Signals and Systems

GATE EC Topic wise QuestionsSignals & Systems

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2011 ONE MARK

Question. 1

The differential equation 100 20 ( )y x tdtd y

dtdy

2

2

− + = describes a system with an input ( )x t and an output ( )y t . The system, which is initially relaxed, is excited by a unit step input. The output y t^ h can be represented by the waveform

Question. 2

The trigonometric Fourier series of an even function does not have

EC Topicwise 1996-2011Signals & Systems

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the

(A) dc term (B) cosine terms

(C) sine terms (D) odd harmonic terms

Question. 3

A system is defined by its impulse response ( ) ( )h n u n2 2n= − . The system is

(A) stable and causal (B) causal but not stable

(C) stable but not causal (D) unstable and non-causal

Question. 4

If the unit step response of a network is (1 )e t− α− , then its unit impulse response is

(A) e tα α− (B) e t1α α− −

(C) (1 )e t1α− α− − (D) (1 )e tα− α−

2011 TWO MARKS

Question. 5

An input ( ) ( 2 ) ( ) ( 6)expx t t u t tδ= − + − is applied to an LTI system with impulse response ( ) ( )h t u t= . The output is

(A) [ ( )] ( ) ( )exp t u t u t1 2 6− − + +

(B) [ ( )] ( ) ( )exp t u t u t1 2 6− − + −

(C) . [ ( )] ( ) ( )exp t u t u t0 5 1 2 6− − + +

(D) . [ ( )] ( ) ( )exp t u t u t0 5 1 2 6− − + −

Question. 6

Two systems ( ) ( )andH Z H Z1 2 are connected in cascade as shown below. The overall output ( )y n is the same as the input ( )x n with a one unit delay. The transfer function of the second system ( )H Z2 is

(A) ( . )

.z z

z1 0 4

1 0 61 1

1

−−

− −

− (B)

( . )( . )

zz z

1 0 41 0 6

1

1 1

−−

−

− −


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(C) ( . )( . )

zz z

1 0 61 0 4

1

1 1

−−

−

− −

(D) ( . )

.z z

z1 0 6

1 0 41 1

1

−−

− −

−

Question. 7

The first six points of the 8-point DFT of a real valued sequence are 5, 1 3, 0, 3 4, 0 3 4andj j j− − + . The last two points of the DFT are respectively

(A) 0, 1 3j− (B) 0, 1 3j+

(C) 1 3, 5j+ (D) 1 3, 5j−

2010 ONE MARK

Question. 8

The trigonometric Fourier series for the waveform ( )f t shown below contains

(A) only cosine terms and zero values for the dc components

(B) only cosine terms and a positive value for the dc components

(C) only cosine terms and a negative value for the dc components

(D) only sine terms and a negative value for the dc components

Question. 9

Consider the z -transform ( ) 5 4 3; 0x z z z z< <2 1 3= + +− . The inverse z - transform [ ]x n is

(A) 5 [ 2] 3 [ ] [ 1]n n n4δ δ δ+ + + −

(B) 5 [ 2] 3 [ ] 4 [ 1]n n nδ δ δ− + + +

(C) [ ] [ ] [ ]u n u n u n5 2 3 4 1+ + + −


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(D) [ ] [ ] [ ]u n u n u n5 2 3 4 1− + + +

Question. 10

Two discrete time system with impulse response [ ] [ 1]h n n1 δ= − and [ ] [ 2]h n n2 δ= − are connected in cascade. The overall impulse response of the cascaded system is

(A) [ 1] [ 2]n nδ δ− + − (B) [ 4]nδ −

(C) [ 3]nδ − (D) [ 1] [ 2]n nδ δ− −

Question. 11

For a N -point FET algorithm N 2m= which one of the following statements is TRUE ?

(A) It is not possible to construct a signal flow graph with both input and output in normal order

(B) The number of butterflies in the mth stage in N/m

(C) In-place computation requires storage of only 2N data

(D) Computation of a butterfly requires only one complex multiplication.

