T2-PSB Academy-Chapter 8 24
Disadvantages of QPSK
• QPSK have constant envelope but with 180° phase changes
leads to wider bandwidth.
minimized by bandlimiting prior to transmission.
• QPSK requires linear power amplifiers
nonlinearities will lead to
spectral regrowth; i.e.
the filtered sidelobes
are restored to their
original unfiltered
levels.
less power efficient.
Po
we
r sp
ec
tra
l d
en
sit
y (
dB
)
0
-20
-40
-60
-80
-100 f c R
b -
Frequency
f c R b + f c
QPSK without limiting
QPSK with limiting (10dB comp)
Page 13
25 ELEC3540-2012-T2-PSB Academy-Chapter 8 25
M-ary PSK : Offset QPSK
• An Offset QPSK signal is given by
(23)
where the possible signal points are
• Thus, the quadrature branch is delayed by half a symbol interval .
S +
- 90 o
Serial-to- Parallel
Converter
In-phase Branch
Quadrature Branch
Pulse Shaping
Filter
Local Oscillator
tf1
I
nd
Q
nd tf2
tπfTtf c2cos/21
tπfTtf c2sin/22
Pulse Shaping
Filter
QPSK Signal
tsm
2bs RR
Offset QPSK Modulator
sR
2
sT
sT
nd
NRZ Data Bit Stream
bR
10
11
26 ELEC3540-2012-T2-PSB Academy-Chapter 8 26
M-ary PSK : Offset QPSK
• Since now and cannot both
change together, the phase changes
are limited to , every
seconds and so the envelope
amplitude cannot go to zero.
• The power density spectrum of
OQPSK is the same as for QPSK
and still requires spectral shaping.
• However, bandpass filtering
will now generate only relative
small envelope variations, which can
be removed by hard limiting with
only partial regeneration of out of
band spectra.
• This gives OQPSK an advantage over QPSK in non-linear satellite
channels.
0 90 2/s
T
Polar NRZ data
I
nd
Q
nd
0d 1d 3d2d 4d
0d
1d
3d
2d
bs TT 2
bT
I
ndQ
nd
Page 14
27 ELEC3540-2012-T2-PSB Academy-Chapter 8 27
M-ary PSK : Offset QPSK
• Depicted below are the signal trajectories of a QPSK signal (left) and an
OQPSK signal (right).
• Both use the same signal space constellation with signal points
and a Nyquist roll-off pulse shape with roll-
off factor α = 0.3.
28 ELEC3540-2012-T2-PSB Academy-Chapter 8 28
M-ary PSK : Offset QPSK
• The effect is a more constant signal envelope for OQPSK than for QPSK,
as shown below.
Page 15
29 ELEC3540-2012-T2-PSB Academy-Chapter 8 29
M-ary Quadrature Amplitude Modulation (M-QAM)
• The M possible signal points lie on a rectangular grid with Euclidean
distance w between adjacent signal points.
• A QAM modulator transmits one of the signals
(12)
(13)
where
30 ELEC3540-2012-T2-PSB Academy-Chapter 8 30
M-ary Quadrature Amplitude Modulation (M-QAM)
S +
- 90 o
Serial-to- Parallel
Converter
In-phase Branch
Quadrature Branch
Pulse Shaping
Filter
Local Oscillator
tf1
I
nd
tf2
tπfTtf c2cos/21
tπfTtf c2sin/22
Pulse Shaping
Filter
QAM Signal
tsm
2bR
QAM Modulator for
Rectangular Signal
Constellation
2bR
Q
ndbitk 2
DAC
Q
md
bitk 2
DAC
I
md
ionconstellatpointM k 2
nd
NRZ Data Bit Stream
bR
10
11
Page 16
31 ELEC3540-2012-T2-PSB Academy-Chapter 8 31
Energy of Signals
• If g(t) is nonzero only on the interval [0;T], the energy of s(t) per symbol is
(14)
• If the data sequence is uncorrelated with zero mean and variance then
(15)
where Eg is the energy of the transmit pulse g(t).
