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6.8 Linear Modulation Techniques

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6.8 Linear Modulation Techniques

linear modulation: carrier’s amplitude carrier varies linearly with m(t)• bandwidth efficient • attractive for systems with limited spectrum (e.g. wireless)

s(t) = sI(t)cos(2πfct) - sQ(t)sin(2πfct)(1) Cartesian basis:

s(t) = a(t)cos( 2πfct + θ(t) )(2) Polar basis:

constructing signal constellations of linear modulated signals

a(t) = )()( 22 tsts QI envelope of s(t) given as

θ(t) =

)()(

tan 1

tsts

I

Qphase of s(t) given as

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A = signal amplitude fc = carrier frequency s(t) = transmitted signalm(t) = modulating digital signalmR + jm1(t) = complex envelope representation of m(t)

s(t) = Re[Am(t) exp(j2fct)]

s(t) = A[ mR(t) cos(2fct) - m1(t) sin(2fct) ]6.65

• generally linear modulation doesn’t have constant envelope

• non-linear modulation has either linear or constant carrier envelope

Provides Basis for any transmitted signal

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6.8.1 BPSK

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6.8.1 BPSK

phase of constant amplitude carrier switched between 2 values• phase switches based on 2 possible symbols, m1 and m2

• normally phase separated by 180o

Assume rectangular pulse shape: p(t) =

b

bTTt

rect2/

b

bTE2• Ac =

• carrier given by Accos(2πfct + θc)

• bit energy Eb = ½ Ac2Tb

• c = phase shift in carrier

For a carrier with frequency = fc and amplitude (volts) = Ac

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1. Transmitted BPSK signal is

6.66sBPSK(t)= bccb

b Tt tfTE

0)2cos(2

binary 1

binary 0 sBPSK(t) = bcc

b

b Tt tfTE

0)2cos(2

= bccb

b TttfTE

0)2cos(2

bE bEBPSK constellation

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Basis Signal Set consists of one waveform (symbol), 1(t)

1(t) = bcb

Tt tfT

0)2cos(2 6.60

BPSK signal set expressed in terms of basis

SBPSK = )(),( 11 tEtE bb 6.61

6.67bcb

b Tt tfTE

0)2cos(2 sBPSK(t)= m(t)

generalize m1 and m2 as binary signal m(t)

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BPSK signal is a double sideband amplitude modulated waveform • suppressed carrier

• applied carrier = Accos(2fct)

• data signal = m(t) BPSK signal can be generated using balanced modulator

Spectrum and Bandwidth of BPSK• BPSK signal can be expressed in complex envelope form • use polar baseband data waveform of m(t),

sBPSK = Re[gBPSK(t) exp(j2 fct)] 6.68

gBPSK(t) = cbb jtmTE exp)(2 6.69

where gBPSK(t) is complex envelope of sBPSK(t) given by:

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PSD of baseband complex envelope can be shown to be

PgBPSK(f) =

2sin2

b

bb fT

fTE

6.70

PSD for BPSK at RF passband can be evaluated by translating baseband spectrum according to 6.41.

22

)()(sin

)()(sin

2 bc

bc

bc

bcb

TffTff

TffTffE

6.71PgBPSK(f) =

Ps(f) = ¼[Pg(f-fc) + Pg(-f-fc)]

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1

0

-1

T 2T 3T 4T 5T 6T

1 1 0 0 1 1

Peak PSD

fc-5f0 fc-3f0 fc-f0 fc fc+f0 fc+3f0 fc+5f0

(occurs with 101010…pattern)

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norm

aliz

ed P

SD (d

B)

fc-3Rb fc-2Rb fc-Rb fc fc+Rb fc+2Rb fc+3Rb

0

-10

-20

-30

-40

-50

-60

rectangular pulsesRC shaping with = 0.5

PSD of BPSK signal with rectangular pulse and RC pulse shaping• null-to-null BW = 2Rb (Rb = bit rate)

pulse shape % pulse energy occupied BWrectangular pulse 90% 1.6 Rb

pulse with RC filter, = 0.5 100% 1.5 Rb

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2. Received BPSK Signal

BPSK Receiver can be expressed as

assumes no multipath impairments induced by channel

ch = phase shift from channel time delay

c = phase shift in carrier

fc = carrier frequency

6.72

sBPSK(t)=

= )2cos(2

)( tfTE

tm cb

b

)2cos(2

)( chccb

b tfTE

tm

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BPSK Receiver uses coherent (synchronous) demodulation

