Ch. 4 Bandpass Modulation and Demodulation...6 BANDPASS MODULATION • Bandpass modulation – The amplitude, frequency, or phase of a radio frequency (RF) carrier, or a combination

Department of Electrical EngineeringUniversity of Arkansas

ELEG5663 Communication Theory

Ch. 4 Bandpass Modulation and Demodulation

Dr. Jingxian Wu

[email protected]

2

OUTLINE

• Introduction

• Bandpass Modulation

• Coherent Detection

• Noncoherent Detection

• Complex Envelope

3

INTRODUCTION

• Bandpass modulation

– Recall: baseband modulation

• Mapping digital symbols onto baseband waveforms (pulse shapes)

that are compatible with channel.

– Bandpass modulation

• The baseband pulse shapes are translated to a higher frequency by

using a carrier wave (a high frequency sinusoid)

• Why bandpass modulate?

– To transmit a signal, antennas are usually ¼ of wavelength

• : higher frequency smaller wavelength smaller

antenna.

– E.g. 3KHz baseband signal,

– 1GHz bandpass signal,

– Translate the signal to a pre-allocated channel

• E.g. frequency division multiple access (FDMA)

fc /

4

INTRODUCTION

• Bandpass modulation and demodulation

5

OUTLINE

• Introduction





6

BANDPASS MODULATION

• Bandpass modulation

– The amplitude, frequency, or phase of a radio frequency (RF) carrier, or a

combination of them, is varied in accordance with the information to be

transmitted.)](2cos[)()( ttftAts c

– Through bandpass modulation, a baseband signal is shifted to a higher

frequency.

• Example, amplitude shift keying.

• Time domain:

• Frequency domain:

tftxtx cc 2cos)()(

)()(2

1)( ccc ffXffXfX

7

BANDPASS MODULATION

• Types of bandpass modulation

– Coherent (the receivers exploits knowledge of the carrier’s phase for

detection)

• Phase shift keying (PSK)

• Frequency shift keying (FSK)

• Amplitude shift keying (ASK)

• Continuous phase modulation (CPM)

• Hybrid: Quadrature amplitude modulation (QAM)

– Noncoherent (the receivers operate without knowledge of the absolute

value of the signal’s phase)

• Differential phase shift keying (DPSK)



• Continuous phase modulation (CPM)

• Hybrid: Quadrature amplitude modulation (QAM)

8

BANDPASS MODULATION: VECTOR REPRESENTATION

• Most bandpass modulated signals can be represented as

tfT

stfT

sts QI 00 2sin2

2cos2

)(

– Orthogonal representation of signal

tfT

t 01 2cos2

)( tfT

t 02 2sin2

)(

• and are orthonormal

– Proof:

)(1 t )(2 t

– The bandpass modulated signal can be represented as

• : there is a one-to-one relationship between and

the two dimensional vector

)()()( 21 tststs QI

],[)( QI ssts )(ts

QI ss ,

)(1 t

)(2 t

Is

Qs

Tt 0

Tt 0

9

BANDPASS MODULATION: VECTOR REPRESENTATION

• Vector representation of bandpass modulated signal

– Inphase component:

– Quadrature component:

– The inphase carrier, , and quadrature carrier,

are orthonomral.

– The bandpass modulated signal can be equivalently represented as vector

Is

Qs

tT

t 01 cos2

)(

],[)( QI ssts s

• Complex number representation of bandpass modulated signal

– There is a one-to-one relationship between a 2D vector and complex

number

)(1 t

)(2 t

Is

Qs

QIQI jssssts ],[)( s

– The bandpass modulated signal can be equivalently represented as a

complex number

QI jssts )(

– The complex number representation is only used

for mathematical convenience.

tT

t 01 sin2

)(

10

BANDPASS MODULATION: PSK

• Phase shift keying (PSK)

)](cos[2

)( 0 ttT

Ets ii Mi ,,1

Tt 0

– : carrier frequency

– Each digital symbol is mapped to a different phase

– Why ?

00 2 f

MiM

iti ,,1,

2)(

T

E2

– An example of BPSK

11

BANDPASS MODULATION: FSK


]cos[2

)( tT

Ets ii

–

– Each digital symbol is mapped to a different frequency.

– The set of signals, , could be orthogonal or non-orthogonal.

– An example of orthogonal FSK.

Mi ,,1

Tt 0

ii f 2

M

ii ts 1)}({

ij

T

jiT

Edttsts 0 )()(

12

BANDPASS MODULATION: ASK


]cos[)(2

)( 0 tT

tEts i

i Mi ,,1

Tt 0

– Each digital symbol is mapped to a different amplitude

– An example of BASK (on-off keying)

13

BANDPASS MODULATION: ASK

• Amplitude phase keying (APK)

]cos[)(2

)( 0 tT

tEts i

i

– Both amplitude and phase are altered by the digital symbol.