• We compute the energy of a PAM signal sm(t) as
(16)
and therefore, if all signals are equally likely,
(17)
32 ELEC3540-2012-T2-PSB Academy-Chapter 8 32
Energy of Signals
• For PSK, all signal waveforms have equal energy,
(18)
• For QAM, the energy of the signal s(t) is
(19)
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33 ELEC3540-2012-T2-PSB Academy-Chapter 8 33
Signal Space
• For 0 ≤ t ≤ T, the signal space of linearly modulated bandpass signals is
spanned by
(20)
(21)
and the signal space of linearly modulated equivalent lowpass signals is
spanned by
(22)
• Thus, the signal space of linearly modulated bandpass signals is two-
dimensional, whereas the signal space of linearly modulated equivalent
baseband signals is only one-dimensional.
34 ELEC3540-2012-T2-PSB Academy-Chapter 8 34
Power Spectra of Linear Modulation Schemes
• Channel bandwidth is a precious commodity.
• Thus, there is great interest in designing bandwidth-efficient modulation
schemes.
• In this section, we will investigate how the system designer can control the
spectral characteristics of linearly modulated signals.
• Recall that the equivalent low pass representation of linearly modulated
signals is (24)
• We assume that the data sequence dn is WSS with mean
(25)
and autocorrelation function
(26)
Page 18
35 ELEC3540-2012-T2-PSB Academy-Chapter 8 35
Power Spectra of Linear Modulation Schemes
• The mean of the linearly modulated signal sl(t) is
(27)
which is periodic with period T.
• The autocorrelation function of sl(t) can be obtained as
(28)
(29)
• We now make the variable substitution κ = n - m, which yields
(30)
36 ELEC3540-2012-T2-PSB Academy-Chapter 8 36
Power Spectra of Linear Modulation Schemes
• It is easy to see that
(31)
which means that sl(t) is a wide-sense cyclostationary process with period
T.
• For cyclostationary processes, we may compute a mean correlation
function, averaged over one period:
(32)
(33)
(34)
(35)
Page 19
37 ELEC3540-2012-T2-PSB Academy-Chapter 8 37
Power Spectra of Linear Modulation Schemes
• Now define the function
(36)
which has Fourier transform
(37)
• With this definition, we finally obtain
(38)
and the average power spectral density is given by the Fourier transform of
(39)
38 ELEC3540-2012-T2-PSB Academy-Chapter 8 38
Power Spectra of Linear Modulation Schemes
• In this equation, Sdd( f ) is the discrete-time Fourier transform of Rdd[κ].
• Drawing upon our knowledge of equivalent lowpass signals, the average
psd of the bandpass signal s(t) is easily obtained as
(40)
• From looking at (39) we see that there are two ways of controlling the
spectral characteristics of sl(t) and thus s(t):
by the design of the pulse shape g(t)
by the design of the correlation characteristics of the data sequence
dn.
• If the data sequence is uncorrelated with zero mean and variance then
(41)
Page 20
39 ELEC3540-2012-T2-PSB Academy-Chapter 8
ELEC3540 Analog and Digital Communications Chapter 8b: Optimal Signal Detection and
Comparison of Digital Modulation Techniques
1. Signal Detection
2. Optimum Detector
3. Error Performance of the Optimum Receiver
4. Binary Anti-Podal Signals
5. Carrier Signals (Binary)
6. Comparison of Digital Modulation
40 ELEC3540-2012-T2-PSB Academy-Chapter 8 40
Basic Detection Problem
• For an AWGN Channel
• Modulator Rule :
• For
• Given decide if or
is present in the received signal,
• The data rate is
• Note : There is no interference between pulses as is time limited to
, i.e., no intersymbol interference (ISI).