• receiver requires information about c & fc

• options to recover fc and c include:

(1) send low level pilot carrier signal & use PLL

(2) synthesize carrier phase & frequency e.g. use Costas loop or - Squaring loop

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BPSK Receiver with squaring loop

sBPSK(t) = m(t)Accos(2fct+)(1) received signal is

(2) sBPSK(t)2 generates dc signal & amplitude varying sinusoid at 4fc

(3) dc signal is filtered using BPF with center frequency = 2fc

(4) frequency divider () used to recreate cos(2fct +)

12 3 4 5 m(t)

bitsynch

integrate & dump

sBPSK(t) square law

frequency f/2

2fc

m2(t)A2ccos2(2fct+)

cos(4fct+2) cos(2fct+)

m(t)Accos2(2fct+)

6

(5) output of mixer

6.73)2(cos2)( 2 tfTEtm cb

b

)24cos(

21

212)( tf

TEtm cb

b

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(6) mixer output applied to integrate & dump • forms LPF segment of detector• if transmit & receive pulse shapes match optimum detection• bit synch facilitates sampling of integrator output at end of Tb

• at end of Tb integrator switch closes & output dumped to decision circuit

(7) decision circuit uses threshold to determine if bit is a 1 or 0• threshold must be tuned to minimize error• if 1 or 0 are equally likely use midpoint of detector voltage output level

Decision Boundary

W

t‘0’

‘1’S

N

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Probability of Bit Error for BPSK• many modulation schemes in AWGN channel – use Q-function of distance between signal points

0

2NEQ bPe,BPSK = 6.74

Bit Error Probability for BPSK from substitution into 6.62

• for BPSK – the distance between points in constellation is given by

bE2d12 =

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6.8.2 DPSK

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6.8.2 DPSK• non-coherent PSK receiver doesn’t need reference signal• simplified receiver - easy & cheap to build, widely used

Let {mk} = input binary sequence

{dk} = differentially encoded output sequence

dk = kth differentially encoded output, generated from compliment of modulo 2 sum of mk and d k-1

dk = mk dk-1

net effect also achieved by following rule: • if mk = 1 dk = dk-1

• if mk = 0 dk = dk-1

no symbol transition with mk = 1 possible synchronization issue

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876543210k

10011011-{dk-1}10101001-{mk}

010011011{dk}

o o = 1 o ō = 0

e.g. for given data stream: {mk}

• less energy efficiency - about 3dB < coherent PSK

• average probability of bit error: PeDPSK =

0exp

21

NEb 6.75

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Transmitter: DPSK obtained by passing dk to product modulator

DPSK signal dkmk

cos(2fct)dk-1

delayTb

DelayTb

DPSK Signal

logic circuit

thresholddevice

integrate & dump

Receiver: • input signal demodulated• original sequence recovered by undoing differential encoding

dk = mk dk-1 mk= dk dk-1

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6.8.3 QPSK

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6.8.3 QPSK• 2 bits transmitted per symbol 2bandwidth efficiency of BPSK • symbol determined from 4 possible phases

• Ts = 2Tb (one symbol time = two bit periods)

• Es = 2Eb bit energy = ½ symbol energy

6.76

2)1(2cos2 itf

TE

cs

ssQPSK(t)= 0 t Ts

QPSK signal sQPSK(t) can be expressed as:

i = 1,2,3,4

s

s

TE2

is the symbol’s amplitude

2)1( i is phase of the symbol (0, 90 , 180 , 270)

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define a Basis for S over interval 0 t Ts: {1(t), 2(t)}

