AE57_AC57_AT57

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AE57/AC57/AT57 SIGNALS AND SYSTEMS JUNE 2013

© IETE 1

Q2 (a) Determine whether the following signals are periodic or not of periodic

then find its fundamental period

(i) ( )2n1)n(x −=

(ii)∑

∞

−∞=

−δ−=

k

k

)k 2t()1()t(x

Answer

(i) ( )2n1)n(x −=

For given signal to be periodic

2 Nisx(n)of period

1)1((-1) 2 N

11(-1) 1 N

)1(12)1(

if periodic)(

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

2)1()1(

2)1(

))(1()(

)()(

442 N

2n1

2

2

2

2

2

2

22

2

=∴

=−==

≠−==

−==+−

∴

+−=+−−=

++−=

+−=+∴

=+

++

+

lFundamenta

for

for

nN

isn x

nN n x

nN

nN N

N n N n x

n x N n x

nnN

m N

N

N n

n

(ii) ∑∞

−∞=

−−=k

k k t t x )2()1()( δ

For x(t) to be periodic x(t+T)=x(t)

∑∞

−∞=

−+−=+∴k

k k T t T t x )2()1()( δ

Performing a change of variables

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© IETE 2

4x(t)of period lfundamenta 4

22

)1(1(-1)

periodic betox(t)

)1)((

)1)(2()1(

)2()1()(

2

2

22

22

T

2

2

2

=∴=

=

−==

−=

−−−=

−−=+

+=

−=−

−=−

∑

∑

∞

−∞=

∞

−∞=

+

mT

mT

for

t x

mt

mt T t x

mT

K

mK T

mT T

m

T

T

k

m

k

m

T

δ

δ

Q2 (b) For each of the following systems determine whether it is Memoryless,

Causal, Stable, Linear and Time invariant.

(i) )]n(x[log)n(y e=

(ii) )n(x)n(y 2=

Answer (i) )]n(x[log)n(y e=

As the present output depends on the present input only ∴ system is memory less,causal• Assuming that input signal x(n) satisfies the condition |x(n)| ∞

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© IETE 3

Invariant-Timeis)(y)(ysin

)]([log)(y

)]([log

)]([log)(yi/pshifted toingcorrespond /

)()(x/

)]([log)(y

Linear - Nonis

)()()]()([log)]([log)(y

linear betosystem

)()()(x

)]([log)()(

)]([log)()(|

012

0101

01

22

012

11

21

2133

213

222

111

systemnnnce

nn xnn Now

nn x

n xn pO

nn xn pshiftedi

n x x

System

nbynay

nbxnaxn xn

for

nbxnaxnlet

n xn yn x

n xn yn x

e

e

e

e

ee

e

e

∴ −=

−=−

−=

=

−=

=

∴

+≠+==

+==

=↔=

=↔

(ii) )()( 2n xn y =

• System has memory since the value of output signal y(x) at time n depends on thefuture inputs

)/)(/)( 2000 n xn ynnn y ===

∴ It is not memory less• For bounded i/p system gives a bounded o/p ∴ System is BIBO stable.• system is Non-Causal since present value of o/p y(n) depends on the future values

of input signal x(n)

• Consider two arbitrary inputs )()()( 2111 n xn yn x =↔ )()()( 2222 n xn yn x =↔

Let )()()( 213 nbxnaxn x +=

If system is linear them

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© IETE 4

Variant-Timeis

)()(

))((x )(y

)()(y

)()(y i/pshifted toingcorrespond o/p)()(xi/pshifted the

)()(y*

linearissystem RHSLHS

)(y)(y

)()()()(y*

linearisSystem

)(

)()()()(

)()()(

012

20101

02

12

222012

211

21

22

21

211

2)(1

22

21

233

213

system

nn yn y

nnnn Now

nn xn

n xnnn xnconsider

n xn

Since

RHS nbna

nbxnaxn xn

RHS LHS Since

RHS nbyay

nbxnaxn xn y LHS

nbynayn y

n

∴

−≠∴

−=−

−=

=−=

=

∴=

=+=

+==

∴=

=+=

+====>

+=

Q3 (a) Find the trigonometric Fourier series for the triangular wave shown in

Fig.1 and hence plot its line spectrum.