• We want a detector that gives
tsts 01 0,1
1 ,0; , 1 itntstrkTtTk i
, 1, kTtTktr ts1 ts0
tr
bits/sec 1
TR
tslTt 0
minerrorP
kα
)(1 ts
)(tn
)(tr
Channel
Modulator Demodulator Binary Source
)(0 ts
0 ,1ˆ kα
0
1
1
0
1
T
T
t
t
0
)(1 ts
)(0 ts
Page 21
41 ELEC3540-2012-T2-PSB Academy-Chapter 8 41
Vector Detection Rule
• Say we have where and are vectors.
• The received vector is
• Let
• We are given and we know and .
• Hence, if is more likely than .
Decode
Optimal Detection Rule
• Let
• Decode if
and otherwise.
• If tie, toss a coin.
01 ss 0,1 1s0s
nsr i
rr oflength
r 1s0s
010 , see 1s
0s
iii andbetweendistanceEuclideand srsrsr ,2
1s
0s
0
2
1
2 , , srsr dd
1s
0s
r0e
1e
42 ELEC3540-2012-T2-PSB Academy-Chapter 8 42
Signal Detection
• Let be an -vector sample of .
• Let be an -vector sample of .
• The Euclidean distance is
• Therefore, we can define the distance between on
at as
• Decode if
and otherwise. Tie Toss a coin.
Nrr , . . . ,1r N tr
iNii ss , . . . ,1s N tsi
2
1
2 2 , i
N
j
ijji srd srsr
tsrtdN
j
ijji 1
2 2 , sr 0 2
0 tasdttstr
T
i
tstr i , Tt 0T
dttstrtstrdT
ii
2
0
2 ,
ts1
tstrdtstrd 0
2
1
2 ,,
ts0
Page 22
43 ELEC3540-2012-T2-PSB Academy-Chapter 8 43
Signal Detection
• Consider the equivalent rule occurs for
Decode
• Consider
• Let the energy of be
• We drop terms in that do not depend on as the operator is
used for detection, i.e. we can ignore the term
as it does not depend on .
• Noting that , decode if
for , and otherwise.
tstrd jj
, min 2
1,01j
ts1
dttsdttstrdttr
dttstrtstrd
T
j
T
j
T
T
jj
0
2
0
0
2
2
0
2
2
,
ts j dttsET
jj
0
2 2d ts j min
j
dttrT
0
2
j
maxmin ts1 max2
0
jT
j
Edttstr
1j ts0
44 ELEC3540-2012-T2-PSB Academy-Chapter 8 44
Optimum Detector
• If (equal energy signals), we can drop the subtraction by
in each branch of the receiver.
• This is called a correlator receiver as in each branch of the receiver, we
correlate against one possibility, say
Integrator 1
ts1
dtT
0
Sample at t = T
Integrator 2
dtT
0
ts0
)(tr
21E
+ _
20E
+ _
+ _
Threshold Th = 0
kα̂Dump
Dump
00
01 ˆ
z
zαk
z
1z
0z
01 EE 2jE
ts j
Page 23
45 ELEC3540-2012-T2-PSB Academy-Chapter 8 45
Optimum Detector
• For signals, , bits per symbol transmission, we use
correlators and choose the signal that has the best correlation.
ts1
dtT
0
Sample at t = T
dtT
0
tsk
)(tr
20E
+ _
2jE
+ _ jα̂
Dump
Dump
0z
kz
dtT
0
tsM 121ME
+ _
Dump
1Mz
C
H
O
O
S
E
M
A
X
M kM 2 k M
j
k
j αMkα of values word,bit 2
46 ELEC3540-2012-T2-PSB Academy-Chapter 8 46
Error Performance of the Optimum Receiver
• Let be sent by the transmitter.
