Ch6 - fju.edu.t · Page 2 Wireless Communication Chapter 6 –Modulation techniques for Mobile...

Chapter 6Modulation Techniques for Mobile Radio

Voltage

Time

Voltage

Time1 0 1 0

1 0 1 0

1 0 1 0

1 0 1 0

Wireless Communication

Chapter 6 –Modulation techniques for Mobile Radio 1 Dr. Sheng-Chou Lin

Modulation by Analog Inputs

For example, let’s use this analogwaveform to modulate a signal.

The basic, unchanging, steady radiosignal without modulation is called a“carrier”Characteristics of the carrierwhich we could modulate:

Amplitude (i.e., strength)example: AM radio broadcasting

FrequencyFM broadcasting,Voice transmissionin AMPS cellular

Phase

Modulation is the process of varying some characteristic of aradio signal in order to convey informationVoltage

Time

Notice thatfrequency and

phase modulationlook very similarwith this kind of

input.



Modulation and Occupied BandwidthThe bandwidth occupied by a signal depends

on:•input information bandwidth•modulation method

Information to be transmitted, called“input”or “baseband”•bandwidth usually is small, much lower

than frequency of carrier Unmodulated carrier

•the carrier itself has Zero bandwidth!! AM-modulated carrier

•Notice the upper & lower sidebands•total bandwidth = 2 x baseband

FM-modulated carrier•Many sidebands! bandwidth is a complex

mathematical function PM-modulated carrier

•Many sidebands! bandwidth is a complexmathematical function

Voltage

Time

Time-Domain(as viewed on an

Oscilloscope)

Frequency-Domain(as viewed on a

Spectrum Analyzer)

Voltage

Frequency0

fc

fc

UpperSideband

LowerSideband

fc

fc



Advanced Modulation ConceptsPerformance of different Modulation Types

Each type of modulation has advantages and drawbacks: Necessary bandwidth

•How wide is the signal? How much spectrum is needed?•How big a “guard band”is needed between channels?

Relative vulnerability to interference•What C/I ratio is required for good system performance?

Relative difficulty of implementation•Is complex equipment required?•Is it costly to implement?•Is it hard to maintain and adjust?

Let’s explore the different modulation methods used inmodern mobile telephony.



A Closer Look at Analog FMHow Much Bandwidth is Required?

Carson RuleBandwidth Required = 2 x (highest input frequency + frequency deviation)

As time passes, the carrier moves backand forth in frequency in exact stepwith the input signal

frequency deviation is proportional to theinput signal voltage

a group of many sidebands is created,spaced from carrier by amounts N x fi

relative strength of each sidebandNdepends on Bessel function of (inputsignal freq./freq. deviation)

strength of individual sidebands faraway from the carrier is proportional to(freq. deviation x input frequency)

Voltage

Frequencyfc

Sidebandsfc+fifc+2fi

fc+3fifc-3fi

fc-fi

fc-2fi

Voltage

Input Signal

fc

frequencydeviation

Voltage

Time

CarrierFrequency

fi =input signalfrequency



Analog FM is used onAMPS Cellular Voice Channels

Two signals simultaneously modulatethe AMPS cellular voice channel:

User’s voice waveform•complex, many frequencies approx.

300Hz to 3500 Hz.•peak deviation limited to +/- 12 KHz

Supervisory Audio Tone (“SAT”)•tone frequency 5970, 6000, or 6030 Hz.•peak deviation set as +/- 2.0 KHz.

The resulting composite FM signal fitswithin the assigned 30 KHz.-widechannel

Signaling Tone at 10 KHz with +/- 8 kHz.deviation is also transmitted inoccasional bursts for call control

VoltageTime

Time-Domain(as viewed on an

Oscilloscope)

Frequency-Domain(as viewed on a

Spectrum Analyzer)

VoltageFrequency

0

fc

VoltageFrequency

0

Voltage

Time

Voltage

30 KHz. ChannelVoltage

6KHz

Voice

SAT



S/N COMPARISON OF BROADCAST AMWITH FM

CNR RATIO (dB) re AM RECEVER

BA

SE

BA

ND

S/N

RA

TIO

(dB

)

0

10

20

30

40

50

0 10 20 30 40 50

*WIDEBAND FM

NARROWBAND FM

*AM

FM CAPTURETHRESHOLD

* measurements from M.G. Crosby, frequency modulation noise characteristics, PROC. IRE, Vol. 25, pp. 472-514, Fig. 10,April 1937

S/N gains from emphasis are included for the wideband fm curve



Analog Frequency Modulation

Can be very resistant to natural or man-made interference, if the RFbandwidth is much larger than the audio bandwidth. FM in analogcellular is 30 kHz bandwidth for a nominal 5 kHz audio bandwidth.

Received RF signal can be amplitude clipped to remove pulses of noisebigger than the desired signal.

Co-channel interference will produce a very high frequency audiooutput, related to the instantaneous frequency difference between thedesired and interfering signal. This can be removed by a low-passaudio filter.

FM has the "capture phenomenon". When a desired FM signal issufficiently stronger than FM co-channel interference, substantially theonly audio output from the detector is the desired audio signal. Theratio of carrier to interference usually specified for analog cellular is 18dB C/I.



MODULATION EQUATIONS FOR CONGESTEDCELLULAR SYSTEMS

AM S/N ratio: baseband S/N ratio for cellular amplitude modulation is

S/NAM = 0.25 C/I where C/I is the RF carrier-to-interference ratio

AMPS FM S/N ratio for voice: baseband S/N ratio for cellular voicechannel

S/NFM = C/I where C/I = the RF carrier-to-interference ratio ( C/I >13dB)= 48S/NAM = modulation index = 4

Assumption: the dominant noise source in a congested cellular system isco-channel interference (C/I)

34

AMPS FM S/N ratio for SAT: baseband s/n ratio for narrowbandcellular FM

S/NFM J1(C/I where C/I = the RF carrier-to-interference ratio= 0.164 C/I = modulation index = 1/3

J1(x) = first order Bessel function (i.e. J1(1/3)0.164)



FSK (Frequency Shift Keying) forAMPS Cellular Control Messaging

Input signal is Manchester-encoded data(no DC component)•10 KB rate

Output Signal is FSK-modulated•+/- 8 KHz deviation•Binary 0 = fc - 8 KHz•Binary 1 = fc + 8 KHz.

