4 - Proakis Chapter 4 - Charaterization of Comm Signals and Systems 1-7-2010 Printable Version

Digital Communication Dr. Mahlab Uri1

Chapter 4

characterization of communication

signals and systems

Digital Communication Dr. Mahlab Uri

contents

4-1 Parts 1,2- Representation of Bandpass Signals4-1-1 Representation of Bandpass Signals

4-1-2 Representation of Linear Bandpass System

4-1-3 Response of a Bandpass Systems to a Bandpass signal

4-1-4 Representation of Bandpass Stationary Stochastic Processes

4-2 Parts 3 signal space representation4-2-1 Vector Space Concept

4-2-2 Signal Space Concept

4-2-3 Orthogonal Expansions of Signals

4-3 Parts 4,5,6 Representation of digitally modulated signals4-3-1 Memoryless Modulation Methods

4-3-2 Linear Modulation with Memory

4-3-3 Nonlinear Modulation Methods with Memory


Part - 1


4-1 Representation of Bandpass Signals

In the next subsections we will represent the

bandpass signals and systems in terms of equivalent

lowpass waveforms and the characterization of

band pass stationary stochastic processes


Representation of 1-1-4Bandpass Signals

Signals and channels that satisfy the condition that their bandwidth is much

smaller than the carrier frequency are termed narrowband bandpass signals

and channels.

Is a real value signal that shown in figure ts (4-1-1)

Figure (4-1-1)


a signal that contains only the positive frequencies in ts

fSfufS 2 (4-1-1) fS Is the Fourier transform of ts

fu Is the unit step function

The Equivalent time-domain expression is:

FSFfuF

dtefSts ftj

11

2

2

(4-1-2)

ts the analytic signal or pre-envelope of ts


t

jtfuF

21

tst

jts

tst

jtts

(4-1-3)

(4-1-4)


tst

jts

tst

jtts

d

t

sts

tts

11

th ts ts

tt

th ,1

Such a filter is called a Hilbert transformer

(4-1-5)

(4-1-4)

(4-1-6)


The frequency response of a Hilbert transformer

0

00

0

11 2

2

fj

f

fj

dtet

dtethfH

ft

ft

(4-1-7)

This filter is basically a phase shifter for all frequencies f


cl ffSfS

tfj

tfj

l

c

c

etsjts

etsts

2

2

1F

(4-1-8)

(4-1-9)

tfjl cetstsjts2

(4-1-10)

frequency translation

equivalent lowpass representation


tjytxtsl

tftytftxts cc 2sin2cos

tftytftxts cc 2cos2sin

The signal is a complex-value and may be expressed as: tsl

(4-1-11)

(4-1-12)

(4-1-13)

ty tx and are called the quadrature componentsof the bass band signal ts


t

cfj

etl

st

cfj

etjytxts 2

Re2

Re

tsl is usually called the complex envelope of the Real signal ts , and it basically the equivalentlowpass signal.

(4-1-14)

II- Another representation is:


tjl etats

tytxta 22

tytx

t 1tan

Finally a third possible representation is:

Where:

(4-1-15)

(4-1-16)

(4-1-17)


is called the envelop of ta ts t is called the phase of ts

ttftaeta

etsts

c

ttfj

tfj

l

c

c

2cos

Re

Re

2

2

Then:

(4-1-18)


Therefore:

tfjl cetsts 2Re

tftytftxts cc 2sin2cos

ttftats c 2cos

Are equivalent representation of bandpass signals

(4-1-12)

(4-1-14)

(4-1-18)


dtedtets

dtetsfS

ftjtfj

l

ftj

c

22

2

Re

The Fourier transform of ts is:

Using of the identity

2

1Re

(4-1-19)

(4-1-20)


clcl

ftjtfj

l

tfj

l

ffSffS

dteetsetsfS cc

2

1

2

1 222

The basic relationship between the spectrum of

the real bandpass signal and the spectrum of

the equivalent lowpass signal

fS fSl

(4-1-21)


dtets

dtts

tfj

lc

22

2

Re

The energy in the signal ts is defined as:

Using the identity we obtain:

dtttfts

dttsE

cl

l

24cos2

1

2

1

2

2

(4-1-22)

(4-1-23)

2

1Re


ta 2 varies slowly relative

the energy in bandpass signal expressed in terms of

the equivalent lowpass signal is:

dttsE l

2

2

1

Where tsl is just the envelop ta of ts

(4-1-24)