1(t) = )2cos(2 tfT c

s 2(t) = )2sin(2 tf

T cs

6.78)(2

)1(sin)(2

)1(cos 21 tiEtiE ss

si(t)=

define QPSK signal set: S = {s1(t), s2(t), s3(t), s4(t)}

rewrite equation 6.76 over 0 t Ts

6.77)2sin(2

)1(sin2)2cos(2

)1(cos2 tfiTEtfi

TE

cs

sc

s

s

sQPSK(t)=

(cos(α + β) = cosα cosβ - sinα sinβ)

QPSK Basis

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Alternate view of QPSKparallel combination of 2 BPSK modulators operating in quadrature phase to each other

m(t) = 110010100011m1(t) = 101101

m2(t) = 100001

demultiplex binary stream m(t) into m1(t) and m2(t)

1 for 0 ≤ t ≤ 2T 0 otherwise

p(t) =

bk,i = +1 for binary ‘1’

bk,i = -1 for binary ‘0’

m1(t) = k

Ik kTtpb )(, m2(t) = k

Qk kTtpb )(,

p(t) = pulse shape, assume a rectangular pulse:

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0

3/2

/2phase shift key of binary streams • 0 and phase shift mI(t)

• /2 and 3/2 phase shift mQ(t)

Ac = s

s

TE2

sI(t) = AcmI(t)cos(2fct)

sQ(t) = AcmQ(t)sin(2fct)

mI(t) =

2)1(cos i for i = 1, 3

mQ(t) =

2)1(sin i for i = 2,4

sQPSK(t) = sI(t) + sQ(t)

= AcmI(t)cos(2fct) + AcmQ(t)sin(2fct)

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sI(t) = mI(t)cos(2fct)bit = 1 sI(t) = cos(2fct) for 0 ≤ t ≤ 2T

bit = 0 sI(t) = -cos(2fct) for 0 ≤ t ≤ 2T

sQ(t) = mQ(t)sin(2fct)bit = 1 sQ(t) = sin(2fct) for 0 ≤ t ≤ 2T bit = 0 sQ(t) = -sin(2fct) for 0 ≤ t ≤ 2T

sQPSK(t) = cos(2fct) sin(2fct)

normalize Ac =1

I bit Q bit sQPSK(t)1 1 cos(2fct) + sin(2fct)0 1 -cos(2fct) + sin(2fct)1 0 cos(2fct) - sin(2fct)0 0 -cos(2fct) - sin(2fct)

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QPSK Constellation Diagram has four points

Q

I

/4

54 7/4

3/4

47,

45,

43,

4

M2 =

sE

sE2

0

3/2

/2Q

I M1 =

23,,

2,0

• rotate constellation by /4obtain new QPSK signal set

Es = 2Eb

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Signal Space Characterization of QPSK Signal Constellations

ith QPSK signal, based on message points (si1, si2) defined in tables

for i = 1,2 and 0 ≤ t ≤ Ts

si(t) = si1 1(t) + si22(t) (3.36)

π/4003π/4015π/4117π/410

si2 si1 grey coded

QPSK signalbinary symbol

bE bEbEbE

bE

bEbE

bE

00000π/201

0π1103π/210

si2si1 grey coded

QPSK signalbinary symbol

bE2

bE2

bE2

bE2

bE20

3/2

/2/4

54 7/4

3/4

bE

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Thus PeQPSK = PeBPSK

• QPSK has 2 spectral efficiency of BPSK & same energy efficiency

• QPSK can be differentially encoded - allows non-coherent detection

• since Ts = 2Tb Es = 2Eb

• assumes AWGN channel

• distance between adjacent points = bs EE 22

Average probability of bit error: PeQPSK

6.79PeQPSK =

0NEQ s =

0

2NEQ b

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S1 =

23,,

2,0

Baseband QPSK Signal in Time Domain• Ts = 0.1s• Tb = 0.05 Rb = 20bps

t

Ac

1 1 0 1

I1

0

-12T 4T 6T 8T

Q1

0

-1 2T 4T 6T 8T

1 1 0 1

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QPSK data stream• Ts = 0.1s• Tb = 0.05 Rb = 20bps

t

Ac

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QPSK Spectrum & Bandwidth

• PSD of QPSK using rectangular pulses given by PQPSK(f)

