Ec8352 signals and systems 2 marks with answers

Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
Question Bank- Two Marks With Answer EC8352/ Signals and Systems
UNIT I - CLASSIFICATION OF SIGNALS AND SYSTEMS
1. Define Continuous time step function and Delta function.
A step function is defined as
1 :t 0
u(t)
0 :t 0

 

A delta function is defined as
1 :t 0
(t)
0 :t 0

  

2. Define energy and power signals.(Apr/May 2015) (Nov/Dec 2015)
The energy of a signal x(t) is defined as
2
E x(t) dt


 
A signal x(t) is called an energy signals if the energy is finite and power is zero.
The average power of a signal x(t) is defined as
T 2
2
T
T 2
1
P lim x(t) dt
T



 
A Signal x(t) is called a power signal if the average power is finite and Energy is infinite.
3. Verify whether
- tx(t) = A e u(t) is an energy signal or not
This signal is non periodic and of infinite duration .It can be an energy signal.
2
-
2
- t
0
2 -2 t
0
2 t 2
2
0
E = x(t) dt
= [Ae ] dt Since u(t) = 1 for 0 t
= A e dt
e A
= A =
2 2






 
  
 
    



The energy is finite and non zero; hence the signal is energy signal

4. Define Causality system and stability of a system with an example for each
Causality: A system is said to be causal if its output at any time depends on the present and past
inputs only. Example:
o( ) = f [ x(t) ; t t ]oy t
Stability: A system is said to be stable,if it produces bounded output for all bounded inputs
Example: y(t) = x(t) sin(100 t)
5. Show that the Complex exponential signal ( ) = is periodic and that
fundamental period is
For a periodic signal x(t) =x(T + t),If
oj t
x(t) = e

Therefore
oj (t + T)
x(t + T) = e

o oj t j T
x(t + T) = e e
 
This condition is satisfied x(t) = x(T + t) only if
oj T
e

=1 from which we have
o
o
2
T 2 or T =

  

6. Check whether the signal ( ) 2cos3 7cos9 x t t t is periodic or not.
1 2
2
31 1
2
92 2
2 2 2
;
3 3 9
18 3 3
;
6
T T
T T
Therefore
T T
 

  
  
   
is not a rational number, therefore the signal is not periodic.
7. Define Signal.
Signal is a physical quantity that varies with respect to time, space or any other independent
variable. Or it is a mathematical representation of the system
E.g. y (t) = t. and x (t) = sin t.
8. Define system.
A set of components that are connected together to perform the particular task.

9. What are the major classifications of the signal?
(i) Discrete time signal
(ii) Continuous time signal
10. Define discrete time signals and classify them.
Discrete time signals are defined only at discrete times, and for these signals, the independent
variable takes on only a discrete set of values.
Classification of discrete time signal:
1. Periodic and Aperiodic signal
2. Even and Odd signal
11. Define continuous time signals and classify them.
Continuous time signals are defined for a continuous of values of the independent variable. In the
case of continuous time signals the independent variable is continuous.
For example:
(i) A speech signal as a function of time
(ii) Atmospheric pressure as a function of altitude
Classification of continuous time signal:
(i) Periodic and Aperiodic signal
(ii) Even and Odd signal
12. Define discrete time unit step &unit impulse.
Discrete time Unit impulse is defined as
1 :n 0
[n]
0 :n 0

  

Unit impulse is also known as unit sample.
Discrete time unit step signal is defined by
1 :n 0
u[n]
0 :n 0

 


13. Define unit ramp signal.
Continuous time unit ramp function is defined by
1 :t 0
r(t)
0 :t 0

 