On voice channels, when systemmessages must be sent, the FM voiceand SAT modulation is briefly mutedand replaced by FSK (this is called “blank and burst”mode)

On control channels, FSK data istransmitted exclusively (no voice)

Time

Voltage

Voltage

Time

fc

Voltage 30 KHz. Channel

Frequency

Input Signal

Output Signal



Digital Modulation

Advancements in VLSI and DSP have made digital modulation morecost effective than analog modulations•Greater noise immunity•Robustness to channel impairments•Easier multiplexing of various forms of information (e.g. voice, data, video)•Greater security•Error control codes (detect and/or correct transmission errors)•Signal conditioning and processing techniques to improve performance of

overall communication link (ex: Source coding, Encryption, Equalization)

Multipurpose programmable DSP processor•Implement digital modulators and demodulators in software•Allow alternations and improvements without having to redesign or replace

the modem.



The previous example showed modulation by an analog waveform.What happens if we use a digital input?

Modulation by Digital Inputs

For example, let this digital waveformmodulate a signal. No more continuousanalog variations, now we’re “shifting”between discrete levels. We call this“shift keying”

The steady radio signal withoutmodulation is called a “carrier”

Amplitude Shift KeyingASK example: digital microwave

Frequency Shift KeyingFSK example: control messages in AMPS

cellular; TDMA cellularPhase Shift Keying

PSK examples: TDMA cellular,GSM & PCS-1900

Voltage

Time1 0 1 0

1 0 1 0

1 0 1 0

1 0 1 0



/4 DQPSKModulation Method for TDMA Cellular

DQPSK Differential Quadrature PhaseShift Keying

Differential: no absolute phasereference; each symbol is referencedonly against the previous symbol•Resolves phase ambiguity•this greatly simplifies the decoder!

Quadrature: four possible phase shiftamounts; therefore, each symbol carriestwo bits (efficient!)

/4: additional phase shift resolvesphase ambiguity of ordinary DQPSK

highly bandwidth-efficient Amplitude variations require linear

amplifiers in DPSK transmitters

4 DQPSK Signal Constellation

/4

0

/2

3/4

5/4

3/2

7/4

0 /2 3/2 2

/2

/4

x

x+

IQ

/4DQPSK



Modified forms of FSK:MSK and GMSK

MSK and GMSK are forms of FSK• input signal is pre-filtered to eliminate abrupt

shifts• this reduces the spectrum occupied by the

output signal MSK

•The frequency shift never produces a phasediscontinuity; this reduces spectrum required

• the output spectrum still contains sidelobes GMSK: Used in GSM, DCS1800, PCS1900

•Side lobes in output spectrum are prevented bythe Gaussian pre-filtering

•Generates narrow power spectrum•Spectrally efficient modulation technique•BER is slightly worse than MSK. This is a

worthwhile tradeoff since error control coding isavailable

XFSKModulator

Carrier

FSK ModulatedOutput

FILTER

1 00 T

0 T/2 T

1 0

1 0 1

NRZData

MSK Minimum Shift Keying

GMSKGaussian Minimum Shift Keying

GaussianFilter

MSKModulator

GMSKOutput

Input:BinaryData



Power Spectral Density QPSK main lobe narrower than FSK

•QPSK better for high capacity applications suchas voice channels of TDMA cellular

FSK side lobes roll of faster than QPSK•FSK better for control channel use

Other Observations:•FSK modulation simpler to implement.•With filtering, both methods meet EIA standard

attenuation >26 db on adj. channel.

BER Performance QPSK approx. 3dB better than FSK

Eb/No (dB) = (S/N)(BNR)Eb = Energy per bit No = Noise per bitN = Total noise power S = Signal powerBN = Noise bandwidth R = Bit Rate

Comparison of QPSK and FSK

0.010.0001

1E-61E-8

1E-101E-121E-141E-161E-181E-20

0

2 4 6 8 10 12 14 16 180 20

BER Performance ComparisonQPSK vs FSK

Eb/No (db)

BER

0

-10

dB

-50

-40

-30

-20

fc+1fbit

Power Spectral DensityQPSK vs. FSK

fc fc+2fb fc+3fb fc+4fbFrequency



Choice of Digital Modulation A desirable modulation scheme:

•Low bit error rates at low SNR•Good performance in an environment with multipath, fading and interference•Sensitivity to detection of timing jitter caused time-varying channel•Minimum of occupied bandwidth•Easy-cost effective to implementUsing simulation to determine relative performance and ultimate selection

Trade-offs are made when selecting a digital modulation depending on thedemands of the particular application•Power efficiency p

–Pe , Power level as possible– Is expressed as Eb/No for a certain error probability (ex: 10-3)

•Bandwidth efficiency B

–Data rate , BW as possible; However, Data rate , BW , some schemesperform better

– Is expressed as R/B bps/Hz, R: data rate, B: occupied BW–with a greater B, more data will be transmitted in a given spectrum allocation–Upper bound on B Shannon’s theorem (Channel capacity)

•There is a trade-off between p andB



Bandwidth Efficiency and Shannon’sChannel Coding Theorem

Shannon’s channel coding: a fundmental upper bound onachievable bandwidth efficiency

NS

1logBC

η 2Bmax

C: channel capacity (bps)B: RF bandwidthS/N: Signal-to-noise ratio

Tradeoff between bandwidth efficiency and power efficiency for aparticular bit error rate• Error control coding: B B required received power p• M-ary keying: B B required received power p

EX: SNR of a wireless link = 20dB, RF bandwidth= 30kHz The maximum theoretical data rate = C = B log2(1+S/N) =30000 log2(1+100) = 199.75

kbps

USDC (US Digital Cellular Standard) data rate = 48.6 kbps ¼ the theoretical limit under20 dB SNR conditions

EX: RF bandwidth= 200kHz, Compare to GSM (279.833kbps) SNR = 10dB, The maximum theoretical data rate = C = 691.886kbps 4 rateGSM

SNR = 20dB, The maximum theoretical data rate = C =1.99Mbps



Bandwidth and Power Spectral Density ofDigital Signals

Power spectral density of a random signal w(t )

PSD of baseband signal g(t ) PSD of bandpass signal s(t )

