Modulation and Demodulation - Iowa State...

1/31/20051

Analog modulationAnalog modulationAMAMPM/FMPM/FM

Digital modulationDigital modulationBinary ModulationBinary ModulationQuadratureQuadrature ModulationModulation

Power Efficiency Power Efficiency

NoncoherentNoncoherent DetectionDetection

Modulation and Demodulation

1/31/20052

Modulation• Important fact: a continuous, oscillating signal will

propagate farther than other signals.• Start with a carrier signal

– usually a sine wave that oscillates continuously. – Frequency of carrier fixed

1/31/20053

Carrier Signal• In analog transmission, the sending device

produces a high-frequency signal that acts as a basis for the information signal. – This base signal is called the carrier signal or

carrier frequency• The receiving device is tuned to the frequency

of the carrier that it expects from the sender.

1/31/20054

Modulation• Signal information is modulated on the carrier

signal by modifying one of its characteristics (amplitude, frequency, phase).

• This modification is called modulation• Same idea as in radio, TV transmission• The information signal is called a modulating

signal.

1/31/20055

Modulation• Modulation: process of changing a carrier

wave to encode information.• Modulation used with all types of media• Why is modulation needed?

– Allows data to be sent at a frequency which is available– Allows a strong carrier signal to carry a weak data signal– Reduces effects of noise and interference

1/31/20056

Types of modulation• Amplitude modulation (used in AM radio) –

strength, or amplitude of carrier is modulated to encode data

• Frequency modulation (used in FM radio) –frequency of carrier is modulated to encode data

• Phase shift modulation (used for data) –changes in timing, or phase shifts encode data

1/31/20057

Analog modulation

1/31/20058

Amplitude Modulation

1/31/20059

DSB-SCAM

Signal: m(t)Modulated signal: A(t)=Acm(t), 1-1 correspondence to m(t)Carrier: Cos ωct

{ {

48476carriert

fixedccos

.signal modulated

)t(A)t(xform General c ω = : ×

Amplitude modulation

1/31/200510

m(t)

t

t

phase reversal

f

f

W- W 0

fc- fc

AcM(f+fc)/2 AcM(f-fc)/2

fc+Wfc-W

M(f)

{ {

48476carriert

fixedccos

.signal modulated

)t(A)t(xform General c ω = : ×

X f12

A M f f12

A M f f

transla tion of M f

f2

c c c c c cc

( ) ( ) ( )

( )

= + + − =1 24 4 34 4 1 24 4 34 4

1 24 4 4 4 4 4 44 34 4 4 4 4 4 4 4

, ω

π

Doubled-Sideband Modulation (Suppressed Carrier): DSB-SC

1/31/200511

Coherent (Synchronous) Demodulator (Detector):The receiver knows exactly the phase and frequency of the received signal.

xc(t) = Acm(t)cosωctd(t) = [Acm(t)cosωct] 2cosωct

= Acm(t) Message we want!+ 2Ac m(t)cosωcty1(t)= Acm(t)

LPFm(t) xc(t) d(t) yD(t)

2cosωctcosωct

xc(t)

Modulator Demodulator

Demodulation

1/31/200512

-4 -2 2 4

-1

-0.5

0.5

1

-4 -2 2 4

-2

-1

1

2X(t) d(t)

-fc fc

Xc(f)

0f

-2fc 2fc

D(f)

0f

1/31/200513

<xc2(t)>=<[A+m(t)]2 (Ac)2 cos2ωc t>

Assume m(t) varies slowly w.r.t. cos2ωct

=<(1/2)(Ac)2 [A +m(t)] 2 >+<(Ac)2 [A+m(t)]2 cos2ωct>

Since <cosX>=0 :=(1/2)(Ac)2< [A2+2A<m(t) +<m2(t)>]

Assume <m(t)>=0:= (1/2)(Ac)2 [ A2 + <m(t) 2>]

Carrierpower

dc biaspower

power of m(t),signal power

Power Analysis

1/31/200514

Efficiency E: the percentage of total power that conveys information.

E=

EAM =

=

=

Power TotalPower Signal

%)100(])t(m)][A

2[1/2(Ac

)t(m)]2(2/1[22

2

><+

><Ac

%)100()t(mA

)t(m22

2

><+

><

2n2

2n2

a m t

1 a m t

< >

+ < >

( )

( )( %)100

1/31/200515

Frequency Modulation• Frequency of the carrier signal is modulated to

follow the changing voltage level (amplitude) of the modulating signal.

