Digital Modulation Techniques

DIGITAL MODULATION TECHNIQUES

Digital Modulation FormatsModulation is defined as the process by which some characteristic of a carrier is varied in accordance with a modulating waveWith a sinusoidal carrier, the feature that is used by the modulator to distinguish one signal from another is a step change in amplitude, frequency, or phase of the carrierThe result of this modulation process is

Amplitude-shift keying (ASK)Frequency-shift keying (FSK)Phase-shift keying (PSK)

Digital Modulation Formats

Digital Modulation Formats

The scheme that attains as many of the following design goals as possible

Maximum data rateMinimum probability of symbol errorMinimum transmitted powerMinimum channel bandwidthMaximum resistance to interfering signalsMinimum circuit complexity

Coherent Binary Modulation Techniques

Coherent binary PSKThe pair of signal s1(t) and s2(t), used to represent binary symbols 1 and 0, respectively, are defined

0≤ t ≤ Tb and Eb is the transmitted signal energy per bit

( )tfTE

ts cb

b π2cos2

)(1 =

( ) ( )tfTEtf

TEts c

b

bc

b

b πππ 2cos22cos2)(2 −=+=


Coherent binary PSKThe basis function

We may expand the transmitted signal s1(t)and s2(t) in terms of Φ1(t)

)2cos(2)(1 tfT

t cb

πφ =

)()( 11 tEts bφ=

)()( 12 tEts bφ−=

bTt ≤≤0

bTt ≤≤0

bTt ≤≤0


Coherent binary PSKThe coordinates of the message point equal

dtttss bT)()( 10 111 φ∫= bE+=

dtttss bT)()( 10 221 φ∫= bE−=


Coherent binary PSK


Coherent binary PSKThe probability of symbol error

⎟⎟⎠

⎞⎜⎜⎝

⎛=

021

NEerfcP b

e


Coherent binary PSK


Coherent binary PSK


Coherent binary FSKIn binary FSK system, symbols 1 and 0 are distinguished from each other by transmitting one of two sinusoidal waves that differ in frequency by a fixed amount

Where i = 1, 2; symbol 1 is represented by s1(t) and symbol 0 by s2(t)Eb is the transmitted signal energy per bitTransmitted frequency

( )tfTE

ts ib

b

i π2cos0

2)(

⎪⎩

⎪⎨

⎧=

elsewhereTt b≤≤0

b

ci T

inf

+=


Coherent binary FSKThe most useful form for the set of orthonormal basis functions is

( )tfTt ibi πφ 2cos0

2)(

⎪⎩

⎪⎨⎧

=elsewhere

Tt b≤≤0


Coherent binary FSKThe coefficient sij for i = 1, 2 and j = 1, 2 is defined

dtttss bT

jiij ∫= 0)()( φ

( ) ( )dttfT

tfTE

ib

i

T

b

bb ππ 2cos22cos2

0∫=

⎪⎩

⎪⎨⎧

=0

bE

jiji

≠=


Coherent binary FSKThe two message points are defined by the signal vectors

⎥⎦

⎤⎢⎣

⎡=

01bEs

⎥⎦

⎤⎢⎣

⎡=

bE0

2s


Coherent binary FSKThe observation vector x has two elements, x1 and x2, are defined by, respectively

( ) dtttxx bT)(101 φ∫=

( ) dtttxx bT)(202 φ∫=


Coherent binary FSK


Coherent binary FSKDefine a new Gaussian random variable Lwhose sample value l is equal to the difference between x1 and x2

The mean value of L depends on which binary symbol was transmitted

21 xxl −=


Coherent binary FSKThe conditional mean of the random variable L, given that symbol 1 was transmitted, is

On the other hands,

[ ] [ ] [ ]1|1|1| 21 XEXELE −= bE+=

[ ] [ ] [ ]0|0|0| 21 XEXELE −= bE−=


Coherent binary FSKThe variance of the random variable L is independent of which symbol was transmittedSince the random variable X1 and X2 are statistical independent, each with a variance equal to N0/2

[ ] [ ] [ ]21 XVarXVarLVar +=

0N=


Coherent binary FSKSuppose that symbol 0 was transmitted, the corresponding value of the conditional probability density function of random variable L equals

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡ +−=

0

2

0 2exp

210|

NEl

NLf b

L π


Coherent binary FSKSince the condition x1 > x2, or, equivalently, L > 0, corresponds to the receiver making a decision in favor of symbol 1, the conditional probability of error, given that symbol was transmitted is given by

( )0)0( >= lPPe

( )

( )dl

NEl

N

dllf

b

L

∫

∫∞

∞

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +−=

=

00

2

0

0

2exp

21

0|

π


Coherent binary FSKPut

We may rewrite

zNEl b =

+

02

( )dzzPNEe

b∫∞

−=02

2exp1)0(π

⎟⎟⎠

⎞⎜⎜⎝

⎛=

0221

NE

erfc b


Coherent binary FSKProbability of symbol error

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

0221

NbE

erfcPe


Coherent binary FSK


Coherent binary FSK


Coherent quadrature-modulation techniques

The quadrature-carrier multiplexing system produces a modulated wave described as

sI(t) is the in-phase component of the modulated wavesQ(t) is the quadrature component

