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Digital Modulation and Detection

Mobile CommunicationsDigital Modulation and Detection

Wen-Shen Wuen

Trans. Wireless Technology LaboratoryNational Chiao Tung University

WS Wuen Mobile Communications 1

Outline Digital Modulation and Detection

Outline

1 Structure of Wireless Communication Link

2 Analog Modulation TechniquesAnalog Modulation Techniques

3 Digital Modulation TechniquesIntroductionLine CodingPulse ShapingGeometric Representation of Modulation Signals

4 Linear Modulations

5 Constant Envelope Modulation

6 Combined Linear and Contstant Envelope Modulation Techniques


Structure of Wireless Communication Link Digital Modulation and Detection

Wireless Transceiver Block Diagram

informationsource

source

coderchannelcoder

modulatormultipleaccess

transmissiontechnique

RFtransmitter

propagationchannel

RFreceiver

diversitycombiner

separation ofdesired user

equalizerdemodulatorchanneldecoder

source

decoder

informationsink



Transmitter

+RX

AnalogSource

SourceADC

Sourcecoder

ChannelCoder

Multiplexer

Signaling

BasebandModulator

TransmitADC

Lowpass

Filter

Upconverter

LocalOscillator

TX filterChannel

NoiseACI, CCI



Receiver

FromTX

RFfilter

DownConverter

Lowpass

FilterReceiveADC

CarrierRecovery

BasebandDemodulator

De/MUX

TimingRecovery

Signaling

ChannelDecoder

SourceDecoder

LocalOscillator

SourceDAC

InformationSink



Descriptions of Wireless Transceiver Block Diagram

Information source: digital or analog signals.

Source coder: reducing redundancy in the source signal toincrease entropy (information per bit) e.g., compression, orensuring security by encryption.

Channel coder: adding redundancy to protect data againsttransmission errors, interleaving to break up error bursts.

Signaling: adding control information for the establishing andending of connections, synchronizations and user authorizedinformation. Signaling information is usually strongly protectedby error correction codes.

Multiplexer: combines user data and signaling information

Baseband modulator: assigns the gross data bits to complextransmit symbols and determines spectral properties,intersymbol interferences, peak-to-average ratio, etc.



Descriptions of Wireless Transceiver Block Diagram,

cont’d

Carrier recovery: determines the frequency and phase of thecarrier of the received signal.

Baseband demodulator: obtains soft-decision data fromdigitized baseband data and may also include equaliztion.

Symbol timing recovery: uses demodulated data to determinean estimate of the duration of symbols and use it to fine tunesampling intervals.

Decoders: use soft estimates from the demodulator to find theoriginal digital source data. Recent RX may perform jointdemodulation and decoding.

Demultiplexer: separates the user data and signalinginformation.


Analog Modulation Techniques Digital Modulation and Detection

Introduction

Modulation

The process by which some characteristic of a carrier wave isvaried in accordance with an information-bearing signal.

Modulating signal: information bearing signal

Modulated signal: the output of the modulation process

Linear modulation: if the input-output relation of the modulatorsatisfies the principle of superposition.

Benefits of modulation

Modulation used to shift the spectral content of a messagesignal so that it lies inside the operating frequency band of thewireless communication channel.

Modulation provides a mechanism for putting the informationcontent of a message signal into a form that may be lessvulnerable to noise or interference.

Modulation permits the use of multiple access.



Introduction, cont’d

Analog and Digital Modulation Techniques

Analog modulation: m(t) and s(t) are continuous function oftime

Digital modulation: mapping of data bits to signal waveformsthat can be transmitted over an analog channel.

modulatorm(t) s(t)

Sinusoidal

Inputmodulating

signal

Outputmodulated

signal

carrierc(t)



Amplitude and Angle Modulation

Amplitude modulation, AMthe amplitude of the carrier, A(t), is varied with linearly with themessage signal m(t)

Angle modulationthe angle of the carrier,

φ(t) = 2πfct +θ

is varied linearly with the message signal m(t)

Frequency Modulation, FMthe frequency of the carrier is varied linearly with the messagesignal m(t)Phase Modulation, PMthe phase of the carrier θ is varied linearly with the messagesignal m(t)



Amplitude Modulation

AM signal

sAM (t) = Ac (1+m(t))cos(2πfct

)=Re{

g(t)ej2πfct}

where g(t) = Ac (1+m(t))Modulation index: the ratio of peak message signal amplitudeto the peak carrier amplitude.If m(t) = Am