• Peak amplitude and phase of the carrier signal remain constant, but as the amplitude of the info signal changes, the frequency of the carrier changes accordingly.

1/31/200516

Frequency Modulation

1/31/200517

Phase-Shift Modulation• vary phase of carrier• may use more than simply 180 degree shift

(binary)• this allows higher bit rate than baud rate

– Eight angles results in 3 bits per signal element. Or 3 bits per baud!

1/31/200518

General form：xc(t)=Accos[ωct+φ(t)]Phase modulation: φ(t)=Kpm(t), Kp: deviation constantFrequency modulation : d φ(t)/dt = Kf.m(t)

Kf: deviation constant; fd: Frequency deviation constant ∫

∫Φ

Φ+=Φt

to d

t

tof

o

o

+)dm(f 2=

d)(mK)t(

ααπ

αα

+=

+=

∫t

todcc

pcc

]d)(mf2t[cosA(t):FM

)]t(mkt[cosAt)(:PM

ααπω

ω

c

c

x

x

Angle Modulation: FM and PM

1/31/200519

FM/PM waveforms

LCfc π2

1=

xBB controls the value of C

1/31/200520

Indirect implementation (Armstrong)using a mixer and summer

)sin)((cos)( tttAtx cccc ωφω −=If φ(t) very small: ))(cos()( ttAtx ccc φω +=

1/31/200521

Direct method using PLL

Crystaloscillator

LPF

VCOFrequency

Divider÷N

m(t)

xc(t)

∑

Vc(t)

Vc controls frequency, so is m(t)

1/31/200522

FM/PM spectrum

Ideally

Reality

1/31/200523

Theorem• If |Φ(t)|<<1,

• Or if

[ ] [ ]{ })()()()(2

)( ccccc

c ffffjffffAfX +Φ−−Φ+++−= δδ

[ ]1

2)(max

>B

tmK f

πwhere B is the bandwidth of m

−−+−= )(2()(2(

2)(

2

cf

cff

cc ff

KMff

KM

KAfX πππ

1/31/200524

Digital Modulation

1/31/200525

• Power efficiency : describes the ability of a modulation technique to preserve the fidelity of the digital message at low power levels.

– ηP : Eb/N0

• Bandwidth efficiency: describes the ability of a modulation scheme to accommodate data within a limited bandwidth.

– ηB : R/B bps/Hz

Design Parameters

1/31/200526

• Channel capacity formula– Rmax <= C = Blog2(1+S/N)

– ηBmax = C/B = log2(1+S/N)• C is the channel capacity (in bps)

• B is the RF bandwidth

• S/N is the signal-to-noise ratio

Channel capacity formula

1/31/200527

• Absolute bandwidth : The range of frequencies over which the signal has a ‘non-zero’ power spectral density.

• Null-to-null bandwidth : equal to the width of main spectral lobe.

• Half-power ( 3-dB) bandwidth : the interval between frequencies at which the PSD has dropped to half power, or 3dB below the peak value.

• 99 percent bandwidth (by Federal Communication Commission) : occupied 99 percent of signal power.

Bandwidth

1/31/200528

• Geometric Representation of Modulation – set of modulation signal :

– vector representation with orthogonal basis functions :

–

–

{ })(),.......(),( 21 tststsS M=

∑=

Φ=N

jjiji tsts

1

)()(

ji 0)()( ≠=ΦΦ∫∞

∞−

dttt ji

∫∞

∞−

=Φ= 1)(2 dttE i

General Digital Modulation

1/31/200529

• BPSK:b1 Tt0 )2cos(2)( ≤≤= tf

TEts cb

b π

b1 Tt0 )2cos(2)( ≤≤=Φ tfT

t cb

π

b2 Tt0 )2cos(2)( ≤≤−= tfTEts cb

b π

{ })(),( 11 tEtES bbBPSK Φ−Φ=I

Q

bEbE−

Geometric Representation of Modulation

1/31/200530

Geometric Representation of Modulation

• Bit error analysis: in AWGN channel with a noise spectral density No.