Answer

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Waveform is periodic write period T=2

& Fundamental FrequencyT

wπ 2

0 =

( ) )

[ ]

n

n

n

n

T

n

T

abnanline

t t t

t nwaat x

na

nn

n

t n

n

t nt

dt t nt

dt t nwt xT a

t t

dt t dt t xT

a

Thet

t t x

=+=

++++=

+=∴

=

==

−=

−−=

−=

==

=−=>

−==

=∴

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© IETE 6

Q3 (b) A continuous time periodic signal is real valued and has a fundamental

period T= 8 .The non zero Fourier series coefficients for x(t) are

2XX 11 == − , j4XX*

33 == − . Express x(t) in the form

∑

∞

=

Φ+ω=

0n nnn)t(CosA)t(x

Answer Given:

)24

3cos(8)

4cos(4)(

43sin8)4cos(4

4422

x(t)have

48

28

4/34/34

33

0

4

00

π π π

π π

π π

π π

π π

++=

−=

−++=

+=

==∴=

−−

∞

−∞=

−−

−∑

t t t x

t t

je jeee

e X e X we

wT

t jt j jt j

n

t w jt jw

n

t

Q3 (c) Find the time domain signal corresponding to following DTFS coefficients

Answer

π+

π=11

6k sin j2

11

4k cosXk

)9(2

11)8(11)3(11)2(

2

11x(n)

equationstwoabove the

)(

11

)1(2

11)8(11)3(11)2(

2

11

11

1

2

1

2

1

2

1

2

1

2

1

2

1

,21

0

11210

0

811

2311

2911

2211

2

1111

23.11

211.11

2211

2211

2

116

114

114 11

6

−+−−−+−==

=

−+−−−+−=

−++=

−++=

−++=

∑

∑−

=

=

−

− −

nnnn

Comparing

en x N X

ennnn

eeee

eeeee

eeee

nk N

j N

K k

nk j

k

k jk jk jk j

k jk jk jk jk j

k jk jk j k j

δ δ δ

δ δ δ δ

π

π

π π π π

π π π π π

π π π π

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© IETE 7

Q4 (a) State and Prove duality property of Continuous Time Fourier Transform.

Using it, find the Fourier Transform of following signals

(i) jt1

1)t(g

+=

(ii) 2t1

1

)t(x +=

Answer

)(2

1

1

)(2 jt1

1

x(-w)2x(t)

duality to

jt1

1 x(t)& )(a)(

jw1

1 x(w)& )(a)(

1

1)(a

1a

1)(e

g(t)of expressionin thet with w& jw1

1 x(w)

jt1

1g(t) (i)

x(-w)2x(t)

)]([x(-w)2

)(x(-w)2

w.and tvariables the

)(x(-t)2

t- bytReplace

)(x(t)2

)(2

1x(t) -: proof

x(-w)2then x(t)

x(w)x(t)of property

w

t-

t-

at-

wue

jt

f

wue

According

wuw x

t ut x

jwt u

ingSubsititut jwa

t u

replace Define

t x F

dt et x

ing Interchang

dwew x

dwew x

dwew x

Duality

w

w

jwt

jwt

jwt

jwt

−=

+

−↔+

↔

+=−=−

+==

+↔

=

+↔

+=

+=

↔

=

=

=

=

=

↔

↔

∫

∫

∫

∫

∞

∞−

−

∞

∞−

−

∞

∞−

∞

∞−

π

π

π

π

π

π

π

π

π

π

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© IETE 8

||

2

||

2

||

2

|-|-w

2

|-|-w

2

|t|a-

2

|t|a-

2|t|a-

22

|t|a-

2

.1

1

.t1

1

2

1.2

t1

1

x(-w)2X(t)

Duality .

e2

1

1

1 x(t)e

2

1)(

1

1 x(w)and e

2

1)(

1

1e

2

1

1 2e 1a

2e ,have we

t1

1 x(t))(

w

w

w

et

F

e

e

to Acc

t w x

wt x

w

wPut

wa

a

ii

−

−

−

=

+=

↔+

=

↔+

=

↔

=

+==−

+==

+↔

+↔=

+↔

+=

π

π

π

π

Q4 (b) Consider a stable LTI system characterized by the differential equation

)t(x2dt

)t(dx)t(y3

dt

)t(dy4

dt

)t(yd 2

2

+=++

(i) Find the frequency response )(H ω and impulse response h(t) of thesystem.