( ) ( )tftstftsts cQcI ππ 2sin)(2cos)()( −=


Quadriphase-shift keying (QPSK)In QPSK, the phase of the carrier takes on one of four equally space values, such as π/4, 3π/4, 5π/4, 7π/4

i = 1, 2, 3, 4; E is the transmitted signal energy per symbolT is the symbol duration, and the carrier frequency fcequals nc/T for some fixed integer nc

( )⎪⎩

⎪⎨⎧

⎥⎦⎤

⎢⎣⎡ −+=

04

122cos2)(

ππ itfTE

ts ci

elsewhere

Tt ≤≤0


Quadriphase-shift keying (QPSK)We may rewrite

( ) ( )

( ) ( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎦⎤

⎢⎣⎡ −−

⎥⎦⎤

⎢⎣⎡ −

=

0

2sin4

12sin2

2cos4

12cos2

)( tfiTE

tfiTE

tsc

c

i ππ

ππ

elsewhere

Tt ≤≤0


Quadiphase-shift keying (QPSK)There are only two orhtonormal basis functions, Φ1(t) and Φ2(t), contained in the expansion of si(t)

Tt ≤≤0( )tfT

t cπφ 2cos2)(1 =

( )tfT

t cπφ 2sin2)(2 = Tt ≤≤0


Quadriphase-shift keying (QPSK)There are four message points, and the associated signal vectors are defined by

( )

( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −−

⎟⎠⎞

⎜⎝⎛ −

=

412sin

412cos

π

π

iE

iEis 4,3,2,1=i


Quadraphase-shift keying (QPSK)


Quadriphase-shift keying (QPSK)




Quadriphase-shift keying (QPSK)The received signal, x(t), is defined by

w(t) is the sample function of a white Gaussian noise process of zero mean and power spectral density N0/2

)()()( twtstx i +=4,3,2,1

0

=

≤≤

i

Tt


Quadriphase-shift keying (QPSK)The observation vector, x, of a coherent QPSK receiver has two elements, x1 and x2

x1 and x2 are sample values of independent Gaussian random variables with mean values equal to

with common variance equal to N0/2 ]4)12cos[( π−iE

]4)12sin[( π−iE

∫=T

dtttxtx0 11 )()()( φ ( ) 14

12cos wiE +⎥⎦⎤

⎢⎣⎡ −=

π

∫=T

dtttxtx0 22 )()()( φ ( ) 24

12sin wiE +⎥⎦⎤

⎢⎣⎡ −−=

π




Quadriphase-shift keying (QPSK)The probability of correct detection , Pc, equals the conditional probability of joint event x1> 0 and x2> 0, fiven that signal s4(t)was transmitted


Quadriphase-shift keying (QPSK)Since the random variables X1 and X2 are (with sample value x1 and x2, respectively) are independent, Pc also equals the product of the conditional probabilities of the events x1> 0and x2> 0, both given s4(t) was transmitted

( ) ( )2

0

2

2

00

10

2

1

00

2exp12

exp1 dxN

ExN

dxN

ExN

Pc ⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−= ∫∫

∞∞

ππ


Quadriphase-shift keying (QPSK)Let

We may rewrite

zN

ExN

Ex=

−=

−

0

2

0

1 22

( )2

2

2

0

exp1⎟⎠

⎞⎜⎝

⎛ −= ∫∞

−dzzP

NEebπ


Quadriphase-shift keying (QPSK)Since

We have

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=−∫

∞

−0

2

2

2211exp1

0 NEerfcdzz

NEbπ

2

02211

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

ENEerfcPc

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

0

2

0 241

21

NEerfc

NEerfc


Quadriphase-shift keying (QPSK)The average probability of symbol error for coherent QPSK is

ce PP −=1

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛=

0

2

0 241

2 NEerfc

NEerfc


Quadriphase-shift keying (QPSK)If E/2N0 >> 1, the average probability of symbol error for coherent QPSK as

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

02NEerfcPe


Quadriphase-shift keying (QPSK)In QPSK system, there are two bits per symbol. This mean that the transmitted signal energy per symbol is twice the signal energy per bit, that is

We may expressthe average probability of symbol error in terms of the ratio Eb/N0

bEE 2=

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

0NEerfcP b

e






Minimum Shift Keying (MSK)Consider a continuous-phase frequency-shift keying (CPFSK) signal, which is defined for the interval 0 ≤ t ≤T, as follows

for symbol 1

for symbol 0

Eb is the transmitted sinal energy per bit, and Tb is the bit duration

[ ]

[ ]⎪⎪⎩

⎪⎪⎨

⎧

+

+=

)0(2cos2

)0(2cos2

)(

2

1

θπ

θπ

tfTE

tfTE

ts

b

b

b

b


Minimum shift Keying (MSK)The phase θ(0), denoting the value of the phase at time t = 0, depends on the past history of the modulation processThe frequencies f1 and f2 are sent in response to binary symbols 1 and 0 appearing at the modulator input, respectively


Minimum Shift Keying (MSK)Signal s(t) may be expressed in the conventional form of an angle-modulated wave

where θ(t) is the phase of s(t)when the phase θ(t) is a continuous function of time, the modulated wave s(t) itseft is also continuous at all times, including the inter-bit switching time

