1 Let g(t) be periodic; period = T o. Fundamental frequency = f o = 1/ T o Hz or o = 2 / T o...

1

Let g(t) be periodic; period = To . Fundamental frequency = fo = 1/ To Hz or o = 2/ To rad/sec.

Harmonics =n fo , n =2,3 4, . . .

Trigonometric forms

Communication Systems : Prof. Ravi Warrier

dt)tnsin()t(gT2

b,dt)tncos()t(gT2

a0nFor

,componentdcdt)t(gT1

a

)tnsin(b)tncos(aa)t(g

o1

1

o1

1

o1

1

Tt

t

oo

n

Tt

t

oo

n

Tt

too

1nonono

)ab

(tanbaC0nFor

Ca

)tncos(CC)t(g

:formCompact

n

n1n

2n

2nn

oo

1nnono

EXAMPLE :g(t)

0.2-0.2 0.6 1-0.6-1 0t

)t(g

dt)nt5.2sin(28.02

dt)tnsin()t(gT2

b

dt)nt5.2cos(28.02

dt)tncos()t(gT2

a0nFor

,dt28.01

dt)t(gT1

a

.sec/rad5.2sec,8.0T;)tnsin(b)tncos(aa)t(g

2.0

2.0

Tt

t

oo

n

2.0

2.0

Tt

t

oo

n

2.0

2.0

Tt

too

oo1n

onono

o1

1

o1

1

o1

1

R

C

+

g(t)

-

+

g o(t)

-

a) For R = 1 M and C=1 µF , what is go(t) ?

b) For R = 1 M and C=0.1 µF , what is go(t) ?

2

FOURIER SERIES

2


EXERCISE :g(t)

3 5-5 0t (sec)

-1 1-3

2

What is go(t) if the frequency response of the filter is as shown ?

FilterH()

g(t) go(t)

EXERCISE : a)Find the Fourier series in trigonometric compact form.

g(t)

2 4-4 0

t (sec)

-1 1-2

2

-2

-2

0 10 200

0.2

0.4

0.6

0.8

1

1.2

1.4MAGNITUDE

Freq. in rad/s

|H(jw

)|

0 10 20-350

-300

-250

-200

-150

-100

-50

0PHASE

Freq. in rad/s

| H(jw

)

3


o1

1

oo

Tt

t

tjn

on

n

tjnn dte)t(g

T1

DeD)t(g

:formlExponentiaComplex

EXAMPLE :g(t)

0t

5.0 5.0 5.15.1

2

21

Tdte221

dte)t(gT1

D o

5.0

5.0

jntTt

t

tjn

o

n

oo1

1

o

Sketch the Fourier spectra.

Suppose that g(t) is passed through a filter of frequency response as shown. What is the output signal ?

(Both positive and negative frequencies are shown here)

H(j)g(t) go(t)

|H(j)|1

0 3.2-3.2

H(j)

4


ENERGY AND POWER

Energy of g(t): g(t) is an energy signal if Eg < .

Power of g(t) :

dt)t(gE 2g

.ldt)t(gT1

P2

T

2

T

2

Tlim intervatimearbitraryanisTwhere

g(t) is a power signal if 0< P < .

EXAMPLES : Let g(t) be as shown . The energy of g(t) is : Eg=54 J.

Let g(t) be a unit step function: g(t) = u(t). Is this a power or an energy signal ?

-1 4 5

-3

3g(t)

t

1

g(t)

t

0

2g (t)dtgE dt Not an energy signal.

w2

1dt

T

1lim

T(t)dtg

T

1lim

TP

2

T

2

T

2

T

2

0

A power signal.

Average Power of sinewaves

2

2A2o

T

2o

Tt)]dtocos(2[1

2

2A

T

12o

T

2o

Tt)dto(cos2A

T

12T

2T

(t)dt2gT

1lim

TP

t)Acos(g(t)Let

o

2

o

o

.Then)tcos(nCCg(t)Let1n

nono

1n 2

2nC2

oCP

2nC

|nD,|oDoC

.TheneDg(t)Let-n

tjnn

o

why?

1n

2nD22

oD-n

2nDP

POWER OF ANY PERIODIC FUNCTION IN TERMS OF FOURIER COEFFICIENTS :

EXAMPLE :

g(t)

0.2-0.2 0

2

t-0.6-1 -0.60.6

W2

...00901.0081206.01

)n5.0sin(n4

21

1PThen

)tn5.2cos()n5.0sin(n4

1)t(gthatfoundWe

2

1n

1n

5


Signal Comparison : CORRELATION

Let g1(t) and g2(t) be two signals. Their correlation is defined as

.dt)t(g)t(g)( 21gg 21

If g1(t) = g2(t) = g(t), this becomes autocorrelation function, given by

.dt)t(g)t(g)(g

We see that g(0)=Eg we ge the signal energy. That is, the signal energy =autocorrelation at = 0.