E - The energy in bandpass signal

ttftats c 2cos


Representation of Linear 2-1-4Bandpass System

In this section we will described a filter or

a system by its impulse response or by its

frequency response.

th


A linear filter or system may be described either

by its impulse response th or by its frequencyresponse fH

Since th is real

fHfH (4-1-25)


We define cl ffH as:

00

0

f

ffHffH cl

Then:

0

00

ffH

fffH cl

(4-1-27)

(4-1-26)


Using we have:

clcl ffHffHfH

(4-1-28)

tfjl

tfj

l

tfj

l

c

cc

eth

ethethth

2

22

Re2

(4-1-29)

is the inverse Fourier transform and its a complex value of thl fH l

fHfH

1F frequency translation


Response of a Bandpass 3-1-4Systems to a Bandpass signal

In this section we demonstrate that output of a bandpass

system to a bandpass input signal is simply obtained

from the equivalent lowpass input signal and the

equivalent lowpass impulse response of the system.


th

ts tr

tsl thl

ts Is a narrowband bandpass signal, with an equivalent lowpass signal

?trl

tsl

Is the impulse response of a narrowband bandpass system, with an

equivalent lowpass impulse response

th

thl


The output of the bandpass system is also a bandpass

signal, and it can be expressed as:

tfjl cetrtr 2Re(4-1-30)

And is related to the input signal and the impulse

response by the convolution integral:

dthstr

(4-1-31)


in the frequency domain expressed by: tr

fHfSfR (4-1-32)

Substituting from: fS

clcl

ftjtfj

l

tfj

l

ffSffS

dteetsetsfS cc

2

1

2

1 222

and from: fH clcl ffHffHfH

for

for


We obtain the result:

clcl

clcl

ffHffH

ffSffSfR

2

1

(4-1-33)

0 cl ffS

0* clcl ffHffS

and

ts narrowband bandpass signal th impulse response of a narrowband system

0* clcl ffHffS

0 cl ffH for 0f


Therefore, simplifies to :

clcl

clcl

clcl

ffRffR

ffHffS

ffHffSfR

2

1

2

1

(4-1-34)

clcl

clcl

ffHffH

ffSffSfR

2

1


The output spectrum of the equivalent lowpass

system exited by the equivalent lowpass signal:

fHfSfR lll (4-1-35)

The time domain relation is given by the

convolution integral:

dthstr lll

(4-1-36)


NOTE 1:The combination of

with gives the relationship between the

bandpass output signal and the equivalent lowpass time

functions and tsl thl

NOTE 2:This simple relationship allows as to ignore any linear

frequency translations encountered in the modulation of signal for

purposes of matching its spectral content to frequency allocation of a

particular channel.

Thus, for mathematical convenience, we shall deal

only with transmission of equivalent lowpass

signals through equivalent lowpass channels.

summary

dthstr lll

tfjl cetrtr 2Re tr


Part - 2


Representation of Bandpass 4-1-4Stationary Stochastic Processes

In this section we extend the representation to sample

function of a bandpass stationary stochastic process.


tn Is a sample function definitions of wide-sense stationary stochastic process with zero mean and

power spectral density fnn

The stochastic process Is said to be a narrow

bandpass process if the width of the spectral

density is much smaller than the carrier frequency

tn

cf

definitions

fnn


under this conditions can be represented

by 3 equivalent forms:

tn

ttftatn c 2cos 1

tfj cetztn 2Re 3

tftytftxtn cc 2sin2cos 2

(4-1-37)

(4-1-38)

(4-1-39)

ta The envelope of the real valued signal.

t The phase of the real valued signal. tx ty tn

The complex envelope of tn tz

The quadrature components of .


yyxx

If tn is zero mean then tx tyand must also havezero mean values. In addition the stationary of tnImplies that the autocorrelation and cross-correlation

function of tx tyand satisfy the followingproperties:

yxxy

(4-1-40)

(4-1-41)


The autocorrelation function nn of tn

tftytftxtftytftxEtntnE cccc 2sin2cos2sin2cos

(4-1-42)

tftf

tftf

tftf

tftf

ccyx

ccxy

ccyy

ccxx

2sin2cos

2cos2sin

2sin2sin

2cos2cos

tftytftxtn cc 2sin2cos


Use of the trigonometric identities:

(4-1-43)

