Chapter 4: Bandpass Modulation and Demodulation/Detection

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Digital Communication 1 Chapter 4Chapter 4

Chapter 4: Bandpass Modulation and Demodulation/Detection

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4.1 Why Modulate?4.1 Why Modulate?

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Digital modulation : digital symbol : waveform compatible with the characteristic

of the channel Why use carrier? ⓐ reduce size of antenna (=3108m/fc)

e.g.) fc = 3kHz : antenna span : /4 = 25km

fc = 900 MHz : antenna diameter : /4 = 9cm

ⓑ frequency-division multiplexing ⓒ minimize the effect of interference : spread spectrum ⓓ place a signal in a frequency band where design requirements are met (e.g.)RF->IF

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4.2 Digital Bandpass Modulation 4.2 Digital Bandpass Modulation TechniqueTechnique

General form of a carrier wave

)](cos[)()(

)()(

)(cos)()(

0

0

tttAts

ttt

ttAts

4.2.1 Phasor Representation of a Sinusoid complex notation of a sinusoidal carrier wave

tjte tj00 sincos0

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Analytical form of transmitted waveform )(),(cos 0 AMt mm

221Re)( 0

tjtjtj

mm eeets

Analytical representation of narrowband FM(NFM)

tjtjtj mm eee

221 Res(t) 0

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4.2.2 Phase Shift Keying

4.2.3 Frequency Shift Keying

4.2.4 Amplitude Shift Keying

MiM

it

Mi

Tttt

T

Ets

i

ii

,...,12

)(

,...,1

0)(cos

2)( 0

Mi

Ttt

T

Ets ii ,...,1

0)cos(

2)(

Mi

Ttt

T

tEts i

i ,...,1

0)cos(

)(2)( 0

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4.3 Detection of signals in Gaussian 4.3 Detection of signals in Gaussian NoiseNoise

Two-dimensional signal space (M=2) Detector decides which of the signals s1 or s2 was transmitted, after receiving r =>Minimum-error decision rule chooses the signal class s.t. distance is minimized Decision region Decision rule

isrd

1 1

2 2

r Region s sent

r Region s sent

4.3.1 Decision Regions

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Step 2 : Choose waveform si(t) that has the largest correlation with r(t) Choose the si(t) whose index corresponds to the max Zi(T)

4.3.2 Correlation Receiver

Received signal Detection process Step 1 : Transform the waveform r(t) into a single random variable(R.V.)

MiTttntstr i ,...,1,0)()()(

),...,1()(..)( ' MiTZVRorTZ i

Matched filter (Correlator) maximizes SNR

T

ii dttstrTZ0

)()()(

Another detection approach (Fig.4.7.(b)) Any signal set can be expressed in terms of some set of basis functions

),...,1()( Mitsi ( ) ( 1,..., )j t j N where N M

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Signal NSignal N symbol M symbol M

Signal N< symbol MSignal N< symbol M

Ex) M-ary PSKEx) M-ary PSK

N=2N=2

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4.3 Detection of signals in Gaussian 4.3 Detection of signals in Gaussian NoiseNoise4.3.2.1 Binary Detection Threshold

Decision stage : choose the signal best matched to the coefficients aij (with the set of output Zj(T))

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4.3 Detection of signals in Gaussian Noise4.3 Detection of signals in Gaussian Noise

1 2

1( ) ( | ) ( | )

2p z p z s p z s

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Minimum error criterion for equally likely binary signals corrupted by Gaussian noise

For antipodal signals,

1 1 2

2

( ) ( ) ( )

( )

or decide s t if z T z T

s t otherwise

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4.4 Coherent Detection4.4 Coherent Detection

4.4.1Coherent Detection of PSK(BPSK) Coherent detector BPSK example

Orthonormal basis function):(

0)cos(2

)cos(2

)(

0)cos(2

)(

0

02

01

symbolperenergysignalE

TttT

E

tT

Ets

TttT

Ets

1 0

1 1

1 11 1 1

2 21 1 1

2( ) cos( ) 0

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

i i

t t t TT

s t a t

s t a t E t

s t a t E t

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When s1(t) is transmitted, the expected values of product integrator