2010 TWO MARKS

Question. 12

Given ( )( )

f t Ls s k s

s4 3

3 113 2=+ + −

+−; E. If ( ) 1lim f t

t=

"3, then the value

of k is

(A) 1 (B) 2

(C) 3 (D) 4

Question. 13

A continuous time LTI system is described by

( )

4( )

3 ( )dt

d y tdt

dy ty t2

2

+ + ( )

( )dtdx t

x t2 4= +

Assuming zero initial conditions, the response ( )y t of the above system for the input ( ) ( )x t e u tt2= − is given by

(A) ( ) ( )e e u tt t3− (B) ( ) ( )e e u tt t3−− −

(C) ( ) ( )e e u tt t3+− − (D) ( ) ( )e e u tt t3+


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Question. 14

The transfer function of a discrete time LTI system is given by

( )H z z z

z

1 43

81

2 43 1

1 2=

− +

−− −

−

Consider the following statements:

S1: The system is stable and causal for ROC: /z 1 2>

S2: The system is stable but not causal for ROC: 1/z 4<

S3: The system is neither stable nor causal for ROC: / /z1 4 1 2< <

Which one of the following statements is valid ?

(A) Both S1 and S2 are true (B) Both S2 and S3 are true

(C) Both S1 and S3 are true (D) S1, S2 and S3 are all true

2009 ONE MARK

Question. 15

The Fourier series of a real periodic function has only

(P) cosine terms if it is even (Q) sine terms if it is even (R) cosine terms if it is odd (S) sine terms if it is odd

Which of the above statements are correct ?

(A) P and S (B) P and R

(C) Q and S (D) Q and R

Question. 16

A function is given by ( ) sin cosf t t t22= + . Which of the following is true ?

(A) f has frequency components at 0 and 21π

Hz

(B) f has frequency components at 0 and 1π

Hz

(C) f has frequency components at 21π

and 1π

Hz

(D) f has frequency components at .20 1

π and 1

π Hz


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Question. 17

The ROC of z -transform of the discrete time sequence

( )x n ( ) ( 1)u n u n31

21n n

= − − −b bl l is

(A) z31

21< < (B) z

21>

(C) z31< (D) z2 3< <

2009 TWO MARKS

Question. 18

Given that ( )F s is the one-side Laplace transform of ( )f t , the Laplace

transform of ( )f dt

0τ τ# is

(A) ( ) ( )sF s f 0− (B) ( )s

F s1

(C) ( )F ds

0τ τ# (D) [ ( ) ( )]

sF s f1 0−

Question. 19

A system with transfer function ( )H z has impulse response (.)h defined as ( ) , ( )h h2 1 3 1= =− and ( )h k 0= otherwise. Consider the following statements.

S1 : ( )H z is a low-pass filter.

S2 : ( )H z is an FIR filter.

Which of the following is correct?

(A) Only S2 is true

(B) Both S1 and S2 are false

(C) Both S1 and S2 are true, and S2 is a reason for S1

(D) Both S1 and S2 are true, but S2 is not a reason for S1

Question. 20

Consider a system whose input x and output y are related by the

equation ( ) ( ) ( )y t x t g d2τ τ τ= −3

3

−# where ( )h t is shown in the graph.


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Which of the following four properties are possessed by the system ?

BIBO : Bounded input gives a bounded output.

Causal : The system is causal,

LP : The system is low pass.

LTI : The system is linear and time-invariant.

(A) Causal, LP (B) BIBO, LTI

(C) BIBO, Causal, LTI (D) LP, LTI

Question. 21

The 4-point Discrete Fourier Transform (DFT) of a discrete time sequence {1,0,2,3} is

(A) [0, j2 2− + , 2, j2 2− − ] (B) [2, j2 2+ , 6, j2 2− ]

(C) [6, j1 3− , 2, j1 3+ ] (D) [6, j1 3− + , 0, j1 3− − ]

Question. 22

An LTI system having transfer function 2 11

s ss

2

2

+ ++ and input

( ) ( )sinx t t 1= + is in steady state. The output is sampled at a rate

sω rad/s to obtain the final output { ( )}x k . Which of the following is true ?

(A) (.)y is zero for all sampling frequencies sω

(B) (.)y is nonzero for all sampling frequencies sω

(C) (.)y is nonzero for 2>sω , but zero for 2<sω

(D) (.)y is zero for 2>sω , but nonzero for 2<2ω

2008 ONE MARK

Question. 23

The input and output of a continuous time system are respectively denoted by ( )x t and ( )y t . Which of the following descriptions corresponds to a causal system ?