∑≠=

≤

ijj

ijis N

dQsP

,1 02)|(ε

dxxxQx

)2/exp(21)( 2−= ∫

∞

π

∑=

==M

iisiiss sP

MsPsPP

1

)|(1)()|()( εεε

dij: distance between the ith and jth signal point in the constellation

x

1/31/200531

Linear Modulation Techniques• Transmitted signal amp. is proportional to modulated signal• Good spectral efficiency • Bad power efficiency since linear AMP is needed. • E.g. QPSK, OQPSK, π/4QPSK…

)]2sin()()2cos()([ )2exp()(Re[)(

tftmtftmAtfjtAmts

cIcR

c

πππ

−==

1/31/200532

• BPSK TransmitterBit 1: ABit 0: -A

cos(2πfct+θc)

XBPSK(t)

)1 ( )2cos(2)( bittfTEts cc

b

bBPSK θπ +=

) 0 ( )2cos(2 )( bittfTEts cc

b

bBPSK θπ +−=

Linear Modulation Techniques

xin

22

BPSKsin)(),()(

=−++=

b

bbcpcpcp fT

fTTAfPffPffPPπ

π

1/31/200533

Demodulation Techniques

2cos(2πfct+θc)

XBPSK(t)Notchfilter xin2fc

xin

1/31/200534


• Waveform of BPSK after adding noise

1/31/200535

Detection with Matched Filter:

Filter impulse response is a square pulse, matching the pulse shapes in xin

1/31/200536

∫∞+

∞−−=

∗=

,)()(

)()()(

τττ dhtp

thtpty

2

0max N

ESNR p=

∫+∞

∞−= dttpEP

2)(∫

∞+

∞−=

∗=

)()()*

)()()(22 dffNfHtn

tntptn

o

o

2/2oo Nn =

1/31/200537

∫+∞

∞−−= )()( )y(Tb τττ dThx b

∫+∞

∞−= ,)()( τττ dpx

∫=

== bT

b dpxTyτ

ττττ

0 )()()(

Optimum Detection:

1/31/200538

=

0,

2NEQP b

BPSKeBit Error Rate :

Demodulation• BPSK Coherent Receiver :

)2cos(2)(

)2cos(2)()(

θπ

θθπ

+=

++=

tfTEtm

tfTEtmts

cb

b

chccb

bBPSK

[ ])2(2cos112)()2(cos2)(2 2cc

b

bc

b

b tfTEtmtf

TEtm θπθπ ++=+

Using Carrier Recovery to get carrier frequency:

1/31/200539

• BPSK Coherent Receiver Architecture

Symbol/clock

recovery

Carrier recovery

AGCReceived signal

Matchedfilter sampler

SignalPulse

generatorDetect

decisioncos(2πfct+φ)

1/31/200540

I

Q

Eb

2E b I

QI

Q

E2 b

Linear Modulation Techniques• From BPSK to QPSK

Constellation diagram

1/31/200541

• Constellation diagram after adding noise


1/31/200542

1,2,3,4i Tt0

]2

)1(2cos[2)(

s =≤≤

−+=π

π itfTEts c

s

bQPSK

)2sin( ]2

)1sin[(2

)2cos( ]2

)1cos[(2)(

tfiTE

tfiTEts

cs

b

cs

bQPSK

ππ

ππ

−−

−=

−−−= )(]

2)1sin[()(]

2)1cos[()( 21 tiEtiEts bbQPSK φ

πφ

π

QPSK transmitter

2

BPSK 22sin2)(),()(

=−++=

b

bbpcpcp fT

fTCTfPffPffPPπ

π

C is average magnitude squared

1/31/200543

QPSK transmitter

2 bits to 4 levels

Map to constellation

an

bn

bit stream

Pulse shaping

Local Oscillator

+90o

s(t)

Pulse shaping

cos(2πfct+φ)

-sin(2πfct+φ)

1/31/200544

=

0,

2NEQP b

QPSKeBit Error Rate :

QPSK Receiver

1/31/200545

• BPSK v.s QPSK– BER is the same !!!

Bit Error Rate :

=

0

2NEQP b

e

But QPSK only need ½ the bandwidth of BPSK !!!

BER & BW efficiency comparison

BPSK: symbol rate = bit rateQPSK: symbol rate = ½ bit rate

1/31/200546

• Offset QPSK– To overcome 180° phase shift in QPSK thus

prevent the overhead to overcome signal envelop distortion caused in QPSK

Reducing the maximum phase jump

1/31/200547

• Offset QPSK Tx


1/31/200548

• Offset QPSK Rx


1/31/200549

• π/4-QPSK – Alternate between the 2 QPSK constellations– If the current two bits corresponds to a point in

the left constellation– Next two bits will be represented by a point on

the right constellation– Vice versa

– Also called π/4-DQPSK

Further reduction of phase jump

1/31/200550

Concerns over linear modulation• On each of the I/Q axis, we are performing AM

modulation• Transmitted signal amplitude changes with

time– Cause receiver challenges– Transmitter power utilization– Sensitive to additive noise