• Then,
• Our goal is to find
• The output of the rail is
Integrator 1
ts1
dtT
0
Sample at t = kT
Integrator 2
dtT
0
ts2
)(tr
21E
+ _
22E
+ _
+ _
Threshold Th = 0
kα̂Dump
Dump
00
01 ˆ
z
zαk
z
1z
2z
ts1
tntstr 1
11 sePsenttserrorP
ts1
210
11 EdttstrzT
Page 24
47 ELEC3540-2012-T2-PSB Academy-Chapter 8 47
Error Performance of the Optimum Receiver
• As , we have
• Also, for the rail,
where signal cross-correlation
• We want in order not to make an error.
• The variable for bit detection is
(A)
where output noise voltage,
210
10
111 EdttstndttstszTT
tntstr 1
dttstnET
011 2
ts2
Signal term + noise term
220
22 EdttstrzT
220
20
21 EdttstndttstsTT
dttstnEρT
02212 2
dttstsρT
0 2112
21 zz
21 zzz
dttststnnT
0 21
nρEE 1221 2
48 ELEC3540-2012-T2-PSB Academy-Chapter 8 48
Error Performance of the Optimum Receiver
• In general, the psd at the output of a filter
is
• If the input is WGN, , with double-sided psd for all , then the
output power is given by
• By Parsavel’s Theorem,
• For ,
the output noise power is
fH XXYY SfHfS
2
tn 20N f
dfSσRtYE YYYYY 0
22
)(tx fH )(ty
dffHN
20
2
dtthdffH
2
2
dtthtnT
0 dttststnn
T
0 21
dttstsN
σ
2
21
02
2
Page 25
49 ELEC3540-2012-T2-PSB Academy-Chapter 8 49
Error Performance of the Optimum Receiver
• Output noise power (B)
where
is the signal distance2 between and
• Upon expansion,
• Now, we have an error when as is sent by transmitter
dttstsd
2
21
2
12
ts1 ts2
1221
2
12 2ρEEd
2
12
02
2d
Nσ
0z ts1
nρEEzzz 122121 2
02 1221 nρEE
1221 2 where ρEEWWn
50 ELEC3540-2012-T2-PSB Academy-Chapter 8 50
Error Performance of the Optimum Receiver
• Now, as is gaussian distributed with zero mean and variance ,
where the Q-function is defined as
• Therefore,
(C)
2σn
-1 -2 -3 -4
0.399
0 1 2 3 4
W x
-W
22
π2
1 xexP
σ
WQArea
σ
WnPWnPseP 1
2
11
2
2
1
x
x dxeπ
xQ
2
2
1 σ
WQseP
σ
WQ
1
1
2
2
:Notex
x dxeπ
xerfc
Page 26
51 ELEC3540-2012-T2-PSB Academy-Chapter 8 51
Error Performance of the Optimum Receiver
• From Eq.(B),
and
• Thus, from Eq.(C),
• Similarly,
122102
1202 2
22ρEE
Nd
Nσ
1221 2 ρEEW
12210
2
12211
22
4
2
ρEEN
ρEEQseP
0
2
12
2N
dQ
0
2
122
2
N
dQseP
52 ELEC3540-2012-T2-PSB Academy-Chapter 8 52
Error Performance of the Optimum Receiver
• The probability of a bit error Pb is
• If both signals are equally likely, the bit error probability is then, due to
symmetry,
• The bit error is related to the signal distance between and .
• This is the general equation to compute the BER for binary case.
• The threshold is set to
2211 sPsePsPsePPb
0
2
121
2
N
dQsePPb
ts1 ts2
2
21 EETh
Page 27
53 ELEC3540-2012-T2-PSB Academy-Chapter 8 53
Error Performance of the Optimum Receiver
Equivalent Detector
• Lower complexity – uses only one multiplier + one integrator.
• Correlate with the difference signal .