Accos

(2πfmt

), k = Am

Ac

Spectrum of AM signal

SAM (f ) = 1

2Ac

(δ

(f − fc

)+M(f − fc

)+δ(f + fc

)+M(f + fc

))RF bandwidth of AM

BAM = 2fm

Total power of AM signal

PAM = 1

2A2

c (1+< m(t) >)2

if m(t) = k cos(2πfmt

)⇒ PAM = 12 A2

c

(1+ k2

2

)WS Wuen Mobile Communications 13


Amplitude Modulation, cont’d



Amplitude Modulation, cont’d



Angle Modulation

FM signal

sFM (t) = Ac cos[2πfct +θ(t)

]= Ac cos

[2πfct +2πkf

∫ t

−∞m(τ)dτ

]

if m(t) = Am cos(2πfmt

)⇒ sFM (t) = Ac cos[

2πfct + kf Am

fmsin

(2πfmt

)]Frequency modulation index βf : defines the relation betweenthe message amplitude and the bandwidth of the transmittedsignal.

βf =kf Am

W= ∆f

W

PM signalsPM (t) = Ac cos

[2πfct +kθm(t)

]Phase modulation index βp

βp = kθAm =∆θ



FM Demodulation: Slope Detection

v1(t) = V1 cos[2πfct +θ(t)

]= V1 cos

[2πfct +2πkf

∫ t

−∞m(τ)dτ

]v2(t) =−V1

[2πfct + dθ

dt

]sin

(2πfct +θ(t)

)vout (t) = V1

[2πfct + dθ

dt

]= V12πfc +V12πkf m(t)



FM Demodulation: Quadrature Detection



FM Demodulation: Zero-Crossing



Bandwidth of FM Signals

Carson’s rule

An FM signal has 98% of the total transmitted power in a RFbandwidth BT is

BT = 2(βf +1

)fm (Upper bound)

BT = 2∆f (Lower bound)

Carson’s rule: for small values of modulation index (βf < 1), thespectrum of an FM wave is effectively limited to the carrierfrequency, and one pair of sideband frequencies at fc ± fm. Forlarge values of modulation index, the bandwidth approachesand is only slightly greater than 2∆f .



Tradeoff Between SNR and Bandwidth in FM

SNR at the output of an FM receiver

SNRout = 6(βf +1

)β2

f

(m(t)

Vp

)2

SNRin

SNR at the input of an FM receiver

SNRin,FM = A2c /2

2N0(βf +1

)B

SNR at the input of an AM receiver

SNRin,AM = A2c

2N0B

for m(t) = Am sinωmt

SNRout = 3β2f

(βf +1

)SNRin,FM = 3β2

f SNRin,AM

FM offers excellent performance for fading signals.



Example 1

How much bandwidth is required for an analog frequencymodulated signal that has an audio bandwidth of 5 kHz and amodulation index of three? How much output SNR improvementwould be obtained if the modulation index is increased to five?What is the tradeoff bandwidth for the improvement?Solution:BT = 2

(βf +1

)fm = 2(3+1)5 = 40 kHz.

The output SNR improvement factor is 3β2f (βf +1)

βf = 3 ⇒ 3 ·32(3+1) = 108 = 20.33 dB.βf = 5 ⇒ 3 ·52(5+1) = 450 = 26.53 dB. The improvement of output SNRis 26.53−20.33 = 6.2 dB. For βf = 5, the required bandwidth is 60 kHz.


Digital Modulation Techniques Digital Modulation and Detection

Digital Modulation Techniques

Advantages of Digital Modulation

greater noise immunity

robustness to channel impairments

easier multiplexing of various forms of information

greater security

Factors that influence the choice of digital modulation

Bandwidth efficiency

Adjacent channel interference

Sensitivity with respect to noise

Robustness with respect to delay and Doppler dispersion

Easy and cost-effective to implement



Power Efficiency and Bandwidth Efficiency

Power efficiency (Energy efficiency), ηp

The ability of a modulation technique to preserve the fidelity ofthe digital message at low power levels.

Expressed as the ratio of the signal energy per bit to noisepower spectral density (Eb/N0) required at the receiver input fora certain probability of error.

Bandwidth efficiency, ηB

The ability of a modulation scheme to accommodate datawithin a limited bandwidth.

Defined as the ratio of the throughput data rate per Hertz in agiven bandwidth.