• similar to PSD of BPSK, replace Tb with Ts

6.80PQPSK(f) =

22

)()(sin

)()(sin

2 sc

sc

sc

scsTff

TffTff

TffE

22

)(2)(2sin

)(2)(2sin

bc

bc

bc

bcb Tff

TffTff

TffE

PQPSK(f) =

•Wnull-QPSK = Rb

•Wnull-BPSK = 2Rb

Wnull= Null to Null Bandwidth

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norm

aliz

ed P

SD (d

B)

fc-Rb fc-½Rb fc fc+½Rb fc+Rb

0-10-20

-30-40-50-60

• rectangular pulse• RC pulse shaping with = 0.5

PSD of QPSK signal

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QPSK Transmitter - based on modulating 2 modulated BPSK signals

(1) m(t) = bi-polar NRZ input with bit rate = Rb

(2) split m(t) into even and odd stream, mI(t) & mQ(t) each with ½ Rb

(3) modulate each stream with quadrature carriers 1(t), 2(t)

(4) sum two resultant BPSK signals to produce QPSK output

(5) band pass filter confines signal to allocated passband*pulse shaping normally done at baseband, prior to modulation

m(t) at Rb Serial -Parallel

mI(t) at ½ Rb

mQ(t) at ½ Rb

QPSKoutput

LO

90o

2(t)

1(t)

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Coherent QPSK Receiver

(1) front end BPF removes out of band noise

(2) filtered output is split

(3) each part coherently demodulated using I & Q carriers

(4) revover carriers coherently from received signals with squaringloop

(5) demodulated output passed to decision circuit which generates I & Q streams

(6) I & Q streams are multiplexed to recover original binary stream

recovered signal

carrier recovery

90o

receivedsignal symbol timing

recovery MUX

decisioncircuit

decisioncircuit

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6.8.5 Offset QPSK

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6.8.5 Offset QPSK (OQPSK)

QPSK is ideally constant envelope (e.g. amplitude is constant)

Pulse shaped (bandlimited) QPSK signals lose constant envelope

• if phase shift = signal envelope can momentarily pass through 0 (zero crossing)

• hardlimiting or non-linear amplification of zero crossings brings back filtered side lobes

- fidelity of signal at small voltages is lost in transmission- sidelobes result in spectral widening

• Use of linear amplifiers to amplify pulses will avoid this but will result in inefficient power use

04/22/23 38

OQPSK (offset QPSK)

• phase (bit) transition instants of mI(t) & mQ(t) are offset by Tb

• phase transitions occur every Tb = ½Ts

• max phase shift = 90o (/2) only one bit stream value changes• ensures smaller phase transitions applied to RF amplifier reduces spectral growth after amplification

QPSK: • phase (bit) transitions of mI(t) & mQ(t) occur at same time instants• phase transitions occur every Ts = 2Tb

• maximum phase transition = 180o () both mI(t) & mQ(t) change• non-linear amplification results in spectral widening

04/22/23 39

OQPSK (offset QPSK)

• bit transitions of mI(t) & mQ(t) are offset by Tb in relative alignment

- phase transistions occur every Tb = ½ Ts

- at any time, only one bit stream can change values

maximum phase shift of transmitted signal limited to 90°

m1 m3 m5 m7 m9 m11 m13

-Tb 0 Tb 2Tb 3Tb 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb 10Tb 11Tb 12Tb13Tb

mI(t)m0 m2 m4 m6 m8 m10 m12

mQ(t)

04/22/23 40

0

3/2

/2

0

3/2

/2

OQPSKpossible phase shifts

OQPSKpossible phase shifts

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s2(t) and s2_offset(t)

s1(t) and s2_offset(t)

Offset QPSK

04/22/23 42

Ts = symbol periodTb = bit period

mI(t) = even bit streammQ(t-Tb) = odd bit streams, offset by Tb

main differences from QPSK is time alignment of mI(t) & mQ(t)

sI(t) = AcmI(t)cos(2fct)

sQ(t) = AcmQ(t-Tb)sin(2fct)

sOQPSK(t) = sI(t) + sQ(t)