[ ])(2cos2)( ttfTEts cb

b θπ +=


Minimum Shift Keying (MSK)The nominal carrier frequency fc is chosen as the arithmetic mean of the two frequencies f1and f2

The phase θ(t) of a CPFSK signal increases or decreases linearly with time during each bit period of Tb seconds

( )2121 fff c +=

tThtb

πθθ ±= )0()( bTt ≤≤0


Minimum Shift KeyingThe phase θ(t) of a CPFSK signal increases or decreases linearly with time during each bit period of Tb seconds

The plus sign corresponds to sending symbol 1, and the minus sign corresponds to sending symbol 0The parameter is referred to as the deviation ratio, measured with respect to the bit rate 1/Tb

tThtb

πθθ ±= )0()(bTt ≤≤0

( )21 ffTh b −=


Minimum Shift KeyingAt the time t = Tb

for symbol 1for symbol 0⎩

⎨⎧−

=−h

hTb π

πθθ )0()(


Minimum Shift Keying (MSK)


Minimum Shift Keying (MSK)Using a well-known trigonometic identity, we may express the CPFSK signal s(t) in terms of its in-phase and quadraturecomponents as follows

[ ] ( ) [ ] ( )tftTEtft

TEts c

b

bc

b

b πθπθ 2sin)(sin22cos)(cos2)( −=


Minimum Shift Keying (MSK)With deviation ratio h = ½

The plus sign corresponds to symbol 1 and the minus sign corresponds to symbol 0A similar result holds for θ(t) in the interval -Tb≤ t ≤ 0Since the phase θ(0) is 0 or π, depending on the past history of the modulation process, in the interval Tb≤ t ≤+Tb , the polarity of cos[θ(t)] depends only on θ(0), regardless of the sequence of 1s or 0s transmitted before or after t = 0

tT

tb2

)0()( πθθ ±=bTt ≤≤0


Minimum Shift Keying (MSK)For this time interval -Tb≤ t ≤ +Tb, the in-phase component, sI(t) consists of a half-cosine pulse defined as follows

the plus sign corresponds to θ(0) = 0, and minus sign corresponds to θ(0) = π

[ ])(cos2)( tTEtsb

b θ=

[ ]

⎟⎟⎠

⎞⎜⎜⎝

⎛±=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

tTT

E

tT

tTE

bb

b

bb

b

2cos2

2cos)(cos2

π

πθ

bb TtT ≤≤−


Minimum Shift Keying (MSK)In the interval 0 ≤ t ≤ 2Tb, the quadrature component, sQ(t),consists of a half-sine pulse, whose polarity depends only on θ(Tb)

the plus sign corresponds to θ(Tb) = π/2 and the minus sign corresponds to θ(Tb) = -π/2

[ ])(sin2)( tTEtsb

bQ θ=

[ ]

⎟⎟⎠

⎞⎜⎜⎝

⎛±=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

tTT

E

tT

TTE

bb

b

bb

b

b

2sin2

2sin)(sin2

π

πθ

bTt 20 ≤≤


Minimum Shift Keying (MSK)With h = 1/2 , the frequency deviation (i.e., the difference between the two signaling frequencies f1 and f2) equals half of bit rateThis is the minimum frequency spacing that allows the two FSK signals representing symbols 1 and 0, to be cohenrently orthogonal in the sense that they do not interfere with one another n the process of detectionCPFSK signal with a deviation ratio of one-half is referred to as minimum-shift keying (MSK)


Minimum Shift Keying (MSK)One of four possibilities can arise, as described

The phase θ(0) = 0 and θ(Tb) = π/2, corresponding to the transmission of symbol 1The phase θ(0) = π and θ(Tb) = π/2, corresponding to the transmission of symbol 0The phase θ(0) = π and θ(Tb) = - π/2 (or, equivalently, 3π/2, modulo 2π), corresponding to the transmission of symbol 1The phase θ(0) = 0 and θ(0) = - π/2, corresponding to the transmission of symbol 0


Minimum Shift KeyingIn MSK signal, the appropriate form for the orthonormal basis functions Φ1(t) and Φ2(t) is as follows

Both Φ1(t) and Φ2(t) are defined for a period equal to twice the bit duration

( )tftTT

t cbb

ππφ 2cos2

cos2)(1 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

bb TtT ≤≤−

( )tftTT

t cbb

ππφ 2sin2

sin2)(2 ⎟⎟⎠

⎞⎜⎜⎝

⎛= bTt 20 ≤≤


Minimum Shift KeyingCorrespondingly, we may express the MSK signal in the form

The coefficients s1 and s2 are related to the phase states θ(0) and θ(Tb), respectively

)()()( 2211 tststs φφ +=bTt ≤≤0


Minimum Shift KeyingThe in phase component of s(t)

The quadrature component of s(t)

∫−=T

Tb

dtttss )()( 11 φ

[ ])0(cos θbE= bb TtT ≤≤−

∫=bT

dtttss2

0 22 )()( φ

[ ])(sin bb TE θ−= bTt 20 ≤≤


Minimum Shift KeyingBoth integrals are evaluated for a time interval equal to twice the bit duration, for which Φ1(t) and Φ2(t) are orthogonalBoth the lower and upper limits of the product integration used to evaluate th coefficient s1 are shifted by Tb seconds with respect to those used to evaluate the coefficient s2The time interval 0 ≤ t ≤ Tb, for which the phase state θ(0) and θ(Tb) are defined is common to both integrals