FOURIER TRANSFORMSFOURIER TRANSFORMS

Definition : F{g(t)} = G() F-1{G()} =g(t)

d)eG(

2

1g(t)dt g(t)e)G( tjtj-

TRANSFORM EXAMPLES :

)t()t(g t,1)(Gg(t)

t0 0

)(G 1

t,A)t(g).(A2)(G

t0

Ag(t)

t0

Ag(t)

2

t

rectA)t(g

0

)(G

).(A2

)(G

A

2

csinA)(G

2

Wtsincg(t)

2W

rectW

)G(

2

2

)(Ag(t) tg(t)

A

2

2 0 t

2

4csin

2A)(G

)(G

x)xsin()x(csin

6


0a),t(ue)t(g at g(t)

ja

1)(G

g(t)

0a,eg(t) |t|a 22a

2a)G(

0

0

tj oeg(t) )2)G(

)(G

o0

)tcos(g(t) o )))G(

)(G

o0o

t

t

t

)tsin(g(t) o ))j)G(

)tsgn(g(t) 1

-10 t

)t(ug(t)

j2)G(

j1)()G(

7

PROPERTIES F{g(t)} = G()

1. Symmetry :

2. Scaling :

3. Time-shifting :

4. Frequency shifting :

)(g2)t(G

)(G)at(gaa

1

)(Ge)tt(g otjo

)(G)t(g

FOR PROOF READ TEXT.

(The term represents a linear phase ; in time domain it is delay).

otje

)(Ge)t(g;)(Ge)t(g otj

otj oo

)(G)(G

)t(g

?)tcos()t(g

oo21

2ee

otojtoj

F

FWhat is

Here g(t) is modulating the sinusoid amplitude - AMPLITUDE MODULATION.

g(t) is the modulating signal, cos(ot) is called the carrier.

EXAMPLE : .trect)t(gLet We will find the Fourier transform of g(t)cos(10t).

1

0

trect)t(g cos(10t)

21

21

t t 0

21

21

t

g(t)cos(10t)

=

TIME

domain

)(G

1

2 2

)t10cos()t(gF

21

10-10

Note : Multiplication in time-domain doesn’t transform to multiplication in frequency domain. 0

?)t10cos()t(gF

Math:

2

10csin

2

10csin)t10cos()t(rect

2csin)t(rect

21FF

EXERCISE : 1. What is

2. Find the spectrum of a)

b)

).t20cos()t2(g(t)

).t20sin()t2(g(t)

Sketch the time functions and the spectra.


FREQUENCY

domain

8


5. Differentiation ).(G)j()(Gj n

dt

)t(gddt

)t(dgn

n

FF

EXAMPLE : We find the Fourier transform of the triangular function shown using this property.

)(Ag(t) t

g(t)A

2

2 0 t 2

2

0

t

A2

dt)t(dg

2

20 t

A2

2

2

dt

)t(gd

A2

A4

)(csin2

A

)(

)(sin

2

A)(sin)(G

1)(sin212)cos(2e2e)(G

:functionsimpulsetheoftransformFouriertheTaking

)(G)(G)j(

42

24

42

24

2A84

2A42

A2jjA22

22

dt

)t(gd

22

2

2

F

6. Integration : )()0(Gd)(gj

)(Gt

F

7. Convolution :

)(G)(Gdx)x(G)x(G)t(g)t(g 2121

2121

21

F

CONVOLUTION IN TIME DOMAIN ( )

MULTIPLICATION IN FREQUENCY DOMAIN

)(G)(Gd)t(g)(g)t(g)t(g

d)t(g)(g)t(g)t(g

212121

2121

FF

CONVOLUTION IN FREQUENCY DOMAIN

MULTIPLICATION IN TIME DOMAIN

9


EXAMPLE: )t(g)t(gFind);t(rect)t(gLet

).()}t(g)t(g{

)(csin)}({tabletransformFourierFrom

)(csin)(G)}t(g)t(g{).(csin)(GhaveWe

2t

22

2t

2222

2

1-F

F

F

0 t

g(t)

0 t2

1

2

Note : g(t) has a pulse width of /2 sec but has pulse width of sec. Convolution increases the width of the function.

EXERCISE : Let g(t) = sinc(50t). What is the spectral width (bandwidth) of g(t) ?

What is the bandwidth of g2(t) ?

EXAMPLE : What is the Fourier transform of a periodic function ?