(ii) What is the response of this system if the input )t(ue)t(x t−=

Answer

)t(x2

dt

)t(dx)t(y3

dt

)t(dy4

dt

)t(yd

2

2

+=++

Taking Fourier Transform on both sides

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© IETE 9

)(2

1)(

2

1h(t)

Transformfouriesinverse3

1.

2

1

1

1.

2

1)(

)3)(1(

2

34)(

2

)(

)()(

)(2)()(3)(4)()(

3

2

2

t uet ue

Jaking

jw jww H

jw jw

jw

jw jw

jw

w x

w yw H

w xw jwxw yw jwyw y jw

t t −− +=

++

+=

+++

=++

+==

+=++

Q5 (a) Suppose that a system has the response )n(u)4

1( n to the input

)n(u)2

1)(2n( n+ .If the output of this system is )n(u)

2

1()n( n−−δ ,what

is the input?

Answer

)(4

1

2

1

4

1y(t)

Transformfouriesinverse

)3(

1

4

1

)1(

1

2

1

1

1

4

1)(

)3()1(

2)3)(1(

2

jw1

1x(w).H(w))(

jw1

1) x(w

)(eGiven x(t) )(

3

2

2

-t

t uetee

Taking

jw jw jww y

jw jw

jw jw jw

jww y

t uii

t t t

−+=

+−

++

+=

+++

==>

++

+

+

==

+=

=

−−−

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© IETE 10

jw

jw

jw

jw

n

jw

jw

jw

jw jw

jw

jw

n

jw

jw

jw jw

jw

nn

n

e

e

e

e y

nuGiven

e

e

e x

e y

e H

e

e y

nuGiven

e

e

ee

e

Taking

nnn

nuGiven

−

−

−

−

−

−

−

−

−−

−

+=

+−=

−=

−

−

==

−=

=

−

−=

−+

−

=

+

=

+=

2

11

2

1

2

11

11)(

)(2

1-(n)y(x)

4

112

2

11

)(

)(

)(

4

11

1)(

)(4

1 y(n)

2

11

4

112

2

11

12

2

11

2

1

)x(e

equationabovetheof DTFT

)(42

12)(4

2

1

)(2

13)(n) that x(n

2

2

2

jw

δ

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© IETE 11

)(2

1

4

1)1(

2

1

8

3)1(

2

1

8

3x(n)

DTFTinverse

2

11

81

2

11

83

2

11

83

2

11

2

11

4

11

2

11

2

1

2

11

4

112

)(

2

11

4

112

)(

11

2

2

2

2

2

2

nunnunu

Taking

e

e

e

e

e

e

ee

ee

e

e

e

e

e y

e

e

e x

nnn

jw

jw

jw

jw

jw

jw

jw jw

jw jw

jw

jw

jw

jw

jw

jw

jw

jw

+−

+−

−=

−

+

−

+

+

=

+

−

−

=

+

=

−

−

=

=

−

−

=

−−

−

−

−

−

−

−

−−

−−

−

−

−

−

−

−

Q5 (b) State and Prove convolution property of Discrete Time Fourier

Transform. Using it determine the convolution )n(x)n(x)n(x21

∗= of

the sequences, where

)1n()n()1n()n(x)n(x 21 −δ+δ++δ==

Answer

Convolution property:

If

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© IETE 12

[ ] [ ]

)

[ ]

[ ]

