Digital Modulation and Detection

Digital Modulation and Detection

Wha Sook Jeon Mobile Computing & Communication Lab.

2

Signal Space Analysis

3

Signal and System Model (1)

System sends bits every T seconds Each bit sequence of length K comprises a message mi ={b1, … , bK} The message i has the probability pi of being selected for transmission

( ) Each message is mapped to a unique analog signal si (t), is transmitted

over the channel during the time interval [0, T], and has energy

The transmitted signal is sent through an AWGN channel (a white Gaussian noise process n(t) of power spectral density N0/2)

Received signal r(t)=s(t)+n(t) The receiver should determine the best estimate of the transmitted

signal and outputs the best estimate of the transmitted message

MK 2log=

11

=∑ =

M

i ip

∫ ==T

iS MidttsEi 0

2 ,,1 ,)(

}ˆ,,ˆ{ˆ 1 Kbbm =

4

Signal and System Model (2)

Goal of the receiver design − minimizing the probability of message estimation error

sent) (sent) | ˆ(1

ii

M

iie mpmmmpP ∑

=

≠=

Communication system model

5

Geometric Representation of Signals (1) By representing the signals geometrically, we can solve for the

optimal receiver design in AWGN based on a minimum distance criterion

Basis function representation − Any set of M real energy signals can be

represented as a liner combination of N (≤ M) real orthonormal basis functions according to Gram-Schmidt orthogonalization procedure

: Modulation − Orthonormal basis function:

− Real coefficient representing the projection of si(t) onto : Demodulation

)}(,),({ 1 tstsS M=

)}(,),({ 1 tt Nφφ

Tttsts j

N

jiji <≤=∑

=

0 ),()(1

φ

dtttss j

T

iij )()(0

φ∫=)(tjφ

≠=

=∫ jiji

dttt j

T

i if0 if1

)()(0

φφ

6

Geometric Representation of Signals (2)

Basis set of linear pathband modulation technique −

− With fcT >> 1 for orthonormality

Transmitted signal

The basis set can include a bandpass pulse-shaping filter g(t) to improve the spectral characteristics of the transmitted signal

The pulse shape g(t) must maintain the orthonormal property of basis functions

)2sin(2)( ),2cos(2)( 21 tfT

ttfT

t cc πφπφ ==

)2sin(2)2cos(2)( 21 tfT

stfT

sts cicii ππ +=

)2sin()()2cos()()( 21 tftgstftgsts cicii ππ +=

0)2sin()2cos()( ,1)2(cos)(0

22

0

2 == ∫∫ dttftftgdttftg cc

T

c

Tπππ

7

Geometric Representation of Signals (3) Signal space representation

− Signal constellation point of the signal si(t) ■ ■ the vector of coefficients in the basis representation of si(t), that is, ■ One-to-one correspondence

between si(t) and si − The distance between two signal constellation points si and sk

NiNii ss R),( 1 ∈= s

∫

∑

−=

−=−=

T

ki

N

jkjijki

dttsts

ssss

0

2

1

2

))()((

)(

)()(1

tsts jN

j iji φ∑ ==

Signal space of MQAM or MPSK is two-dimensional

8

Receiver Structure and Sufficient Statistics (1)

Given the channel output r(t)=si(t)+n(t), 0 ≤ t < T, the receiver determines which constellation point (or message) is sent over time interval [0, T )

9


In this receiver structure −

− − nr(t) is the “remainder” noise which is orthogonal to signal

space

)()()()()(

)()()(

)()()(

11

11

tntrtntns

tntnts

tntstr

rjN

j jrjN

j jij

rjN

j jjN

j ij

i

+=++=

++=

+=

∑∑∑∑

==

==

φφ

φφ

dtttnndtttss j

T

jj

T

iij )()( ,)()(00

φφ ∫∫ ==

10


)]()()()([ ),(

1111

21

tntrtrtrrrr

rnrφφφ ++==

),,( 21 rnrr

11


Goal of receiver design is to minimize the error probability

Maximizing

r =(r1, …, rN) is a sufficient statistic for r(t) in optimal detection of transmitted message