Periodic functions can be expressed in time domain as sum of a dc term and sinusoids of fundamental frequency and harmonics. Fourier transform of a sinusoid is a pair of impulse functions. Therefore, the Fourier transform of a periodic function is a sum of impulse functions centered at zero frequency, fundamental frequency and harmonics.

non

n

tjnn

n

tjnn

)n(D2eD)}t(g{

eD

o

o

FF

g(t)

EXAMPLE : We find the Fourier transform of the periodic function shown.

1

0

)t(g)t(g )t(g)t(g

2

2 2 2

g(t)

t

The Fourier series of g(t) is given (in page 51, text) by

The Fourier transform of g(t) is

)7()7(

5()5()3()3()1()1(2)()(G

71

51

31

)t7cos(

)t5cos()t3cos()tcos()t(g

142

102

622

21

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

|G()|

G()

)))}t{cos( o FWe have

10


-4To -3To -2To -To 0 To 2To 3To 4To t

g(t)

EXAMPLE : Consider the periodic function g(t) consisting of impulse functions at equal spaces of To sec.

We find the Fourier transform of g(t). We can express g(t) as

noo

noT

2

n

tjnT1

n

tjnT1

no

)n()n(e)}t(g{

.seriesFouriere

)nTt(

oo

o

oo

FF

g(t)

-4o -3o -2o - o 0 o 2o 3o 4o

G()

o

SIGNAL ENERGY AND ENEGY SPECTRAL DENSITY

We define signal energy as .

If g(t) is complex we can express energy as

(t)dtgE 2g

(t)dtg(t)gdtg(t)E *2g

Parseval’s theorem : Signal energy is

d)G()d()GG(dtg(t)E2

21*

212

g

Energy Spectral Density (ESD): is called the energy spectral density of g(t). The signal energy is the integral of the energy spectral density ( multiplied by 2).

2g )G()(

)d(E g21

g

ESD provides a way of computing energy from the Fourier transform of g(t).

.J1

)(tand)d(E

)(G)()(G

.J1dtedtedtg(t)E

5.0x21

5.01

5.0x21

25.01

21

g21

g

25.012

g5.0j1

0

t-

0

20.5t-2g

2

2

EXAMPLE : Consider g(t)=e-0.5t u(t). Find the energy and the ESD of g(t).

25.0

2)G(

0

11


ENERGY OF MODULATED SIGNAL

Let g(t) be a baseband energy signal bandlimited to B Hz.

Let .B2where,signalulatedmodamplitudean)tcos()t(g)t( oo

.EEthatfindweThen

)()(

)(G)(G)(

,B2Since

.)(G)(G)(is)t(ofESD

.)(G)(G)(

g21

ogog41

2o

2o4

1o

2oo4

1

oo21

.)(G)(G)( oo21

Suppose that G() is as shown. ThenG()

B2B2 0

()

oB4

o

The signal energy is reduced by 1/2 when it is multiplied by a sinewave of unit amplitude.

ESD OF A SYSTEM INPUT AND OUTPUT

H()G() Y()

d)()(Hd)(E)()(H)(

)(G)(H)(Y)(G)(H)(Y

G2

21

Y21

yG2

Y

222

EXAMPLE : Find the input and output energies. R=200 and C=0.01 F. g(t)=sinc(t).

R

C

+

g(t)

-

+

g o(t)

-

J87.0)2(tan2

ddd)(E

)()(H)(

)(rectx)(G)(H)(Y

)(G)(H)(Y)(H

JdE)(rect )(G

1

11

8

1

1

1

1

181

1421

Y21

y

G2

Y

22

141222

12j1

1jRC1

1

1

221

g2

2

212

2

22

2

s/rad 1for124

2

Otherwise0

EXERCISE : Redo the Example problem for R=200 and C=0.001 F.

12


Autocorrelation Function and ESD : For g(t) a real function

.)G()()GG()(

)(G)G(-de)g(de)g()}F{g(-)G()}F{g(

)}.)}F{g(-F{g()(

)()dxg(xg(x)x))dx(g(g(x))g(-)g(

x)dxg(g(x))g()g( : of functionaasnConvolutio

case).thisin(lagtimetheoffunctionaisnatioautocorrelTime

operation.timeaisnConvolutio.operationsdifferentrepresenttheybut

? nconvolutio to related this isHow

function.evenan)()(Thus

TEXT)READtlettingbythisshowcan(We)dtg(tg(t))dtg(tg(t))(

2*g

*jj

g

g

gg

g

Energy Spectral Density is the Fourier Transform of the autocorrelation function.