Minimum Shift KeyingIn a case of an AWGN channel, the received signal is given by

s(t) is the transmitted MSK signal, and w(t) is the sample function of a white Gaussian noise process of zero mean and power spectral density N0/2

)()()( twtstx +=


Minimum Shift KeyingIn order to decide whether symbol 1 or symbol 0 was transmitted in the interval 0 ≤ t ≤ Tb, we have to establish a procedure for the use of x(t) to detect the phase states θ(0) and θ(Tb)For optimum detection of θ(0), we have to determine the projection of the received signal x(t)onto the reference signal Φ1(t) and Φ2(t)


Minimum Shift Keying

dtttxx b

b

T

T)()( 11 ∫−= φ

11 ws +=bb TtT ≤≤−

dtttxx bT))(

2

0 22 ∫= φ

22 ws += bTt 20 ≤≤


Minimum Shift KeyingThe average symbol error for the MSK is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛=

0

2

0 41

NEerfc

NEerfcP bb

e

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

0NEerfcP b

e





Noncoherent Binary Modulation Techniques

Noncoherent Orthogonal ModulationConsider a binary signaling scheme that involves the use of two orthogonal signal s1(t) and s2(t), which have equal energyDuring the interval 0≤ t ≤ T, one of these two signals is sent over an imperfect channel that shifts the carrier phase by an unknown amount


Noncoherent Orthogonal ModulationLet g1(t) and g2(t) denote the phase-shifted versions of s1(t) and s2(t), respectivelyIt is assumed that g1(t) and g2(t) remain orthogonal and of equal energy, regardless of the unknown carrier phaseWe refer to such a signaling scheme as noncoherent orthogonal modulation


Noncoherent Orthogonal ModulationThe channel also introduces an AWGN w(t) of zero mean and power spectral density N0/2We may express the received signal x(t) as

x(t) is used to discriminate between s1(t) and s2(t) , regardless of the carrier phase

⎩⎨⎧

++

=)()()()(

)(2

1

twtgtwtg

tx TtTt

≤≤≤≤

00


Noncoherent Orthogonal ModulationThe receiver consists of a pair of filters matched to the basis function Φ1(t) and Φ2(t) that are scaled versions of the transmitted signal s1(t) and s2(t), respectivelyBecause the carrier phase is unknown, the receiver relies on amplitude as the only possible discriminantAccordingly, the matched filter outputs are envelope detected, sampled, and then compared with each other.


Noncoherent Orthogonal Modulation


Noncoherent Orthogonal ModulationThe quadrature receiver itself has two path

In in-phase path, the receiver signal x(t) is correlated against the basis function Φi(t), representing a scaled version of the transmitted signal s1(t) or s2(t) with zero carrier phase.In the quadrature path, signal x(t) is correlated against another basis function , representing the version of Φi(t) that results from shifting the carrier phase by -900

Naturally, Φi(t) and are orthogonal to each other

)(ˆ tiφ

)(ˆ tiφ




Noncoherent Orthogonal ModulationThe average probability of error for the noncoherentreceiver will be calculated by making use of the equivalence depicted previous pictureSince the carrier phase is unknown, noise at the output of each matched filter has two degrees of freedom, namely, in-phase and quadrature. Accordingly, the noncoherent receiver has a total of four noise parameters that are statistical independent and identically distributed, denoted by xI1, xQ1, xI2, xQ2

The first tow account for degrees of freedom associated with the upper pathThe latter two account for degrees of freedom associated with the lower path


Noncoherent Orthogonal ModulationSince the receiver has a symmetric structure, the probability of choosing s2(t), given that s1(t) was transmitted, is the same as the probability of choosing s1(t), given that s2(t) was transmittedThis means that the average probability of error may be obtained by transmitting s1(t) and calculating the probability of choosing s2(t), or vice versa


Noncoherent Orthogonal ModulationSuppose that signal s1(t) is transmitted for the interval 0≤ t ≤ T, an error occurs if the receiver noise w(t) is such that the output l2 is greater than the output l1

The receiver makes a decision in favor of s2(t) rather than s1(t)


Noncoherent Orthogonal ModulationTo calculate the probability of error, we must have the probability density function of the random variable L2 (represented by sample value l2).Since the filter in the lower path is matched to s2(t), and s2(t) is orthogonal to the transmitted signal s1(t), it follows that the output of this matched filter is due to noise alone.