ηB = R

Bbps/Hz



Shannon’s Channel Capacity Theorem

For an arbitrary small probability of error, the maximum possiblebandwidth efficiency is limited by the noise in the channel,

ηB,max = C

B= log2

(1+ S

N

)whereC is the channel capacity in bps,B is the RF bandwidth, andSN is the signal to noise ratio.



Example 2

If the SNR of a wireless communication link is 20 dB and the RF bandwidthis 30 kHz, determine the maximum theoretical data rate that can betransmitted.Solution:SNR=20 dB=100, Bandwidth B=30000 Hz,C = B log2 (1+S/N) = 30000log2(1+100) = 199.75 kbps

Example 3

What is the theoretical maximum data rate that can be supported in a200 kHz channel for SNR=10 dB and 30 dB? How does this compare to theGSM standard?Solution:SNR=10 dB=10, B=200 kHz,C = B log2 (1+S/N) = 200000log2(1+10) = 691.886 kbpsGSM data rate is 270.833 kbps, which is about 40 % of the theoretical limitfor 10 dB SNR conditions.For SNR=30 dB=1000, B=200 kHz,C = B log2 (1+S/N) = 200000log2(1+1000) = 1.99 Mbps



Bandwidth and Power Spectral Density of Digital

Signals

Various bandwidth definitions are based on the power spectraldensity of the signal.

Power spectral density (PSD) of a random signal w(t)

Pw(f ) = limT→∞

∣∣WT (f )∣∣2

T

where WT (f ) is the Fourier transform of wT (t) which is thetruncated version of the signal w(t)

wT (t) ={

w(t) −T/2 < t < T/20 otherwise

Power spectral density of a modulated (bandpass) signals(t) =Re

{g(t)ej2πfct

}Ps(f ) = 1

4

[Pg (f − fc)+Pg (−f − fc)

]where Pg (f ) is the PSD of g(t).



Bandwidth and Power Spectral Density of Digital

Signals, cont’d

Absolute bandwidth: the range of frequencies over which thesignal has a non-zero power spectral density.

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

Half-power bandwidth (3-dB bandwidth): the interval betweenfrequencies at which the PSD has dropped to half power

Occupied bandwidth (FCC): the band which leaves exactly 0.5%of the signal above the upper band limit and exactly 0.5% ofthe signal power below the lower band limit, i.e. 99% of thesignal power is contained within the occupied bandwidth



Line Coding

Mapping of binary information sequence into the digital signal thatenters the channel.



Properties of Line Codes

Self-Synchronization There is enough timing informationbuilt into the code so that bit synchronizers can extract thetiming or clock signal.

Low Probability of Bit Error Receivers can be designed thatwill recover the binary data with a low probability of bit errorwhen the input data is corrupted by noise or ISI.

A Spectrum that is Suitable for the Channel For example,if the channel is AC coupled, the PSD of the line code signalshould not have DC component. In addition, the signalbandwidth needs to be sufficiently small compared to thechannel bandwidth, so that ISI will not be a problem.

Transmission Bandwidth This should be as small as possible.

Error Detection Capability It should be possible toimplement this feature easily by the addition of channelencoders and decoders, or the feature should be incorporatedinto the line code.



Power Spectral Density of Line Codes



The Need for Pulse Shaping

The most simple basis pulse is a rectangular pulse with duration T .

gR(t,T) ={

1 0 ≤ t ≤ T0 otherwise

GR(f ,T) = F{gR(t,T)

}= Tsinc(πfT

)e−jπfT

where sinc(x) = sin(x)/x.

When rectangular pulses are passed through a bandlimitedchannel, the pulses will spread in time, and the pulse for eachsymbol will smear into time intervals of succeeding symbols. ⇒inter-symbol interference (ISI)

In frequency domain the rectangular pulse creates largeadjacent channel interference. ⇒ the transmitting pulse shouldbe shaped to limit the bandwidth as well as to minimize ISI.



Nyquist Criterion for ISI Cancellation

The effect of ISI could be completely nullified if the overall responseof the communication system is designed so that at every samplinginstant at the receiver, the response due to all symbols except thecurrent symbol is equal to zero.