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(A) ( ) ( ) ( )y t x t x t2 4= − + + (B) ( ) ( ) ( )y t t x t4 1= − +

(C) ( ) ( ) ( )y t t x t4 1= + − (D) ( ) ( ) ( )y t t x t5 5= + +

Question. 24

The impulse response ( )h t of a linear time invariant continuous time system is described by ( ) ( ) ( ) ( ) ( )exp exph t t u t t u tα β= + − where ( )u t− denotes the unit step function, and α and β are real constants.

This system is stable if

(A) α is positive and β is positive

(B) α is negative and β is negative

(C) α is negative and β is negative

(D) α is negative and β is positive

2008 TWO MARKS

Question. 25

A linear, time - invariant, causal continuous time system has a rational transfer function with simple poles at s 2=− and s 4=− and one simple zero at s 1=− . A unit step ( )u t is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response of this system is

(A) [ ( ) ( )] ( )exp expt t u t2 4− + −

(B) [ ( ) ( ) ( )] ( )exp exp expt t t u t4 2 12 4− − − − − −

(C) [ ( ) ( )] ( )exp expt t u t4 2 12 4− − + −

(D) [ . ( ) . ( )] ( )exp expt t u t0 5 2 1 5 4− − + −

Question. 26

The signal ( )x t is described by

( )x t t1 1 1

0

for

otherwise

# #=

− +)

Two of the angular frequencies at which its Fourier transform becomes zero are(A) , 2π π (B) 0.5 , 1.5π π

(C) 0, π (D) 2 , 2.5π π


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Question. 27

A discrete time linear shift - invariant system has an impulse response

[ ]h n with [ ] , [ ] , [ ] ,h h h0 1 1 1 2 2= =− = and zero otherwise The system is given an input sequence [ ]x n with [ ] [ ]x x0 2 1= = , and zero otherwise. The number of nonzero samples in the output sequence

[ ]y n , and the value of [ ]y 2 are respectively

(A) 5, 2 (B) 6, 2

(C) 6, 1 (D) 5, 3

Question. 28

Let ( )x t be the input and ( )y t be the output of a continuous time system. Match the system properties P1, P2 and P3 with system relations R1, R2, R3, R4

Properties Relations

P1 : Linear but NOT time - invariant R1 : ( ) ( )y t t x t2=

P2 : Time - invariant but NOT linear R2 : ( ) ( )y t t x t=

P3 : Linear and time - invariant R3 : ( ) ( )y t x t=

R4 : ( ) ( )y t x t 5= −

(A) (P1, R1), (P2, R3), (P3, R4)

(B) (P1, R2), (P2, R3), (P3, R4)

(C) (P1, R3), (P2, R1), (P3, R2)

(D) (P1, R1), (P2, R2), (P3, R3)

Question. 29

{ ( )}x n is a real - valued periodic sequence with a period N . ( )x n and ( )X k form N-point Discrete Fourier Transform (DFT) pairs. The

DFT ( )Y k of the sequence ( ) ( ) ( )y n N x r x n r1r

N

0

1

= +=

−

/ is

(A) ( )X k 2 (B) ( ) ( )N X r X k r1r

N

0

1

+=

−

/

(C) ( ) ( )N X r X k r1r

N

0

1

+=

−

/ (D) 0


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Statement for Linked Answer Question 30 & 31:

In the following network, the switch is closed at 0t = − and the sampling starts from t 0= . The sampling frequency is 10 Hz.

Question. 30

The samples ( ), (0, 1, 2, ...)x n n = are given by

(A) 5(1 )e . n0 05− − (B) 5e . n0 05−

(C) 5(1 )e n5− − (D) 5e n5−

Question. 31

The expression and the region of convergence of the z −transform of the sampled signal are

(A) ,z e

z55−

z e< 5− (B) ,z e

z5.0 05− − z e< .0 05−

(C) ,z e

z5.0 05− − z e> .0 05− (D)

z ez5

5− − , z e> 5−

Statement for Linked Answer Question 32 & 33:

The impulse response ( )h t of linear time - invariant continuous time system is given by ( ) ( 2 ) ( )exph t t u t= − , where ( )u t denotes the unit step function.

