Modulation Techniques

Dr. J. Martin Leo Manickam

Professor

Challenges in Mobile Communication

• Channel– fading

• Energy

• Bandwidth

Digital Modulation

• Digital signal is converted into analog bit stream

)cos()( ttaType Constant variable

ASK a(t)

FSK

PSK

QAM

,

),(ta),(ta

),(ta

Classification

Linear modulation

• Amplitude of the transmitted signal varies linearly with message signal

• Bandwidth efficient• QPSK, OQPSK

Non linear (constant envelope) modulation

• Amplitude of the transmitted signal does not vary with the amplitude of the message signal

• Power efficient class C amplifiers can be used

• Low out of band radiation• Limiter-discriminator can

be used for demodulation• FSK,MSK, GMSK

BPSK

b

b

b

Tt0 ),2cos(2

)()(

thensymbols,binary for the 1-or 1 represents m(t) if

0)(binary Tt0 ),2cos(2

)(

1)(binary Tt0 ),2cos(2

)(

ccb

bBPSK

ccb

bBPSK

ccb

bBPSK

tfT

Etmts

tfT

Ets

tfT

Ets

BPSK is equivalent to a DSB/SC

Binary PSK

PSD of a BPSK signal

Observation

Pulse shape Null-to-null bandwidth

90 percent energy

Rectangular 2Rb 1.6Rb

Raised cosine pulse (0.5) < 2Rb 1.5Rb

QPSK

PSD of QPSK signal

Constant Envelope modulation

Binary FSK

t

fcb

b

cb

bFSK

cb

bLFSK

cb

bHFSK

cb

bLFSK

cb

bHFSK

dmktfT

E

ttfT

Ets

binarytfT

Etvts

binarytfT

Etvts

binarytffT

Etvts

binarytffT

Etvts

))(22cos(2

)(2cos(2

)(

FSK generates modulationFrequency

)0 (Tt0 ),2cos(2

)()(

)1 (Tt0 ),2cos(2

)()(

bygiven isFSK usdicontinioA

)0 (Tt0 ,)22cos(2

)()(

)1 (Tt0 ,)22cos(2

)()(

b2

b1

b

b

Minimum Shift Keying (MSK)• Phase information is used to improve the noise

performance of the receiver

• CPFSK signal (0≤t ≤Tb)

• Eb – transmitted signal energy• Tb – Bit duration• Θ(0) – value of the phase at t=0, sums up the past history

of the modulation process upto t=0.

0 - symbolfor ))0(2cos(2

1 - symbolfor ))0(2cos(2

)(

2

1

tfT

E

tfT

E

ts

b

b

b

b

• DPSK signal can also be represented as

bTt0 ,)0()(

)(2cos(2

)(

tT

htwhere

ttfT

Ets

b

cb

b

)(

)(2

1

2

2

21

21

2

1

ffTh

fff

fT

hf

fT

hf

b

c

bc

bc

Phase tree

• At time t =Tb,

0 - symbolfor

1 - symbolfor )0()(

h

hTb

Phase trellis (h=1/2)

Signal space diagram

bb

Tt0 ,2T

(0) (t)

)2sin()(sin2

)2cos()(cos2

)(

twhere

tftT

Etft

T

Ets c

b

bc

b

b

bbb

b

b

b

TT- ,2

cosT

2E

2cos)0(cos

T

2E

)(cos2

)(

ttT

tT

tT

EtS

b

b

b

bI

bb

b

bb

b

T20 ,2

sinT

2E

2sin)T(sin

T

2E

)(sin2

)(

ttT

tT

tT

EtS

b

b

b

bQ

In-phase component will be half cycle cosine wave

Quadrature component will be half cycle sine wave

+ sign corresponds to- sign corresponds to

0)0( )0(

+ sign corresponds to - sign corresponds to

2/)( bT2/)( bT

• As and can each assume two possible values, any one of four possible values can arise

• Orthonormal basis functions

)0( )( bT

symbol

0 1

0

1

0 0

)0( )( bT

2/

2/2/

2/

bcbb

2

bcbb

1

Tt0 t),fsin(2πt2T

πsin

T

2(t)φ

Tt0 t),fcos(2πt2T

πcos

T

2(t)φ

• The MSK signal can represented by

– Where s1 and s2 are related to the phase states

and , respectively.