èuse nonlinear modulation

1/31/200551

• Class C Amp. can be used – saving power

• Limiter-discriminator detection can be used– easy and simple architecture

• Good performance against random noise and signal fluctuation due to Rayleigh fading– good performance

• But! BW is larger than linear modulation

Constant Envelope Modulation

1/31/200552

• BFSK– General Form

– Discontinuous phase FSK

1)(bit 0)22cos(2)()(FSK bcb

bH Tttff

TEtts ≤≤∆+== ππυ

0)(bit 0)22cos(2)()(FSK bcb

bL Tttff

TEtts ≤≤∆−== ππυ

1)(bit 0)2cos(2)()( 1FSK bHb

bH Tttf

TEtts ≤≤+== θπυ

0)(bit 0)2cos(2)()( 1FSK bLb

bL Tttf

TEtts ≤≤−== θπυ

Binary frequency shift keying

1/31/200553

• Discontinuous phase FSK– can be combined by two OOK– cause spectral spreading and spurious

FSK

OOKH

OOKL

Discontinuous phase FSK

1/31/200554

[ ])(sin2sin)(cos2cos2

])(22cos[2

))(2cos(2)(FSK

ttfttfTE

dmftfTE

ttfTEts

ccb

b

t

dcb

b

cb

b

θπθπ

ηηππ

θπ

⋅−⋅=

+=

+=

∫∞−

• Continuous phase FSK– similar to FM except that m(t) is binary

Continuous phase FSK

1/31/200555

• Coherent detection of BFSK

=

0 FSK, N

EQP be


1/31/200556

−=

0NC FSK, , 2

exp21

NEP b

e

• Non coherent detection of BFSK


1/31/200557

• Modulation Index of FSK:

rate bits theis anddeviationfrequency RFpeak theis where

, )2(

b

b

RF

RFh

∆

∆=

Example:

5.0412

41

=×

=⇒

=−=∆

b

b

bc

R

Rh

RffF H

Modulation Index

1/31/200558

CPFSK and modulation index

h = ¼ h = 1/3

h = ½ h = 2/3

I

Q

Q Q

Q

I

II

1/31/200559

– a special type of Continuous phase FSK– modulation index – peak RF frequency deviation=Rb/4– coherently orthogonal. i.e.

– MSK=fast FSK– MSK=OQPSK with baseband rectangular being

replaced with half-sinusoidal– MSK = FSK with binary signaling freq. of fc±1/4Tb

∫ =bT

LH dttt0

0)()( υυ

5.0=h

Minimum shift keying

1/31/200560

• Advantage of MSK: particularly attractive for use in mobile radio communication systems: – constant envelope

– spectral efficiency ?

– good BER

– self-synchronizing capability


1/31/200561

• MSK as OQPSK:

• MSK as CPFSK :

• MSK power spectrum

)2sin()2

sin()()2cos()2

cos()()( tfTttmtf

Tttmts c

bQc

bIMSK π

ππ

π+=

+−= k

bQIc T

ttmtmtfts φπ

π2

)()(2cos)(MSK

2

222

2

MSK 1612cos16)(),()(

−

=−++=b

bcpcpcp Tf

fTAfPffPffPP ππ


Pulse shape: half period cos

1/31/200562

• MSK Transceiver

)(cos2)(

−+= ∫ ∑∞−

t

mbmccMSK dtmTtpbtAtx ω


1/31/200563

I

Q

MSK


1/31/200564

Comparison of MSK, OQPSK, QPSK90o phase

shift in each

symbol

1/31/200565

• Receiver of MSK

MSK

1/31/200566

Why is MSK more spectrally efficient? • Power spectral density of MSK

– 99% BW of MSK = 1.2/Tb

– 99% BW of QPSK or OQPSK = 8/Tb

1/31/200567

Even better spectral efficiency? Use Gaussian MSK

22lnexp()( 2

2

BffH −=

1/31/200568

• GMSK & GFSK– GMSK reduces the sidelobe levels of MSK

– GMSK = Gaussian filter + MSK

– MSK = GMSK with B = ∞

– Important parameter : 3dB-BW – bit duration product (B3dBTb)