Matched Filter Realization
• MF Impulse response,
Integrator 1
tststg 21
dtT
0 )(tr
Threshold
kα̂
Dump
h
h
kTz
Tzα
0
1 ˆ
z
2
21 EETh
tr tg
)(tr
Threshold
kα̂
h
h
kTz
Tzα
0
1 ˆ
z
2
21 EETh
th
Sample at t = kT
Sample at t = kT
tTstTstTgth 21
54 ELEC3540-2012-T2-PSB Academy-Chapter 8 54
Binary Anti-Podal Signals
• Let and
• Signal cross-correlation
• Signal distance
• Bit error probability
where is the average energy per bit.
• For ,
• Power Spectrum
tsts 1 tststs 10
EdttsdttstsρTT
0
2
0 0110
E
EEEρEEd
4
22 1001
2
10
000
2
10 22
2 N
EQ
N
EQ
N
dQP b
b
bE
Tt
0
)(tp
A tpts TAEb
2
fTTAtsFTfS 2222sinc
)( fS
T
1
T
2f
T
3
T
1
T
2
T
3
22TA
Page 28
55 ELEC3540-2012-T2-PSB Academy-Chapter 8 55
Carrier Signals (Binary)
• All signals have duration , bit rate bits/s
• The basic performance measure is
T TRb /1
0
2
10
2N
dQPb
1001
2
10 2ρEEd
dttstsρT
0 0110
2
01 EEEb
5.001 PP
56 ELEC3540-2012-T2-PSB Academy-Chapter 8 56
Carrier Signals (Binary)
• ASK – On-Off Keying
0,cos 01 tstωAts c
ETA
dtA
dttωAdttωAE
T
T
c
T
c
2
2
12cos2
1 cos
2
0
2
0
2
0
22
1 (Drop 2wc terms for
wcT large)
0
0 0110 dttstsρ
T
(When r10 = 0, we have orthogonal signals)
bEρEEd 22 1001
2
10
00 E
22
01 EEEEb
00
2
10
2 N
EQ
N
dQP b
b ( 3-dB loss compared to Anti-Podal signals)
Page 29
57 ELEC3540-2012-T2-PSB Academy-Chapter 8 57
Carrier Signals (Binary)
• BPSK
λθθ tωAtstωAts cc cos,cos 01
ETA
EE 2
2
01 EEE
Eb
2
01
EETA
dttωAtωA
dttstsρ
T
cc
T
λλ
λθθ
coscos2
coscos
2
0
0 0110
(Best l p)
bEEρEEd 442 1001
2
10
00
2
10 2
2 N
EQ
N
dQP b
b (same as for Anti-Podal signals)
58 ELEC3540-2012-T2-PSB Academy-Chapter 8 58
(3-dB loss compared to BPSK)
Carrier Signals (Binary)
• Orthogonal
θθ tωAtstωAts cc sin,cos 01
ETA
EE 2
2
01 EEE
Eb
2
01
0 22sin2
sincos
0
2
0
0 0110
dttωA
dttωAtωA
dttstsρ
T
c
T
cc
T
θ
θθ
bEEρEEd 222 1001
2
10
00
2
10
2 N
EQ
N
dQP b
b
Page 30
59 ELEC3540-2012-T2-PSB Academy-Chapter 8 59
Carrier Signals (Binary)
• BFSK
ωωωtωAtstωAts 010011 sin,cos θθ
ETA
EE 2
2
01 EEE
Eb
2
01
fTE
fπ
fTπTA
ω
ωTA
dtωtωttωA
dtωtωAtωAdttstsρ
T
TT
2sinc
2
2sin
2
sin
2
cos2cos2
coscos
2
2
0 0
2
0 00
0 0110
60 ELEC3540-2012-T2-PSB Academy-Chapter 8 60
Carrier Signals (Binary)
• BFSK (con’t)
• We want r10 as negative as possible to maximize the distance.