A ramp signal starts at t=0 and increases linearly with time‘t’.
14. Define periodic signal. and non periodic signal.
A signal is said to be periodic, if it exhibits periodicity .i.e., X (t +T) =x (t), for all values of t.
Periodic signal has the property that it is unchanged by a time shift of T.
A signal that does not satisfy the above periodicity property is called an aperiodic signal.
15. Define even and odd signal?
A discrete time signal is said to be even when, x [-n] =x[n].
The continuous time signal is said to be even when, x (-t) = x (t)
The discrete time signal is said to be odd when x [-n] = -x[n]
The continuous time signal is said to be odd when x (-t) = -x (t)
Odd signals are also known as nonsymmetrical signal.
Sine wave signal is an odd signal.
16. Define unit pulse function.
Unit pulse function (t) is obtained from unit step signals
(t)=u (t+1/2) - u (t-1/2)
The signals u (t+1/2) and u (t-1/2) are the unit step signals shifted by 1/2units in the time axis
towards the left and right, respectively.
17. Define continuous time complex exponential signal.
The continuous time complex exponential signal is of the form
x (t) = C eat
where c and a are complex numbers.
18. What is continuous time real exponential signal?
Continuous time real exponential signal is defined by
x (t) = Ceat

where c and a are complex numbers. If c and a are real, then it is called as real exponential.
19. What is continuous time growing exponential signal?
Continuous time growing exponential signal is defined as
x (t) = Ceat
where c and a are complex numbers. If a is positive, as t increases, then x (t) is a growing
exponential.
20. What is continuous time decaying exponential?
Continuous time growing exponential signal is defined as
x (t) = Ceat
where c and a are complex numbers. If a is negative, as t increases, then x (t) is a decaying
exponential.
21. Give the mathematical representation of a continuous time and discrete time unit
impulse functions. (Nov/Dec 2016)
A continuous time impulse function is defined as
1 :t 0
(t)
0 :t 0

  

A discrete time impulse function is defined as
1 :n 0
[n]
0 :n 0

  

22. State the difference between causal and Non-causal systems. (Nov/Dec 2016)
S.No Causal Systems Non-Causal Systems
1 A System is said to be causal if the
output of the system at any time t
depends only on the present input,
past inputs and past outputs but
does not depend on the future
inputs and outputs
A System is said to be Non-
causal if the output of the
system at any time t depends
on the future inputs or outputs
23. Define Static (memory Less) and Dynamic (memory) system?
A system is said to be static or memory less if its output depends upon the present input
only.
The system is said to be dynamic with memory if its output depends upon the present
and past input values.

24. Sketch the following signals: ; ( . ) (May/Jun 2016)
Shift the ( ) by 0.25 units to the left and then do amplitude scaling
( . )
25. Give the relationship between continuous time unit impulse function f(t), step function
u(t) and ramp function r(t)? (Nov/Dec 2015)
Relationship between Ramp and Step function
( ) = ( ) = 1 =
Relationship between Ramp and Step function
( ) = ( )
δ(t)

⎯⎯⎯⎯⎯⎯⎯⎯ ( ) ⎯⎯⎯⎯⎯⎯⎯ ( ) ⎯⎯⎯⎯⎯⎯⎯
parabolic ⎯⎯⎯⎯⎯⎯⎯⎯⎯ ( ) ⎯⎯⎯⎯⎯⎯⎯⎯⎯ ( ) ⎯⎯⎯⎯⎯⎯⎯⎯⎯ δ(t)
26. Given [ ] =
, , , − ,
↑
.Plot the signal x(n-1)(Nov/Dec 2015)

27. Define random signals.(Nov/Dec 2016)
A random signal is characterized by uncertainty before its actual occurrence.
28. What are the different types of representation of discrete time signals?(Nov/Dec 2016)
Graphical Representation
Functional Representation
Tabular Representation
Sequence Representation
29. State two properties of impulse function. (Nov/Dec 2014)
1. ∫ ( ) ( ) = (0)
2. ( ) ( − ) = ( ) ( − )
30. Draw the following signals: (Nov/Dec 2014)
) ( ) − ( − )
) ( − )
a)
b)

UNIT II - ANALYSIS OF CONTINUOUS TIME SIGNALS
1. Define Fourier series.
Continuous Time Fourier series is defined as
0
0
jk n
k
k
2
k N
T
x(n) a e (Synthesis Equation)
P x[k] (Analysis Equation)
x(t) dt <










Discrete Time Fourier series is defined as
0
0
jk n
k
k
jk n
k k
n N
x(n) a e (Synthesis Equation)
1
a a e (Analysis Equation)
N