BABABA

BABABA

BABABA

sinsin2

1cossin

coscos2

1sinsin

coscos2

1coscos


In (4-1-42) yields the result:

tf

fy

tf

f

tntnE

cxyyx

cxyyx

cyyxx

cyyxx

22cos2

1

2sin2

1

22cos2

1

2cos2

1

(4-1-44)


Must be independent of t .

cyxcxxnn ff 2sin2cos (4-1-45)

the right-hand side of:

yyxx yxxy

tn Is stationary.

yesno

?

tf

fy

tf

f

tntnE

cxyyx

cxyyx

cyyxx

cyyxx

22cos2

1

2sin2

1

22cos2

1

2cos2

1


The autocorrelation function of the equivalent

lowpass process:

tjytxtz (4-1-46)

Is defined as:

tztzEzz2

1

(4-1-47)


yxxyyyxxzz jj 2

1

Substituting into

we obtain:

(4-1-48)

If symmetry properties given in and in

are used in

we obtain:

yxxxzz j(4-1-49)

tjytxtz tztzEzz2

1

yxxy yxxyyyxxzz jj 21

yyxx


Finally, we incorporate the result given by

into

cfjzznn e 2Re(4-1-50)

Thus, the autocorrelation function nn of thebandpass stochastic process is uniquely determined

from the autocorrelation function zz of the equivalent lowpass process tz and the carrier frequency

cf .

yxxxzz j cyxcxxnn ff 2sin2cos

and we have:


czzczz

fjfj

zznn

ffff

deef c

2

1

Re 22

The power density spectrum of the stochastic process tn

(4-1-51)

fzz is the power density spectrum of the equivalent

lowpass process tz

Since the autocorrelation function of tz satisfies thethe property *zzzz , it follows that

fzz Is a real-valued function of frequency.


Properties of the Quadrature Components

The cross-correlation function of the quadrature

components of tn

Furthermore, any cross-correlation function

satisfies the condition:

xyyx(4-1-52)

yxxy


xyxy(4-1-53)

is an odd function of xy . 00 xy

tx tyand uncorrelated (for 0 only). If 0xy for all than zz Is real and the power spectral density fzz satisfies the condition

ff zzzz (4-1-54)And vice versa. That is, fzz Is symmetric about 0f


222

222

1,

yx

eyxp

in the special case in which the stationary stochastic

process tn is Gaussian the quadrature components tx tyand are jointly Gaussian.

Moreover for 0 They are statistically independent,

hence, their joint probability density function is:

(4-1-55)

The variance is defined as: 0002 nnyyxx


Representation of White Noise

NOTE: White noise is a stochastic process that is defined to have a flat (constant) power spectral density over the entire frequency range. This

type of noise cannot be expressed in terms of quadrature components, as

result of its wideband character.

The power spectral density of bandpass white noise

resulting from passing the white noise process through

a spectrally flat (ideal) bandpass filter

Figure 4-1-3


Bandpass white noise can be represented by :

tz

Bf

BfN

fzz

2

10

2

10

(4-1-56)

ttftatn c 2cos 1

tfj cetztn 2Re 3

tftytftxtn cc 2sin2cos 2

The equivalent lowpass noise has a power

spectral density:


BNzz

sin0

The autocorrelation function is:

(4-1-57)

The limiting form of zz As B approaches infinityIs

:

0Nzz (4-1-58)


yyxxzz

The power spectral density for white noise and

bandpass white noise is symmetric about 0fso 0xy for all Therefore:

(4-1-59)

That is, the quadrature components tx tyand are uncorrelated for all time shifts are the

autocorrelation functions of tz tx tyand , are allequal.


Part - 3


4-2 signal space representation

In the next subsection we demonstrate that

signals are similar to vectors and we will

develop a vector representation for signal

waveform


Vector Space Concept1 -2-4

A vector with n-dimension

can be represent as a linear combination

of unit vectors or basis vectors

nvvv ......21

n

i

iievv1

Where a unit vector has length of unity and

Is the projection of vector v on ivie

(4-2-1)


The inner product of two vectors is defined as:

n

i

iievvv1

2121

The vectors are orthogonal if 021 vv

The norm of a vector (length) v is:

m

i

jvvvv1

2/1 2

For all : ji i1 mj

(4-2-2)

(4-2-4)

(4-2-3)

v


NOTE: A set of m vectors are defined as orthonormal if

the vectors are orthogonal and each one of them

are unit norm

NOTE: A set of m vectors are said to be linearly independent

if none of the vectors can be represented as a linear

combination of the remaining vectors


The triangle inequality of Two n-dimensional vectors

in the same direction:

2121 vvvv

2121 vvvv

If then we can say that the two

n-dimensional vectors are also satisfied the

Cauchy Schwartz inequality:

21 avv

(4-2-5)

(4-2-6)


The norm square (length) of the vectors:

212121 2222

vvvvvv

021 vvAnd if they orthogonal ( ) then:

2222121 vvvv

(4-2-7)

(4-2-8)


Let us take vectors:

nvvvv 1 12111 .....