Question. 32

The frequency response ( )H ω of this system in terms of angular frequency ω, is given by ( )H ω

(A) j1 2

1ω+

(B) sinω

ω

(C) j2

1ω+

(D) j

j2 ω

ω+


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Question. 33

The output of this system, to the sinusoidal input ( ) cosx t t2 2= for all time t , is

(A) 0 (B) 2 (2 0.125 )cos t.0 25 π−−

(C) 2 (2 0.125 )cos t.0 5 π−− (D) 2 (2 0.25 )cos t.0 5 π−−

2007 ONE MARK

Question. 34

If the Laplace transform of a signal ( )( )

Y ss s 1

1=−

, then its final value is

(A) 1− (B) 0

(C) 1 (D) Unbounded

2007 TWO MARKS

Question. 35

The 3-dB bandwidth of the low-pass signal ( )e u tt− , where ( )u t is the unit step function, is given by

(A) 21π

Hz (B) 21 2 1π

− Hz

(C) 3 (D) 1 Hz

Question. 36

A 5-point sequence [ ]x n is given as [ 3] 1,x − = [ 2] 1,x − = [ 1] 0,x − = [0] 5x = and [ ]x 1 1= . Let ( )X eiω denoted the discrete-time Fourier

transform of [ ]x n . The value of ( )X e dj ωω

π

π

−# is

(A) 5 (B) 10π

(C) 16π (D) j5 10π+

Question. 37

The z −transform ( )X z of a sequence [ ]x n is given by [ ]X z .z1 2

0 51= − − .

It is given that the region of convergence of ( )X z includes the unit circle. The value of [ ]x 0 is

(A) .0 5− (B) 0

(C) 0.25 (D) 05


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Question. 38

A Hilbert transformer is a

(A) non-linear system (B) non-causal system

(C) time-varying system (D) low-pass system

Question. 39

The frequency response of a linear, time-invariant system is given by

( )H f j f1 105= π+ . The step response of the system is

(A) 5(1 ) ( )e u tt5− − (B) 5 ( )e u t1t5− −

6 @

(C) (1 ) ( )e u t21 t5− − (D) ( )e u t5

1 1t5− −^ h

2006 ONE MARK

Question. 40

Let ( ) ( )x t X j* ω be Fourier Transform pair. The Fourier Transform

of the signal ( )x t5 3− in terms of ( )X jω is given as

(A) e X j51

5j53 ωω

−b l (B) e X j

51

5j53 ωω

b l

(C) e X j51

5j3 ωω−

b l (D) e X j51

5j3 ωω

b l

Question. 41

The Dirac delta function ( )tδ is defined as

(A) ( )tt1 0

0 otherwiseδ =

=)

(B) ( )tt 0

0 otherwise

3δ =

=)

(C) ( )tt1 0

0 otherwiseδ =

=) and ( )t dt 1δ =

3

3

−#

(D) ( )tt 0

0 otherwise

3δ =

=) and ( )t dt 1δ =

3

3

−#


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Question. 42

If the region of convergence of [ ] [ ]x n x n1 2+ is z31

32< < then the

region of convergence of [ ] [ ]x n x n1 2− includes

(A) z31 3< < (B) z

32 3< <

(C) z23 3< < (D) z

31

32< <

Question. 43

In the system shown below, ( ) ( ) ( )sinx t t u t= In steady-state, the response ( )y t will be

(A) sin t2

14π−` j (B) sin t

21

4π+` j

(C) sine t2

1 t− (D) sin cost t−

2006 TWO MARKS

Question. 44

Consider the function ( )f t having Laplace transform

( )F s [ ]Res

s 0>202

0

ωω=+

The final value of ( )f t would be(A) 0 (B) 1

(C) ( )f1 13# #− (D) 3

Question. 45

A system with input [ ]x n and output [ ]y n is given as [ ] ( ) [ ]siny n n x n65 π=

. The system is

(A) linear, stable and invertible

(B) non-linear, stable and non-invertible

(C) linear, stable and non-invertible

(D) linear, unstable and invertible


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Question. 46

The unit step response of a system starting from rest is given by ( ) 1c t e t2= − − for t 0$ . The transfer function of the system is

(A) s1 2

1+

(B) s2

2+

(C) s2

1+

(D) s

s1 2

2+

Question. 47

The unit impulse response of a system is ( ) , 0f t e tt $= − . For this system the steady-state value of the output for unit step input is equal to

(A) 1− (B) 0

(C) 1 (D) 3

2005 ONE MARK

Question. 48

Choose the function ( );f t t< <3 3− for which a Fourier series cannot be defined.