•EX: PSD of symbols represented as rectangular baseband = (sin f )2/ f2

,lim

2

TfW

fP f

Tw fW f : Fourier transform of w(t )

cgcgsc ffPffPfPtfjtgts 41

2expRe

Bandwidth• Absolute Bandwidth: range over which signal has a non-zero

PSD•Null-to-null bandwidth: width of main spectral density•Half power bandwidth: 3dB bandwidth• 99% bandwidth: 0.5 % above upper band and lower band• PSD below a certain level: typically 45~60dB attenuation

specified

power

Frequency

Absolute BW

null-to-null BW

0.5%0.5%99% BW

3dB3dB BW

30dB30 dB BW



Line Coding

Use line coding to provide particular spectral characteristics of a pulse train• Return-to-zero (RZ): Spectral widening; better timing synchronization• Non-return-to-zero (NRZ): More spectrally efficient; poor synchronization• Manchester codes: Two pulses to represent each binary symbol; suited for signaling (that must

pass through phone and other dc blocking circuits)All of these may be either polar or unipolar



Pulse Shaping Techniques Techniques that reduce modulation bandwidth and suppress out-of-band

radiation while reducing ISI within maximal bandwidth•Rectangular pulse will spread in time while passing a bandlimited channel•Out-of-band radiation in adjacent channel should be 40dB to 9-dB below the desired

passband•Spectral shaping is done through baseband or IF processing, since it is difficult to

manipulate the transmitter spectrum at RF frequencies•Nyquist criterion for ISI cancellation

00

0

n

nK)nT(h seff

)t(h*)t(h*)t(p*)t()t(h rceff

•Rectangular “brick-wall”filter–Difficult in implementing it–Slope is 1/t (1/t2 or 1/t3 is more desirable to

minimize ISI due to timing jitter

• Nonlinear RF amplifier• Linear amplifiers are required

–Not power efficient–Real-time feedback t offer more power

efficient



Raised Cosine Rolloff Filter

•The most popular pulse shaping filter used in mobile communications•The class of filters which satisfy Nyquist criterion• Impulse response decays much faster at the zero-crossings ( ~ 1/t3 for t>>Ts)

: rolloff factor



Raised Cosine Filter As , Bandwidth , time sidelobe levels in adjacent symbol slots

• Increasing decreases sensitivity to timing jitter, but increases occupied bandwidth

11 BT

Rs

s

fHRC

•Symbol rate Rs for baseband

B: absolute filter bandwidth•Symbol rate Rs for RF band

•Ex: Ts = 41.06s.first zero-crossing RF BW of a rectangular

pulse (nullto-null) = 2/ Ts= 48.71kHzFor a raised cosine filter with =0/35, BW =

(1+ )/Ts=32.88kHz• filters at TX and RX•hRC(t): noncausal, truncated for 6Ts

about t = 0

121 B

TR

ss

BPSK

Decision points

•Store several symbols at a time using pulseshaping Ex: = ½, store three bitsTime span of discrete-

time waveform = 14Ts, data sequence 1,0,1



Gaussian Pulse-Shaping

•A non-Nyquist technique•Particularly effective when used with Minimum Shift Keying (MSK) modulation•Other modulations which are suited for power efficient nonlinear amplifiers

–Does not accurately preserve transmitted pulse shape–Narrow absolute bandwidth, sharp cut-off, low overshoot, pulse area preservation

•Gaussian filter has a smooth transfer function with no zero-crossings•Transfer function is highly dependent upon 3-dB bandwidth•The transfer function is given by

•Is related to B

• Impulse response

)exp()( 22ffHG

BB

5887.0

2

2ln

2

2

2

exp)( tthG

•Used when cost and power efficiency aremajor factors, and bit error rate due to ISI islower than what is normally required



Geometric Representation ofModulation signals

Modulation signal set with a total of M possible signals )(),(),( 21 tststsS M log2M bits of information per symbol

Example: consider a set of BPSK with a rectangular pulse shape

N

jjiji tsts

1)()(

Elements of S as points in a vector space•Any point can be represented as a linear combination of basis signals

{j(t)j=1,2,3, ……, N}, which are orthogonal to one another, such that

•Each is normalized to have unit energy

jidttt ji ,0)()(

1)(2 dttE i

bcb

b TttfTE

tS 0)2cos(2

)(1

bcb

b TttfTE

tS 0)2cos(2

)(2

Eb: energy per bit Tb:bit period

bb

poweravgbb

TEA

ASTE

2

22.

)(),(0)2cos(2

)( 111 tEtESTttfT

t bbBPSKbcb

MN N: Dimension

A



Probability of Symbol Error Some of properties of a modulation can be

inferred from its constellation diagram•BW as number of signal point/dimension

Densely packed is more efficient than sparselyconstellation

•BW occupied by modulation signal as number ofsignal point/dimension

– BW occupied by a modulated signal with dimension N

BPSK constellation diagram

Probability of bit error is proportional to distance between the closest pointsin the constellation•Densely packed constellation is less energy efficient

•A simple upper bound for error symbol probability in an AWGN channel is

jij o

ijis N

dQsP

1 2)(

M

iisiiss sP

MsPsPP

1)(

1)(

)( is sP For symmetric constellations and dij areequivalent Ps(si) is the same for all i• Example: consider a BPSK,

00,

22

22

NE

QNE

QPEd bbBPSKebij



Linear Modulation

Digital modulation classification•Linear: the amplitude of transmitted signal, s(t), varies linearly with the

modulating digital signal signal, m(t)– In general, linear modulation schemes do not have a constant envelope.–Bandwidth efficient and hence are very attractive for use in wireless communication

system

•Nonlinear: may have linear or constant carrier envelopes, developing on whetheror not the baseband waveform is pulsed shaped.

In a linear modulation, the transmitted signal is given by

tfjtmtfjtmAtfjtAmts cIcRc 2sin2cos2exp)(Re)(

• A: amplitude, fc: carrier frequency, m(t) –mR(t)+jmI(t): complex envelope of modulated signal• Must be transmitted using linear RF amplifiers which have poor power efficiency

–Nonlinear amplifiers leads to regeneration of filtered sidelobes ADJ–Most popular linear modulation: QPSK, 0QPSK, /4 QPSK



ASK and PSK

• Binary amplitude shift keying (on-off keying) OOK

• Binary phase shift keying(phase reversal keying)



Binary Phase Shift Keying (BPSK)

Transmitted BPSK signal•A double sideband suppressed carrier amplitude modulation (DSB)•Two possible signal m1 and m2. The two phases are separated by 180o

)1(0)2cos(2

)( BinaryTttfTE

tS bcb

bBPSK

Eb: energy per bit

Tb:bit period

bbbcbcpoweravgbb TEATAEASTE 2,21

2 22.