1

1

1

v

vu

2v

Normalizing

subtracting the projection of onto

nvvvv 222212 .....

1u

11222 uuvvu

(4-2-11)

(4-2-12)

1v

nvvvv 232313 .....


Normalizing the vector to unit length:

2

2

2

u

uu

22311333 uuvuuvvu

3vSubtracting projection on and 2u1u

Normalized it and we will get the next

orthonormal vector :

3

3

3

u

uu

(4-2-13)

(4-2-14)

(4-2-15)


Signal Space Concept2-2-4

developing a parallel treatment to two generally

complex value signals and on some

interval of

tx2 tx1

ba,


The inner product of the complex signals is:

dttxtxtxtxb

a

2121 ,

If the inner product of the complex signals is

zero then the signals are orthogonal.

The norm of the signals is:

2/1

2

dttxtx

b

a

(4-2-16)

(4-2-17)


NOTE: As in the vector representation A set of m signals

are orthonormal if they are orthogonal and their

norms are all unity.

NOTE: A set of m signals is linearly independent, if no

signal can be represent as the linear combination of

the remaining signals.


The signals satisfy the triangle inequality:

txtxtxtx 2121

and the Cauchy-Schwartz inequality when

2/1

2

2/1

2

2 211

dttxdttxdttxtx

b

a

b

a

b

a

taxtx 12

(4-2-18)

(4-2-19)


Orthogonal Expansions of 3-2-4Signals

In this section we will develop a vector representation for

signal waveform and the equalities between a signal

waveform and its vector representation


deterministic, real-valued signal with finite energy: ts

dttss2

a set of orthonormal functions Nntfn ,......2,1,

nm

nmdttftf mn

1

0

(4-2-20)

(4-2-21)


The approximate signal by weighted linear

combination of:

ts

k

k

kk tfsts1

the coefficients ks

The error approximation:

Kksk 1,

tstste

(4-2-22)

(4-2-23)


we will like to select an to minimize the energy

of the approximate error

kse

dttfstsdttsts

k

k

kke

1

2

Two option to find the optimum of

1.differeniaing the series for each coefficient and setting the

firs derivative to zero

2.Multiply with a orthonormal function and base of the

mean-square-error criteria saying that the minimum will

obtain when the result will be zero .

ks

(4-2-24)

tfn

e


01

dttftfsts n

k

k

kk

Kn ,.....,2,1

dttftss nn

k

k

ks sdttsdttste1

22

min

(4-2-25)

(4-2-26)

(4-2-27)


if 0min

dttssk

k

tv

2

1

2

k

k

kk tfsts1

(4-2-28)

(4-2-29)


Appendix

Signal space Signal Space

Inner Product

Norm

Orthogonality

Equal Energy Signals

Distance

Orthonormal Basis

Vector Representation

Signal Space Summary

71


Energy dttsEsT

)(0

2

ONLY CONSIDER SIGNALS, s(t)

Tt

tifts

00)( T

t

72


T

dttytxtytx0

)()()(),(

Similar to Vector Dot Product

x

yyx

cosyxyx

(x(t), y(t))-Inner Product

73


A

-A2A

A/2

T

Tt

t

Example

TAT

AATA

Atytx 2

4

3

2)2)((

2)

2)(()(),(

74


||x(t)||-Norm

T

ExEnergydttxtxtxtx0

22 )()(),()(

Extx )(

Similar to norm of vector

T

A

-A

x

xxx 2

ExT

AdttT

AtxT

2

)2

cos()(0

2

75


Orthogonality

0)(),( tytx T

dttytx0

0)()(

Similar to orthogonal vectors

T

A

-Ax

0 yx

T

Y(t)B

y

76


ORTHONORMAL FUNCTIONS

{

1)()(

0)(),(

tytx

and

tytx

TT

T

dttydttx

dttytx

0

2

0

2

0

1)()(

0)()(

T

T

X(t)