(A) ( )sin t3 25 (B) ( ) ( )cos sint t4 20 3 2 710+ +

(C) ( ) ( )exp sint t25− (D) 1

Question. 49

The function ( )x t is shown in the figure. Even and odd parts of a unit step function ( )u t are respectively,

(A) , ( )x t21

21 (B) , ( )x t

21

21−

(C) , ( )x t21

21− (D) , ( )x t

21

21− −


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Question. 50

The region of convergence of z − transform of the sequence

( ) ( 1)u n u n65

56n n

− − −b bl l must be

(A) z65< (B) z

65>

(C) z65

56< < (D) z

56 < < 3

Question. 51

Which of the following can be impulse response of a causal system ?

Question. 52

Let ( ) ( ) ( ), ( ) ( )x n u n y n x nn21 2= = and ( )Y ejω be the Fourier transform

of ( )y n then ( )Y ej0

(A) 41 (B) 2

(C) 4 (D) 34

Question. 53

The power in the signal ( ) ( ) ( )cos sins t t8 20 4 152π π= − +π is

(A) 40 (B) 41

(C) 42 (D) 82


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2005 TWO MARKS

Question. 54

The output ( )y t of a linear time invariant system is related to its input ( )x t by the following equations

( )y t . ( ) ( ) . ( )x t t T x t t x t t T0 5 0 5d d d= − + + − + − +

The filter transfer function ( )H ω of such a system is given by(A) (1 )cos T e j tdω+ ω− (B) (1 0.5 )cos T e j tdω+ ω−

(C) (1 )cos T e j tdω− ω− (D) (1 0.5 )cos T e j tdω− ω−

Question. 55

Match the following and choose the correct combination.

Group 1

E. Continuous and aperiodic signalF. Continuous and periodic signalG. Discrete and aperiodic signalH. Discrete and periodic signal

Group 2

1. Fourier representation is continuous and aperiodic2. Fourier representation is discrete and aperiodic3. Fourier representation is continuous and periodic4. Fourier representation is discrete and periodic

(A) E 3, F 2, G 4, H 1− − − −

(B) E 1, F 3, G 2, H 4− − − −

(C) E 1, F 2, G 3, H 4− − − −

(D) E 2, F 1, G 4, H 3− − − −

Question. 56

A signal ( ) ( )sinx n n0ω φ= + is the input to a linear time- invariant system having a frequency response ( )H ejω . If the output of the system ( )Ax n n0− then the most general form of ( )H ej+ ω will be

(A) n0 0ω β− + for any arbitrary real

(B) n k20 0ω π− + for any arbitrary integer k

(C) n k20 0ω π+ for any arbitrary integer k

(D) n0 0ω φ−


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Statement of linked answer question 57 & 58 :

A sequence ( )x n has non-zero values as shown in the figure.

Question. 57

The sequence ( )y n

( ),

,

x n

n

1

0

For even

For odd

n2

=

−

* will be


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Question. 58

The Fourier transform of ( )y n2 will be

(A) [ 4 2 2 2]cos cose j2 ω ω+ +ω− (B) cos cos2 2 2ω ω+ +

(C) [ 2 2 2]cos cose j ω ω+ +ω− (D) [ 2 2 2]cos cose j2 ω + +ω−

Question. 59

For a signal ( )x t the Fourier transform is ( )X f . Then the inverse

Fourier transform of ( )X f3 2+ is given by

(A) x t e21

2j t3π

` j (B) x t e31

3j t34π-

` j

(C) 3 (3 )x t e j t4π− (D) ( )x t3 2+

2004 ONE MARK

Question. 60

The impulse response [ ]h n of a linear time-invariant system is given

by [ ] [ ] [ ) [ ]h n u n u n n n3 2 2 7= + + − − − where [ ]u n is the unit step

sequence. The above system is

(A) stable but not causal (B) stable and causal

(C) causal but unstable (D) unstable and not causal

Question. 61

The z -transform of a system is ( )H z .zz0 2= − . If the ROC is .z 0 2< ,

then the impulse response of the system is

(A) ( . ) [ ]u n0 2 n (B) ( . ) [ ]u n0 2 1n − −

(C) ( . ) [ ]u n0 2 n− (D) ( . ) [ ]u n0 2 1n− − −

Question. 62

The Fourier transform of a conjugate symmetric function is always

(A) imaginary (B) conjugate anti-symmetric

(C) real (D) conjugate symmetric


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2004 TWO MARKS

Question. 63

Consider the sequence [ ]x n [ ]j j4 51 25= − − +-

. The conjugate anti-symmetric part of the sequence is

(A) [ 4 2.5, 2, 4 2.5]j j j− − − (B) [ 2.5, 1, 2.5]j j−

(C) [ 2.5, 2, 0]j j− (D) [ 4, 1, 4]−

Question. 64

A causal LTI system is described by the difference equation

[ ]y n2 [ 2] 2 [ ] [ ]y n x n x n 1= α β− − + −The system is stable only if(A) 2α = , 2<β (B) ,2 2> >α β