• Evaluating s1 and s2:

b2211 Tt0 ),t(s)t(s)t(S

)0( )( bT

bb

T2

0 b22

bb

T

T b11

T200 ,)T(sinEdt)t()t(Ss

T0T- ,)0(cosEdt)t()t(Ss

b

b

b

• Observation– Both integrals are evaluated for a time interval equal

to twice the bit duration– Both lower and upper limits of the product integration

used to evaluate s1 are shifted by Tb w.r.t those used to evaluate the s2.

– The time interval , for which the phase states and are defined are common to both intervals

• Signal space of MSK is two dimensional with four message points

bTt0 )0( )( bT

Signal space characterization of MSK

Transmitted binary symbol

Phase states

(radians)

Coordinates of message points

s1 s2

0 0

1

0

1 0

bTt0 )0( )( bT

2/2/

2/2/

bE bE

bE

bE

bE

bE bE

bE

• Optimum detection of

• If x1 > 0, receiver choose the estimate

• If x1 < 0, receiver choose the estimate

• Optimum detection of

• if x2 > 0, receiver choose the estimate

• If x2 < 0, receiver choose the estimate

)0(

b

T

T

Ttwsdtttxxb

b

b1111 T- )()(

0)0(ˆ

)0(ˆ

b

T

Ttwsdtttxxb

20 )()(2

0

2222

2/)(ˆ bT

2/)(ˆ bT

Estimates

• Symbol – 0

• Symbol – 1

• Probability of error

• Same as that of the BPSK and QPSK

2/)(ˆ&)0(ˆ

)(

2/)(ˆ&0)0(ˆ

b

b

T

or

T

2/)(ˆ&0)0(ˆ

)(

2/)(ˆ&)0(ˆ

b

b

T

or

T

02

1

N

EerfcP b

e

PSD of MSK

• MSK has lower sidelobes than QPSK and OQPSK

• Faster roll off• Less spectrally efficient

– Main lobe of MSK is wider

• Bandlimiting is easier– Continuous phase

• Amplified using non linear amplifiers– Constant phase

• Simple modulation and demodulation

GMSK modulation• Sidelobe levels of the spectrum are further

reduced • Pulse shaping filter requirements

– Narrow bandwidth– Sharp cutoff frequencies– Low overshoot– Carrier phase must be ±π/2 at odd multiples and two

values 0 and π at even multiples

Gaussian LPF FM TransmitterNRZ data GMSK output

• Impulse response

• Transfer function

• parameter related to B, the 3dB baseband bandwidth by

• GMSK filter may be completely specified by B and the basedand symbol duration T

2

2

2

Q exp)(h tt

)exp()(H 22Q tf

B

0.5887

B2

2 ln

PSD of a GMSK signal

• When BT = ∞, GMSK is equivalent to MSK• When BT decreases, sidelobe levels falls off

rapidly– At BT = 0.5, peak of the sidelobe level is 30 db and

20 db below the main lobe for GMSK and MSK– Reducing BT increases the error rate produced by the

LPF due to ISI

Table 6.3 Occupied RF bandwidth (for GMSK and MSK as a fraction of Rb). Containing the

given percentage of power

• GMSK is spectrally tighter than MSK• GMSK spectrum is compact at smaller values of

BT but degradation due to ISI increases.