))(|ˆ(1))(|ˆ( trmmptrmmpP iie =−=≠=

)),,(|sent ),,(( ))(()),,((

))(()),,(sent, ),,((

))(, ),,(())(, ),,(sent, ),,((

))(, ),,(|sent ),,(())(|sent (

11

1

11

1

11

11

iNiiNi

riNi

riNiiNi

riNi

riNiiNi

riNiiNii

rrssptnprrp

tnprrssptnrrp

tnrrssptnrrssptrsp

=

=

=

=

12

Decision Region Optimal receiver selects corresponding to constellation si

that satisfies Design region:

imm =ˆijrsprsp ji ≠≥ allfor )|sent ()|sent (

} )|sent ()|sent (:{ ijrsprsprZ jii ≠∀>=

13

Maximum Likelihood Decision Criteria (1)

Optimal receiver selects corresponding to constellation si that maximizes

Likelihood function: A maximum likelihood receiver outputs corresponding to

constellation si that maximizes L(si)

imm =ˆ

)()()|()|(

rpspsrprsp ii

i =

)|(max arg

)()|(max arg)(

)()|(max arg

i

ii

s

ss

i

iiii

srp

spsrprp

spsrp

=

=

Let p(si)=1/M

)|()( ii srpsL =

imm =ˆ

14


Conditional distribution of r − Since n(t) is a Gaussian random process, r(t)=si(t)+n(t) is also a

Gaussian random process and n(t) has a zero mean. −

− rj is a Gaussian random variable that is independent of with mean sij and variance N0/2.

≠=

=

=

−−=

==−+=−=

=+==

kjkjN

nn

srrsrr

Nnssnsr

ssnssr

kj

irkrjikj

jijijjijsrjsr

ijijjijijjsr

kj

ijjij

ij

02

]E[

]|))([(E]|cov[

2][E]|)[(E])[(E

]|[E]|[E

0

0222

||2

|

µµ

µσ

µjijj nsr +=

)( kjrk ≠

15


Likelihood function L(si): conditional distribution of r

Log likelihood function

−−== ∑∏

==

N

jijjNijj

N

ji sr

NNsrpsrp

1

2

02/

01 )(1exp

)(1 )|()|(

π

2

01

2

0

1)(1)( i

N

jijji sr

Nsr

Nsl −−=−−= ∑

=

minimizing

16


A maximum likelihood receiver outputs corresponding to constellation si that satisfies

Decision Region

Constellation point si is determined from the decision Zi that contains r

imm =ˆ

2

1

2 minarg)(minarg is

N

jijjs

srsrii

−=−∑=

},,,1 :{ ijMjsrsrrZ jii ≠=∀−<−=

17

Union Bound of Error Probability (1)

Aik: the event that given that the constellation point si was sent

If the event Aik occurs, the constellation will be decoded in error.

ik srsr −<−

)(sent) (11

ik

M

ikk

ik

M

ikk

ie ApApmP ∑≠=≠

=≤

=

))((

))()((

sent) |()(

nssnp

snssnsp

ssrsrpAp

ik

iiki

iikik

<−−=

−+<−+=

−<−=

The probability that n is closer to the vector sk-si than to the origin

a one-dimensional Gaussian r.v. ik ss −

18

Union Bound of Error Probability (2)

The event Aik occurs if

kiikik ssddn −=> where,2

∑

∫

≠=

∞

≤

=

−=

>=

M

ikk

ikie

ikd

ikik

NdQmP

NdQdv

Nv

NdnpAp

ik

1 0

00

2

2/0

2sent) (

2exp1

2)(

π

∑∑∑≠===

≤=

M

ikk

ikM

iiei

M

ie N

dQmPmpP1 011 2M

1sent) ()(

19

Approximation of Error Probability

The minimum distance of constellation: dmin

− : looser bound

The number of neighbors at the minimum distance: −

− In case of binary modulation (M=2):

