ELEC3540 Lecture 8 Digital Modulation

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


Linear Modulation Schemes

M-PAM, M-PSK, and M-QAM

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


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].


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)


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


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)



• 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



• 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


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



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)



• 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



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


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



• 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



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



• 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


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


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


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


Time Domain Waveform of Filtered QPSK

RF

am

plit

ude

Time

180 phase shift o

90 phase shift o


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


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


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)


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



• 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



• 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.



• The effect is a more constant signal envelope for OQPSK than for QPSK,

as shown below.


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


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


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)


Energy of Signals

• For PSK, all signal waveforms have equal energy,

(18)

• For QAM, the energy of the signal s(t) is

(19)


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.


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)



• 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)



• 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)



• 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)



• 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)


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


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


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


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


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


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


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


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



• 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



• 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



• 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



• 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



• 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



• 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



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


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


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



• 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)



• 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)


(3-dB loss compared to BPSK)


• 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



• 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



• 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



• 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)


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


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



• We can evaluate this probability as

(74)

(75)

(76)



• 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.



• 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





• 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



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



• 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.





• 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)



• 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:



• 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)



• 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.

ELEC3540 Lecture 8 Digital Modulation

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