Impulse response of the overall communication system, heff (t)

heff (nTs) ={

K n = 00 n 6= 0

heff (t) = δ(t)∗p(t)∗hc(t)∗hr(t)

heff (t) should have a fast decay with a small magnitude nearthe sample values for n 6= 0



Nyquist Ideal Pulse Shaping

heff (t) =sin

(πtTs

)πtTs

= sinc

(πt

Ts

),Heff (f ) = 1

fsrect

(f

fs

)where fs is the symbol rate.Ideal Nyquist pulse shaping filter is difficult to implement

Noncausal system (heff (t) exists for t < 0) and is difficult toapproximate

Sensitive to jitter (error in sampling time of zero-crossings willcause significant ISI)



Nyquist Filter

Nyquist proved that any filter with a transfer function having arectangular filter of bandwidth f0 ≥ 1/2Ts, convolved with anyarbitrary even function Z(f ) with zero magnitude outside thepassband, satisfied the zero ISI condition.

Heff (f ) = rect

(f

f0

)∗Z(f )

where Z(f ) = Z(−f ) and Z(f ) = 0 for |f | ≥ f0 ≥ 1/2Ts.

Nyquist criterion

Any filter with an impulse response

heff (t) = sin(πt/Ts)

πtz(t)

can achieve ISI cancellation. Filters which satisfies the Nyquistcriterion are called Nyquist filters.



Nyquist Filter, cont’d

Heff is often achieved by using filters with transfer function√

Heff atboth the transmitter and receiver.⇒ providing a matched filter response for the system while at thesame time minimizing the bandwidth and ISI.



Raised Cosine Rolloff Filter

HRC (f ) =

1 0 ≤ |f | ≤ 1−α

2Ts12

[1+cos

(π(2|f |Ts−1+α)

2α

)]1−α2Ts

≤ |f | ≤ 1+α2Ts

0 |f | > 1+α2Ts

where α is the rolloff factor which ranges between 0 and 1.



Raised Cosine Rolloff Filter, cont’d

hRC (t) =sin

(πtTs

)πt

cos(παtTs

)1−

(4αt2Ts

)2

The symbol rate Rs that can be passed through a baseband raisedcosine rolloff filter is

Rs = 1

Ts= 2B

1+αwhere B is the absolute filter bandwidth.



Gaussian Pulse-Shaping Filter

Gaussian pulse-shaping filter is particularly effective when used inconjunction with minimum shift keying (MSK) modulation or othermodulation well suited for power efficient nonlinear amplifiers.

Transfer functionHG(f ) = e−α

2f 2

where α is related to B the 3-dB bandwidth of the basebandGaussian shaping filter,

α=p

ln2p2B

= 0.5887

B

Impulse response

hG(t) =pπ

αe−

π2

α2 t2

Gaussian pulse-shaping filter does not satisfy the Nyquistcriterion for ISI cancellation, reducing the spectral occupancycreates degradation in performance due to increased ISI.



Gaussian Pulse-Shaping Filter, cont’d

Impulse response of Gaussian pulse-shaping filter



Example 4

Find the first zero-crossing RF bandwidth of a rectangular pulsewhich has Ts = 41.06 µs. Compare this to the bandwidth of a raisedcosine filter pulse with Ts = 41.06 µs and α= 0.35.Solution:The first zero-crossing filter (null-to-null) bandwidth of a rectangularpulse is equal to

2

Ts= 2

41.06×10−6= 48.71 kHz

and that for a raised cosine filter with α= 0.35 is

1

Ts(1+α) = 1

41.06×10−6(1+0.35) = 32.88 kHz



Geometric Representation of Modulation Signals

M-ary modulation signals

Choosing a particular signal waveform si(t) from a finite set ofpossible signal waveforms (or symbols) based on theinformation bits applied to the modulator.

S = {s1(t),s2(t), . . . ,sM (t)}

where M represents a total of M possible signals in themodulation set S. ⇒ a maximum of log2 M bits of informationper symbol.

Represent the modulation signals on a vector space with a setof N orthonormal signals that form a basis for the vector space.Any point in the vector space can be represented as a linearcombination of the basis signals.



Signal Space for Modulation Signals

Basis {φ1(t),φ2(t), . . . ,φN (t)

}Modulation Signals

si(t) =N∑

j=1sijφj(t).