• From tables, the most negative sinc(x) is
1001
2
10 2ρEEd
00
2
10 11.1
2 N
EQ
N
dQP b
b
4.122.0sinc xx
EfTEρ 22.02sinc 10
TffTx
7.04.12 (Best frequency spacing)
bEE
ρEEd
22.222.2
2 1001
2
10
Page 31
61 ELEC3540-2012-T2-PSB Academy-Chapter 8 61
Carrier Signals (Binary)
• BFSK (con’t)
• Loss relative to BPSK (anti-podal) =
• Gain relative to orthogonal =
• Note that for the case of orthogonal signals, r10 = 0
• For the smallest bandwidth,
00
2
10
2 N
EQ
N
dQP b
b
dB 5.211.1
2log10
dB 45.011.1log10
02sinc 10 fTEρ
T
mfmfT
2integer2
Tf
2
1
(3-dB loss compared to BPSK)
62 ELEC3540-2012-T2-PSB Academy-Chapter 8 62
Bit Error Probabilities (Alternative Method)
• We shall now evaluate the probability of error
for the detection of BPSK signals.
• For BPSK, the sufficient statistic
(69)
is a real scalar and Gaussian with mean ±
and variance N0.
• The energy per transmitted data bit is Eb = Eg.
• A random variable Y has a Gaussian
distribution if its probability density function
has the form
2
2
2exp
2
1)(
Y
Y
Y
Yσ
μy
σπyf
P(r|s ) 1
P(r|s ) 2
likelihood of s 1
likelihood of s 2
r
bE
bE
0
Page 32
63 ELEC3540-2012-T2-PSB Academy-Chapter 8 63
Bit Error Probabilities
• The two possible conditional pdfs are
(70)
(71)
• The probability of a bit error Pb is
(72)
• If both signals are equally likely, the bit error
probability is then, due to symmetry,
(73)
P(r|s ) 1
P(r|s ) 2
likelihood of s 1
likelihood of s 2
r
bE
bE
0
64 ELEC3540-2012-T2-PSB Academy-Chapter 8 64
Bit Error Probabilities
• We can evaluate this probability as
(74)
(75)
(76)
Page 33
65 ELEC3540-2012-T2-PSB Academy-Chapter 8 65
Bit Error Probabilities
• Interestingly enough, Pb only depends on the signal to noise ratio Eb/N0,
but not on other signal characteristics.
• Moreover, Pb can be expressed in terms of the Euclidean distance w
between the two signals s1 and s2.
• Since w = |s1 - s2| = we have
(77)
• In order to compare different modulation schemes fairly, we have to
compare them in terms of their bit error probability (also called BER = bit
error rate), rather than their symbol error probability.
• Keep in mind that the energy per symbol, Es, is related to the energy per
bit, Eb, as
(78)
for an M-ary signalling scheme.
• For instance, for QPSK we have Es = 2Eb.
66 ELEC3540-2012-T2-PSB Academy-Chapter 8 66
Bit Error Probabilities
• The required Eb/N0, at a data rate of Rb, to achieve a specified level of Pe
(i.e. BER) is related to the received power level, Pr, by
• Deriving analytical expressions for the BER of different modulation
schemes is often difficult.
• The figure on the following page shows a typical BER plot for different
modulation schemes with logarithmic Pb-axis.
• We can clearly see the penalty that noncoherent detection occurs
compared to the corresponding coherent detection scheme (DPSK vs.
BPSK, and noncoherent FSK vs. coherent FSK).
00000
1log10(dB)
1
N
P
RN
E
N
P
RN
PT
N
E r
b
br
b
rbb
Page 34
67 ELEC3540-2012-T2-PSB Academy-Chapter 8 67
Bit Error Probabilities
68 ELEC3540-2012-T2-PSB Academy-Chapter 8 68
Comparison of Digital Modulation Techniques
• As we have seen above, QPSK has the same bit error probability as
BPSK.
• Does this mean that BPSK is “as good as” QPSK? Not quite, because we
have not considered the bandwidth efficiency yet.