)2cos(2

)0(0)2cos(2

)(

ccb

b

bccb

bBPSK

tfTE

BinaryTttfTE

tS

ccBPSK tftmtS )2cos()(

• To generate m1 and m2 as a binary data signal as abinary data signal m(t)



Spectrum and Bandwidth of BPSK

)2cos(Re)( tfgtS cBPSKBPSK cj

b

bBPSK etm

TE

tg 2)(

(complex envelope of signal)• Power spectral density (PSD) of gBPSK(t)

22

sin2sin

2 bbb

bbgBPSK fTcE

fTfT

EfP

• Power spectral density (PSD) ofBPSK at RF is then given by

22sinsin

2

41

bc

bc

bc

bcb

cBPSKcBPSKBPSK

TffTff

TffTffE

ffPffPfP

• BPSK signal in complex form

for rectangular pulse shape

• Null-to-null bandwidth = 2Rb = 2/Tb• 90% BW = 1.6Rb (rectangular pulse)• 90% BW = 1.5Rb (= 0.5 Raised-cosine)

½ for amplitude



BPSK Receiver BPSK uses coherent or synchronous demodulation

•Phase and frequency of the carrier is required at the receiver•A low level pilot carrier signal is transmitted along with BPSK signal

–Phase and frequency may be recovered using phase luck loop (PLL)–A Costas loop or squaring loop may be used, if no pilot is transmitted

A bit synchronizer is used to facilitate sampling ofthe integrator output precisely at the end of eachbit period• An optimum threshold level such that error

probability is minimized• Midpoint between detector output voltage levels is

used

tfTE

tm

tfTE

tmts

cb

b

chccb

bBPSK

2cos2

2cos2

)(

– No multipath impairments are induced– ch : phase delay corresponding to channel time delay



Differential PSK

Differential PSK is a noncoherent form of PSK•Noncoherent receivers are cheap and easy to build, and hence are widely used in

wireless communications reduced receiver complexity•The input binary sequence is first differentially encoded and the modulated using a

BPSK modulator

DPSK transmitter

DPSK Receiver

•(Energy efficiency)DPSK is 3dBinferior to (Energy efficiency)cPSK

– Given the same error rate, requiredpower is 3dB higher for DPSK

1 kkk dmd

coherent

o

bDPSKe N

EP exp

21

,



Quadrature Phase-Shift Keying (QPSK)

QPSK has twice bandwidth efficiency of BPSK•Two bits are transmitted in a single modulation symbol•Four equally spaced-value phase: 0, /2, and 3/2

4,3,2,102

12cos2

)(

iTtitf

TE

tS scs

sQPSK

Ts: Symbol duration = 2Tb (Bit period)

)2(sin2

)1(sin2

)2(cos2

)1(cos2

)( tfiTE

tfiTE

tS cs

sc

s

sQPSK

)2cos(2

)(1 tfT

t cs

)2sin(2

)(2 tfT

t cs

• Two-dimensional constellation diagram

22

/powe.avgss ArSTE

Es/Ts=S avg. Power=A2/2

A= 2Es/ Ts

)(2

)1(sin)(2

)1(cos)( 21 tiEtiEtS ssQPSK

j = 1, 2



Symbol Error Probability and Spectrum

•The distance between adjacent points = 2Es = 2 Eb, where Es = 2Eb

00

2

1,

22

22

22

)(NE

QNE

QN

dQPsymbolP bb

j o

ijsQPSKe

• Bit error rate of QPSK = BPSK• QPSK provides twice spectral efficiency with the same energy efficiency,

compared to BPSK• QPSK can also be differentially encoded

Spectrum and Bandwidth

22

22

22sin

22sin

sinsin2

bc

bc

bc

bcb

sc

sc

sc

scsQPSK

TffTff

TffTff

E

TffTff

TffTffE

fP

• Rs = Rb/2• Null-to-null bandwidth = 2Rs = 2/Ts= Rb

j = 1,2 for adjacentpoints

0,,

22/)()(

NE

QsymbolPbitP bQPSKeQPSKe 1 symbol = 2 bits



Transmission•Unipolar binary message stream bipolar

NRZ sequence•Bit stream m(t) mI(t) and m Q(t) (I

and Q streams), Rs = Rb/2

•The two binary sequences are modulated bytwo carriers 1(t) and 2(t)

•The two modulated signals are summed•RF filter confines the PSD within allocated

band: prevents signal adjacent channel,and remove out-of-band spurious duringmodulation

Detection•BPF removes out-of-band noise and ADJ

(adjacent channel interference)•Coherently demodulated using I,Q carriers•Decision circuit•The two components are multiplexed to

produce original sequence

BPSK Transmission and Detection



Offset QPSK The occasional phase shift of radians can cause signal envelope to pass

through zero for just an instant•Any kind of hardlimiting or nonlinear amplification of zero-crossings brings back filtered

sidelobes– QPSK signals use pulse shaping use pulse shaping be amplified only using linear amplifiers

which are less efficient– OQPSK ensures power baseband signal transitions applied to RF amplifier, which helps

eliminate spectrum regrowth after amplification (more efficient amplification)

OQPSK is similar to QPSK, except for time alignment of even and odd bit streams

• Even abd odd bit streams, mI(t) andmQ(t), are offset bye one bit period(alf-symbol period)

• Maximum phase shift is limited to 90o (180o for QPSK) envelopevariations are considerably less

• OPSK perform better than QPSK inthe presence of phase jitter due tonoisy reference signals



/4 QPSK Transmission

•mk mI,k, mQ,k I,k, Q,k

•I,k, Q,k are determined by their previous values

•k: a function of the current symbols mI,k, mQ,k

kkk

kkkkkk

kkkkkk

QIQ

QII

1

11

11

sincossin

sincoscos

ttQtcontIts ccQPSK sin)(4/

2/cos2/)(1

0

1

0ss

N

kkss

N

kk TkTtpTkTtpItI

2/sin2/)(1

0

1

0ss

N

kkss

N

kk TkTtpTkTtpQtQ

All possible states

k-1=n/4 k-1=n/2

P(t): Pulse shape

• Ik, Qk are usually passed through raised cosine rolloff pulseshaping filters

• I(t) and Q(t) can take one of five possible values, 0,+1,-1,+1/2, -1/ 2

•The information is completely contained in the phasedifference, it is possible to use noncoherent differentialdetection even in the absence of differential encoding