Basis signals are orthogonal to one another in time∫ ∞

−∞φi(t)φj(t)dt = 0 i 6= j

Each basis signal is normalized to have unit energy

E =∫ ∞

−∞φ2

i (t)dt = 1



Signal-Space for Digital Modulation

Consider two-dimensional signal constellations, the normalizedbasis of unite energy are φ1(t) and φ2(t)

φ1(t) =√

2

Tcos(2πfct), 0 ≤ t ≤ T

φ2(t) =√

2

Tsin(2πfct), 0 ≤ t ≤ T

where ∫ T

0φ1(t)φ2(t)dt = 0

and ∫ T

0φ2

1(t)dt =∫ T

0φ2

2(t)dt = 1

si(t) = si1φ1(t)+ si2φ2(t) or si = (si1,si2)

||si −sk|| =√√√√ N∑

j=1

(sij − skj

)2 =√∫ T

0(si(t)− sk(t))2 dt



Example: BPSK Signals

s1(t) =√

2Eb

Tbcos(2πfct), 0 ≤ t ≤ Tb

s2(t) =−√

2Eb


where Eb is the energy per bit, Tb is the bit period, and a rectangularpulse p(t) = rect((t −Tb/2)/Tb) is assumed.

φ1(t) =√

2


BPSK signal set

SBPSK ={√

Ebφ1(t),−√

Ebφ1(t)}



Constellation



Error Probability

Upper bound for the probability of symbol error in an additive whiteGaussian noise channel (AWGN) with a noise density N0 for anarbitrary constellation

The average probability of error for a particular modulationsignal

Ps(ε|si) ≤∑

j=1,j 6=iQ

(dijp2N0

)where dij is the Euclidean distance between the i-th and j-thsignal point in the constellation.

If all the M modulation waveforms are equally likely to betransmitted, the average probability of error for a modulation

Ps(ε) = 1

M

M∑i=1

Ps(ε|si)P(si) = 1

M

M∑i=1

Ps(ε|si)


Linear Modulations Digital Modulation and Detection

Binary Phase Shift Keying (BPSK)

sinusoidal carrier amplitude Ac, Energy per bit Eb = 12 A2

c Tb

sBPSK (t) =√

2Eb

Tbcos(2πfct +θc) for binary 1

sBPSK (t) =−√

2Eb

Tbcos(2πfct +θc) for binary 0

or

sBPSK (t) = m(t)

√2Eb

Tbcos(2πfct +θc)

orsBPSK (t) =Re

{gBPSK (t)ej2πfct

}where

gBPSK (t) =√

2Eb

Tbm(t)ejθc



Spectrum and BPSK Bandwidth

Power spectral density of the complex envelope gBPSK (t)

PgBPSK (f ) = 2Eb

(sinπfTb

πfTb

)2

PSD of a BPSK signal at RF

PBPSK (f ) = Eb

2

[(sinπ(f − fc)Tb

π(f − fc)Tb

)2

+(

sinπ(−f − fc)Tb

π(−f − fc)Tb

)2]



BPSK Receiver

If no multipath impairments are introduced by the channel, thereceived BPSK signal

rBPSK (t) = m(t)

√2Eb

Tbcos(2πfct +θc +θch) = m(t)

√2Eb

Tbcos(2πfct +θ)

The output of the multiplier after the frequency divider

m(t)

√2Eb

Tbcos2(2πfct +θ) = m(t)

√2Eb

Tb

[1

2+ 1

2cos(4πfct +2θ)

]Probability of bit error

Pe,BPSK = Q

(√2Eb

N0

)



BPSK Receiver, cont’d



Differential Phase Shift Keying (DPSK)

Noncoherent form of phase shift keying ⇒ avoids the need for acoherent reference signal at the receiver.Input binary sequence is first differentially encoded and thenmodulated using BPSK modulator.Differentially encode: dk = mk ⊕dk−1. Leave dk unchanged fromthe previous symbol if the incoming binary symbol mk is 1 andto toggle dk if mk is 0.

mk 1 0 0 1 0 1 1 0dk−1 1 1 0 1 1 0 0 0

dk 1 1 0 1 1 0 0 0 1



Differential Phase Shift Keying (DPSK), cont’d

DPSK signaling has about 3 dB worst energy efficency thancoherent PSK.

Average probability of error for DPSK in AWGN channel is

Pe,DPSK = 1

2e− Eb

N0



Quadrature Phase Shift Keying (QPSK)

QPSK has twice the bandwidth efficiency of BPSK, since two bits aretransmitted in a single modulation symbol.