Noncoherent Orthogonal ModulationLet xI2 and xQ2 denote the in-phase and quadrature components of the matched filter output in the lower pathFor i = 2

22

222 QI xxl +=


Noncoherent Orthogonal ModulationThe random variables XI2 and XQ2 are both Gaussian distributed with zero mean and variance N0/2

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

22

02 exp1)(

2 Nx

Nxf I

IX I π

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

22

02 exp1)(

2 Nx

Nxf Q

QX Q π


Noncoherent Orthogonal ModulationRandom variable L2 has the following probability density function

⎪⎩

⎪⎨

⎧⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

exp2)(

0

22

0

2

22 Nl

Nl

lf L

elsewherel 02 ≥


Noncoherent Orthogonal ModulationThe conditional probability that l2 > l1, given the sample value l1, is defined

( ) 222112 )(|1

dllflllPl L∫∞

=>

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=>

0exp|

21

112llllP


Noncoherent Orthogonal ModulationSince the filter in the path is matched to s1(t),and it is assumed that s1(t) is transmitted, it follows that l1 is due to signal plus noiseLet xi1 and xQ1 denote the components at the output of the matched filter (in the upper path) that are in-phase and quadrature to the received signal

21

211 QI xxl +=


Noncoherent Orthogonal ModulationSince is orthogonal to s1(t), it is obvious that xI1 is due to signal plus noise, whareas xQ1 is due to noise alone.

XI1 represented by sample value xI1 is Gaussian distributed with mean and variance N0/2,where E is the signal energy per symbolXQ1 represented by sample xQ1 is Gaussian distributed with zero mean and variance N0/2

)(1̂ ts

E


Noncoherent Orthogonal ModulationThe probability density functions of these two independent random variable

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−=

0

2

1

01 exp1)(

1 NEx

Nxf I

IX I π

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

21

01 exp1)(

1 Nx

Nxf Q

QX Q π


Noncoherent Orthogonal ModulationSince the tow random variable XI1 and XQ1are independent, their joint probability density function is simply the product of the probability density functions of two random variable


Noncoherent Orthogonal ModulationTo find the average probability of error, we have to average the conditional probability of error over all possible values of l1

Given xI1 and xQ1, an error occurs when the lower path’s output amplitude l2 due to noise alone exceeds l1 due to signal plus noise


Noncoherent Orthogonal ModulationThe probability of such an occurrence is

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

0

21

21

11 exp),(N

xxxxerrorP QI

QI

( ) )()(, 111111 QQXIIXQI xfxfxxerrorP

[ ]⎭⎬⎫

⎩⎨⎧

+−++−= 2121

21

21

00

)(1exp1QIQI xExxx

NNπ


Non-coherent Orthogonal ModulationSince

22

22)( 2

1

2

12

12

12

121

ExExxExxx QIQIQI ++⎟⎟⎠

⎞⎜⎜⎝

⎛−=+−++


Noncoherent Orthogonal ModulationThe average probability of error

( ) 11111111 )()(, QIQQXIIXQIe dxdxxfxfxxerrorpP ∫ ∫∞

∞−

∞

∞−=

10

21

1

2

1000

2exp

22exp

2exp1

QQ

II dxNx

dxExNN

EN ∫∫

∞

∞−

∞

∞− ⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

π


Noncoherent Orthogonal ModulationSince

222exp 0

1

2

10

πNdxEx

N II =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−∫

∞

∞−

22

exp 01

0

21 πN

dxNx

QQ =⎟⎟⎠

⎞⎜⎜⎝

⎛−∫

∞

∞−


Noncoherent Orthogonal ModulationAccordingly

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

02exp

21

NEPe


Noncoherent Binary FSKIn the case of binary FSK

( )⎪⎩

⎪⎨

⎧=

0

2cos2

)( tfTE

ts ib

b

iπ

bTt ≤≤0




Noncoherent Binary FSKThe noncoherent binary FSK described is a special case of noncoherent orthogonal modulation with

The probability of error

bTT =

bEE =

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

02exp

21

NEP b

e


Differential Phase-shift Keying (DPSK)DPSK eliminates the need for a coherent reference signal at the receiver by combining two basic operations at the transmitter

Differential encoding of the input binary wavePhase-shift keying

In effect, to send symbol 0 we phase advance the current signal waveform by 1800, and to send symbol 1 we leave the phase of the current signal waveform unchanged


Differential Phase-shift Keying (DPSK)


Differential Phase-shift Keying (DPSK)The differential encoding process at the transmitter input starts with an arbitrary first bit, serving as reference, and thereafter the differentially encoded sequence {dk} si generated by

Bk is the input binary digit at time KTb

Dk-1 is the previous value of the differentially encoded bit

kkkkk bdbdd 11 −− +=




Differential Phase-shift Keying (DPSK)The receiver is equipped with a storage capability, so that it can measure the relative phase difference between the waveforms received during two successive bit intervalsAssumed that the unknown phase θ contained in the received wave varies slowly (that is, slow enough for it to be considered essentially constant over two bit intervals), the phase difference between waveforms received in two successive bit intervals will be independent of θ




Differential Phase-shift Keying (DPSK)Let s1(t) denote the transmitted DPSK signal in the case of symbol 1 at the transmitter input

( )

( )⎪⎪⎩

⎪⎪⎨

⎧

=tf

TE

tfTE

ts

ib

b

ib

b

π

π

2cos22

2cos22

)(1

bb

b

TtTTt2

0≤≤≤≤


Differential Phase-shift Keying (DPSK)Let s2(t) denote the transmitted DPSK signal in the case symbol 0 at the transmitter input

( )