}1,2,,3,2,1{

)2()1(2)(3)1(2)2()(

232

)()2(3

(2w)cos2wcos43

wcos21

)().()()(xF

property,conditionUsing

wcos21

1

)()()(

)1()()1(}"{")()()()()()(

)()()()(

term bracketed theo property tshiftingtime

)()(

summationof orderthe

()(

)(x)()(x)(f -: proof

)()()(x)(then

)()( x)()(

22

22

2

2121

1

1

121

21

2121

2112

21

21

2121

2121

2211

=−+−+++++==

++++=

++++=

++=+=

=+

+=++=

=

==

−+++=↑==↔×=

==

−=

−=

+=+

↔+

↔↔

−−

−−

−

−=

−

−∞

−∞=

∞

−∞=

−∞

−∞=

−∞

−∞=

∞

−∞=

−∞

−∞=

∑

∑

∑∑

∑ ∑∑

nnnnnn x

eeee

eeee

e xe xn xn

ee

e

en xe xe x

nnnn xn x

e xe xn xn x

e xe xe xe x

Apply

emn xm x

ing Interchang

emn xm x

enn xnn x

e xe xnn x

e xne xn x

w j jw jww j

w jw j jw jw

jw jw

jwn jwn

n

jwn

jwn

n

jw jw

jw jw

jw jw jw jw

n

jwn

m

jwn

n m

jwn

n

jw jw

jw jw

δ δ δ δ δ

δ δ δ

Q6 (a) Determine the conditions on the sampling interval TS so that each x(t) is

uniquely represented by the discrete time sequence x(n) = x(nTS). (i) )t4sin()t2sin(3)tcos()t(x π+π+π= (ii) )t2(csin)t6sin(3)t(csin)t2cos()t(x π+π=

Answer )4sin()2sin(3)cos()( t t t t x π π π ++=

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© IETE 13

Comparing it with

4

1T intervalsampling

4max2f

sec/82wfrequencySampling

2

2

f or4

4,2,

)sin()sin(),cos(,)(

s

s

maxs

maxmox3

321

3322

≤

=≥

=≥∴

====

===

++=

Hence

Hz f or

rasw

wwW

www

t w At w At w At x

mox

π

π π

π π π

[ ]

[ ] [ ]

8

1T IntervalSampling

82f freq.Sampling

42

wf or8wisfreq.Maximum

)8cos()4cos(4

3)sin()3sin(

t2

1

2

)2sin()6sin(3)sin()3sin(

)2(

1

2

)2sin()6sin(3

)sin()2cos(

)2(sin)6sin(3)(sin)2cos()(Given)(

s

maxs

moxmaxmox

≤

=≥

===

−+−=

+−=

+−

=

+=

Hz f

t t t

t t

t

t t t t

t

t

t t

t

t t

t ct t ct t xii

π π

π π π

π π π

π

π π π π

π

π

π π

π

π π

π π

Q6 (b) A causal LTI system is described by the differential equation

)t(x)t(y2dt

)t(dy=+

Determine:

(i) The frequency response of the system

(ii) The group delay associated with the system

(iii) Output of the system to the input x(t)=e-tu(t)

(iv) Output of the system if the input has its fourier transform

)2 j(1 j) jw(X

+ω +ω=

Answer (i) Taking F.transform both the sides of given equ

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© IETE 14

{ }

[ ]

[ ] )(4ey(t) transformfouriesinverse

)2(

1)().()(

)(eey(t)

transformfouriesinverse

2)(jw

1

1)(jw

1

2)1)(jw(jw

1(jw)H(jw)y(jw)

1 jw

1 x(jw)

)(e x(t))(

4

2

2

1.

21

1

2tan)()(

2-tanH(jw) )(

2

1

x(jw)

y(jw)H(jw)

x(jw)2y(jw)y(w)

22t-

2

2t-t-

t-

22

1

1-

t te

Taking

jw

jw jw X jw H iv

t u

Taking

t uiii

ww

w

dw

d jw H

dw

d wT

wii

jw

jw

t −

−

−=

+

+=

−=

+−

+=

++=×=

+=

=

+=

+

=

−−=∠−=

=∠

+==

=+

Q7 (a) Consider the signal )1t(ue)t(x t5 −= − and its Laplace Transform be X(s)

(i) Evaluate X(s) and find its ROC

(ii) Determine the values of the finite numbers A and t 0 such thatthe Laplace transform G(s) of )tt(uAe)t(g 0

t5

−−=

−

has thesame algebraic form as X(s).What is the ROC corresponding

to G(s)?

Answer

(i) By definition

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© IETE 15

{

[ ]

-5R{s}R.O.C 5

)55(

)55(0)5(

1)55(

)55(

1

)55(1

1)(

1t 01t 1

)1(

)1()(

)()5(

>+

+−

=

+−−+−=

∞

+−

+−=

∫∞

+−=∫∞

−−=

<>=−

∫∞

∞−

−−−=

∫∞

∞−

−=

s

t e

t es

t e

dt t

edt st

est

es

x

t u

dt st

et ust

es x

dt st

et x x

{ } 5sR R.O.C

-1.to-1,A

)8()8(..........5

)(

5

)(

00)0(0

00)0(1)(

)(

)()(

)(

0

0

0

)5(

0)5(

0

)5(5

0

0

5

−<

==

=+

−=

+

−=

==

−>→ 1

(ii) Re{s} < - 2

(iii) -2 < Re{s} < 1

Answer

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© IETE 16

{ } 1)(

12)(

)1(

1

)1(

1

)1)(2(

3)(

>

−

−−

+=

−+

−=

s ROCR

iand haspolesat s X

ss

sss X

Is to the right of the eight most poles so both poles correspond to casual signals.