4.4.1Coherent Detection of PSK(BPSK)

Decision stage Choose the signal with largest value of zi(T)

21 1 1 1

0

22 1 1 1 1 0

0

21 1 0 0

0

22 1 0 0

0

| ( ) ( ) ( )

2| ( ) ( ) ( ) , ( ) cos

2| cos ( ) cos )

2| cos ( ) cos )

T

T

T

T

E z s E E t n t t dt

E z s E E t n t t dt then t tT

E z s E E t n t t dt ET

E z s E E t n t t dt ET

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Example 4.1 Sampled Matched Filter Consider the BPSK waveform set

Illustrate how a sampled matched filter or correlator can be used to detect a received signal, say s1(t), from the BPSK Waveform set, in the absence of noise.

4.4.2 Sampled Matched Filter

ttsandtts cos)(cos)( 21

sec)1sec,25.0,10002..( mTmTge s

Sampled MF (N samples per symbol)

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Ex) Sampled MF (4 samples per symbol)

sampled s1

sampled s2

1( 3) 2z k

2 ( 3) 2z k

3

0212 ][]3[]3[

n

ncnskz

)(1 tTs

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Signal space for QPSK(quadri-phase shift keying), M=4 (N=2)

4.4.3 Coherent Detection of Multiple Phase Shift Keying

For typical coherent MPSK system, Orthonormal basis function

MiTtM

it

T

Etsi ,...,1,0)

2cos(

2)( 0

tT

ttT

t 0201 sin2

)(,cos2

)(

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Signal can be written as Received signal

),...,1,0(

)()2

sin()()2

cos(

)()()(

21

2211

MiTt

tM

iEt

M

iE

tatats iii

)arctan(ˆ

)()(:

)()(:

0

2

0

1

XY

dtttrYcorrelatorlower

dtttrXcorrelatorupper

T

T

Demodulator

4.4.3 Coherent Detection of Multiple PSK

)()()( tntstr i

decision i

)(2 t

)(1 t

8M 1i

8i

7i5i

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Demodulator of multiple-PSK

)arctan(ˆ

)()(:

)()(:

0

2

0

1

XY

dtttrYcorrelatorlower

dtttrXcorrelatorupper

T

T

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Typical set of FSK signal waveforms

Orthonormal set

Distance between any two prototype signal vectors is constant

2( ) cos( ) 0 1,...,i i

Es t t t T where i M

T

otherwise

jiforEta

tdtT

tT

Eta

NjtT

Et

ij

j

T

iij

jj

0

)(

cos2

cos2

)(

),...,1(cos)(

0

( , ) 2i j i jd s s s s E for i j

The ith prptotype signal vector is located on the ith coordinated axis a displacement from originE

4.4.4 Coherent Detection of FSK

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Example:3-ary FSK signal

Mi

i 2

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4.5 Non-coherent 4.5 Non-coherent DetectionDetection

Non-coherent detection : actual value of the phase

of the incoming signal is not required

4.5.1 Detection of Differential PSK

• For coherent detection, MF is used

• For non-coherent detection, this is not possible because MF

output is a function of unknown angle α

),...,1,0()(])(cos[2

)(:

)](cos[2

)(:

0

0

MiTttnttT

EtrsignalR

ttT

EtssignalT

ix

iix

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Differential encoding : information is carried by the difference in phase between two successive waveforms. To sent the i-th message (i=0,…,M), the present signal must have its phase advanced by over the previous signal Differential coherent detection : non-coherent because it does not require a reference in phase with received carrier Assuming that αvaries slowly relative to 2T, phase difference is independent of α as

Mi

i 2

)()()(])([])([ 21212 TTTTT ijkjk


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DPSK Vs. PSK DPSK : 3dB worse than PSK PSK compares signal with clean reference DPSK compares two noisy signals, reducing complexity