1/31/200569

)(sin2sin)(cos2cos))(2cos()(

ttfttfttfts

cc

c

φπφπφπ

⋅−⋅=+=

GLPF ( )∫ ∞−

tk

)(cos tφ

)(sin tφ

)(tφ⊗

o90π2cos tf c ⊕

s(t)NRZ

⊗

• Transmitter of GMSK : QUAD architecture

)(tm

1/31/200570

• Deciding frequency modulation index Kf

∫

∫∫

∫ ∑∫

∞

∞−

∞

∞−

∞

∞−

∞−

∞

−∞=∞−

∗Π=∴

=∗Π⇒

=⇒

−==

dtthtK

dtthtK

dttrK

dnTraKdmKt

G

f

Gf

f

t

nn

t

ff

)()(

2/

2/)()(

2/)(

)()()(

π

π

π

ηηηηφ

])(2cos[2

))(2cos(2)(FSK

ηηπ

φπ

dmKtfTE

ttfTEts

t

fcb

b

cb

b

∫∞−

+=

+=

1/31/200571

• Power spectral density of a GMSK signal

1/31/200572

• Receiver of GMSK

=

0

2N

EQP be

α

∞==

≅)( MSK simplefor 85.0

25.0 GMSK withfor 68.0BT

BTα

1/31/200573

• Occupied RF Bandwidth

1/31/200574

Simulated power spectral density• GSM GMSK v.s MSK

1/31/200575

• Bluetooth v.s GSM

Simulated power spectral density

1/31/200576

M, 2, 1, 0 , )1(22cos2)( K=≤≤

−+= iTti

Mtf

TEtS sc

s

si

ππ

M , 2, 1,

t)2sin(2)1(sin2t)2cos(2)1(cos2

K=

−−

−=

i

fM

iTEf

Mi

TE

cs

sc

s

s ππ

ππ

M , 2, 1,

)(φM2π)1(sin)(φ

M2π)1(cos)( 21PSK-M

K=

−−

−=

i

tiEtiEtS ss

M-ary PSK• Two or more bits are grouped to form symbols and

one of the possible M symbols may be sent

1/31/200577

• Constellation diagram of 8PSK

φ1(t)

φ2(t)

M-ary PSK

1/31/200578

Quadrature amplitude modulation

• Emin : energy of the signal with the lowest amplitude• ai, bi, index of the signal point

, M, , iTt

fbTEfa

TEtS ci

s

minci

s

mini

K21 0

t)2sin(2t)2cos(2)(

=≤≤

−= ππ

scs

TttfT

t ≤≤= 0)2cos(2)(1 πφ scs

TttfT

t ≤≤= 0)2sin(2)(2 πφ

1/31/200579

φ2(t)

φ1(t)

Constellation diagram of 16QAM

1/31/200580

Digital QAM Modulator

Serial/Parallel Map to 2-D constellation

Impulse modulator

Impulse modulator

Pulse shaping gT(t)

Local Oscillator

+90o

Pulse shaping gT(t)

d[n]an

bn

a*(t)

b*(t)

s(t)

1 l

MatchedDelay

Matched delay matches delay through 90o phase shifter

bit stream

1/31/200581

Phase Shift by 90 Degrees• 90o phase shift performed by Hilbert

transformer

cosine => sine

sine => – cosine

• Frequency response of idealHilbert transformer:

)(21)(

21) 2cos( 000 fffftf −++⇒ δδπ

)(2

)(2

) 2sin( 000 ffjffjtf −−+⇒ δδπ

)sgn()( fjfH −=

<−

=

>

=

0 if1

0 if0

0 if1

)sgn(

x

x

x

x

1/31/200582

Hilbert Transformer• Magnitude response

All pass except at origin

• For fc > 0

• Phase responsePiecewise constant

• For fc < 0

)2sin()2

2cos( tftf cc πππ =+

))(2sin()2

)(2cos(

))2

2(cos()2

2cos(

tftf

tftf

cc

cc

−=+−=

+−=−

ππ

π

ππ

ππ

f

)( fH∠

-90o

90o

f

|)(| fH

)sgn()( fjfH −=

1/31/200583

Hilbert Transformer• Continuous-time ideal

Hilbert transformer• Discrete-time ideal

Hilbert transformer

h(t) = 1/(π t) if t ≠ 0

0 if t = 0h[n] =

if n≠0

0 if n=0

nn )2/(sin2 2 π

π

Even-indexed samples are zero

t

h(t)

)sgn()( fjfH −= )sgn()( ωω jH −=

n

h[n]