1}{2)(

)()(-e)(

1

1)(e-

2

1)(e-

signalsanticasualtocorrespond poles bothso poleleftmostof leftthetois2)(

)(

)()()(

1

1)(

2

1)(

2t-

t-

2t-

2

2

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© IETE 17

Q8 (a) Determine the signal x(n) whose z-transform is given

by az),az1log()z(X 1 >+= −

Answer

11

1)

)((

1

)(),1log()(

1

2

1

−+

−−=−

+

−=

>+=

−

−

−

az

az

dz

zdx Z

az

az

dz

zdx a zaz z X

Take inverse Z-transform

||||1

)()(

||||1

1)()(

1

1)(

1)(

11

1)()(

1

1

1

1

11

11

a zazanuaa

a Z

aznua

aznua

knowwe

az

az znnx

az

az z

dz

zdx z

n

n

n

>+↔−

>+

↔−

−↔

+

−=

−+

−−=

=−=

−

−

−

−

−−

−−

Using time –shifting property,

)1()1(1

)1()(

)(

)1()()(

||||1

)1()(

||||1

)1()(

1

1

1

1

11

−−=−−−

=

−−−=

>+

↔−−−

>+

↔−−

+

−

−

−

−−

nuan

nun

an x

nuannx

lyconsequent

a zaz

aznua

a zaz

aznuaa

nnn

n

n

n

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© IETE 18

Q8 (b) Find the inverse z-transform of1

1

z)3/1(1

z1)z(X

−

−

−

+=

When,

(i) ROC: |z| > 1/3

(ii)

ROC : |z| < 1/3, using power series expansion

Answer

......27

4

9

4

3

41

....,.........27

4,

9

4,

3

4,0)(

......27

4

9

4

3

41)(

27

4

27

4

9

4

94

9

4

3

4

3

11

13/11

321

321

3

32

2

21

1

11

++++

=∴

+++=∴

−

−

−

+−

−−

−−−

−

−−

−

−−

−

−−

z z z

n x

z z z z x

z

z z

z

z z

z

z z

.....36z-12z-3-

36z

36z-12z

12z

12z-4

4

3

113

1

)(

2

2

2

1

11

+

−

++−−

−−

z

z z

ii

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© IETE 19

Q9 (a) A random variable X has the uniform distribution given by

( )

π≤≤π=

otherwise0,

2x0for ,2

1xf X

Determine its mean and variance

Answer

[ ]

33

4

][][var

34

321

21

2

1.)(][

22

1.

2

1

2

1.)(

222

222

2

2

0

32

0

2

2

0

222

2

0

22

0

2

0

π π π

π π π

π

π π π

π

π

π

π

ρ π

π

=−=

−=

===

==

===

==

∫

∫∫

∫

∫∫

∞

∞−

∞

∞−

x E x E iancex

xdx x

dx xdx x fx x x E

xdx x

dx xdx x xfxm x

Q9 (b) A WSS random process X(t) with autocorrelation function ||aX e)(R τ−=τ

where a is a real positive constant is applied to the input of LTI system

with impulse response h(t) = e-bt

u(t) where b is real positive constant.

Find the autocorrelation function of the output Y(t) of the system.

Answer Frequency response H/W of the system

H (W) =F [h(t)]=1/jw+b

Power spectral density of X(t) isSx (w) = F [Rx(z)]=

( )

+−−

+−=

+

+==

+

22222222

2222

2

22

22

)(

21)(|)(|)(

2

aw

a

bba

b

bw

b

bba

a

bw

a

bwwsxw H wsr

aw

Taking inverse Fourier transform on both sides

[ ]|||/22 )(

1)( eaeb beae

bbac R

−− −+

=λ

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© IETE 20

Text Books

1. Signals and Systems, A.V. Oppenheim and A.S. Willsky with S. H. Nawab,

Second Edition, PHI Private limited, 2006.

2. Communication Systems, Simon Haykin, 4th Edition, Wiley Student Edition, 7th

Reprint 2007.

Documents

AE57_AC57_AT57