Gray code: mistaking a constellation point for one of its nearest neighbors results in a single bit error −

−≤

0

min

2)1(

NdQMPe

mindM

≈

0

min

2min NdQMP de

( )0min 2NdQPb =

MPP e

b2log

≈

20

Amplitude and Phase Modulation

21

Amplitude and Phase Modulation (1)

Over the time interval Ts, K (log2M) bits are encoded in the amplitude and/or phase of the transmitted signal − Signal: − In signal space:

There are three main types of amplitude/phase modulation: − Pulse Amplitude Modulation (MPAM) : Uses amplitude only − Phase Shift Keying (MPSK) : Uses phase only − Quadrature Amplitude Modulation (MQAM)

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

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

02

01

2211

φπφφπφφφ

+−=+=

+=

tftgttftgt

ttsttsts

c

c

ii

22


Amplitude/Phase

Modulator

23


0φφ =

2212

12110

)cos( )sin( and )sin( )cos( ,0

nssrnssr

ii

ii

+∆+∆−=+∆+∆=≠∆=−

φφφφφφφ

Amplitude/Phase

Demodulator

Coherent detection ( )

if => performance degradation Synchronization or timing recovery: the sampling function is synchronized to the start of every symbol period.

24


Pulse Amplitude Modulation

Phase-Shift Keying

Quadrature Amplitude Modulation

Differential Modulation

Modulator with Quadrature Offset

25

Pulse Amplitude Modulation (MPAM) (1)

Encodes all of the information into the signal amplitude (Ai) Transmitted signal over time:

− Signal constellation:

− Parameterization distance d is typically a function of a signal energy − Minimum distance : − Constellation mapping is usually done by Gray encoding

Amplitude of each transmitted signal has M different values − Each pulse conveys K = log2M bits per symbol time Ts

}...,,2,1 ,)12({ MidMiAi =−−=

{ } ( ) cscitfj

i fTttftgAetgAts c /10,2cos)()(Re)( 2 ⟩⟩≤≤== ππ

dAAd jiji 2min ,min =−=

Gray Coding

26


Over each symbol period, the MPAM signal associated with the ith constellation has energy − Average energy:

Decision region associated with signal amplitude dMiAi )12( −−=

( )∫ ∫ === s s

i

T T

iciis AdttftgAdttsE0 0

22222 2cos)()( π

∑=

=M

iis A

ME

1

21

( )[ )[ )

=∞−−≤≤+−

=+∞−=

MidAMidAdA

idAZ

i

ii

i

i

,,12,

,1,

27


Coherent MPAM demodulator

28

Phase Shift Keying (MPSK) (1)

Encodes information in the phase of the transmitted signal Transmitted signal over one symbol time:

Constellation points:

Minimum distance: where A is typically function of signal energy

All possible signals have equal energy A2

Constellation mapping usually uses Grady encoding

),( 21 ii ss

{ }( )

( ) ( ) tfMitgAtf

MitgA

MitftgA

eetgAts

cc

c

tfjMiji

c

ππππ

ππ

ππ

2sin12sin)(2cos12cos)(

122cos)(

)(Re)( 2/)1(2

−

−

−

=

−

+=

= −

MiMiAsMiAs ii ,...,1for ]/)1(2sin[ ],/)1(2cos[ 21 =−=−= ππ),/sin(2min MAd π=

29


Gray Coding for MPSK

30


Decision Regions for MPSK

( ) ( ){ }MiMireZ ji 212232: −<<−= πθπθ

31


Coherent Demodulator for BPSK

32

Quadrature Amplitude Modulation (MQAM) (1)

Information bits are encoded in both amplitude and phase of the transmitted signal

MPSK and MPAM have only one degree-of-freedom, but MQAM has two degree-of-freedom. Thus, MQAM is more spectral-efficient

