Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK)...

45
PB.19 Coherent PSK The functional model of passband data transmission system is Modulator Signal transmission encoder i m Carrier signal ) (t s i Channel ) (t x Detector i s x Signal transmission decoder m ˆ i m is a sequence of symbol emitted from a message source. The channel is linear, with a bandwidth that is wide enough to transmit the modulated signal and the channel noise is Gaussian distributed with zero mean and power spectral density 2 / o N .

Transcript of Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK)...

Page 1: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.19

Coherent PSKThe functional model of passband data transmission systemis

ModulatorSignal

transmissionencoder

im

Carrier signal

)(tsi Channel)(tx

Detectoris x Signaltransmission

decoder m̂

• im is a sequence of symbol emitted from a messagesource.

• The channel is linear, with a bandwidth that is wideenough to transmit the modulated signal and thechannel noise is Gaussian distributed with zeromean and power spectral density 2/oN .

Page 2: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.20

Coherent PSK

The following parameters are considered for a signaling scheme:

Probability of error

A major goal of passband data transmission systems is the optimum design of the receiver so as to minimize the average probability of symbol error in the presence of additive white Gaussian noise (AWGN)

eP

Page 3: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.21

Coherent PSK

Power spectra

Use to determine the signal bandwidth and co-channel interference in multiplexed systems.

In practice, the signalings are linear operation, therefore, it is sufficient to evaluate the baseband power spectral density.

B 2B

Multiplexer

interference

Page 4: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.22

Coherent PSK

Example:– Raised cosine spectrum (α=0.5)– Binary Phase shift keying (BPSK)

=(1+α)Rb=1.5Rb

2Rb

Page 5: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.23

Coherent PSK

Bandwidth Efficiency

– Bandwidth efficiency bits/s/Hz

where is the data rate and B is the used channel bandwidth.

Example: Nyquist channel for baseband data transmission

Bandwidth B = W = 1/2Tb.

BRb=ρ

bR

W

bits/s/Hz 22/1/1

===∴b

bb

TT

BR

ρ

Page 6: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.24

Coherent PSK

In a coherent binary PSK system, the pair of signals and

used to represent binary symbols 1 and 0, respectively, is defined by

where , and is the transmitted signal energy per bit

)(1 ts

)(2 ts

)2cos(2

)(1 tfTE

ts cb

b π=

)2cos(2

)2cos(2

)(2 tfTE

tfTE

ts cb

bc

b

b πππ −=+=

bT

bTt ≤≤0 bE

Page 7: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.25

Coherent PSK

Example:

To ensure that each transmitted bit contains an integral number of cycles of the carrier wave, the carrier frequency

is chosen equal to for some fixed integer n.

[ ] bb

b

bT

cb

bTE

TTE

dttfTE

dttsE bb =⋅=== ∫∫ 22

)2(cos2

)(0

2

0

21 π

cf bTn /

cb fT /2=

Page 8: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.26

Coherent PSK

The transmitted signal can be written as

and

where

)()(1 tEts bφ=

)()(2 tEts bφ−=

bcb

TttfT

t <≤= 0 )2cos(2)( πφ

1 )2cos(2)( :Note0

2

2 =

= ∫

bT

cb

dttfT

t πφ

Page 9: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.27

Generation of coherent binary PSK signalsTo generate a binary PSK signal, the first step is representing the input binary sequence in polar form with symbols 1 and 0 represented by constant amplitude levels of and , respectively.

This signal transmission encoder is performed by a polar nonreturn-to-zero (NRZ) encoder.

Signaltransmission

encoder

101011 is 101011

−+

=0 is symbolinput 1 is symbolinput

b

bi E

Es

Page 10: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.28

Generation of coherent binary PSK signals

The second step is multiplying the carrier encoder output with the carrier

ProductModulator

is

)2cos(2)( tfT

t cb

πφ =

)(tsi

−=−=

===

bicb

b

bicb

b

i

EstfTE

ts

EstfTE

tsts

if)2cos(2

)(

if)2cos(2

)()(

2

1

π

π

bc Tnf /=

Page 11: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.29

Detection of coherent binary PSK signals

To detect the original binary sequence of 1s and 0s, we apply the noisy PSK signal to a correlator. The correlatoroutput is compared with a threshold of zero volts.

∫bT

0)(tx

)(tφ

Correlator

X1x Decision

device

0

0 if 00 if 1

1

1

<>

xx

Page 12: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.30

Detection of coherent binary PSK signals

Example

If the transmitted symbol is 1,

and the correlator output is

Similarly, if the transmitted symbol is 0, .

)2cos(2

)( tfTE

tx cb

b π=

b

T

cb

b

T

cb

cb

b

T

E

dttfT

E

dttfT

tfTE

dtttxx

b

b

b

=

⋅=

⋅=

=

0

2

0

01

)2(cos2

)2cos(2)2cos(2

)()(

π

ππ

φ

bEx −=1

Page 13: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.31

Error probability of binary PSK

We can represent a coherent binary system with a signalconstellation consisting of two message points.