( )⎪⎪⎩

⎪⎪⎨

⎧

+=

ππ

π

tfTE

tfTE

ts

ib

b

ib

b

2cos22

2cos22

)(2bb

b

TtTTt2

0≤≤≤≤


Differential Phase-shift Keying (DPSK)s1(t) and s2(t) are orthogonal over the two-bit interval 0≤ t ≤ 2Tb

DPSK is a special case of noncoherentorthogonal modulation with

bTT 2=

bEE 2=


Differential Phase-shift Keying (DPSK)The probability of symbol error

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

exp21

NEP b

e

M-Ary Modulation TechniquesIn M-ary signaling scheme, we may send one of M possible signals, s1(t), s2(t), …, sM(t),during each signaling interval of duration T

In almost applications M = 2n

The symbol duration T = nTb

These signal are generated by changing amplitude, phase, or frequency of a carrier in M discrete steps. Thus we have:

M-ary ASKM-ary PSKM-ary FSK

M-Ary Modulation Techniques

M-ary PSKIn M-ary PSK, the phase of the carrier takes one of M possible values, θi = 2iπ/M, where i = 0, 1, …, M-1Accordingly, during each signaling interval of duration T, one of possible signals

E is the signal energy per symbol

Carrier frequency fc = nc/T for some fixed integer nc

⎟⎠⎞

⎜⎝⎛ +=

Mitf

TEts ci

ππ 22cos2)( 1,...1,0 −= Mi


M-ary PSKEach si(t) may be expanded in terms of two basis function

)2cos(2)(1 tfT

t cπφ =

)2sin(2)(2 tfT

t cπφ =

Tt ≤≤0

Tt ≤≤0


M-ary PSKThe signal constellation of M-ary PSK is two dimensional.The M message points are equally spaced on a circle of radius , and center at the origin

E


M-ary PSK

M-Ary Modulation TechniquesM-ary PSK

The optimum receiver for coherent M-ary PSK includes a pair of correlators with reference signals in phase quadratureThe two correlators outputs, denoted as xI and xQ, are fed into a phase discriminator that first computes the phase estimate

The phase discriminator then selects from the set {si(t), i = 0, …, M-1} that particular signal whose phase is closet to the estimate

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

I

Q

xx1tanθ̂

θ̂


M-ary PSK


M-ary PSKIn the presence of noise, the decision-making process in the phase discriminator is based on the noisy input

Where wI and wQ are samples of two independent Gaussian random variables WI and WQ whose is mean zero and common variance equals N0/2

II wM

iEx +⎟⎠⎞

⎜⎝⎛=π2cos

QQ wM

iEx +⎟⎠⎞

⎜⎝⎛−=π2sin

1,...1,0 −= Mi

1,...1,0 −= Mi


M-ary PSKThe message points exhibit circular symmetrBoth random variables WI and WQ have a symmetric probability density functionThe average probability of symbol error Peis independent of the particular signal si(t)is transmitted


M-ary PSKWe may simplify the calculation of Pe by setting θi = 0, which corresponds to the message point whose coordinates along the Φ1(t)- and Φ2(t)-axes are and 0, respectivelyThe decision region pertaining to this message point is bounded by the threshold

below the Φ1(t)-axis and the threshold above the Φ1(t)-axis

E

Mπθ −=ˆMπθ +=ˆ


M-ary PSKThe probability of correct reception is

is the probability density function of the random variable whose sample value equals the phase discriminator output produced in response to a received signal that consists of the signal s0(t) plus AWGN

∫− Θ=M

Mc dfPπ

πθθ ˆ)ˆ(

)ˆ(θΘf

Θ θ̂

⎟⎟⎠

⎞⎜⎜⎝

⎛

+= −

I

Q

WE

W1tanθ̂


M-ary PSKThe probability density function has a known value. Especially, for we may write

)ˆ(θΘfπθπ ≤≤− ˆ

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=Θ θθθ

ππθ ˆcos

211ˆsinexpˆcosexp

21ˆ

0

2

000 NEerfc

NE

NE

NEf


M-ary PSKA decision error is made if the angle falls outside

θ̂MM πθπ +≤≤− ˆ

ce PP −=1

∫− Θ=M

Mdf

π

πθθ ˆ)ˆ(


M-ary PSKFor large M and high values of E/N0, we may derive an approximate formula for Pe

For high values of E/N0 and for , we may use the approximation

2ˆ πθ <

⎟⎟⎠

⎞⎜⎜⎝

⎛−≈⎟⎟

⎠

⎞⎜⎜⎝

⎛− θ

θπθ ˆcosexpˆcos

1ˆcos 2

0

0

0 NE

EN

NEerfc


M-ary PSKWe get

⎟⎟⎠

⎞⎜⎜⎝

⎛−≈Θ θθ

πθ ˆsinexpˆcos)ˆ( 2

00 NE

NEf

2ˆ πθ <

∫−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−≈

M

Me d

NE

NEP

π

π

θθθπ

ˆˆsinexpˆcos1 2

00

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

MNEerfc πsin

0


M-ary PSK

M-Ary Modulation TechniquesM-ary QAM

In an M-ary PSK system, in-phase and quadraturecomponents of the modulated signal are interrelated in such a way that the envelope is constrained to remain constant. This constraint manifests itself in a circular constellation for the message pointsHowever, if this constrained is removed, and the in-phase and quadrature components are thereby permitted to be independent, we get a new modulation scheme called M-ary quadratureamplitude modulation (QAM)


M-ary QAMThe signal constellation for M-ary QAM consists of a square lattice of message points.