= AcmI(t)cos(2fct) + AcmQ(t-Tb)sin(2fct)

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OQPSK vs QPSK

• OQPSK switching occurs at Tb vs 2Tb for QPSK

• OQPSK eliminates 180° phase transition

• spectrum of OQPSK = spectrum of QPSK – unaffected by offset alignment of bit streams

• OQPSK retains bandlimited nature even with non-linear amplification

- critical for low power operations

• OQPSK appears to perform better than QPSK with phase jitter from noisy reference signals

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6.8.6 /4 QPSK

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6.8.6 /4 QPSK

• compromise between OQPSK & QPSK• either coherent or non-coherent demodulation• maximum phase change limited to 135o

- 180o for QPSK- 90o for OQPSK

• preservation of constant BW property in between 2 variants• performs better than both in multipath spread & fading

/4 DQPSK differential encoded version• facilitate differential detection or coherent modulation

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/4 QPSK modulation • modulated signal selected from 2 QPSK constellations shifted by /4 • for each symbol switch between constellations –total of 8

symbols states 4 used alternately

• phase shift between each symbol = nk = /4 , n = 1,2,3

- ensures minimal phase shift, k = /4 between successive symbols

- enables timing recovery & synchronizationQ

Iall possible signal transitions

= possible states for k for k-1 = n/4= possible states for k for k-1 = n/2

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6.8.7 /4 Transmitter

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6.8.7 /4 QPSK Transmitter

(i) partition input mk into symbol stream mIk, mQk

(ii) produce pulses Ik and Qk by signal mapping during [kT,(k+1)T]

(iii) form I(t) & Q(t) from p(t), Ik, Qk & modulate by quadrature carriers

(iv) pre-modulation pulse shaping

/4 QPSK Transmittercos wct

sin wct

I(t)

Q(t)

/4QPSKoutput

amplifier

mkSerial -Parallel

Signal Mapping

mIk

mQk

Ik

Qk

04/22/23 49

(i) input bit stream m(t) partitioned into symbol streams mIk, mQk by serial-parallel conversion

• each symbol stream with symbol rate, Rs = ½Rb

• for symbol at k+1, phase shift = k is a function of mIk, mQk

-/400-3/4103/401/411

phase shift kinputs

mIk, mQk

04/22/23 50

Ik = cosk

= Ik-1cos k - Qk-1sin

k

6.81

Qk = sink

= Ik-1sin k - Qk-1cos k

6.82

k = k-1 + k 6.83

(ii) signal mapping circuit produces Ik & Qk during kT t (k+1)T

• Ik = kth in-phase pulse

• Qk = kth quadrature pulse

• k = phase of kth symbol is a function of k

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(iii) Ik & Qk bit streams are separately modulated by carriers in quadrature to each other (same as QPSK)

6.86

1

0)2/(

N

kssk TkTtpQQ(t) =

1

0)2/(sin

N

kssk TkTtp=

6.85

1

0)2/(

N

kssk TkTtpII(t) =

1

0)2/(cos

N

kssk TkTtp=

s/4(t) = I(t)coswct – Q(t)sinwct 6.84

• transmitted /4 wave form given by:

• p(t) = pulse shape

• Ts = symbol period

04/22/23 52

(iv) Ik & Qk often filtered by RC pulse shaping filters, pre-modulation

• reduce bandwidth occupancy• reduces spectral restoration problem

- significant in fully saturated non-linear amplifier systems

Ik , Qk , & peak values of I(t), Q(t) limited to 5 values:

• 0

• 1

• 21

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6.8.8 /4 Detection

04/22/23 54

non-coherent differential detection can be used

• even without differential encoding

• bit information is completely contained in k - k = relative phase difference of carrier between 2 adjacent symbols

• easier to implement, compared to coherent detection

• BER performance of non-coherent differential detection ≈ 3dB less than BER performance of QPSK

coherent detection BER performance of = QPSK

6.8.8 /4 QPSK Detection

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types of detection techniques (1) baseband differential detection (2) IF differential detection (3) FM discriminator detection