1/31/200584

QAM Receiver• Channel has linear distortion, additive noise,

and nonlinear distortion• Adaptive digital FIR filter used to equalize

linear distortion (magnitude/phase distortion in channel)Channel equalizer coefficients adapted during

startupAt startup, transmitter sends known PN training

sequence

1/31/200585


Symbol/clock

recovery

Carrier recoveryAGC

90o

Phaseshift

Received signal


SignalPulse

generator

Detectdecision

cos(2πfct+φ)

-sin(2πfct+φ)

1/31/200586

In-Phase/Quadrature Demodulation• QAM transmit signal• QAM demodulation by modulation then filtering

– Construct in-phase i(t) and quadrature q(t) signals– Lowpass filter them to obtain baseband signals a(t) and b(t)

) sin( )() cos( )()( ttbttatx cc ωω +=

)cos()(2)( ttxti cω= )cos()sin()(2)(cos)(2 2 tttbtta ccc ωωω +=)2sin()()2cos()()( ttbttata cc ωω ++=

)sin()(2)( ttxtq cω= )(sin)(2)sin()cos()(2 2 ttbttta ccc ωωω +=)2cos()()2sin()()( ttbttatb cc ωω −+=

baseband high frequency component centered at 2 ωc

baseband high frequency component centered at 2 ωc

1/31/200587

Performance Analysis of QAM• Received QAM signal

• Information signal s(nT)

where i,k ∈ { -1, 0, 1, 2 } for 16-QAM• Noise, vI(nT) and vQ(nT) are independent

Gaussian random variables ~ N(0; σ2/T)

)()()( nTvnTsnTx +=

dkjdibjanTs nn )12( )12( )( −+−=+=

)( )()( nTvjnTvnTv QI +=

1/31/200588

Performance Analysis of QAM• Type 1 correct detection

))(&)(()(1 dnTvdnTvPcP QI <<=

))(())(( dnTvPdnTvP QI <<=

)))((1))()((1( dnTvPdnTvP QI >−>−=

2))(21( TdQσ

−=

3

3

3

3

2 2

2

2

2 2

2

211

1 1

I

Q

16-QAM

)(2 TdQσ

)(2 TdQσ

1/31/200589

Performance Analysis of QAM• Type 2 correct detection

• Type 3 correct detection

))(&)(()(2 dnTvdnTvPcP QI <<=

))(())(( dnTvPdnTvP QI <<=

))(1))((21( TdQTdQσσ

−−=

))(&)(()(3 dnTvdnTvPcP QI −><=

))(())(( dnTvPdnTvP QI −><=

2))(1( TdQσ

−=

3

3

3

3

2 2

2

2

2 2

2

211

1 1

I

Q

16-QAM

1/31/200590

Performance Analysis of QAM• Probability of correct detection

• Symbol error probability

22 ))(1(164))(21(

164)( TdQTdQcP

σσ−+−=

))(1))((21(168 TdQTdQ

σσ−−+

)(49)(31 2 TdQTdQ

σσ+−=

)(49)(3)(1)( 2 TdQTdQcPeP

σσ−=−=

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Average Power Analysis• PAM and QAM signals are deterministic• For a deterministic signal p(t), instantaneous power

is |p(t)|2• 4-PAM constellation points: { -3 d, -d, d, 3 d }

– Total power 9 d2 + d2 + d2 + 9 d2 = 20 d2

– Average power per symbol 5 d2

• 4-QAM constellation points: { d + j d, -d + j d,d – j d, -d – j d }– Total power 2 d2 + 2 d2 + 2 d2 + 2 d2 = 8 d2

– Average power per symbol 2 d2

1/31/200592

Summary of QAM.

{ }

+−−+−+−+−+−

−−−+−−+−−−−+−−+−

=

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

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

,

LLLLLL

LLLLLLLLLLLL

ba ii

L

L

L

{ }

−−−−−−−−−−−−

−−−−

=

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(

, ii ba

−≅

0

2114NEQ

MP min

e

−

−≅

0)1(3114

NMEQ

MP av

e

1/31/200593

– error probability under coherent detection

– error probability under non-coherent detection

M, 2, 1, 0 )(cos2)( K=≤≤

+= iTttin

TTEtS sc

ss

si

π

−≤

0

2log2)1(N

MEQMP be

+

−

−

+

−= ∑

−

=

+

0

1

1

1

)1(exp

11

)1(Nk

kEk

Mk

P sM

k

k

e

M-ary FSK

1/31/200594

• M_ary FSK– BW of coherent MFSK :

– BW of noncoherent MFSK :

MMRB b

2log2)3( +

=

MMRB b

2log2=

M-ary FSK

Modulation and Demodulation - Iowa State...

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