M-ary QAMThe corresponding signal constellations the in-phase and quadaraturecomponents of the amplitude phase modulated wave are shown


M-ary QAMThe general form of M-ary QAM is defined by the transmitted signal

E0 is the energy of the signal with the lowest amplitudeai and bi are a pair of independent integers chosen in accordance with the location of the pertinent message point

( ) ( )tfbTE

tfaTE

ts cicii ππ 2(sin2

2(cos2

)( 00 += Tt ≤≤0


M-ary QAMThe signal si(t) can be expanded in terms of a pair of basis functions

)2cos(2)(1 tfT

t cπφ =

)2sin(2)(2 tfT

t cπφ =

Tt ≤≤0

Tt ≤≤0


M-ary QAMThe coordinates of the ith message point are and , where (ai, bi) is an element of the L-by-L matrix ( )

Eai Ebi

{ }⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−−+−+−+−+−

−−−+−−+−−−−+−−+−

=

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

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

,

LLLLLL

LLLLLLLLLLLL

ba ii

L

MMM

L

L

ML =


M-ary QAMFor example, for the 16-QAM whose signal constellation is

{ }

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−−−−−−−

−−−−

=

3,33,13,13,31,31,11,11,3

1,31,11,11,33,33,13,13,3

, ii ba


M-ary QAM


M-ary QAMSince the in-phase and quadraturecomponents of M-ary QAM are independent, the probability of correct detection for such a scheme may be written as

Where is the probability of symbol error for either component

( )2'1 ec PP −=

'eP


M-ary QAMThe signal constellation for the in-phase or quadrature component has a geometry similar to that for discrete pulse-amplitude modulation (PAM) with a corresponding number of amplitude levelsWe may write

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −=

0

0' 11NEerfc

LPe


M-ary QAMThe probability of symbol error for M-aryQAM is given

where it is assumed that

ce PP −=1

( )'

2'

2

11

e

e

P

P

≈

−−=

1' <<eP


M-ary QAMThe probability of symbol error for M-aryQAM may be written

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −≈

0

0112NEerfc

MPe


M-ary QAMThe transmitted energy in M-ary QAM is variable in that its instantaneous value depends on the particular symbol transmittedIt is logical to express Pe in terms of the average value of the transmitted energy rather than E0

Assuming that the L amplitude levels of the in-phase or quadrature component are equally likely, we have

( ) ⎥⎦

⎤⎢⎣

⎡−= ∑

=

2

1

20 122

2L

iav i

LE

E


M-ary QAMThe limits of the summation take account of the symmetric nature of the pertinent amplitude levels around zeroWe get

( )3

12 02 EL

Eav−

=

( )3

12 0EM −=


M-ary QAMAccordingly, we may rewrite probability of symbol error in terms of Eav

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎠⎞

⎜⎝⎛ −≈

0)1(23112

NMEerfc

MP av

e


M-ary QAM


M-ary FSKIn an M-ary FSK scheme, the transmitted signals are defined by

i = 1, 2, …, MThe carrier frequency fc = nc/2T for some fixed integer nc

The transmitted signals are of equal duration T and have equal energy E

( ) ⎥⎦⎤

⎢⎣⎡ += tinTT

Ets ciπcos2)( Tt ≤≤0


M-ary FSKSince the individual signal frequencies are separated by 1/2T Hz, the signals are orthogonal

0)()(0

=∫ dttsts j

T

i


M-ary FSKFor coherent M-ary FSK, the optimum receiver consists of a bank of M correlatorsor matched filtersAt the sampling time t = KT, the receiver makes decisions based on the largest matched filter output


M-ary FSKAn upper bound for the probability of symbol error

For fixed M, this bound becomes increasingly tight as E/N0 is increasedFor M = 2, the bound becomes an equality

⎟⎟⎠

⎞⎜⎜⎝

⎛−≤

02)1(

21

NEerfcMPe


M-ary FSKThe probability of symbol error for noncoherent of M-ary FSK is given

The upper bound on the probability of symbol error for noncoherent detection of M-ary FSK

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−= ∑

−

=

+

0

1

1

1

1exp

11

)1(Nk

kEk

Mk

PM

k

k

e

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−≤

02exp

21

NEMPe


Power spectraThe description of a band-pass signal s(t)contains the definitions of ASK, PSK, and FSK signals, depending on the way in which the in-phase component sI(t) and the quadrature component sQ(t) are defined


Power SpectraWe may express s(t) in the form

Where Re[.] is real part of the expression contained inside the bracket

)2sin()()2cos()()( tftstftsts cQcI ππ −=

( )[ ]tfjts cπ2exp)(~Re=


Power SpectraWe also have

The signal is called the complex envelope of the band-pass signal s(t)The component sI(t) and sQ(t) and thereforeare all low-pass signal

)()()(~ tjststs QI +=

( ) ( )tfjtftfj ccc πππ 2sin2cos)2exp( +=

)(~ ts

)(~ ts


Power SpectraLet SB(f) denote the baseband power spectral density of complex envelope We refer to SB(f) as the baseband power spectral densityThe power spectral density, SS(f), of the original band-pass signal s(t) is a frequency-shifted version of SB(f), except for a scaling factor