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4.5.2 Binary Differential PSK Example

)()()1()(

)()1()(

hereusedkmkckc

orkmkckc

Sample index k

Original message

Differential message

Correspondng phase

decoder

encoder

1 Arbitrary setting

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4.5 .3 Non-coherent Detection of 4.5 .3 Non-coherent Detection of Binary Differential FSK

• Just an energy detector without phase measurement• Twice as many channel branches• Quadrature receiver

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Three different cases :

)()cos()()3

)(sin)()2

)(cos)()1

1

1

1

tnttr

tnttr

tnttr

Another implementation for non-coherent FSK detection Envelop detector : rectifier and LPF Looks simpler, but (analog) filter require more complexity

4.5 .3 Non-coherent Detection of 4.5 .3 Non-coherent Detection of Binary Differential FSK

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In order for the signal set to be orthogonal, any pair of adjacent tones must have a frequency separation of a multiple of 1/T[Hz] cf) Nyquist filter

( ) (cos 2 ) ( )

1 2 2( )0 2

{ ( )} ( )

i i

i i

ts t f t rect TT Tfor t

twhere rect T Tfor t

Fourier transform

F s t Tsinc f f T

Minimum tone separation:1/T[Hz]

4.5.4 Required Tone Spacing for Non-coherent Orthogonal FSK Signaling

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4.5 Non-coherent Detection : 4.5 Non-coherent Detection : Example 4.3

⊙ Non-coherent FSK signal :

⊙ Coherent FSK signal :

1 2 1 20

1 2

1cos(2 )cos 2 0

. . 10,000 11,000 ?

1,000 / ,

1,000 / ,

Tf t f tdt for orthogonality f f

Te g two tones f Hz and f Hz orthogonal

if rate symvols s then orthogonal

if rate symvols s then not orthogonal

1 2

10

2f f

T

1 2 1 2cos(2 ) cos 2f t f t where f f

1 20

1 2 1 20 0

1 2 1 2

1 2 1 2

cos(2 )cos 2

cos cos 2 cos 2 sin sin 2 cos 2

sin 2 ( ) cos 2 ( ) 1cos sin

2 ( ) 2 ( )

T

T T

f t f tdt

f t f tdt f t f tdt

f f T f f T

f f f f

1 2 1 2

1 2 1 2

1 2

sin 2 ( ) cos 2 ( )0

2 ( ) 2 ( )

si 0

f f T f f T

f f f f

nce f f

Non-coherent이면 둘 다 0

이어야 함

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4.7 Error Performance for Binary 4.7 Error Performance for Binary SystemsSystems

4.7.1Probability of Bit Error for Coherently Detected BPSK

Antipodal signals

Basis function Decision rule is

TttT

E

tEtats

tEtatst

T

Ets

tT

Ets

0)cos(2

)()()(

)()()()cos(

2)(

)cos(2

)(

0

11212

1111102

01

TtfortT

t 0cos2

)( 01

otherwisets

Tzifts

)(

0)()(

2

01

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4.7 Error Performnace for Binary 4.7 Error Performnace for Binary SystemsSystems

dzaz

dzszPP

pdfofsymmetrysHPsHPP

sHPsHPP

sPsPsPsHPsPsHPP

u

aau

aaB

B

B

B

2

)(

2

0

2

0

2

)(2

2112

2112

21221112

21

210

2

1exp

2

1

)|(

)|()|(

)|(2

1)|(

2

12

1)()()()|()()|(

x

azudu

uXQ

0

22

,2

exp2

1)(

The same a priori

probability

a1 a2

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4.7 Error Performnace for Binary 4.7 Error Performnace for Binary SystemsSystems

02

2

00

020

21

0

21

2

)(

2

2

2exp

2

1

))(2)(2:)((

2

:,,

22exp

2

1

0

21

N

EQdu

uP

NRNPSDwithnoisewhitetn

NSince

symbolbinaryperenergysignalEEaEa

aaQdu

uP

b

NE

B

n

bbb

u

aau

B

b

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4.7 Error Performance for Binary 4.7 Error Performance for Binary SystemsSystems Another approach (1)