BT 90% 99% 99.9% 99.99%

0.2 GMSK 0.52 0.79 0.99 1.22

0.25GMSK 0.57 0.86 1.09 1.37

0.5 GMSK 0.69 1.04 1.33 2.08

MSK 0.78 1.20 2.76 6.00

GMSK Bit error rate

0

be N

E2P

Q

• is a constant related to BT by

) (BTMSK simple for 85.0

0.25 BT GMSK with for 0.68

GMSK Receiver (Fig. 6.43)

Combined Linear and Constant Envelope Modulation Techniques

• Varying envelope and phase (or frequency) of an RF carrier (M-ary modulation)

• Two or more bits are grouped together to form symbols and one of M possible symbols are is transmitted during each symbol period

• Bandwidth efficient• Power inefficient• Poor error performance (closely located message

points

M - ary Phase Shift Keying (MPSK)• Carrier phase takes on one of M possible values

• Where Es is the energy per symbol = (log2M)Eb

Ts is the symbol period = (log2M)Tb

• In quadrature form

• Orthonormal basis function:

1,i Tt0 , 1-iM

2f2cos

T

E2)(S sc

s

si

tt

1,2,3,...Mi )f2sin(M

21)-i(sin

T

2E - )f2cos(

M

21)-i(cos

T

2E )(S c

s

sc

s

si

ttt

scs

2cs

1 Tt0 , )f2sin(T

2 )( , )f2cos(

T

2 )( tttt

• M – ary PSK can be expressed as

• Signal space is 2D and the M-ary message points are equally spaced on a circle of radius at the origin

• Distance between the adjacent

symbols is equal to • Average symbol error prob.

• Q function defined as

1,2,3,...Mi (t) M

21)-i(sinE -(t)

M

21)-i(cosE )(S 2s1si

t

sE

1

2

M

sinE2 s

M

sin N

MlogE22P

0

2be

Q

x

2 dx )2/exp(2

1Q(x) x

8 - PSK

Bandwidth and power efficiency of M-ary PSK signals

M 2 4 8 16 32 64

0.5 1 1.5 2 2.5 3

Eb / No for

BER = 10-6

10.5 10.5 14 18.5 23.4 28.5

*B /BRb

B: First null bandwidth of M-ary PSK signals

M-ary PSK PSD, for M = 8,16(PSD for both rectangular and RCF pulses

for fixed Rb • When M increases

(fixed Rb) – First null bandwidth

decreases– Bandwidth efficiency

increases– Constellation is

densely packed– Power efficiency

decreases– More sensitive to the

timing jitter

M-ary Quadrature amplitude modulation (QAM)

• Amplitude and phase of the transmitted signal are varied

Where Emin - energy of the signal with the lowest amplitude and ai and bi are a pair of independent integers chosen according to the location of the particular point

• No constant energy per symbol• No constant distance between possible symbol

states

• Si(t) is detected with higher probability than others

1,2,...Mi T,t0 )fsin(2 b T

E2 )fcos(2 a

T

E2)(S ci

s

minci

s

mini ttt

• Ortho normal basis functions

• Coordinates of ith message point:• Where is an element of L by L matrix given

by

• Where

Tt0 )fsin(2 T

2 )( );fcos(2

T

2)( sc

s2c

s1 tttt

)Eb ,Ea( minimini

)b ,a( ii

1)L- 1,-L(-1)L- ,3L(1)L- 1,L(

----

----

3)-L 1,-L(-3)-L ,3L(3)-L 1,L(

1)-L 1,-L(-1)-L ,3L(1)-L 1,L(

b,a ii

M L

• Constellation Diagram for 16-QAM:

• LXL matrix is given by

• Probability of Error:

)3,3()3,1()3,1()3,3(

)1,3()1,1()1,1()1,3(

)1,3()1,1()1,1()1,3(

)3,3()3,1()3,1()3,3(

b,a ii

0

mine N

E2M1

14P Q

Bandwidth and Power efficiency of QAM

M 4 16 64 256 1024 4096

1 2 3 4 5 6

Eb / No for

BER = 10-6

10.5 15 18.5 24 28 33.5

*B /BRb

• Power spectrum and bandwidth efficiency of QAM is identical to M-ary PSK

• Power efficiency is superior than M-ary PSK

M-ary Frequency Shift Keying (MFSK)