/4 DQPSK Transmission

Example: Bit stream 0 0 1 0 1 1 is to be sent using /4 DQPSK, o=0o. Determine k, and Ik, Qk

•First two bits: 0 0 1 = -3 /4 1= o + 1 = -3 /4 I1, Q1= (-0.707, -0.707)

•Second two bits: 1 0 2 = -/4 2= 1 + 2 = -I2, Q2 = (-1,0)

•Third two bits: 1 1 3 = /4 2=1 + 2 = -3 /4 I3, Q3 = (-0.707, -0.707)

•BER of /4 DQPSK is 3dB inferior toQPSK, while

– BER/4 CQPSK = BERCQPSK

– In low bit rate, fast Rayleigh fadingchannels, differential detection offersa lower error rate

/4 DQPSK Transmitter



/4 DQPSK Detection

• Baseband differential detection

• IF differential detection

• FM discriminator detection

Baseband differential detection

IF differential detection

FM discriminator detection



Constant Envelope Modulation

Nonlinear modulation methods•Amplitude of carrier is constant, regardless of variation in modulating signal,•Used in many practical mobile radio communication•Advantage

–Power efficient Class C amplifier can be used without introducing degradation–Low out-of-band radiation of the order –60dB to –70dB–Limiter-discriminator detection can be used high immunity against random FM

noise fluctuation due to Rayleigh fading

•Occupy a larger bandwidth than linear modulation schemes– Is not well-suited in situations where bandwidth efficiency is more important than power

efficiency

Nonlinear modulation schemes•Frequency Shift Keying (FSK)•Minimum Shift Keying (MSK)

–Gaussian Minimum Shift Keying (GMSK)



FSK and MSK

Signal space diagram forCPFSK (continuous phaseFSK)

MSK transmitter

MSK phasor diagram

FSK transmitter



Binary Frequency Shift Keying (BFSK)•Carrier signal is switched between two values according to the two possible

message states (high and low tones)

)1(0,22cos2

binaryTttffTE

tts bcb

bHFSK

)0(0,22cos2

binaryTttffTE

tts bcb

bLFSK

2f : a constant offset

•FSK signals is switched between two independent oscillators–Waveform is discontinuous at switching time discontinuous FSK

)binary(Tt,tfcosTE

tts bHb

bHFSK 102

21

)binary(Tt,tfcosTE

tts bLb

bHFSK 002

22

– Phase discontinuous pose spectral spreading and spurious transmissions– Is generally not used in highly regulated wireless systems

t

fcb

bc

b

bFSK dmktf

TE

ttfTE

ts 22cos2

)(2cos2

• More common method for generating FSK signal ( similar to analog FM)



Spectrum and detection of BFSK

Spectrum and Bandwidth•Power spectral density consists of

discrete frequency components atfc, fc+nf, fc-nf, where n is aninteger–Continuous phase FSK falls off

inverse fourth power–Discontinuous phase FSK falls off

inverse square power•Transmission bandwidth BT is given by

Carson’s rule as

BT = 2 f + 2B; B is bandwidth ofdigital baseband signal–B = R for rectangular pulse

assuming first null bandwidth isused

–B = (1+)R/2 for raised cosinepulse-shaping

Detection• Coherent Detection (optimum detector)

Correlator # 1

Correlator # 2

0, N

EQP b

FSKe

• Noncoherent Detection: matched filters

t = kTb

Matched filter

o

bDPSKe N

EP

2exp

21

,



Minimum Shift Keying (MSK) MSK is a special type of CPFSK

•CPFSK with peak frequency deviation = ¼ Rb(bit rate); modulation index of 0.5; kFSK=(2F) / RbF= peak RF

•The minimum frequency spacing allows two FSK signals to be coherently orthogonaland orthogonal detection.

Representation of an MSK signal as a form of two staggered binary PSKsignals, each with a sinusoidal envelope

MSK

Offset QOSK(rectangular)

Conventional QPSK(rectangular)

Odd bit

Even bit



Minimum Shift Keying (MSK) Advantages and properties

•Spectrally efficient modulation and is particular attractive for use in mobileradio communication systems

•Constant envelope, spectral efficiency, good BER performance, self-synchronizing capability

•Can be seen as a special form of 0QPSK; baseband rectangular pulses half-sinusoidal pulses of a period of 2Ts

1

0

1

02sin22cos2

N

i

N

icbbQcbIMSK tfTiTtptmtfiTtptmts

ii

elsewhere

TtTt

tp bb

0

202

cos mIi(t): odd bits; in-phase

mQi(t): Even bits; quadratureFor bipolar series dataFeed at a rate of Rb/2

k

bQIc

b

bMSK T

ttmtmtf

TE

tsii

2

2cos2k: 0 or depending on

mI (t) is 1 or -1

• MSK is an FSK signal with binary signaling frequencies fc +1/4T and fc-1/4T• With constant amplitude, MSK can be amplified using efficient nonlinear amplifier• Continuous phase makes it highly desirable for highly reactive loads



MSK power Spectrum

• MSK has lower sidelobes than QPSK: smoother pulsefunctions are used

• 99% B= 1.2/T for MSK and B= 8/T for QPSK• Mainlobe of MSK is wider: less specially efficient

,0

2cos

elsewhere

TtTt

tp

2

222

2

222 16.1)(2cos16

16.1)(2cos16

Tf

TffTf

TfffP cc

MSK

• Simple demodulation andsynchronization circuits



Gaussian Minimum Shift Keying (GMSK)

A derivative of MSK•Sidelobe levels are further reduced by passing NRZ data waveform through a

premodulation Gaussian pulse-shaping filter•More attractive for its excellent power efficiency (constant envelope) and its

spectral efficiency•If the 3dB-bandwidth-bit duration product (BT) of filter is greater than 0.5,

degradation from ISI is not sever

2222

2

expexp ffHtth GG

~ B: 3dB bandwidth of HG(f)BB

5887.02

2ln

– MSK is equivalent to GMSK with a BT product of infinity– BT , sidelobe levels very rapidly– BT = 0.5, second lobe is more than 30dB below main lobe– ISI caused by filtering is minimum for a BT=0.5887, degradation

in Eb/No is only 0.14dB from case of no ISI.