Y(t)

T/2

T/2

1

1

x

y

1)()(

0)(),(

tytx

tytx

77


Correlation Coefficient

EyEx

dttytx

tytx

tytx

T

0

)()(

)()(

)(),(

In vector presentation

1 -1

=1 when x(t)=ky(t) (k>0)

yx

yx cos

x

y

78


Example

T

TAdttytxtytx0

2

4

5)()()(),(

Now,

14.0

)8

7)(10(

45

)(),(2

TATA

TA

EyEx

tytx

shows the real correlation

t tA

-AT/2 7T/8

T

10A

X(t) Y(t)

79


Distance, d

ExEy2EyExd

dt)t(y)t(x)t(y)t(xd

2

T

0

222

For equal energy signals

)1(E2d2

=-1 (antipodal) E2d

3dB better then orthogonal signals

=0 (orthogonal) E2d

80


Equal Energy Signals

)1(2 Ed

E2d

E

y

x

PSK (phase Shift Keying)

tfAty

Tt

tfAtx

0

0

2cos)(

)0(

2cos)(

To maximize d

)()(

1

tytx

(antipodal signals)

E2d

81


EQUAL ENERGY SIGNALS

ORTHOGONAL SIGNALS (=0)

Ed 2

E

y

x

Ed 2FSK (Frequency Shift Keying)

tfAty

Tt

tfAtx

0

1

2cos)(

)0(

2cos)(

(Orthogonal if ...),2

3,1,

2

1)(

01 Tff

82


Signal Space summary Inner Product

T

dttytxtytx0

)()()(),(

Norm ||x(t)||

EnergydttxtytxtxT

0

22 )()(),()(

Orthogonality

)(1)()(

0)(),(

Orthogonaltytx

if

tytx

83


Corrolation Coefficient,

ExEy

dttytx

tytx

tytx

T

0

)()(

)()(

)(),(

Distance, d

ExEy2EyExd

dt)t(y)t(x)t(y)t(xd

2

T

0

222

84


Orthonormal Basis

Suppose we try to approximate x(t) by writing,

Xa(t) where

Suppose we have a function x(t) and we are

given a set of orthonormal functions, Nii

t,...,2,1

)(

N

iiia

txtatx1

)()()(

Question:

What are the best ai that we can select

such that Xa(t) is close to X(t) ?

85


We want to minimize the distance between x(t)

and xa(t) .

N

iiia

tatxtxtxd1

)()()()(Distance

The best ai are

...

)()()(),(

2

11

22

min

0

N

N

ii

T

iii

aaExd

dtttxttxa

opt

opt

86


In general is an othonormal basis in

if any function in can be written as

,...2,1

)(ii

t

T

iii

iii

dtttxttxa

where

Tttatx

0

1

)()()(),(

0)()(

2L

Sometimes called COMPLETE ORTHONORMAL SET (COS)

OTHONORMAL BASIS

2L

87


A Set of Orthogonal functions - Nii t ,...2,1)(

iallfor

dttt

jiif

dttttt

T

i

T

jiji

i

0

22

0

1)()(

0)()()(),(

May be simply written as

ji

ji

if

iftt

ijji

1

0{)(),(

Is the Kronecker Deltaij

88


EXAMPLE

Fourier Series

tT

22sin

T

2)t(

tT

22cos

T

2)t(

tT

2sin

T

2)t(

tT

2cos

T

2)t(

T1)t(

5

4

3

2

1

T

0

T

0

tdtT

2sin

T

2)t(x3a

tdtT

2cos

T

2)t(x2a

89


. encecorrespond one-to-one a have

,...)a,a(a vector,

theand (t)function x The*

)t(x a

a)t(x

)t(),t(xa

)t(a)t(x

21

ii

1i

i

Space RepresentationSignal

90


a1

a2

a3

a

The vector )3,2,1(a aaa

Represents one and only one function

(using a given basis) ,...2,1 ),( iti

3

1

)()(i

iitatx

Vector Representation

91


)(1

t

)(3

t

)(2

t

X(t)

...)t(a )t(a )t(a x(t)

:as drepresente becan x(t) thusbasis theIs -

)t( and )t(),t( :use we

3a and 2a,1a,a using of Instead

332211

i

321

Vector as a Signal

92


Ex)t(xa

Therefore,

a...aaaa

,...)a,a,a(a

a vector at thelook weifBut

a)t(x

)t(a,)t(aEx)t(x

)t(a)t(x

1i

2

i

2

3

2

2

2

1

2

321

1i

2

i

2

1j

jj

1i

ii

2

1i

ii

X(t) -

Norm (or Energy) In Signal Space

93


. .