Transmitted signal:

Signal energy in si(t):

Distance between constellation points:

{ }( ) ( ) sciicii

tfjjii

TttftgAtftgAetgeAtS ci

≤≤−==

0,2sin)()sin(2cos)()cos()(Re)( 2

πθπθ

πθ

∫ == sT

iisi AdttsE0

22 )(

( ) ( )2222

11 jijijiij ssssssd −+−=−=

33


4-QAM and 16-QAM constellations

For square signal constellation, − Values on (2i-1-L)d

for i = 1, …, L − dmin= 2d

Good constellation mapping can be hard to find for QAM signal

34


Decision Regions for 16-QAM

35

Differential Modulation (1)

Information in MPSK and MQAM signals is carried in the signal phase − MPSK and MQAM require coherent detection − Phase recovery mechanism required in receiver − Coherent demodulation

■ makes receiver complex ■ is hard in rapidly fading channel ■ is more susceptible to phase drift of the carrier

The principle of differential modulation is to use the previous symbol as a phase reference for current symbol for avoiding the need for a coherent phase modulation − Differential BPSK (DPSK)

■ 0-bit: no change in phase, 1-bit: a phase change of π − Differential QPSK (DQPSK)

■ 00: no change in phase, 01: a phase change of π/2 ■ 10: a phase change of - π/2, 11: a phase change of π

36


Phase Comparator − Transmitted signal: − Received signal at time k: − Received signal at time (k-1): − Phase difference

modulation with memory less sensitive to random drift in the carrier phase With non-zero Doppler frequency, previous symbol is not good for

phase reference

( ) )()()()( 0)(21 knAekjrkrkr kj +=+= −+ φφθ

( ) ( ) ( ) ( ) ( )1111 0)1(21 −+=−+−=− −+− knAekjrkrkr kj φφθ

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )1

11*)1(

*)()1()(2*

0

0

−++

−+=−−+−−

−+−−

knknkneAknAeeAkrkr

kj

kjkkj

φφθ

φφθθθ

Phase difference in the absence of noise (n(k) = n(k-1) = 0)

( )0)()( φθ += kjAeks

37


DPSK demodulator

38

Modulation with Quadrature Offset (1)

Linearly modulated signal may cause transition to symbol which makes phase change up to π, and signal amplitude to cross zero point − Abrupt phase transition and amplitude variations can be distorted

by non-linear filters and amplifiers To avoid the above problem

− Offsetting the quadrature branch pulse g(t) half a symbol period − Phase can change maximum π/2

39

Modulation with Quadrature Offset (2)

Modulator with quadrature offset

40

Pulse Shaping (1)

Bandwidth of the baseband and passband modulated signal is a function of the bandwidth of the pulse shape g(t).

The effective received pulse: − c(t): the channel impulse response − g*(-t): the matched filter − In AWGN channel (c(t)=δ(t)),

To avoid ISI between the received pulses, p(t) must satisfy Nyquist criterion, which requires the pulse to equal zero at the ideal sampling point associated with past or future symbols.

Pulse shapes that satisfy the Nyquist criterion − Rectangular pulse − Cosine pulse − Raised Cosine pulse

)()()()( * tgtctgtp −∗∗=

)()()( * tgtgtp −∗=

41

Pulse Shaping (2)

Raised Cosine Pulse − These pulses are designed in the frequency domain

− The pulse p(t) in the time domain:

factor rolloff a is where

21

21

21sin1

2

210

)(

β

βββπ

β

+≤≤

−

−−

−≤≤

=

sss

ss

ss

Tf

TTfTT

TfT

fP

222 /41/cos

//sin)(

s

s

s

s

TtTt

TtTttp

ββπ

ππ

−×=

42

Pulse Shaping (3)

Raised Cosine Pulse in frequency domain

P(f)

43

Pulse Shaping (4)

Raised Cosine Pulse in time domain

p(t)