GMSK Bit Error Rate

Occupied RF bandwidth•Example: 0.25GMSK, channel data rate Rb=270 kbps

– T = 1/Rb = 1/(270103) = 3.7 s, BT =0.25 B= 0.25/T= 0.25/ (27010-6) =67.567 kHz

– For 90%, power bandwidth = 0.57Rb 90% RFBW= 0.57Rb =0.57270103 =153.9 kHz

Bit Error Rate:•A function of BT, since pulse shaping impact ISI•Offer performance within 1dB of optimum MSK as BT = 0.25

0,

2N

EQP b

GMSKe Is a constant related to BT

)(85.0

25.068.0BTMSKsimplefor

BTwithGMSKfor



GMSK Transmitter and Receiver

Transmitter•NRZ Gaussian baseband filter

FM modulatorUS cellular•Used in Digital Packet Data (CDPD)

and Global System for Mobile (GSM)system

GMSK receiver

Logic logic circuit for GMSK demodulation

GMSK transmitter

Detection•Orthogonal coherent detectors

–Carrier recovery is similar toCoseas loop

–A PLL with a frequency doubler:can be easily implemented usingdigital logic

•Noncoherent detectors: standard FMdiscriminators



M- ary Modulation

Combined linear and constant Envelop Modulation•Digital baseband data may be sent by varying both envelope and phase (or

frequency) of an RF carrier•M possible signals, s1(t), s2(t), …, sM(t) is transmitted during each symbol

period of duration Ts, M = 2n, one symbol = log2M bits•M-ary ASK (amplitude), M-ary PSK (frequency), M-ary FSK(frequency), M-ary

QAM (amplitude and phase)

•Particularly attractive for use in bandlimited channels– M-ary modulation schemes can achieve better bandwidth efficiency at expense of

power efficiency– Example: BW(8-PSK)= 1/3 BW(BPSK), Log28 = 3; BER is significantly worse than

BPSK

•Limited in applications due tosensitivity to time jitter– Smaller distance between signals

timing errors error performance

QPSK8-PSK



M-ary Phase Shift (MPSK)

The carrier phase takes on one of M possible values

MiTtM

itfTE

tS scs

si ,,3,2,10

212cos

2)(

Es/Ts= S avg. Power= A2/2 A = 2Es/ Ts

Es = (log2M) Eb: symbol Energy; Ts = (log2M) Tb: Symbol period

)2(sin2

)1(sin2

)2(cos2

)1(cos2

)( tfM

iTE

tfM

iTE

tS cs

sc

s

si

)2cos(2

)(1 tfT

t cs

)2sin(2

)(2 tfT

t cs

• Two-dimensional constellation diagram

)(2

)1(sin)(2

)1(cos)( 21 tM

iEtM

iEtS ssMPSK

j = 1, 2sE

– M-ary message points are equally spaced on a circle of radius 2Escentered at the origin

– MPSK is a constant envelope signal when no pulse shaping is used

MEs

sin2



Error Probability and Power Spectra

• Distance between adjacent symbols = 2 Es sin(/ M); Average symbol error probability

MNME

QN

MEQ

Nd

QsymbolP bs

j o

ijMPSKe

sin

log22

2)/sin(2

22

)(0

2

0

2

1,

Es = (log2M) Eb

MNE

QsymbolP sMPSKe

sin

42)(

0,

For a differential M-ary PSK system

Power Spectra of M-ary PSK

2

2

2

2

2

22

22

log2log2sin

log2logsin

2log

sinsin2

MTffMTff

MTffMTffME

TffTff

TffTffE

fP

bc

bc

bc

bcb

sc

sc

sc

scsQPSK

• These values assume no timing jittering orfading; a large negative effect on bit error rateas M increases.

• Simulation must be used to determine bit errorin wireless channels (interference andmultipath)

Ts = (log2M) Tb: Symbol period



Bandwidth and Power Efficiency

As M , first null bandwidth of M-ary PSK while Rb is heldconstant•Bandwidth efficiency B = Rb/B

As M , constellation is more densely packed•Error probability , Eb/No is held constant•Power efficiency (noise tolerance)

– Ideal Nyquist Pulse Shaping– First null bandwidth

• Pilot symbols or equalization must beused; This has not been a popularcommercial practice



M-ary Quadrature Amplitude Modulation (QAM)QAM: allow amplitude to also vary with phase

)2(sin2

)1(sin2

)2(cos2

)1(cos2

)( minmin tfM

ibTE

tfM

iaTE

tS cis

cis

i

Emin energy of signal with lowest amplitude

)2cos(2

)(1 tfT

t cs

)2sin(2

)(2 tfT

t cs

ai, bi: independent integers = - (L-1), -(L-3), …, 0, ...(L-1), (L-3)MiTt s ,,3,2,10

• energy per symbol is not constant• Distance between states is also not constant

ML

-3 3-1

1

1-1

3

-3

0

0

min,

131

14

2114)(

NME

QM

NE

QM

symbolP

av

MQAMe

min132

EMEav

– Power spectrum, bandwidth efficiency: QAM = M-ary PSK

– Power efficiency: QAM is superior to M-ary PSK

– Pilot symbols or equalization must be used



M-ary Frequency-Shift Keying (MFSK)

)1(0,cos2

binaryTttinTT

Etts bc

ss

sHFSK

• fc = nc / 2Ts for some fixed integers nc• M transmitted signals are of equal energy and equal duration• Signal frequencies are separated by 1/2Ts Hz, making signals orthogonal to one another

Coherent M-ary FSK: optimum receiver consists of a bank of M correlators, or matchedfilters

o

be N

MEQMP 2log

1

Coherent M-ary FSK: matched filters followed by envelope detectors

o

s

o

sM

k

k

e NEM

NkkE

kM

kP

2exp

21

1exp

11

11

1

1 Binomial expansion

M

MRB b

2log23

MMR

B b

2log2

• M- FSK are bandwidth inefficient: As M , bandwidth efficiency of M-ary FSK • M- FSK are power efficient: As M , power efficiency of M-ary FSK • Can be amplified using nonlinear amplifiers with no performance degradation



DCPSK on nonfading channel

Differently coherent phase shift keying (DCPSK)•If we define then the average error rate isSNR

ePr 21

powerNoisepowerSignal

(DCPSK)

erfcPr 21

(BPSK)

The received signal is given by kctfiAeRe 2 Signal power = A2 / 2

The received signal under cellular propagation is given by

kctfiAreRe 2 Signal power = A2 r2 / 2 = 2 / 2

N22

othewrise

eP0

022 2

2

(Rayleigh Distribution)