– An example of a 8-ary APK

Mi ,,1

Tt 0

14

OUTLINE

• Introduction





15

COHERENT DETECTION: BPSK

• Recall: correlation receiver of binary baseband signal

– the output of correlation receiver is the same as the output of matched

filter and sampler

2

210

aa

T

dttststsa0

2111 )()()( T

dttststsa0

2122 )()()(

– Bit error probability

02)(

N

EQEP d

T

d dttstsE0

2

21 )]()([

16


• Recall: correlation receiver of binary baseband signal (Cont’d)

– Bit error probability with vector signal representation

• is the squared distance between the two modulation points.dE

)()()( 2121111 tatats )()()( 2221212 tatats

T

d dttstsE0

2

21 )]()([

– Bipolar signal

0

2)(

N

EQEP b

– Orthogonal signal

0

)(N

EQEP b

17


• Binary Phase Shift Keying (BPSK)

]cos[2

)( 01 tT

Ets ]cos[

2]cos[

2)( 002 t

T

Et

T

Ets

– Orthogonal representation

tT

t 01 cos2

)( tT

t 02 sin2

)( Tt 0

)()( 11 tEts )()( 12 tEts

– Correlation receiver

tT

t 01 cos2

)(

2

210

aa

T

dtttsa0

111 )()(

T

dtttsa0

122 )()(

18


• Bit error probability of BPSK

02)(

N

EQEP d

T

d dttstsE0

2

21 )]()([

T

d dttstsE0

2

21 )]()([

0

2)(

N

EQEP b

19


• BPSK: revisit maximum likelihood detector

)()()( tntstr i

nadtttntsdtttrTz i

T

i

T

01

01 )()()()()()(

2

2

2

)(exp

2

1)|(

ii

azszp

bEa 1 bEa 2

– Likelihood function

– Maximum likelihood detection rule:

• If , detect

• If , detect

• Equivalently:

– In the orthogonal representation of signal, choose the signal that has the smallest Euclidean distance with the received sample

21 azaz )(1 ts

21 azaz )(2 ts

20


• Example

– Find the expected number of bit errors made in one day by the following

continuously operating coherent BPSK receiver. The data rate is 1 kbps.

The input waveforms are and where A =

2mV and the double-sided noise PSD is .

tAts 01 cos)( tAts 02 cos)(

HzW /10 10

COHERENT DETECTION: MAXIMUM LIKELIHOOD

• Maximum Likelihood Detection

– Signal:

– Rx Signal:

– Correlation receiver:

– Likelihood function

21

n

j

jiji tats1

)()(

)()()( tntstr i

…

)(1 t

)(tn

Max

imu

m L

ikel

ihoo

d

1z

nz

is

COHERENT DETECTION: MAXIMUM LIKELIHOOD

• Maximum Likelihood Detection

– Choose that minimizes

22

)(tsi

n

k

ikk az1

2)(

• Graphical Interpretation

– M-ary modulation, there are M constellation points:

– After correlation detector, there is an n-dimension point:

– Detection: choose such that the Euclidean distance is minimized

),,( 1 mnmm aa s

),,( 1 nzz z

is

zs i

23

COHERENT DETECTION

• Maximum Likelihood detection: 2-Dimension

– Choose such that the Euclidean distance between the vector and is

22 )()(|||| iQQiIIi srsr sr

ris

are minimized.

– Decision region example

• Whenever the received signal is located in region 1, choose signal

• Whenever the received signal is located in region 2, choose signal

r 1s

r 2s

24

COHERENT DETECTION: MPSK

• Multiple Phase Shift Keying (MPSK)

M

it

T

Etsi

2cos

2)( 0

Tt 0

Mi ,,1

– Orthogonal representation of MPSK signals

tTM

iEt

TM

iEtsi 00 sin

22cossincos

22cos)(

)(2

cossin)(2

cos)( 21 tM

iEt

M

iEtsi

– Inphase component

M

iEsI

2cos

– Quadrature component

M

iEsQ

2sin

25


• MPSK coherent detection

– Structure of receiver

– Likelihood functions

IiI

T

i nsdtttntsX 0 1 )())()(( QiQ

T

i nsdtttntsY 0 2 )())()((

2

2

2

)(exp

2

1)|(

iIi

sXsXp

2

2

2

)(exp

2

1)|(

QI

i

sYsYp

2

22

2 2

)()(exp

2

1)|,(

iQiI

i

sYsXsYXp

– Maximum likelihood detection

• Choose that minimizes )(tsi22 )()( iQiI sYsX

26


• MPSK coherent detection (Cont’d)

– The distance between and

– Maximum likelihood decision rule

T

dtttrX0

1 )()( T

dtttrY0

2 )()(

• Choose that minimizes the distance between and

• Equivalently, choose with phase that is closest to the phase

of the signal at the output of the correlator:

– Find that minimize

)(tsi

)(tsi i

X

Yarctanˆ )(tsi

|ˆ|

),( iQiIi sss

22 )()( iQiIi sYsXd

),( YXr

),( YXr

),( iQiIi sss

27


• Example:

– For a system with 8PSK, if the received symbols are:

28

COHERENT DETECTION: MFSK

• Multiple frequency shift keying (MFSK)

tT

Ets ii cos

2)(

– The value of can be chosen such that are mutually

orthogoanal orthogonal MFSK

• Orthogonal MFSK is a special case of MFSK

• We are only going to examine orthogonal MFSK

– Orthogonal MFSK

Tt 0

Mi ,,1

i M

iit 1cos

tT

t ii cos2

)( ij

T

ji dttt 0 )()(

– Example: Show the following system is orthogonal BFSK if

Tff c

2

11

Tff c

2

12

Tfc

1

29


• Coherent receiver structure of MFSK

– Structure of a receiver

– Output of the correlation detector for MFSK

mim

T

miim nsdtttntsr 0 )())()((

)()()()( 2211 tstststs MiMiii

• Output of the correlation detector of MFSK is coordinate in

orthogonal signal representation

),,,( 21 iMii rrr r),,,( 21 iMiii sss s

30


• Maximum Likelihood detection

– Choose that minimizes the Euclidean distance between)(tsi

),,,( 21 iMii rrr r),,,( 21 iMiii sss s

• Bit error probability of BFSK

31

COHERENT DETECTION

• Vector representation of bandpass communication system

– Recall vector representation of bandpass modulated signal

)()()( 21 tststs iQiIi

tfT

t 01 2cos2

)( tfT

t 02 2sin2

)( Tt 0

)()()( 21 tntntn QI

)()()()()()( tnstnstntstr QQQIIIi )()()( trtrtr QQII

IiII nsr

QiQQ nsr

– The bandpass communication system is equivalently represented as the

summation of signal vector and noise vector

nsr i

],[ QI rrr ],[ iQiIi sss ],[ QI nnn

32

COHERENT DETECTION

• Maximum likelihood detection

– For AWGN with two-sided PSD

• The noise variance per dimension is

– Likelihood functions

2

2

2 2

)(exp

2

1)|(

iIIiII

Srsrp

2

0N

2

02 N

2

2

2 2

)(exp

2

1)|(

iQQ

iQQ

Srsrp

– and are independent In Qn

2

22

4 2

)()(exp

2

1),|,(

iQQiII

iQiIQI

srsrssrrp

– Maximize Minimize ),|,( iQiIQI ssrrp ),|,( iQiIQI ssrrp 22 )()( iQQiII srsr

22 )()(|||| iQQiIIi srsr sr

– Minimize the Euclidean distance between the vector and ris

33

OUTLINE

• Introduction





34

NONCOHERENT DETECTION

• Why noncoherent detection?

– Coherent detection requires the exact knowledge of the phase of the received signal

– Example: BPSK

• The distance between Tx and Rx is d

• At Tx, the signal is tfT

Ets b

02cos2

)(

• At Rx, the signal is

• : the amount of time the signal travels from Tx to Rx

• : the phase of the signal at the receiver

• In order to perform coherent detection, the Rx needs to know

– can be estimated through a circuitry called phase locked loop (PLL)

)(2cos2

)(2cos2

)( 00 ttfT

ETtf

T

Etr b

db

cdTd /

dTft 02)(

)(t

)(t

35


• Why noncoherent detection? (Cont’d)

– What if the Rx doesn’t have the knowledge of ?

– Example:

• Assume a system operates a 1GHz. If the distance between Tx and Rx

is 24.075m, find out the phase of the Rx signal.

)(t

• Noncoherent detection

– The Rx doesn’t require the knowledge of the absolute phase of the Rx

signal.

36

NONCOHERENT DETECTION: DPSK

• Differential PSK

– The information is carried by the phase difference between the current

symbol and the previous symbol

• Recall: for coherent PSK, the information is carried by the absolute

phase of one symbol.

– Example:

• The kth Rx symbol is

• The (k+1)th Rx symbol is

• The phase difference between the two consecutive symbols

ktfT

Etr 02cos

2)(

102cos2

)( ktfT

Etr

ikk 1

• The information is carried by the phase difference

– The Rx doesn’t need the knowledge of the absolute phase . The

information is carried by the phase difference.

M

ii

2

37


• Binary DPSK

– The essence of differential detection is that the data is carried in the phase

difference between two consecutive symbols

– Tx: differential encoding; Rx: differential decoding.