)(1

t

)(3

t

)(2

t

X(t)

21a

53a

32a

Ex

JoulesEx

Ex

Extx

38

38

532)( 222

94


The Distance d, between x(t) and y(t) is

1

22

2

a

1

22

2

1

2

2

1 1

22

:is vectorsebetween th d distance the

b and a vectorsat thelook weifBut

)(

)()(

)()(

iiiba

b

iii

iiii

i iiiii

babad

bad

tbad

tbta

tytxd

Distance, d, In Signal Space

95


The Distance between the functions equals the distance between the

vectors

X(t)

Y(t)

51a

61b

22b

43a

03b 82 ad

Y(t)=(2,6,0)

x(t)=(8,5,4)

53

)04()28()65()()( 222

d

tytxd

96


bay(t)-x(t)Distance

ax(t)Energy

,...)a,a(a

a)t(x

Tt0

)t(),t(xa

)t(a)t(x

22

21

i

1i

ii

Signal Space Summary

97


Gram-Schmidt Procedure

NOTE: This procedure construct a set of orthonormal

vectors from a set of n-dimensional vectors by

normalize zing its length

98


Part - 4


4-3 Representation of digitally modulated signals

In the next subsections we will describe.

memoryless modulation methods.

Linear modulation with memory.

Nonlinear modulation methods with memory.

100


ASK

FSK

PSK

DSB


Memoryless Modulation 1-3-4Methods

Pulse-amplitude-modulated (PAM) signals

2( ) Re ( ) ( ) 2cj f tm m m cS t A g t e A g t Cos f t

1,2,....., , 0 t T 4.3-1m M

The denote the set of M possible amplitudes

corresponding to possible k-bit blocks of symbols.

, 1 m MmA 2kM

2d - distance between adjacent signals amplitudes.

(2 1 ) , m=1,2,...,M 4.3-2mA m M d

(4-3-1)

(4-3-2)

102


R/k - The symbol rate for the PAM signals

bT 1/ R bit interval

/ bT k R kT symbol interval

The M-PAM signals have energies:

2 2 2 20 0

1 1( ) ( ) 4.3-3

2 2

T T

m m m m gS t dt A g t dt A

g - the energy in the pulse g(t).

Digital PAM is also called amplitude-shift keying (ASK).

(4-3-3)

103


The one-dimensional (N=1) signal:

The signal waveform

f(t) - the unit-energy

( ) ( ) 4.3-4m mS t S f t

2

( ) ( ) 2 4.3-5cg

f t g t Cos f t

1

, m=1,2,....,M 4.3-62

m m gS A

(4-3-4)

(4-3-5)

(4-3-6)

104


The Euclidian distance between any pair of signal points:

The minimum Euclidean dist:

2(e)

mn

1d 2 4.3-7

2m n g m n gS S A A d m n

(e)mind 2 4.3-8gd

(4-3-7)

(4-3-8)

105


Signal space diagram for digital PAM signals

0 1

00 01 11 10

000 001 011 010 110 111 101 100

(a) M=2

(b) M=4

(c) M=8

106


Phase-modulated signalsThe M signals waveforms are represented as:

g(t) - the signal pulse shape

- the M possible phases of the carrier.

22 ( 1) /( ) Re ( ) , m=1,2,...,M, 0 t Tcj f tj m MmS t g t e e

2

( ) ( ) 2 ( 1) 4.3-11

2 2( ) ( 1) 2 ( ) ( 1) 2

m c

c c

S t g t Cos f t mM

g t Cos m Cos f t g t Sin m Sin f tM M

m

Phase-modulated signals is usually called phase-shift keying (PSK).

107


The energy of the signal waveforms:

Represented as a linear combination of two orthonormal

signals:

The two-dimensional vectors are:

2 20 0

1 1( ) ( ) 4.3-12

2 2

T T

m gS t dt g t dt

1 1 2 2( ) ( ) ( ) 4.3-13m m mS t S f t S f t

1 22 2

( ) 2 ( ) 2 4.3-14,15c cg g

f t Cos f t f t Sin f t

2 2

( 1) ( 1) , m=1,2,...,M 4.3-162 2

g g

mS Cos m Sin mM M

(4-3-12)

(4-3-13)

(4-3-14,15)

(4-3-16)

108


The Euclidean distance between signal point:

The minimum Euclidean distance: (|m-n|=1)

The preferred mapping or assignment of k information bits to the possible

phases is Gray encoding, so that the most likely errors caused by noise will result in

a single bit error in the k-bit symbol.