Modulation Performance in fadingand multipath channels

Radio channel impairments: Fading; Multipath; Doppler spread•A transmitted signal will suffer deep fades Outage, loss of sign

–Bit error rate gives a good indication, it does not provide incidents ofbursty errors

–Probability of Outage: Another means to judge effectiveness of signaling

BER and Outage evaluation methods under varioustypes of channel impairments

–Analytical techniques Slow flat-fading channel–Computer Simulation Frequency-selective channel

–Input bit stream channel counting at theoutput of receiver decision circuit

Outage

1010010 1010110

ErrorCounting

TX Channel RX

Decision



Performance corrupted by Fading

Slow Rayleigh Flat-fading•Pe =10-3 ~10-6

–SNR = 30 ~60dB (fading)–SNR = 20 ~50dB (nonfading)

Frequency-Selective fading•Pe normalized rms delay spread (d =

Ts), d Pe • Irreducible BER (floor)



Performance corrupted by Frequency-Selective Fading (delay spread)

Pe normalized rms delay spread (d= Ts) to symbol period

Pe normalized rms delay spread (d= Ts) to bit period



Computer Simulation BER evaluation normalized parameters

•Doppler Spread (vehicle speed): BDTs, BD/RS

•Muttipath Delay (delay of the second multipath ): /T•Ratio of average energy to noise power spectral density: Eb/No dB•Average carrier to interference power ratio : C/I dB•Average main-path to delayed-path power ratio: C/D dB

BERSIM Concept• Actual digital

communication• Baseband digital

hardware simulatorwith softwaresimulation as a driverfor real-time BERcontrol

CD

C/D



Performance corrupted by CCI

/4 DQPSK system•C/N for different CCI•Slow Rayleigh flat-fading•Multipath time dispersion and

Doppler spread are negligible•Errors are caused mainly by

fades and CCI•For C/I > 20dB, errors are

primarily due to fading,interference has little effect

•C/I < 20dB, interferencedominates the linkperformance

•High-capacity mobile systemsare interference limited, notnoise limited.

C/I = 20dB

C/I = 30dB

C/I = 40dB

C/I = 50dBC/I = dB

fc = 850 MHzfs = 24kspsRoll-off = 0.2



Performance corrupted by Speed IF no time dispersion and C/N , BER does not decrease below a

certain irreducible floor•Irreducible is caused is due to random FM (caused by Doppler spread)•Doppler-induced fading (Ex: /4 DQPSK fc = 850 MHz, fs = 24ksps, Roll-off = 0.2)

•Velocity error floor , C/D error floor

Eb/No= 100dBC/I= 100dB

BER in Raleigh flat-fading channelfor various mobile speed

BER in two-ray Raleigh flat-fading channelfor various time delay

C/D = 0dB

C/D = 10dB



DCPSK on Fading Channel Average SNR and average error probability on fading Channel

2222 2 yxE 0

222

212

NNN

(average signal) (average SNR)

0

2

222

0

2

2

22 11

ee

NNe

ddP

P NN

ePr 21

Average probability of error due to Rayleigh fading is

000

11

00

121

111

21

211

21 00

dedeedPPP rb

N

2

0

where

(SNR distribution)

Given

• To average the error probability in AWGN over the possible ranges of signalstrength due to fading



Performance on Fading Channel

If Error performance criteria us such that the error rate does notexceed 10-3(Typically the criteria hosen for conventionaltransmission)

Without Rayleigh fading

dB..ePr 97214661021 3

o

b

o

b

o

b

NE

)T/(NT/E

BNT/E

NS

1

1

With Rayleigh fading

dB.Pb 98264991012

10

3

0

(DCPSK)

Require SNR is much higherover a fading channel



BPSK Performance under fading (1) BPSK under coherent detection scheme (cophasing is assumed)

erfcPr 21 0

0

1

eP

00

21

0

00

21

0

11

0

00

111

1211

111

121

111

121

211

121

22

0121

121

0

0

0

0

de

de

de

de

eerfce

deerfcdPPP rb

(BPSK) < ePr 21

(DCPSK)

dtexerfc

xerfcxQ

xx

2222

1

21

110

1

nn

dxexx xn

Average error rate

Gamma Function

Gamma Function

Q=function

011



BPSK Performance under fading (2)

0

0

210

0

00

41

211121

11121

111

121

121

0

deerfcdPPP rb

111 x,nxx n

Gamma Function

111

00

1

n,nn

n,dxexx xn

n Can be evaluated from the recursive relation

11 nnxProvided one knows the value of 10 nforn

!nx 1 In n is an integer

Without Rayleigh fading

dB..

erfcPr

856844

1021 3

With Rayleigh fading

dB.

Pb

9823250

1041

0

3

0

As



Digital Modulation under fading (1)

Performance in slow, flat fading channels in AWGN

•= [ Eb/ No] 2 is the averagevalue of signal-to-noise ratio, has a Rayleigh distribution

•Mean SNR is significantly largerthan that required when operatingover a nonfading channel (~20-50dB)

Fading v.s. nonfading



Digital Modulation under fading (2)Rayleigh fadingAWGN



Digital Modulation under fading (3)Rayleigh fadingNakagami fading BPSK



Diversity techniques for fading channels Diversity Reception: Different methods of receiving “independent”fading

signals

•Space Diversity•Frequency Diversity•Polarization Diversity•Angular Diversity•Time Diversity

Different was of combining thesignals (space diversity)•Pure Selection•Equal Gain Combining•Maximal-Ratio Combining

Independent of what type of Diversity isused, it has

•M branch diversity, m different signals•Those signals are statistically independent•The same signal power and noise power

Model of digital comm. With diversity

Space diversity



Diversity techniques

Rake receiver time diversity

Polarization diversity

OFDM Frequency diversity

Diversity implementation

• Antenna polarization diversity• OFDM (Frequency diversity)• Rake receiver (time diversity)



Pure Selection Diversity (SL)

Pure selection techniques chose the signal with the largestinstantaneous power•The selected signal•The signal-to-noise ratio of the selected signal•Find pdf of

mr,.....,r,rmaxz 21 m,.....,,max 21

32121 ,...,,FF

m...,

im

i,r im...F,...,,FP 132121

If they are independent

ji jiFF For all i

x

y

011

00

0

iii ,eFeP i

i

i

i

ofpdfeem

P

ddF

PeF

m

m

,1

1

00

0

1

0

zY,zXP

zy,xmaxPzZP

z



PDF& CDF with SL Diversity

A

t

............................