)(~ ts

[ ])()(41)( cBcBS ffSffSfS ++−=


Power Spectra of Binary PSKBaseband power spectral density of binary FSK wave equals

Power spectra of binary PSK

( ) )(sin22

)(sin2)( 2

2

fTcEfT

fTEfS bb

b

bbB ==

ππ


Power spectra of binary FSKThe power spectral densities of SB(f) is given

The power spectral density of binary FSK

( )2222

2

14)(cos8

21

21

2)(

−+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

fTfTE

Tf

Tf

TEfS

b

bb

bbb

bB

ππδδ



Power Spectra of QPSKThe baseband power spectral density of QPSK signal

The power spectral density of QPSK signal

)2(sin4)(sin2)( 22 fTcETfcEfS bbB ==


Power spectra of MSK signalThe baseband power spectral density of MSK signal

The power spectral density of MSK signal

( )( )

2

222 116

2cos322

)(2)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−=⎥

⎦

⎤⎢⎣

⎡Ψ=

fT

fTET

ffs

b

bb

b

gB

ππ



Power spectra of M-ary signalBinary PSK and QPSK are special cases of M-ary PSK signalsThe symbol duration of M-ary PSK is defined by

Where Tb is the bit duration

MTT b 2log=


Power spectra of M-ary PSKThe baseband power spectral density of M-ary PSK signal is given by

)(sin2)( 2 TfcEfSB =

)log(sinlog2 22

2 MfTcME bb=



The spectral analysis of M-ary FSK signals is much more complicated than that of M-aryPSKA case of particular interest occurs when the frequencies assigned to the multilevels make the frequency spacing uniform and the frequency deviation k= 0.5That is, the M signal frequencies are separated by 1/2T, where T is symbol duration


Power spectra of M-ary FSKFor k = 0.5, the baseband power spectral density of M-ary FSK signals is defined by

( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∑ ∑∑

= = =

M

i

M

i

M

j j

jji

i

ibB i

iMM

EfS1 1 1

22

2

sinsincos1sin214)(

γγ

γγγγ

γγ

)1(24

4

+−=

⎟⎠⎞

⎜⎝⎛ −=

Mi

bfT

i

ii

α

παγ

Mi ,...,2,1=


Power spectra of M-ary FSK


Bandwidth efficiencyBandwidth efficiency is defined as the ratio of data rate to channel bandwidth; it is measured in units of bits per second per hertzBandwidth efficiency is also referred to as spectral efficiencyWith the data rate denoted by Rb and the channel bandwidth by B, we may express the bandwidth efficiency, , as

BRb=ρ

ρ


Bandwidth efficiency of M-ary PSKThe channel bandwidth required to pass M-ary PSK signals (more precisely, the main spectral lobe of M-ary PSK signals) is given

where T is the symbol durationT

B 2=


Bandwidth efficiency of M-ary PSKSince

Channel bandwidth

Channel efficiency of M-ary PSK signals is

MTT b 2log=

MR

B b

2log2

=

2log2 M

BRb ==ρ


Bandwidth efficiency of M-ary PSK


Bandwidth efficiency of M-ary FSKChannel bandwidth required to transmit M-ary FSK signals as

Channel bandwidth of M-ary FSK signals is

Bandwidth efficiency of M-ary FSK signals is

TMB2

=

MMRB b

2log2=

MM

BRb 2log2

==ρ


Bandwidth efficiency of M-ary FSK


Bit versus symbol error probabilitiesThus far, the only figure of merit we have used to assess the noise performance of digital modulation schemes has been the average probability of symbol errorWhen the requirement is to transmit binary data, it is often more meaningful to use another figure of merit called the probability of bit error or bit error rate (BER)

M-Ary Modulation TechniquesBit versus symbol error probabilities

Case 1:The mapping from binary to M-ary symbols is performed in

such a way that the two binary M-tuples corresponding to any pair of adjacent symbols in the M-ary modulation scheme differ in only one bit position (Gray code)When the probability of symbol error Pe is acceptably small, we find that the probability of mistaking one symbol for either of the two nearest (in-phase) symbols is much greater than any other kind of symbol errorMoreover, given a symbol error, the most probable number of bit errors is one, subject to the aforementioned mapping constraint


Bit versus symbol error probabilitiesCase 1

Since there are log2M bits per symbol, it follows that the bit error rate is related to the probability of symbol error by a formula

Mp

BER e

2log= 2≥M


Bit versus symbol error probabilitiesCase 2:

We assume that all symbol errors are equally likely and occur with probability

where Pe is the average probability of symbol errorK = log2M

121 −=

− Kee P

MP


Bit versus symbol error probabilitiesCase 2:

There are ways in which k bits out of K may be in errorThe average number of bit errors per K-bit symbol is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛kK

eK

KK

kK

eKk KPPk

122

12)(

1

1 −=

−

−

=∑


Bit versus symbol error probabilitiesCase 2

The bit error rate is obtained by dividing the result by K

eK

K

PBER12

12−

=−

ePM

M

BER⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−=

12

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Digital Modulation Techniques