• The bandwidth efficiency is defined as
(79)
or
Hz
bit/s
bandwidth rf
rate dataη
rf
b
W
R
Page 35
69 ELEC3540-2012-T2-PSB Academy-Chapter 8 69
Comparison of Digital Modulation Techniques
Example
• Compute the spectral efficiency for BPSK and QPSK using raised cosine
pulse shaping.
• For raised-cosine baseband pulse shaping,
• For BPSK, M = 2
For = 0, = 1.0 For = 1, = 0.5.
• For QPSK, M = 4
For = 0, = 2.0 For = 1, = 1.0
bits/s/Hz1
log
12
log/2 1
22
η
2
2
α
M
αMR
R
αR
R
b
b
s
b
70 ELEC3540-2012-T2-PSB Academy-Chapter 8 70
Comparison of Digital Modulation Techniques
• For BPSK with raised cosine roll off-pulse, 0.5 ≤ η ≤ 1, and for QPSK with
raised cosine roll off-pulse, 1.0 ≤ η ≤ 2.0.
• So while QPSK and BPSK have the same BER, QPSK has twice the
spectral efficiency of BPSK.
• Therefore, comparison of modulation techniques simply on the basis of
SNR required to achieve a specified bit error probability is not meaningful.
• There are three parameters that a true comparison must consider:
the signal to noise ratio (SNR) Eb/N0 per bit
the bit error probability Pb
the bandwidth efficiency η = R/W
• We can fix only one of these parameters at a time, say the bit error
probability. The remaining two parameters can then be plotted in a two-
dimensional graph, as shown on the next page.
• In this graph, the bit error probability is fixed at Pb = 10-5.
Page 36
71 ELEC3540-2012-T2-PSB Academy-Chapter 8 71
Comparison of Digital Modulation Techniques
72 ELEC3540-2012-T2-PSB Academy-Chapter 8 72
Comparison of Digital Modulation Techniques
• We need to say a few more words about how to compute bandwidth
efficiency for different modulation schemes.
• For PSK and QAM, the bandwidth W is the bandwidth of the equivalent
lowpass pulse g(t).
• If g(t) has duration T, its bandwidth W is approximately W = 1/T.
• With the information rate R = k/T, we obtain for the bandwidth efficiency of
PSK and QAM,
(80)
Page 37
73 ELEC3540-2012-T2-PSB Academy-Chapter 8 73
Comparison of Digital Modulation Techniques
• For M-ary orthogonal signals we require a minimum frequency separation
of 1/(2T) to ensure orthogonality.
• Thus, the bandwidth efficiency for M-ary orthogonal signals is
(81)
• So how good is, say, 32-ary orthogonal FSK?
• How good is 16-QAM?
• How much room for improvement is there?
• The beauty of communication theory lies in the fact that there is indeed an
answer to these questions.
• It was given by Claude E. Shannon in his 1948 landmark paper, in which
he showed the following:
74 ELEC3540-2012-T2-PSB Academy-Chapter 8 74
Comparison of Digital Modulation Techniques
• It is theoretically possible to achieve reliable communication, with as small
an error probability as desired, if the transmission rate R is less than C, the
channel capacity.
• If R >C, it is not possible to make the probability of error tend toward zero.
• The channel capacity C (depicted as the solid line) is sometimes also
referred to as Shannon’s limit.
• It is given as
(82)
Page 38
75 ELEC3540-2012-T2-PSB Academy-Chapter 8 75
Comparison of Digital Modulation Techniques
• For a wireless communication system like cellular, the following are
important considerations for the choice of a suitable digital modulation
scheme:
Minimum modulated spectrum bandwidth
gives higher spectral (bandwidth) efficiency.
Minimum Eb/N0 for a given Pe
better power efficiency.
Minimum energy outside main lobe
lower adjacent channel interference.
Continuous phase of modulated carriers
smaller sidelobe power.
use of efficient non-linear power amplifiers.
Performance under various types of channel impairments
e.g. Rayleigh, Rican, multipath, interference.