(C) 2<α , any value of β (D) 2<β , any value of α

Question. 65

The impulse response [ ]h n of a linear time invariant system is given as

[ ]h n

,

,

n

n

2 2 1 1

4 2 2 2

0 otherwise

=

− = −

= −*

If the input to the above system is the sequence e /j n 4π , then the output is(A) e4 2 /j n 4π (B) 4 e2 /j n 4π−

(C) e4 /j n 4π (D) e4 /j n 4− π

Question. 66

Let ( )x t and ( )y t with Fourier transforms ( )F f and ( )Y f respectively be related as shown in Fig. Then ( )Y f is


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(A) ( /2)X f e21 j f− π− (B) ( / )X f e

21 2 j f2− π

(C) ( / )X f e2 j f2− π (D) ( /2)X f e j f2− π−

2003 ONE MARK

Question. 67

The Laplace transform of ( )i t is given by

( )I s ( )s s1

2=+

At t " 3, The value of ( )i t tends to(A) 0 (B) 1

(C) 2 (D) 3

Question. 68

The Fourier series expansion of a real periodic signal with fundamental frequency f0 is given by ( )g tp c e

nn

j f t2 0=3

π

=-/ . It is given that c j3 53 = +

. Then c 3− is

(A) j5 3+ (B) j3 5− −

(C) j5 3− + (D) j3 5−

Question. 69

Let ( )x t be the input to a linear, time-invariant system. The required output is ( )t4 2π − . The transfer function of the system should be

(A) e4 j f4π (B) 2e j f8π−

(C) 4e j f4π− (D) e2 j f8π

Question. 70

A sequence ( )x n with the z −transform ( ) 2 2 3X z z z z z4 2 4= + − + − − is applied as an input to a linear, time-invariant system with the impulse response ( ) ( )h n n2 3δ= − where

( )nδ ,

,

n1 0

0 otherwise=

=)

The output at n 4= is(A) 6− (B) zero

(C) 2 (D) 4−


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2003 TWO MARKS

Question. 71

Let P be linearity, Q be time-invariance, R be causality and S be stability. A discrete time system has the input-output relationship,

( )y n ( ) 1

0, 0

( )

x n n

n

x n n1 1

$

#

==

+ −*

where ( )x n is the input and ( )y n is the output. The above system has the properties(A) P, S but not Q, R (B) P, Q, S but not R

(C) P, Q, R, S (D) Q, R, S but not P

Common data for Question 72 & 73 :

The system under consideration is an RC low-pass filter (RC-LPF) with R 1= kΩ and .C 1 0= μF.

Question. 72

Let ( )H f denote the frequency response of the RC-LPF. Let f1 be the

highest frequency such that ( )( )

.f fHH f

00

0 9511

# # $ . Then f1 (in Hz) is

(A) 324.8 (B) 163.9

(C) 52.2 (D) 104.4

Question. 73

Let ( )t fg be the group delay function of the given RC-LPF and f 1002 = Hz. Then ( )t fg 2 in ms, is

(A) 0.717 (B) 7.17

(C) 71.7 (D) 4.505

2002 ONE MARK

Question. 74

Convolution of ( )x t 5+ with impulse function ( )t 7δ − is equal to

(A) ( )x t 12− (B) ( )x t 12+

(C) ( )x t 2− (D) ( )x t 2+


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Question. 75

Which of the following cannot be the Fourier series expansion of a periodic signal?