• The coordinates of the message points are all thepossible correlator output under a noiselesscondition.

• The coordinates for BPSK are bb EE − and .

)(tφ

Decisionboundary

bEbE−

Page 14: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.32

Error probability of binary PSK

There are two possible kinds of erroneous decision:

– Signal is transmitted, but the noise is such that the received signal point inside region with and so thereceiver decides in favor of signal .

– Signal is transmitted, but the noise is such that the received signal point inside region with and so the receiver decides in favor of signal .

)(2 ts01 >x

)(1 ts

01 <x)(1 ts

)(2 ts

∫ bT

0

)(tφ

X 1x Decisiondevice

0

0if00if1

1

1

<>

xx

)()( twtsi +

Page 15: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.33

Error probability of binary PSK

For the first case, the observable element is related to the received signal by

is a Gaussian process with mean :

1x)(tx

[ ]

∫∫∫

+−=

+=

=

b

b

b

T

b

T

i

T

dtttwE

dtttwts

dtttxx

0

0

01

)()(

)()()(

)()(

φ

φ

φ

1x

b

T

b

ii

E

dtttwEE

xExb

−=

+−=

=

∫ ])()([

][

Page 16: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.34

Error probability of binary PSK

Variance is

2

)(2

)()()(2

)()()]()([

)()()()(

)()(

])[(

0

2

0 0

0 0

0 0

2

0

22

o

To

T T o

T T

T T

T

ii

N

dttN

dtduututN

dtduutuwtwE

dtduutuwtwE

dtttwE

xxE

b

b b

b b

b b

b

=

=

−=

=

=

=

−=

∫ ∫

∫ ∫

∫ ∫

φ

φφδ

φφ

φφ

φ

σ

Page 17: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.35

Error probability of binary PSK

Therefore, the conditional probability density function of

, given that symbol 0 was transmitted is1x

+−=

−−=

o

b

o NEx

N

xxxf

21

2

211

1

)(exp1

2)(exp

21)0|(

π

σσπ

Page 18: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.36

Error probability of binary PSKand the probability of error is

∫∞

+−=

=

0 1

21

0 1110

)(exp1

)0|(

dxN

ExN

dxxfp

o

b

Putting )(1b

o

ExN

z += , we have

[ ]

=

−= ∫∞

o

b

NE

NE

dzzpob

erfc21

exp10/

210 π

Page 19: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.37

Error probability of binary PSK

Similarly, the error of the second kind

==

o

b

NEpp erfc

21

1001 and hence

=

o

be N

Ep erfc21

Page 20: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.38

Error probability

The probability of bit error rate is proportional to the distance between the closest points in the constellation.

BPSK Binary FSK

=

=

oo

be N

derfcNEerfcP

221

21

bEbE−

=

=

oo

be N

derfcNEerfcP

221

221

bE

bEd d

Page 21: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.39

Transmission Bandwidth

The power spectral density (PSD) of the BPSK for both rectangular and raised cosine rolloff pulse shapes are plotted.

null-to-null bandwidth = 2Rb

=(1+α)Rb=1.5Rb

bps/Hz 5.02

===b

bb

RR

BRρ

Page 22: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.40

Quadriphase-shift keying (QPSK)

QPSK has twice the bandwith efficiency of BPSK, since 2 bits are transmitted in a single modulation symbol. The data input is divided into an in-phase stream , and a quadrature stream .

)(tdk )(td I)(tdQ

)(tdk

)(td I

)(tdQ

:1001

:10

:01

Page 23: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.41

QPSK

t

1 0 0 1

t

1 0

t

0 1

)(tdk

)(tdI

)(tdQ

bT

bTT 2=

Page 24: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.42

QPSKThe phase of the carrier takes on one of four equallyspaced values, such as π/4, 3π/4, 5π/4, and 7π/4.

≤≤−+=elsewhere0

0]4/)12(2cos[2)( Ttitf

TE

ts ci

ππ

where .4,3,2,1=iE is the transmitted signal energy per symbol;T is the symbol duration;

Tnfc /= ;

)2 :(Note bTT =

Page 25: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.43

QPSK

The transmitted signal can be written as

)()(

]4/)12sin[(]2sin[2

]4/)12cos[(]2cos[2

]4/)12(2cos[2)(

2211 tsts

itfTE

itfTE

itfTEts

ii

c

c

ci

φφ

ππ

ππ

ππ

+=

−−

−=

−+=

where

]2sin[2)(;]2cos[2)( 21 tfT

ttfT

t cc πφπφ ==

Page 26: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.44

QPSK

1is = 2/E or 2/E−

2is = 2/E or 2/E−

1000

01 11

2/E2/E

)(1 tφ

)(2 tφ

Page 27: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.45

QPSKEach possible value of the phase corresponds to a unique dibit. For example: Gray code

• only a single bit is change from one dibit to the next

1000

01 11

Page 28: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.46

QPSKDifferent QPSK sets can be derived by simply rotating the constellation.