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4.7 Error Performance for Binary 4.7 Error Performance for Binary SystemsSystems Another approach (2)

s1s2

s1

s2

bE bEbE

bE

2

1

bd EE 42)2( bd EE

BPSK BFSK

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4.7 Error Performance for Binary 4.7 Error Performance for Binary SystemsSystems Probability of bit error for several types of binary systems

TABLE 4.1 Probability of Error for SelectedBinary Modulation Schemes

0

2

N

EQ b

0

exp2

1

N

Eb

0N

EQ b

02exp

2

1

N

Eb

Modulation

PSK(coherent)

DPSK(dfferentially coherent)

Orthogonal FSK(coherent)

PB

Orthogonal FSK(noncoherent)

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4.8 M-ary Signaling and 4.8 M-ary Signaling and PerformancePerformance

(R, Eb/No, BER, BW) : fundamental “trade-off”

M-ary orthogonal

k↑ BER↑ BW↑

4.8.2 M-ary Signaling(M=2k k:bits, M=# of waveforms)

M-ary PSK

k↑ BER↑

same BWShannon

Limit

-1.6dB

k=∞

①

②

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4.8.3 Vectorial View of MPSK Signaling

① (M=2k↑, the same Eb/No)

bandwidth efficiency (R/W) ↑, PB ↑ ② (M=2k↑, the same PB)

bandwidth efficiency (R/W) ↑, Eb/No ↑

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4.8.4 BPSK and QPSK : the same bit error probability

General relationship

R

W

N

S

R

W

N

S

N

Eb 22/

0

QPSK = two orthogonal BPSK channel (I stream, Q stream)

Magnitude

(A)

I stream ( A/root(2) )

Q stream ( A/root(2) )

Power/bit Half Half

Bit rate Half Half

QPSKBPSK

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4.8 M-ary Signaling and Performance4.8 M-ary Signaling and Performance

If original QPSK is given by R[bps], S[watt],

RN

S

R

W

N

S

N

EBPSKeach b 1

2

2:

000

Same BER, BW efficiency : BPSK=1,QPSK=2[bit/s/Hz] Eb/N0 vs. SNR

0

0 0 2 0

2

( ):

1

log

log: , : , 1( )

b

b b b

E N Normalized SNR the most meaningful way of comparing one digital system with another

E E ES W S WT S WT S

N N R N N M N k N N k

M kwhere W Detection BW R data rate WT typical

T TEff

:ect of normalized SNR noise increases as k increases

0bS RE N WN

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4.8 M-ary Signaling and Performance4.8 M-ary Signaling and Performance

Fig. 4.34 : M-ary orthogonal signaling at PE=10-3 in dB(decibel, nonlinear), factor(linear) k=10 (1024-ary symbol), 20SNR(factor)→2SNR per bit(factor); each bit require 2.

10:][10][][

)8(3:][77.4][][

)(2:][3][][

)(1:][][

log10][][

0

0

0

0

0

kdBdBN

SdB

N

E

PSKkdBdBN

SdB

N

E

QPSKkdBdBN

SdB

N

E

BPSKkdBN

SdB

N

E

kdBN

SdB

N

E

b

b

b

b

b

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4.9 Symbol Error Performance for M-ary 4.9 Symbol Error Performance for M-ary System(M>2)System(M>2)

4.9.4 Bit Error Probability vs. Symbol Error Probability for Multiple Phase Signaling

Assume that the symbol(011) is transmitted If an error occur, (010) or (100) is likely→3bit errors Gray code : neighboring symbols differ from one another in only one bit position

yprobabiliterroersymbolPPfork

P

M

PP EE

EEB ),1(

log2

BPSK vs. QPSK

QPSKforPP

BPSKforPP

BE

BE

2

1 (1 ) .kE BNote that P P

Chapter 4: Bandpass Modulation and Demodulation/Detection

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