44

Error Probability of Digital Modulation over AWGN Channel

BPSK and QPSK MPSK MPAM and MQAM

45

Signal-to-Noise Power Ratio (SNR)

In an AWGN channel − Modulated (transmitted) signal: − Received signal: − n(t): a white Gaussian random process with mean zero and power spectral

density N0/2

SNR − Ratio of the received signal power to the power of the noise within the

bandwidth of the transmitted signal −

− In system with interference

■

})(Re{)( 2 tfj cetuts π=

b

b

s

sr

BTNE

BTNE

BNPSNR

000

===

)()()( tntstr +=

rP

I

r

PBNPSINR+

=0

46

Bit/Symbol Errors

For pulse shaping with Ts=1/B (e.g., raised cosine pulse with β=1), SNR=Es/N0

For general pulse, Ts= k/B and Define

− SNR per symbol: − SNR per bit:

We are interested in bit error probability Pb as a function of

Approach − First, compute the symbol error probability Ps as a function of − Then, obtain bit error probability as a function of SNR per bit using assumptions.

■ The symbol energy is divided equally among all bits, ■ Gray encoding is used

− These assumptions for M-ary signaling lead to the approximations and

bγ

0NEss =γ

0NEbb =γ

Ms

b2logγγ ≈

MPP s

b2log

≈

kNEs 1SNR 0 ×=

sγ

47

Error Probability for BPSK

For binary modulation (M=2), −

−

=

0

min

2NdQPb

( )bb

sb QNEQPP γ22

0

=

==

2

0

222

0

21

01min

)2(cos)()(

,2

AdttftgAdttsE

Assdbb T

c

T

b ∫∫ ===

=−=

π

48

Error probability for QPSK

The QPSK system is equivalent to the system consisting of BPSK modulation on both the in-phase and quadrature components of the signal.

The bit error probability on each component is the same as for BPSK −

The symbol error probability is

Since the transmitted symbol energy is split between each branch, the signal energy per branch is

The symbol error probability is the probability that either branch has a error −

For the same and therefore the same average probability of bit error, QPSK

system transmits data at twice the bit rate of a BPSK system for the same channel bandwidth. −

( )[ ]2211 bs QP γ−−=

sE)2( sE

( )ss QP γ2≈

0NEb

)2( bb QP γ=

( )[ ]211 ss QP γ−−=

( )[ ] ( ) ( )bsss QQQP γγγ 222112

=≈−−=

49

Error Probability for MPSK

Signal-space diagram for 8PSK

1φ

2φ

sE

sE

When using the union bound of error probability and the nearest neighbor approximation,

For MPSK,

2,)sin(2

min

min

=

=

d

s

MMEd π

( ))sin(22 MQP ss πγ=

( )0min 2min

NdQMP ds ≈

50

Error Probability for MPAM

The constellation for MPAM is

Since each of the M-2 inner constellation points has two nearest neighbor at distance 2d,

For outer constellation points, there is only one nearest neighbor.

The average energy per symbol for MPAM is

The symbol error probability in terms of the average energy as

MidMiAi ,...,2,1 ,)12( =−−=

1,...,2 ),n()( −=>= MidpsP is

22

1

2 )1(311 dMA

ME

M

iis −== ∑

=

sP

( ) ( )001

2222222 )(1 NdQM

NdQM

MsPM

PM

iiss ×+××

−== ∑

=

=

−

−

1

6)1(22MM

M sQPsγ

51

Error Probability for MQAM

MQAM system can be viewed as two MPAM systems with signal constellations of size transmitted over the in-phase and quadrature signal components, each with half the energy of the original MQAM system.