1000

01

11

Page 29: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.47

Page 30: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.48

Generation of coherent QPSK signals

The incoming binary data sequence is first transformed intopolar form by a nonreturn-to-zero level encoder. The binarywave is next divided by means of a demultiplexer into twoseparate binary sequences.

The result can be regarded as a pair of binary PSKsignals, which may be detected independently due tothe orthogonality of )(1 tφ and )(2 tφ .

Page 31: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.49

Demulti-plexerPolar NRZ

10101 is

)2cos(2)(1 tfT

t cb

πφ =

)(ts

is1

is2

X

X

+

)2sin(2)(1 tfT

t cb

πφ =

Page 32: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.50

Detection of coherent QPSK signals

∫T

0)(tx

)(1 tφIn-phase channel

X1x Decision

device

0

multiplexer

∫T

0

)(2 tφ

X2x Decision

device

0

0 if 00 if 1

1

1

<>

xx

0 if 00 if 1

2

2

<>

xx

Quadrature channel

Page 33: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.51

Error probability of QPSK

The received signal is)()()( twtstx i +=

and the observation elements are

∫∫

+±=

=T

T

dtttwE

dtttxx

0 1

0 11

)()(2/

)()(

φ

φ

∫∫

+±=

=T

T

dtttwE

dtttxx

0 2

0 22

)()(2/

)()(

φ

φ

Page 34: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.52

As a coherent QPSK is equivalent to two coherent binaryPSK systems working in parallel and using two carriers thatare in phase quadrature.

Hence, the average probability of bit error in each channelof the coherent QPSK system is

=

=

oo NE

NEp

2erfc

212/erfc

21

Page 35: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.53

Error probability of QPSK

As the bit error in the in-phase and quadrature channels ofthe coherent QPSK system are statistically independent, theaverage probability of a correct decision resulting from thecombined action of the two channels is

+

−=

−=

−=

oo

o

c

NE

NE

NE

pp

2erfc

41

2erfc1

2erfc

211

)1(

2

2

2

Page 36: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.54

The average probability of symbol error for coherent QPSKis therefore

12/if2

erfc

2erfc

41

2erfc

1

2

>>

=

−=

oo

oo

ce

NENE

NE

NE

pp

Page 37: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.55

In a QPSK system, since there are two bits per symbol, thetransmitted signal energy per symbol is twice the signalenergy per bit,

bEE 2=

and then

o

be N

Ep erfc t

1 0 0 1

t

1 0

t

0 1

)(tdk

)(tdI

)(tdQ

Page 38: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.56

With Gray encoding, the bit error rate of QPSK is

Therefore, a coherent QPSK system achieves the sameaverage probability of bit error as a coherent binary PSKsystem for the same bit rate and the same ob NE / but usesonly half the channel bandwidth.

=

o

b

NEerfc

21BER

Page 39: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.57

Note: The probability of bit error rate is also proportional to the distance between the closest points in the constellation.

=

=

oo

b

Nd

NE

2erfc

21erfc

21BER

b

b

E

E

Ed

2

2/22

2/2

=

=

=

Page 40: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.58

Transmission Bandwidth

Power spectral density (PSD) of the QPSK for both rectangular and raised cosine rolloff pulse shapes:

null-to-null bandwidth = Rb

=(1+α)Rb/2=0.75Rb

bps/Hz 1===b

bb

RR

BRρ

Page 41: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.59

M-ary PSK

During each signaling interval of duration T, one ofthe M possible signals

iiM

tfTEts ci ,...,2,1)1(22cos2)( =

−+=

ππ

is sent.

MTTMEE

b

b

2

2

loglog

==

Page 42: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.60

M-ary PSK

( ) ( ) )(2

1cos)(2

1cos)( 21 tiEtiEtsi φπφπ

−−

−=

tfT

t cπφ 2cos2)(1 =

tfT

t cπφ 2sin2)(2 =

Page 43: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.61

M-ary PSK

The signal constellation of M-ary PSK consists of Mmessage points which are equally spaced on a circle ofradius E . For example, the constellation of8-ary phase-shift keying is

MNEP

oe

πsinerfc 4≥ME

The average probability of symbol error is

=

oNderfc

2d( )ME /sin π

M/π

Page 44: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.62

Transmission Bandwidth

Power spectral density (PSD) of the M-ary PSK for both rectangular and raised cosine rolloff pulse shapes:

Page 45: Coherent PSK - EIEem/dtss05pdf/00g Passband2.pdf · – Binary Phase shift keying (BPSK) =(1+α)Rb=1.5Rb 2Rb. PB.23 Coherent PSK Bandwidth Efficiency – Bandwidth efficiency bits/s/Hz

PB.63

Transmission Bandwidth

Null-to-null bandwidth efficiency of a M-ary PSK signal:

28.523.418.51410.510.5

32.521.510.5

643216842M

bps/Hz log21

log/2 22

MMR

RBR

b

bb ===ρ

ρ

)10BER(/ 6−=ob NE