SQPSK (t) =√

2Es

Tscos

[2πfct + (i−1)

π

2

],0 ≤ t ≤ Ts, i = 1,2,3,4

where Ts = 2Tb. or

SQPSK (t) =√

2Es

Tscos

[(i−1)

π

2

]cos(2πfct)−

√2Es

Tssin

[(i−1)

π

2

]sin(2πfct)

if φ1 =p

2/Ts cos(2πfct) and φ2 =p

2/Ts sin(2πfct) are defined over0 ≤ t ≤ Ts,

SQPSK (t) ={√

Es cos[

(i−1)π

2

]φ1(t)−

√Es sin

[(i−1)

π

2

]φ2(t)

}i = 1,2,3,4



Quadrature Phase Shift Keying (QPSK), cont’d

The average probability of bit error in the AWGN channel is

Pe,QPSK = Q

( p2Esp2N0

)= Q

(√2Eb

N0

)



QPSK Power Spectral Density

The PSD of a QPSK signal using rectangular pulses can be expressedas

PQPSK (f ) = Es

2

[(sinπ(f − fc)Ts

π(f − fc)Ts

)2

+(

sinπ(−f − fc)Ts

π(−f − fc)Ts

)2]or

PQPSK (f ) = Eb

[(sin2π(f − fc)Tb

2π(f − fc)Tb

)2

+(

sin2π(−f − fc)Tb

2π(−f − fc)Tb

)2]



QPSK Transmission and Detection

QPSK Transmitter



QPSK Transmission and Detection, cont’d

QPSK Receiver



Offset QPSK

To ensure fewer baseband signal transitions applied to the RFamplifier ⇒ supports more efficient amplification and helpseliminate spectrum regrowthEven and odd bit streams, mI (t) and mQ(t) are offset in theirrelative alignment by one bit period (half-symbol period.)Only one of the two bit streams can change values ⇒ maximumphase shift of the signal is limited to ± 90◦



QPSK v.s. Offset QPSK



π/4 QPSK

Maximum phase change is limited to ± 135◦

Switching between two constellations, every successive bitensures that there is at least a phase shift nπ

4 for every symbol



π/4 QPSK Transmission Techniques

Ik = cosθk = Ik−1 cosφk −Qk−1 sinφk

Qk = sinθk = Ik−1 sinφk +Qk−1 cosφk

where θk = θk−1 +φk.

sπ/4QPSK (t) = I(t)cos2πfct −Q(t)sin2πfct

where

I(t) =N−1∑k=0

Ikp(t −kTs −Ts/2) =N−1∑k=0

cosθkp(t −kTs −Ts/2)

Q(t) =N−1∑k=0

Qkp(t −kTs −Ts/2) =N−1∑k=0

sinθkp(t −kTs −Ts/2)

and the peak amplitude of I(t),Q(t) can take one of the five values{0,1,−1, 1p

2,− 1p

2

}WS Wuen Mobile Communications 64


π/4 QPSK Transmission Techniques, cont’d

information bits mIk ,mQk Phase shift φk11 π/401 3π/400 −3π/410 −π/4



π/4 QPSK Detection Techniques

Baseband Differential Detection



π/4 QPSK Detection Techniques, cont’d

Baseband Differential Detection

wk = cos(φk −γ),zk = sin(φk −γ)

xk = wkwk−1 +zkzk−1,yk = zkwk−1 −wkzk−1

The output of differential decoder,

xk = cos(φk −γ)cos(φk−1 −γ)+ sin(φk −γ)sin(φk−1 −γ) = cos(φk −φk−1)

yk = sin(φk −γ)sin(φk−1 −γ)+cos(φk −γ)cos(φk−1 −γ) = sin(φk −φk−1)

The output of the decision device,

SI = 1, if xk > 0 or SI = 0, if xk < 0

SQ = 1, if yk > 0 or SQ = 0, if yk < 0




IF Differential Detector




FM Discriminator


Constant Envelope Modulation Digital Modulation and Detection

Nonlinear Modulation

Nonlinear modulation: the amplitude of the carrier is constant,regardless of the variation in the modulating signal

Advantages

Efficient power amplifier can be used ⇒ without introducingdegradation in the spectrum occupancy of the transmitted signal.Low out-of-band radiation of the order of −60 dB to −70 dB can beachieved.Limiter-discriminator detection can be used ⇒ simplifyingreceiver designs and providing high immunity against random FMnoise and signal fluctuation due to Rayleigh fading.