2kM

1/2

(e)

mn

2d 1 ( ) 4.3-17m n gS S Cos m n

M

(e)min2

d 1 4.3-18g CosM

(4-3-17)

(4-3-18)

109


Signal space diagrams for PSK

0 1 11

01

00

10

011001

000

100

101

111

110

010

M=2

(BPSK)

M=4

(QPSK)

M=8

(Octal PSK)

110


Quadrature Amplitude Modulation

quadrature PAM or QAM

The waveforms:

the information-bearing signal amplitudes

the signal pulse.

Alternative expression waveforms may be:

2( ) Re ( ) , m=1,2,...,M, 0 t Tcj f tm mc msS t A jA g t e

( ) ( ) 2 ( ) 2 4.3-19m mc c ms cS t A g t Cos f t A g t Sin f t

msA

( )g t

mcA

2( ) Re ( ) ( ) (2 ) 4.3-20cj f tj mm m m c mS t V e g t e V g t Cos f t

2 2 1 tan /m mc ms m ms mcV A A A A

(4-3-19)

(4-3-20)

111


Representation as a linear combination of two

orthonormal signal waveforms:

1 1 2 2( ) ( ) ( ) 4.3-21m m mS t S f t S f t

1 22 2

( ) ( ) 2 ; ( ) ( ) 2 4.3-22c cg g

f t g t Cos f t f t g t Cos f t

1 21 1

4.3-232 2

m m m mc g ms gS S S A A

(4-3-21)

(4-3-22)

(4-3-23)

112


The Euclidean distance between any pair of signal

vectors:

In the special case signal amplitudes take the set of

discrete values {(2m-1-M)d,m=1,2,..,M}

the Euclidean distance reach the minimum

which the same results as for PAM

2 2( ) 1 4.3-24

2

e

mn m n g mc nc ms nsd S S A A A A

( ) 2 4.3-25emin gd d

(4-3-24)

(4-3-25)

113


M=4

M=8

M=16

M=32

M=64

Several signal space diagrams for rectangular QAM.

114


FSK - orthogonal multi-

dimensional signalsM equal-energy orthogonal signal waveforms:

The equivalent low-pass signal waveform:

2( ) Re ( ) , m=1,2,...,M, 0 t Tcj f tm lmS t S t e

2

S ( ) 2 2 4.3-26m ct Cos f t m ftT

22( ) , m=1,2,...,M, 0 t T (4.3-27)c

j m f t

lmS t eT

(4-3-26)

(4-3-27)

115


The cross-correlation coefficients:

The real part of :

2 ( ) ( )

km0

2 / (4.3-28)

2

T j m k ft j T m k fSin T m k fT e dt eT m k f

km

km kmRe( )

2 4.3-29

2

Sin T m k fCos T m k f

T m k f

Sin T m k f

T m k f

kmRe( ) 0 when 1/ 2 and m kf T

(4-3-28)

(4-3-29)

116


For the case in which the M FSK signals are equivalent to the N-dimensional vectors:

The distance between pairs of signals:

1/ 2f T

1

1

1

s 0 0 .... 0 0

s 0 0 .... 0 0

s 0 0 0 .... 0 4.3-30

( ) 2ekmd

(4-3-30)

(4-3-31)

117


How to

generate

signals


0 T 2T 3T 4T 5T 6T

0 T 2T 3T 4T 5T 6T

+

tfEb 0

2cos2

tfEb 0

2sin2

tf2sinT

2Atf2cos

T

2A)t(s cmscmcm


0 T 2T 3T 4T 5T 6T

0 T 2T 3T 4T 5T 6T

+

tfEb 0

2cos2

tfEb 0

2sin2

tf2sin)t(Qtf2cos)t(I)t(s ccm

)t(sm


0 T 2T 3T 4T 5T 6T

0 T 2T 3T 4T 5T 6T

+

tfEb 0

2cos2

tfEb 0

2sin2

tf2sin)t(Qtf2cos)t(I)t(s ccm

)t(sm

)t(I

)t(Q


+

tfEb 0

2sin2

)t(sm

)t(I

)t(Q

tfEb 0

2cos2

IQ Modulator


+

tfEb 0

2sin2

)t(sm

)t(I

)t(Q

tfEb 0

2cos2

IQ ModulatorPulse shaping filter


Part - 5


Nonlinear Modulation 3-3-4Methods with Memory

In this section we consider a class of digital modulation

methods in witch the phase of the signal is constructed to

be continuous

125


n

n nTtgItd )((4-3-50)