Differential Detection doesn’t rely on phase synchronization

- offers low error floor for low bit rate, fast fading Rayleigh channels

• in IF & baseband detection - k decision based on determination of cosk & sink

• FM discrimination detects k non-coherently

• all 3 have similar BER performance

• important implementation issues with each

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(1) /4QPSK Base Band Differential Detector

(i) signal quadrature demodulated using local oscillators (LO)

(ii) LPF I & Q arms of demodulator output wk & zk

(iii) pass wk & zk through decoders output xk & yk

(iv) decision circuit used to determine SI from xk & SQ from yk

zk

wk

recovered signal

receivedsignal

MUX-2sin wct

2cos wct

decisioncircuitwkwk-1+zkzk-1

xk

SI

decisioncircuitzkwk-1+wkzk-1

yk

SQ

Ts sample at max output

04/22/23 57

(i) incoming /4QPSK signal quadrature demodulated using 2 LOs- LO signal frequency = unmodulated transmitter carrier- LO signals can be out of phase with transmit carrier

wk = cos(k-) 6.87

zk = sin(k-) 6.88

= phase shift due to noise, propagation, & interference

• if Δ << Δk is essentially constantΔ = change in Δk = change in k

• let phase of carrier due to kth bit be: k = tan-1(Qk/Ik) then:(ii) LPF I & Q arms of demodulator output wk & zk

04/22/23 58

xk = wkwk-1 + zkzk-1 6.89

yk = zkwk-1 + wkzk-1 6.90

(iii) sequences wk and zk are passed through decoders that perform

• output of differential decoders

= cos(k - k-1)

xk = cos(k-) cos(k-1-) + sin(k-) sin(k-1-) 6.91

= sin(k-k-1)

yk = sin(k-) cos(k-1-) + cos(k-) sin(k-1-) 6.92

• negates relative phase difference with channel effects

04/22/23 59

(iv) decision circuit uses table to determine

if xk < 0 SI = ‘0’

if xk > 0 SI = ‘1’6.93

if yk < 0 SQ = ‘0’

if yk > 0 SQ= ‘1’6.94

SQ = detected bits in quadrature arms

SI = detected bits in in-phase arms

• critical that LO frequency = transmit carrier frequency• if LO frequency drifts output phase drifts & BER increases

04/22/23 60

(2) IF Differential Detector• uses delay & 2 phase detectors• doesn’t need LO• received signal first converted to IF then bandpass filtered

BPF designed to match transmitted pulse shape• preserves carrier phase• minimizes noise power

• found that passband of BPF specified at 0.57/Ts minimizes ISI & noise

received IF signal • differentially decoded using delay & 2 mixers• passband at decoder output = 2 based band signal at transmit end

04/22/23 61

MUXdemodulatedsignal

decisioncircuit

Ts sample at max output

modulated IF signal

Ts

90 o

decisioncircuit

/4QPSK IF Differential Detector

04/22/23 62

FM Discriminator

(1) BPF: input signal put through BPF that is matched to transmitted signal

(2) Limiter: filtered signal is hard-limited to remove envelope fluctuations

- retains phase changes no information lost

(3) FM discriminator determines instantaneous frequency

deviation of received signal

(4) Integrate and Dump: integration of instantaneous frequency deviation over Ts gives phase difference between 2 sampling instants

(5) Threshold Detector: phase difference detected by 4-level threshold comparator obtain original signal

04/22/23 63

0BPSK

QPSK

0Acm(t)DSB-Suppressed Carrier0Ac(1+kam(t))AM

SQ(t)SI(t)modulation

k

kc )kTt(pbA

k

,kc )kTt(pbA 1 k

,kc )kTt(pbA 2

Specific Canonical Equations

AM: s(t) = Ac(1+Kam(t))cos(2fct)

s(t) = Acm(t) cost(2fct)DSB-SC:

BPSK: m (t) = k

k )kTt(pb s(t) = Acm(t)cos(2fct)and

s(t) = Acm1(t)cos(2fct) + Acm2(t)sin(2fct)

mi(t) = k

ik kTtpb )(,QPSK: and