Binary Frequency Shift Keying (BFSK)

sFSK (t) = vH (t) =√

2Eb

Tbcos

(2πfc +2π∆f

)t, 0 ≤ t ≤ Tb (binary 1)

sFSK (t) = vL(t) =√

2Eb

Tbcos

(2πfc −2π∆f

)t, 0 ≤ t ≤ Tb (binary 0)

Discontinuous FSK

sFSK (t) = vH (t) =√

2Eb

Tbcos

(2πfH t +θ1

), 0 ≤ t ≤ Tb (binary 1)

sFSK (t) = vL(t) =√

2Eb

Tbcos

(2πfLt +θ2

), 0 ≤ t ≤ Tb (binary 0)

Continuous FSK

sFSK (t) =√

2Eb

Tbcos

(2πfct +θ(t)

)=√2Eb

Tbcos

(2πfct +2πkf

∫ t

−∞m(τ)dτ

)WS Wuen Mobile Communications 72


BPSK, cont’d

Transmission bandwidth BT of an FSK signal BT = 2∆f +2B

Rectangular pulses is B = R ⇒ BT = 2(∆f +R)

Raised cosine pulse-shaping filter BT = 2∆f + (1+α)R

Coherent Detection of Binary FSK

Pe,FSK = Q

(Eb

N0

)



Noncoherent Detection of Binary FSK

Noncoherent Detection of Binary FSK

Pe,FSK ,NC = 1

2e− Eb

2N0



Minimum Shift Keying (MSK)

A special type of continuous phase frequency shift keying(CPFSK) wherein the peak frequency deviation is equal to 1/4the bit rate.

Modulation index: kFSK = (2∆F)/Rb, where ∆F is the peak RFfrequency deviation and Rb is the bit rate.

Minimum shift keying: the minimum frequency separation (i.e.,bandwidth) that allows orthogonal detection.

MSK signal can be thought of as a special form of OQPSK wherethe base band rectangular pulses are replaced withhalf-sinusoidal pulses.

SMSK (t) =N−1∑i=0

mIi (t)p(t−2iTb)cos2πfct+N−1∑i=0

mQi (t)p(t−2iTb−Tb)sin2πfct

where

p(t) ={

cos(πt

2Tb

)0 ≤ t ≤ 2Tb

0 elsewhere



Minimum Shift Keying (MSK), cont’d

MSK signal can be seen as a special type of a continuous phaseFSK

SMSK (t) =√

2Eb

Tbcos

[2πfct = mIi (t)mQi (t)

πt

2Tb+φk

]where phik is 0 or π depending on whether mIi (t) is 1 or −1.

Phase continuity at the bit transition period is ensured bychoosing the carrier frequency to be an integral multiple ofRb/4 = 1/(4T)

MSK signal is an FSK signal with binary signaling frequencies offc + 1

4T and fc − 14T .

MSK signal varies linearly during the course of each bit period.



MSK Power Spectrum Density

The baseband pulse shaping function for MSK is

p(t) ={

cos(πt2T

) |t| < T0 elsewhere

PMSK (f ) = 16

π2

(cos2π(f + fc)T

1.16f 2T 2

)2

+ 16

π2

(cos2π(f − fc)T

1.16f 2T 2

)2



MSK Transmitter



MSK Receiver



Gaussian Minimum Shift Keying (GMSK)

Passing the modulating NRZ data waveform through apremodulation Gaussian pulse-shaping filterConsiderably reducing the sidelobe levels in the transmittedspectrumExcellent power efficiency (due to the constant envelope) andspectrum efficiencyISI degradation is not sever if the 3-dB bandwidth bit durationproduct BT > 0.5.As long as GMSK irreducible error rate is less than thatproduced by the mobile channel, no penalty in using GMSK.

GMSK premodulation filter impulse response

hG(t) =pπ

αe−

π2

α2 t2

GMSK premodulation filter transfer function

HG(f ) = e−α2f 2

α=p

ln2p2B

= 0.5887

B



Gaussian Minimum Shift Keying (GMSK), cont’d

GMSK bit error rate

Pe = Q

(√2γEb

N0

)where

γ'{

0.68 GMSK with BT = 0.250.85 MSK (BT =∞)



GMSK Transmitter and Receiver



Example 5

Find the 3-dB bandwidth for a Gaussian low pass filter used toproduce 0.25 GMSK with a channel data rate of Rb = 270 kbps. Whatis the 90% power bandwidth in the RF channel? Specify theGuassian filter parameter α.Solution:

T = 1

Rb= 1

270×103 = 3.7µs

∵ BT = 0.25 ∴ B = 0.25

T= 0.25

3.7×10−6 = 67.567kHz

RF bandwidth

BW = 0.57Rb = 0.57×270×103 = 153.9kHz


Combined Linear and Contstant Envelope Modulation Techniques Digital Modulation and Detection