• Transmitted signal

Where for some fixed integer nc

• Transmitted signals Si(t) themselves can be used as a complete ortho normal basis functions

• M-dimensional signal space, minimum distance is

1,2,3...Mi ;Tt0 , i)tn(T

cosTE2

)t(S scss

si

s

cc 2T

n f

1,2,3...Mi T;t0 ),(SE1

)( is

i tt

sE

• Average probability of symbol error:

Coherent detection:

Non coherent detection:

• Channel Bandwidth:

Coherent MFSK:

Non Coherent MFSK:

0

2be N

MlogEQ 1)-(M P

0

se 2N

Eexp

21-M

P

Mlog 23)M(R

B2

b

Mlog 2MR

B2

b

Bandwidth and Power Efficiency of Coherent MFSK

M 2 4 8 16 32 64

0.4 0.57 0.55 0.42 0.29 0.18

Eb / No for

BER = 10-6

13.5 10.8 9.3 8.2 7.5 6.9

• Can be amplified by using non-linear amplifiers with no performance degradation

• When M increases, – bandwidth efficiency decreases– Power efficiency increases

B

Performance of Digital modulation in Slow Flat-fading channels

• Multiplicative gain variation• Received signal can be expressed as

- channel gain

- phase shift of the channel

n(t) - is the additive gaussian noise• Assumptions

– Attenuation and phase shift remains constant

Tt0 n(t) (t))s(t)exp(-j (t) r(t) )(t

θ(t)

Probability of Error in slow flat-fading channel

• Choose the possible range for signal strength due to fading

• Average the probability of error of the particular modulation in AWGN channel over possible range of signal strength

• - Probability of error for an arbitrary modulation at a specific value of SNR X,

• - pdf of X due to the fading channel• Eb and No are the average energy per bit and noise PSD

in a non-fading AWGN channel•

- instantaneous power values of the fading channel, w.r.t the non fading

0

ee (X)p(X)dXPP

)X(Pe

)X(p0b

2 N/EX

20b N/E

• For rayleigh fading channels ,fading amplitude has rayleigh distribution

is the average value of the signal to noise ratio

0X , )X

exp(1

)X(p

2

0

b

NE

BFSK) orthogonal coherent (non 2

1 P

BPSK) ial(different )1(2

1 P

BFSK) (coherent 2

121

P

BPSK) (coherent 1

121

P

NCFSK,e

DPSK,e

FSK,e

PSK,e

• For large values of Eb/N0:

• For GMSK

BFSK) orthogonal coherent (non 1

P

PSK) ial(different 21

P

FSK) (coherent 21

P

BPSK) (coherent 41

P

NCFSK,e

DPSK,e

FSK,e

PSK,e

BT for 85.0

0.25 BT for 0.68 where

GMSK) (coherent 4

11

121

P GMSK,e

Observations

• Flat-fading channel– Error rate is inversely

proportional to SNR

– Higher power required• BER of 10-3 to 10-6, 30-

60 dB SNR

• AWGN channel– Exponential

relationship between error rate and SNR

– Lower power required• BER of 10-3 to 10-6, 20-

50 dB SNR

Digital Modulation in frequency selective channels

• Caused by multipath time delay spread or time varying Doppler spread

• Produces ISI • Impose bounds on data rate and BER• Errors occurs due to ISI when

– Main (undelayed) signal component is removed through multipath cancellation

– Non-zero value of delayed spread (d)– Sampling time shifted as a result of delay spread

• Small delay spread results flat fading• Large delay spread results timing errors and ISI

BPSK

QPSKOQPSKMSK

BPSK

MSK

OQPSKQPSK