SIGNAL POWER AND POWER SPECTRAL DENSITY(PSD)

Energy and energy spectral density are useful for energy signals. For power signals we define power and power spectral density as follows :

de)(S)}({S)( Or)(S})({

P)0( )()(

)dtg(tg(t)lim)(

as defined is signal power real aoffunctionaisnatioautocorrelTime

d)(SP THEN .lim)(Sdensity spectral power The

d)(GlimE lim dt(t)glimP )t(g Defining

dt(t)glimP

jg2

1gggg

gggg

T1

Tg

g21

gT

)(G

Tg

2T2

1T1

TgT

1

T

2T1

Tg2

T t 2T- for )t(g

otherwise 0

2T1

Tg

2T

2T

2T

2T

2T

TTT

1-FRRF

RRR

R

13


Input signal power , output signal power Let g(t) be a power signal applied to a system.

H() Y()

d)(SP)(S)(H)(S

)(G)(H)(Y)(G)(H)(Y

Y21

yG2

Y

222

G()

EXAMPLE : Let g(t)=A cos(ot) , a power signal.

)0( P Also

d)-()-( d)(SP THEN

)-()-( )}( )(Sdensity spectral power The

)cos(

dt)])t2(cos(dt)cos(

dt)])t2(cos()[cos(

dt))t(cos()tcos(A)dtg(tg(t))dtg(tg(t)lim)(

isfunctionaisnatioautocorrelTime

periodT dt(t)gP

gg

2A

oo2A

21

g21

g

oo2A

gg

o2A

o2A

T1

o2A

T1

oo2A

T1

oo2

T1

T1

T1

Tg

o2A2

T1

g

22

2

2

2oT

2oT

2

o

2oT

2oT

2

o

2oT

2oT

2

o

2oT

2oT

o

2oT

2oT

o

2T

2T

22oT

2oT

o

R

RF{

R

oo

22A

)(gS

EXAMPLE : Consider a noise signal n(t) with PSD is the input to a differentiator.

What is the output noise power ? bb- ,K)(nS

bb- 2)H( K)(nS 2)H()(Sy

22)H( ,j)(H

H()=j y(t)n(t)

)(nS K

bb - 0 bb - 0

2)(H

bb - 0

)(Sy

3

3bK

d)(SP Y21

y

0

14


EXERCISE : 1 Consider a noise signal n(t) with PSD applied to a RC filter with

RC=1 sec. Determine the input noise power and output noise power. (Input power = 2W, Output power=0.25 W)

1010- ,1.0)(nS

R

C

+

n(t)

-

+

n o(t)

-

2. Suppose that the input to RC filter is g(t)=2cos(0.5t), what is the input and out put signal powers ? (Ans: power in=2W, power out=1.79W)

3. Next consider the input filter to be g(t)+n(t). Find the (signal power)/(noise power) at the input and output. This is called the signal-to-noise ratio.

REVIEW : 1) Fourier transform inverse Fourier transform definitions 2) Properties : Important ones :- Symmetry, time-delay /phase shift, Modulation 3) Results : Fourier transform of periodic functions, Energy, ESD, autocorrelation, power, PSD, autocorrelation, input energy-output energy, input power-output power.

What is the autocorrelation function of a sinewave ? What is the PSD of a sinewave ? What is the average power of a sinewave ? Does phase shift affect the power and autocorrelation ? What is the autocorrelation function of g(t) =cos(20t) ? What is the autocorrelation function of g(t) =sin(20t) ?

READ TEXT BOOK A LOT.

Distortionless Transmission : The ideal goal of a communication system is to make sure the received and transmitted signals are the same. That is, the received signal is not distorted. This means the communication system transfer function should have a constant magnitude and linear phase characteristics, in the frequency region of interest.

H() Y()G()

dt-je K)(H

)(H

K

0

)(H

0

dt

EXERCISE : Find the Fourier transform and sketch the spectra of : i)g(t) = sinc(20t)cos(100t)

ii) g(t) = sinc(20t)cos2(100t).

15

t

2bT

2bT

bT For

t

2bT

t

)t(g

)t(g

g(t)=1 for binary 1, g(t) = -1 for binary 0

2bT

is )-g(t)g(t under area the 2bT

writecan weSo

2bT

is )-g(t)g(t under area the 02bT

for that,show can weSimilarly

2bT

0 For

2bT

region shaded the of Area

1)-g(t)g(t product the region shaded the In

2bT


16

)()N( )dtg(tg(t)lim)( For

)N( is area the pulses N For

)dtg(tg(t)lim)dtg(tg(t)lim)(

b

b

b

2bNT

2bNT

b

b

b

2bNT

2bNT

b

2T

2T

T21

2T

NT1

NT1

Ng2

T

2T

NT1

NT1

Tg

R

R


1 Let g(t) be periodic; period = T o. Fundamental frequency = f o = 1/ T o Hz or o = 2 / T o...

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Transcript of 1 Let g(t) be periodic; period = T o. Fundamental frequency = f o = 1/ T o Hz or o = 2 / T o...