M-ary Phase Shift Keying (MPSK)

The carrier phase takes on one of M possible values

θi = 2π(i−1)

M, i = 1,2, . . . ,M

MPSK signal waveform

si(t) =√

2Es

Tscos

(2πfct + 2π

M(i−1)

),0 ≤ t ≤ Ts, i = 1,2, . . . ,M

Es: Energy per symbol, Es = (log2 M)Eb

Ts: symbol period, Ts = (log2 M)Tb

MPSK in quadrature form

si(t) =√

2Es

Tscos

(2π

M(i−1)

)cos(2πfct)−

√2Es

Tssin

(2π

M(i−1)

)sin(2πfct)

Letφ1(t) =

√2/Ts cos(2πfct),φ2(t) =

√2/Ts sin(2πfct),

SMPSK (t) ={√

Es cos

(2π

M(i−1)

),−

√Es sin

(2π

M(i−1)

)}WS Wuen Mobile Communications 85


Error Probability of M-ary PSK

Distance between adjacent symbols: 2p

Es sin(πM

)Average symbol error probability of an M-ary PSK is

Pe ≤ 2Q

(√2Eb log2 M

N0sin

( πM

))Average symbol error probability of a differential M-ary PSK inAWGN channel for M ≥ 4 is approximately

Pe ≈ 2Q

(√4Es

N0sin

( πM

))



Power Spectra of M-ary PSK

PSD of the M-ary PSK signal with rectangular pulses is

PMPSK = Es

2

[(sinπ(f − fc)Ts

π(f − fc)Ts

)2

+(

sinπ(−f − fc)Ts

π(−f − fc)Ts

)2]

PMPSK = Eb log2 M

2

[(sinπ(f − fc)Tb log2 M

π(f − fc)Tb log2 M

)2

+(

sinπ(−f − fc)Tb log2 M

π(−f − fc)Tb log2 M

)2]



Power Spectra of M-ary PSK, cont’d



M-ary Quadrature Amplitude Modulation (QAM)

si(t) =√

2Emin

Tsai cos(2πfct)+

√2Emin

Tsbi sin(2πfct),

0 ≤ t ≤ T , i = 1,2, . . . ,M

where Emin is the energy of the signal with the lowest amplitude andthe coordinate of the ith point are ai

pEmin and bi

pEmin.



M-ary Quadrature Amplitude Modulation, cont’d

(ai,bi) is an element of the L×L matrix

{ai,bi} =

(−L+1,L−1) (−L+3,L−1) · · · (L−1,L−1)(−L+1,L−3) (−L+3,L−3) · · · (L−1,L−3)

......

. . ....

(−L+1,−L+1) (−L+3,−L+1) · · · (L−1,−L+1)

where L =p

M

16-QAM: L =p16 = 4 4×4 matrix

{ai,bi} =

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

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



M-ary QAM Error Probability

Average probability of error in AWGN channel for M-ary QAM,using coherent detection

Pe ≈ 4

(1− 1p

M

)Q

(√2Emin

N0

)

In terms of average signal energy Eav

Pe ≈ 4

(1− 1p

M

)Q

(√3Eav

(M −1)N0

)



M-ary Frequency Shift Keying (MFSK)

M-ary FSK signal

si(t) =√

2Es

Tscos

[π

Ts(nc + i)t

]0 ≤ t ≤ Ts, i = 1,2, . . . ,M

where fc = nc/2Ts for some integer nc

The M transmitted signals are of equal energy and equalduration and the signal frequencies are separated by 1/2Ts Hz.For coherent M-ary FSK, the optimum receiver consists of abank of M correlators or matched filters tuned to the M distinctcarriers.⇒ Average probability of error

Pe ≤ (M −1)Q

(√Eb log2 M

N0

)For non-coherent detection using matched filters followed byenvelope detectors, the average probability of error

Pe =M−1∑k=1

((−1)k+1

k+1

(M −1

k

)e

−kEs(k+1)N0

)and using only the leading terms of the binomial expansion, theprobability of error is bounded as

Pe ≤ M −1

2e− Es

2N0



M-ary Frequency Shift Keying (MFSK), cont’d

Channel bandwidth of a coherent M-ary FSK signal

B = Rb(M +3)

2log2 M

Channel bandwidth of a non-coherent M-ary FSK

B = RbM

2log2 M


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