(A) ( ) cos cosx t t t2 3 3= + (B) ( ) cos cosx t t t2 7π= +

(C) ( ) .cosx t t 0 5= + (D) ( ) . .cos sinx t t t2 1 5 3 5π π= +

Question. 76

The Fourier transform { ( )}F e u t1− is equal to j f1 21

π+. Therefore,

Fj t1 21

π+' 1 is

(A) ( )e u ff (B) ( )e u ff−

(C) ( )e u ff − (D) ( )e u ff −−

Question. 77

A linear phase channel with phase delay Tp and group delay Tg must have

(A) T Tp g= = constant

(B) T fp \ and T fg \

(C) Tp = constant and T fg \ ( f denote frequency)

(D) T fp \ and Tp = constant

2002 TWO MARKS

Question. 78

The Laplace transform of continuous - time signal ( )x t is ( )X s s ss

25

2= − −−

. If the Fourier transform of this signal exists, the ( )x t is

(A) ( ) 2 ( )e u t e u tt t2 − − (B) ( ) 2 ( )e u t e u tt t2− − + −

(C) ( ) 2 ( )e u t e u tt t2− − − − (D) ( ) 2 ( )e u t e u tt t2 − − −

Question. 79

If the impulse response of discrete - time system is

[ ] [ ]h n u n5 1n=− − − ,

then the system function ( )H z is equal to

(A) z

z5−

− and the system is stable


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(B) z

z5−

and the system is stable

(C) z

z5−

− and the system is unstable

(D) z

z5−

and the system is unstable

2001 ONE MARK

Question. 80

The transfer function of a system is given by ( )( )

H ss s 2

12=

−. The

impulse response of the system is

(A) ( * ) ( )t e u tt2 2− (B) ( * ) ( )t e u tt2

(C) ( ) ( )te t u t2− (D) ( ) ( )te u tt2−

Question. 81

The region of convergence of the z − transform of a unit step function

is

(A) z 1> (B) z 1<

(C) (Real part of z ) 0> (D) (Real part of z ) 0<

Question. 82

Let ( )tδ denote the delta function. The value of the integral

( )cost t dt23δ

3

3

−b l# is

(A) 1 (B) 1−

(C) 0 (D) 2π

Question. 83

If a signal ( )f t has energy E , the energy of the signal ( )f t2 is equal to

(A) 1 (B) /E 2

(C) 2E (D) E4


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2001 TWO MARKS

Question. 84

The impulse response functions of four linear systems S1, S2, S3, S4 are given respectively by

( )h t 11 = , ( ) ( )h t u t2 = ,

( )( )

h ttu t

13 =+

and

( ) ( )h t e u tt4

3= −

where ( )u t is the unit step function. Which of these systems is time invariant, causal, and stable?

(A) S1 (B) S2

(C) S3 (D) S4

2000 ONE MARK

Question. 85

Given that [ ( )]L f tss

12

2=++ , [ ( )]

( )( )L g t

s ss3 2

12

= + ++ and

( ) ( ) ( )h t f g t dt

0τ τ τ= −# .

[ ( )]L h t is

(A) ss

312

++ (B)

s 31+

(C) ( )( )s s

sss

3 21

122

2+ ++ +

++ (D) None of the above

Question. 86

The Fourier Transform of the signal ( )x t e t3 2

= − is of the following form, where A and B are constants :

(A) Ae B f− (B) Ae Bf2−

(C) A B f 2+ (D) Ae Bf−


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Question. 87

A system with an input ( )x t and output ( )y t is described by the relations : ( ) ( )y t tx t= . This system is

(A) linear and time - invariant

(B) linear and time varying

(C) non - linear and time - invariant

(D) non - linear and time - varying

Question. 88

A linear time invariant system has an impulse response ,e t 0>t2 . If the initial conditions are zero and the input is e t3 , the output for t 0> is

(A) e et t3 2− (B) e t5

(C) e et t3 2+ (D) None of these

2000 TWO MARKS

Question. 89

One period ( , )T0 each of two periodic waveforms W1 and W2 are shown in the figure. The magnitudes of the nth Fourier series coefficients of W1 and W2, for ,n n1$ odd, are respectively proportional to

(A) n 3− and n 2− (B) n 2− and n 3−

(C) n 1− and n 2− (D) n 4− and n 2−

Question. 90

Let ( )u t be the step function. Which of the waveforms in the figure corresponds to the convolution of ( ) ( )u t u t 1− − with ( ) ( )u t u t 2− − ?