The constellation points in the in-phase and quadrature branches take values

The symbol error probability for each branch is

The probability of symbol error the for MQAM system is If we take a conservative approach and set the number of nearest neighbors to be

four,

ML =

LidLiAi ,...,2,1 ,)12( =−−=

−−

=1

3)1(2branch, M

QM

MP ss

γ

−×≈

134

MQP s

sγ

2branch, )1(1 ss PP −−=

52

Summary in Error Probability for Coherent Modulation (1)

Many of the exact or approximation values for derived for coherent modulation are in the following form:

− αM : the number of nearest neighbors at the minimum distance dmin − βM : a constant that relates the minimum distance to average symbol energy

Performance specifications are generally most concerned with the bit error rate as a function of the bit energy. − With Gray coding and high SNR, and

≈ bMMbb QP γβαγ ˆˆ)(

sP

( )sMMss QP γβαγ ≈)(

MMM 2logˆ αα = MMM 2logˆ ×= ββ

53

Summary in Error Probability for Coherent Modulation (2)

Modulation

BPSK

QPSK

MPAM

MPSK

MQAM

)( ssP γ )( bbP γ

( )bb QP γ2=

( )bb QP γ2≈

( )

−

−≈

1log6

log12

22

2 MMQ

MMMP b

bγ

≈

MMQ

MP bb

πγ sinlog2log

22

2

−

≈1

log3log

4 2

2 MMQ

MP b

bγ

−

≈1

34M

QP ss

γ

≈

MQP ss

πγ sin22

( )

−−

=1

6122M

QM

MP ss

γ

( )ss QP γ2≈

54

Flat Fading Channel

Outage Probability Average Probability of Error

55

Performance Criteria (1)

In a fading environment, the received signal power varies randomly over distance or time due to shadowing and/or multipath fading. − In fading is a random variable with distribution , and therefore

is also random. Performance criteria

− The outage probability, , defined as the probability that falls below a given value corresponding to the maximum allowable

− The average error probability, , averaged over the distribution of

sγ )(γγ sf )( ssP γ

outP

sγsP

sP

sγ

56

Performance Criteria (2)

When the fading coherence time is on the order of a symbol time − The signal fading level is roughly constant over a symbol period − The error correction coding techniques can recover from a few bit errors − An average error probability is a reasonably good figure

When the signal fading is changing slowly − A deep fade affects simultaneously many symbols − Large error bursts that cannot be corrected for with coding of reasonable

complexity − Outage probability − When the channel is modeled as a combination of fast and slow fading (e.g.,

log-normal shadowing with fast Rayleigh fading), outage and average error probability is often combined

When , the fading will be averaged out by the matched filter in the demodulator − For very fast fading, performance is the same as in AWGN

)( cs TT ≈

)( cs TT <<

cs TT >>

57

Outage Probability (1)

Probability that is below a target , which is the minimum SNR required for acceptable performance. −

sγ Outage Ts

sγ

0γ

0γ

γγγγγ

γ dppPssout )()( 0

00 ∫=<=

58

Outage Probability (2)

In Rayleigh fading with mean zero and variance (dB) − The received signal power is exponentially distributed with average − The received SNR also has an exponential distribution with average

■

■ The probability density function of : − Outage probability

− Average SNR ■

0

2

0

2N

TNE ss

sσγ ==

2σ

s

sep

s

γγγ γγ /1)( −=

sγ

22σsγ

sγ

ss edeP ss

outγγ

γγγ γ

γ/

0

/ 0

0

11 −− −== ∫

)1ln(0

outs P−

−=

γγ

59

Average Error Probability

The averaged probability of error is computed by integrating the error probability in AWGN over the fading distributions. − An error probability in AWGN with SNR :

In Rayleigh fading,

− BPSK:

γγγ γ dpPPsss )()(

0∫∞

=

( )sM

M

sM

sMM

sMMs deQP s

γβα

γβγβαγ

γγβα γγ

25.015.01

21

0

/ ≈

+−=⋅= ∫

∞−

( )sMMs QP γβαγ ≈)(γ

)2()( bbb QP γγ =

+−=

b

bbP

γγ

11

21

60

Average Pb for MQAM in Rayleigh Fading and AWGN

Digital Modulation and Detection

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