PAM signal

In sequence of amplitudes obtained by mapping k-bit blocksof binary digits.

g(t) rectangular pulse of amplitude 1/2T and duration T

seconds.

d(t) frequency modulate the carrier.

126


Equivalent lowpass waveform

t

d ddTfjT

tv 0)(4exp2

(4-3-51)

- peak frequency deviation.

- initial phase of the carrier.df

0

127


time varying phase of the carrier.

Carrier modulated signal

0);(2cos2

IttfT

tS c(4-3-52)

);( It

128


time varying phase of the carrier

(4-3-53)

t

d ddTfIt )(4;

dnTgITft

n

nd

)(4

The integral of d(t) is continuous. Hence we have continuousphase signal.

The phase of the carrier in the interval is determined by integrating

TntnT )1(

It;

129


(4-3-54)

ndn

k

kd InTtfITfIt )(22;1

)(2 nTtqhInn

Integrating

- accumulation (memory) of all symbols up totime (n-1)T.

h - modulation index.n

It;

130


(4-3-57)

Tt

TtT

tt

tq

,2

1

0,2

0,0

)(

h, ,and q(t) are defined as:n

Tfh d2 (4-3-55)

1n

k

kn Ih (4-3-56)

131


Continuous-Phase Modulation (CPM)

When expressed in the form of CPFSK becomes a special case of a general class of CPM

signals in which the carrier phase is:

(4-3-58)

n

k

kk kTtqhIIt ),(2; TntnT )1(

- sequence of M-ary information symbols.

sequence of modulation indices.

- some normalized waveform shape.

kI

kh

)(tq

It;

132


Waveform q(t) as integral of g(t) pulse.

t

dgtq0

)()( (4-3-59)

a

b

FIGURE 4-3-16 Pulse shapes for full response CPM (a,b)

ttg

2cos1

2

1

1

0

tg

2

1

0

tq

2

1

0

tq

2

1

0

133


c

d

ttg

cos1

4

1

tg tq

tq

2

1

2

1

2

1

4

1

0

0 0

0

FIGURE 4-3-16 Pulse shapes for partial response CPM (c,d).

dc, CPM. response partial Tfor t 0 tg ba, M.CP response full Tfor t0 tg

134

GMSK pulses


FIGURE 4-3-17 Phase trajectory for binary CPFSK.

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2 3 4 5

0

h

h2

h3

h4

h5

h

h2

h3

h4

h5

CPFSK with binary symbols .1nI Set of phase trajectories beginning at time t=0.

Phase trajectory for binary CPFSK

135


Phase Tree diagram for CPFSK.

d d d d

FIGURE 4-3-18 Phase trajectory for quaternary CPFSK.

3

1

1

3

3

1

1

3

3

1

1

3

3

1

1

3

3

1

1

3

3

1

1

3

3

1

3

1

1

3

3

1

1

3

3

1

1

3

3

1

1

3

1

33

1

1

3

3

1

1

3

3

1

1

3

3

1

1

3

3

1

1

3

3

1

1

3

0

h

h2

h3

h4

h5

h

h2

h3

h4

h5

h6

h6

h6

136


FIGURE 4-3-19 Phase trajectories for binary CPFSK (dashed) and binary partial response CPM

based on raised cosine pulse of length 3T (solid).

1

1

1 1

1 11 1 1

1

1

1

1

1

11

1 1

1 1t Ii,

Phase trajectory generated by the sequence:

(1,-1 , -1,-1 , 1,1 , -1,1)

137


Phase trellis or phase cylinder with binary modulation

FIGURE 4-3-20 Phase cylinder for binary CPM with h=1/2 and a raised cosine

pulse of length 3T.[From sundberg (1986) ,(C) 1986 IEEE.]

138

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4 - Proakis Chapter 4 - Charaterization of Comm Signals and Systems 1-7-2010 Printable Version