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Question. 91

A system has a phase response given by ( ),φ ω where ω is the angular frequency. The phase delay and group delay at 0ω ω= are respectively given by

(A) ( )

,( )

dd

0

0

0ωφ ω

ωφ ω− −

ω ω= (B) ( ),

( )d

do 2

20

o

φ ωω

φ ω−ω ω=

(C) ( )

,( )( )

dd

o

o

oφ ω

ωω

φ ω−ω ω=

(D) ( ), ( )o oo

ω φ ω φ λ3

ω

−#

1999 ONE MARK

Question. 92

The z -transform ( )F z of the function ( )f nT anT= is

(A) z a

zT−

(B) z a

zT+

(C) z a

zT− − (D)

z az

T+ −

Question. 93

If [ ( )] ( ), [ ( )]thenf t F s f t T= − is equal to

(A) ( )e F ssT (B) ( )e F ssT−

(C) ( )e

F s1 sT−

(D) ( )e

F s1 sT− −


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Question. 94

A signal ( )x t has a Fourier transform ( )X ω . If ( )x t is a real and odd

function of t , then ( )X ω is

(A) a real and even function of ω

(B) a imaginary and odd function of ω

(C) an imaginary and even function of ω

(D) a real and odd function of ω

1999 TWO MARKS

Question. 95

The Fourier series representation of an impulse train denoted by

( )s t ( )d t nTn

0= −3

3

=−/ is given by

(A) expT Tj nt1 2

n0 0

π−3

3

=−/ (B) expT T

j nt1n0 0

π−3

3

=−/

(C) expT Tj nt1

n0 0

π3

3

=−/ (D) expT T

j nt1 2n0 0

π3

3

=−/

Question. 96

The z -transform of a signal is given by

( )C z ( )

( )z

z z4 1

1 11 2

1 4

=−

−−

− −

Its final value is

(A) 1/4 (B) zero

(C) 1.0 (D) infinity

1998 ONE MARK

Question. 97

If ( )F ss2 2ω

ω=+

, then the value of ( )Limf tt"3

(A) cannot be determined (B) is zero

(C) is unity (D) is infinite


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Question. 98

The trigonometric Fourier series of a even time function can have only

(A) cosine terms (B) sine terms

(C) cosine and sine terms (D) d.c and cosine terms

Question. 99

A periodic signal ( )x t of period T0 is given by

( )x t ,

,

t T

T t T1

0 2

<

< <

1

10

= *

The dc component of ( )x t is

(A) TT

0

1 (B) TT2 0

1

(C) TT20

1 (D) TT

1

0

Question. 100

The unit impulse response of a linear time invariant system is the unit step function ( )u t . For t 0> , the response of the system to an excitation ( ),e u t a 0>at− will be

(A) ae at− (B) ( / )( )a e1 1 at− −

(C) ( )a e1 at− − (D) e1 at− −

Question. 101

The z-transform of the time function ( )n kk 0

δ −3

=/ is

(A) zz 1−

(B) zz

1−

(C) ( )z

z1 2−

(D) ( )

zz 1 2−

Question. 102

A distorted sinusoid has the amplitudes , , , ....A A A1 2 3 of the fundamental, second harmonic, third harmonic,..... respectively. The total harmonic distortion is

(A) ....A

A A1

2 3+ + (B) .....

AA A

1

22

32+ +


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(C) ....

.....A A A

A A12

22

32

22

32

+ + ++ +

(D) .....A

A A1

22

32+ +

c m

Question. 103

The Fourier transform of a function ( )x t is ( )X f . The Fourier transform

of ( )

dfdX t

will be

(A) ( )

dfdX f

(B) ( )j fX f2π

(C) ( )jfX f (D) ( )jf

X f

1997 ONE MARK

Question. 104

The function ( )f t has the Fourier Transform ( )g ω . The Fourier Transform

( ) ( ) ( )ff t g t g t e dtj t=3

3ω−

−e o# is

(A) ( )f21π ω (B) ( )f2

1π ω−

(C) ( )f2π ω− (D) None of the above

Question. 105

The Laplace Transform of ( )cose tt αα is equal to

(A) ( )

( )s

s2 2α αα

− +−

(B) ( )

( )s

s2 2α αα

− ++

(C) ( )s

12α−

(D) None of the above

1996 ONE MARK

Question. 106

The trigonometric Fourier series of an even function of time does not have the

(A) dc term (B) cosine terms

(C) sine terms (D) odd harmonic terms


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Question. 107

The Fourier transform of a real valued time signal has

(A) odd symmetry (B) even symmetry

(C) conjugate symmetry (D) no symmetry

***********

Signals and Systems

Documents

expx t t u t t

u n u n u n5

exp t u t u t0

b exp t u t u t1

impulse response h t

n n b

output isa exp t u t

n n question