– Binary differential encoding

)()1()( kmkckc

• m(k): information

• c(k): differentially encoded bit

• : modulo-2 addition

• – : complement

• The information, m(k), is carried by the difference between c(k) and

c(k-1)

38


• Binary DPSK

– Binary differential decoding

39


• DPSK: pros and cons

– Pro:

• Doesn’t require the absolute value of the signal phase simpler

receiver

– Con:

• Two noisy signal are compared to detect the signal there are twice

as much noise as in coherent detection the performance is worse

compared to coherent detection

– Trade-off between complexity and performance

40

NONCOHERENT DETECTION: FSK

• Non-coherent detection of FSK



– If has been transmitted

• What are the values at the output of the non-coherent detector of FSK?

41



– The minimum tone space for non-coherent orthogonal FSK

42


• Minimum Tone Spacing for Coherent Orthogonal FSK

43


• Coherent detection v.s. non-coherent detection

44

45

OUTLINE

• Introduction





46

COMPLEX ENVELOPE

• Complex representation of bandpass modulated signal

tfT

tstfT

tsts QI 00 2sin2

)(2cos2

)()( tfj

QI eT

tjststs 022)]()([)(~

)(~Re)( tsts

– There is a one to one relationship between and )(ts )(~ ts

• Complex envelope

– The complex baseband signal is called complex envelope

• The envelope of the bandpass signal.

– The complex envelope is the same as the vector representation of the

signal up to a scaling factor

)()()( tjststg QI

)(tg

)()()(),()( tjststststs QIQI

47

COMPLEX ENVELOPE

• Complex envelope

)(|)(|)( tjetgtg – Polar representation

)()()( tjytxtg

• Amplitude )()(|)(| 22 tytxtg

• Phase)(

)(tan)( 1

tx

tyt

48

COMPLEX ENVELOPE

• Bandpass modulation can be divided into two steps (Constellation)

– 1. Baseband modulation:

• Transfer information (‘1’s and ‘0’s) into complex envelope

• Example: 8PSK

1 0 0 0 1 0 1 1 1 0 0 1

– 2. Frequency upconversion

• Multiply the complex envelope with tfje 02

})(Re{)( 02 tfjetgts

49

COMPLEX ENVELOPE

• Quadrature implementation of a modulator

– Baseband modulation:

• Mapping ‘0’s and ‘1’s to the values of x(t) and y(t).

– Frequency upconversion:

• Upconverting the frequency of the baseband signal through

quadrature modulation.

50

COMPLEX ENVELOPE

• Quadrature implementation of a demodulator

– Frequency downconversion:

• Downconverting the frequency of the bandpass signal.

– Baseband demodulation:

• Mapping the baseband signal to ‘0’s and ‘1’s

51

COMPLEX ENVELOPE: D8PSK

• D8PSK (Differential 8PSK)

– Baseband modulation

– Frequency upconversion

52

COMPLEX ENVELOPE: D8PSK

• D8PSK Demodulation

– Frequency downconversion

– Baseband demodulation

53

OUTLINE

• Introduction





• Error Probability

54

ERROR PROBABILITY: BINARY MODULATION

• Comparison of error performance of binary system

– BER of BPSK

0

2

N

EQP b

B

– BER of coherently detected, differentially encoded binary PSK

00

21

22

N

EQ

N

EQP bb

B

– BER of differentially detected, differentially encoded binary PSK (DPSK)

0

exp2

1

N

EP b

B

55


• Comparison of error performance of binary system

– BER for coherently detected binary orthogonal FSK

0N

EQP b

B

– BER for non-coherently detected binary orthogonal FSK

02exp

2

1

N

EP b

B

56


Coherent detection PSK > coherent detection of differentially encoded PSK > Differential detection of differentially encoded PSK (DPSK) > Coherent detection of orthogonal FSK > Noncoherent detection of orthogonal FSK.

57

ERROR PROBABILITY: MPSK

• Symbol error rate for MPSK (Fig. 4.35)

– Symbol error rate (SER): # of error symbols/# of symbols transmitted

MN

EQMP s

E

sin

22)(

0

– : symbol energy MEE bs 2log

58

ERROR PROBABILITY: MFSK

• SER for MFSK

0

)1()(N

EQMMP s

E

59

ERROR PROBABILITY: BER V.S. SER

• Relationship between BER and SER for MPSK (Fig. 4.39)

– Gray encoding: two adjacent symbols differ in 1 bit

• At high SNR, Most of the errors are the confusion between adjacent symbols

• At high SNR, 1 symbol error approximately corresponds to 1 bit error

EB PM

P2log

1

60

ERROR PROBABILITY: BER V.S. SER

• Relationship between BER and SER for orthogonal signals

EB PM

MP

1

2/

– The relationship is exact

Documents

Ch. 4 Bandpass Modulation and Demodulation...6 BANDPASS MODULATION • Bandpass modulation – The amplitude, frequency, or phase of a radio frequency (RF) carrier, or a combination