.. .... ...

path 1

path 2

.maximum amplitude

rP



DCPSK Error rate with SL Diversity

Consider Binary DCPSK ePr 21

Average probability of error due to Rayleigh fading

0100000

212

0000

2

000

11

0

1

0

11

0

1

00

11

1

00

!211

43

32

21

121

11

21

111

11

1121

01

1121

121

121

00

000

00

00

00

i

mm

mmmm

deemm

deemem

eem

dem

e

deem

edPPP

m

i

m

mm

m

m

rb

dxvuuvdxvu 0



Error rate improvement with SL Diversity

011

21

1

bP,m

00 22

11

21

2

bP,m

dBPb 271012

10

3

0

dB.Pb 5141021

10

3

00

000 33

22

11

21

3

bP,m

dB.Pb 11010321

30

3

000

0 5 10 15 20 25 30 35

SNR (dB)

10-5

10-4

10-3

10-2

10-1

100

Ave

rag

eE

rro

rP

rob

abili

ty

m =1

m =2

m =3No diversty

DCPSK with Selection Diversity

Usually, most diversity schemes provide mostimprovement for m = 2 to 4.



Maximal-Ratio Diversity

The signals from all of the M branches are weighted according to theirindividual signal voltage to noise power ratios and then summed.•Output SNR is equal to the sum of the individual SNR•The advantage of generating an output with an acceptable SNR even when none of the

individual signals are acceptable•The best statistical reduction of fading of any know linear diversity•DSP and digital receivers are now making this optimal form of diversity practical

The combined signal

•The signal-to-noise ratio

m

iiirar

1

m

iii

m

iii

m

iii Na

ra

Na

r

1

2

2

1

1

2

2 22



PDF with Maximal-Ratio Combining (1)• We set iii Nsa

m

ii

m

i i

im

i i

im

iii

m

iii

m

iiii

m

iiii

Nr

Nr

Nr

Nr

NrN

Nrr

11

2

1

2

1

2

2

1

2

1

22

2

1 2212

12

• Find pdf of

21

02

0

01

PPPeP iii

i

If n = 2 21 And 21 and Are I.I.d.

01

011

00

00

00

20

020

000

2

,ede

deeP

1P

2P



PDF of Maximal-Ratio Combining (2)

01

011

00

00

00

20

020

000

2

,ede

deeP

otherwise,

,!m

eP m

m

m

0

01

0

0

1

otherwise,

,ede

deePPP

0

02

1

1

00

00

3

30

2

030

00

20

23

rP



DCPSK Error rate of MR Diversity

Consider DCPSK ePr 21 (binary)

m

mm

m

m

m

m

mrb

em

dm

ee

dm

eePPP

0

1

0

11

0

00

111

00

1

0

11

21

011!11

21

!121

!121

0

0

0

dB.Pb 31310

11

21

10 03

20

3

Without diversity

Pure selection diversity

m = 2

dB270

dB.8140

If

20

20 2

11

121

bP

10 15 20 25 30 35 40 45

SNR (dB)

10-6

10-5

10-4

10-3

10-2

10-1

Ave

rag

eE

rro

rP

rob

abili

ty

L = 1

L = 2

L = 4

No fading

DCPSK

PSK



BPSK Error rate with MR Diversity (1)

erfcPr 21

(BPSK)

otherwise,

,!m

eP m

m

m

0

012

0

0

1

21

!21

!21

!21

22

!21

22

!121

0!121

01sin,!12

1

!121

00

121

00

21

0

00

000

0

1

0

00

1

0

0

0

mm

dem

dem

de

md

emm

erfcmm

forecedm

erfc

dm

eerfcPPP

mm

mm

m

m

m

m

m

m

m

m

m

m

m

mrb

m

xn

mm///m/m/n

/,nnndxexn

213212

2121232121

21110

1

0



BPSK Error rate with MR Diversity (2)

mmmmmmb m

mmmmmm

mP

01

01

0 !2!!12

!213212

213212

!21

Two-branch diversity dB.!!

Pb 36111016

3223

03

20

20

3

No diversity dB.Pb 97231041

21

03

20

20

2

dB.P mb 31310

11

21

03

0

dB.Pb 98264991012

10

3

0

Two-branch diversity

No diversity

• BPSK

• DCPSK

CPSKPDCPSKP bb

1355133

!!!!



Comparison between Precise andapproximate error rate with ML

10 15 20 25 30 35 40 45

SNR (dB)

10-6

10-5

10-4

10-3

10-2

10-1

Ave

rag

eE

rro

rP

rob

abili

ty

L = 1

L = 2

L = 4

No fading

DCPSK

PSK



Comparison between SL and ML

0 5 10 15 20 25 30 35 40 45

SNR (dB)

10-6

10-5

10-4

10-3

10-2

10-1

Ave

rag

eE

rro

rP

rob

abili

ty

0 5 10 15 20 25 30 35 40 45

SNR (dB)

10-6

10-5

10-4

10-3

10-2

10-1

Ave

rag

eE

rro

rP

rob

abili

ty

Selection diversity Maximal-Ration diversity

DCPSK

PSK

DCPSK

No fading

L = 1

L = 2L = 4

L = 1

L = 2

L = 4

Nofading



Equal-Gain Diversity It may not always be convenient or desirable to provide the variable

weighting capacity required for true maximal-ratio combining.•The gains may be set to a constant value of unity equal-gain combining

Cophase &Summing

• The combined signal

m

iirr

1 mNr

N

rm

ii

22 2

1

2

• The signal-to-noise ratio

othewrise

er

rPiri

i

0

022 2

2

242

00

22

22

22

20 2

21

2

2

2

2

2

2

21

rerfc

re

rde

re

rforrPrPrPrPrrr

rr

rrrr

rPrPrPrPrrrr rrrr 321321 No closed form



No closed form or other techniques exist to evaluateexplicitly Pr(r) for m > 2. Numerical techniques can beused to evaluate Pr(r) for any value of m.

PDF with Equal-Gain Diversity

m

m

m

m

mP

me

P

0

1

0

1

!11

0,!1

0

(maximal-ratio)

012

2

0

11

,!m

mP m

mmm

• Same Type of approx. for equal-gain combining

!m

!mm

ratioimalmaxPgainequalP mm

b

b 112

2 1

For m = 2 3311

3222

.!!ratioimalmaxP

gainequalP

b

b



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