 





2. State Dirichlet condition for Fourier series.(Nov/Dec 2014,2013)
i)The function x(t) should be within the interval T0
ii) The function x(t) should have finite number of maxima and minima in the
interval T0
iii) The function x(t) should have finite number of discontinuities in the interval T0
iv ) The function x(t) should be absolutely integrable
0T
x(t) dt < 
3.State Parsevals theorem for discrete time signal.
The average power of the sequence x[n] is equal to the summation of its squared components
2
k N
P x (k)

 
4. State the time shifting property of Fourier series.

If the Fourier series co-efficients of ( )x t are nc then the Fourier co-efficient of the shifted signal
0( )x t t are
0
2
( )
0[ ( )]
 
 
jn
T
nFS x t t e c

5. Find the Fourier transform of ( ) 
Inverse Fourier transform is given as
1
( ) ( ) dt
2
j t
x t X e 




 
1
( ) ( ) dt
2
j t
x t e 
 



 
1 : = 0
( )
0 : 0

 


 

01
( ) (0)
2
x t e


1
2

6. Find the Fourier Transform of
at
e u (-t)
0
( ) ( ) dt
dt
j t
at j t
X x t e
e e












=
1
s a
 
  
7. Find the Laplace Transform of a unit step function.(Nov/Dec 2015)
st
st st
0 0
1 for t 0
X(s) x(t) e dt u(t)
0 for t 0
e 1
= e dt =
s s


 
  
   
 
 
   



8. Determine the Laplace transform of -at
x(t) = e sin(ωt) u(t)
   
 a j a j tt t1
L e e
2j
     
 

1 1 1
2j s (a j ) s (a j )
 
  
      
2 2
, ROC : Re(s) >-a
(s a)


  
9. Find the Laplace transform of -at
x(t) t e u(t), where a > 0.
L-at 1
e u(t) ROC : Re (s) > -a
s a '


Differentiation in s-domain property gives,
L d
-t x(t) X(s)
ds

L-at d 1
t e u(t)
ds s a
 
    
L-at
2'
1
t e u(t) ROC : Re (s) > -a
(s a)


10. Do Fourier transform and Laplace transform exist? What is the condition for its
existence?
Existence of Fourier Transform
a. x(t) should be absolutely integrable.
b. x(t) should be single valued in any finite interval.
c. x(t) should have finite number of maxima and minima in any finite interval.
d. x(t) should have finite number of maxima and minima over any finite interval T.
Convergence of Laplace Transform
(i)
t
x(t) e
must be absolutely integrable.
The remaining conditions are same as those for Fourier transform to exist.
11. Define Fourier transform pair for continuous time signal.(No/Dec 2015)
( ) ( )

 

  
j t
x j x t e dt for all  and
1
( ) ( )
2



  
j t
x t x j e d

for all t.
12. State the frequency shifting property of Laplace transform.

If [ ( )] ( )L x t X s Then [ ( )] ( )
 at
L e x t X s a
13.Define Laplace transform.
The Laplace transform is given by ( ) ( )



 
st
X s x t e dt
14. Find the Laplace transform of ( ) = ( ). (Nov/Dec 2016)
st
(s a)t
at st
0 0
1 for t 0
X(s) x(t) e dt u(t)
0 for t 0
e 1
= e .e dt =
(s a) s+a


  
 
  
   
 
 
    



15. What is the inverse Fourier transform of ) ) ( − ) ? May/Jun 2016
) = ( − )
) [ ( − )] =
1
2
16. Give the Laplace transform of ( ) = ( ) − ( ) with ROC.(May/Jun 2016)
-2 1 2
X(s) =
5 s 4 5(s 1)

 
There are two poles Re(s) <-4 and Re(s) >1 The overall ROC Re(s) >1
-4 0 1 σ
s-plane
1
jΩ

17.Draw the spectrum of CT rectangular pulse.(Apr/May 2015)
18. Given ( ) = ( ).Find ( ) ( ) .(Apr/May 2015)
A delta function is defined as
1 :t 0
(t)
0 :t 0

  

2 2
1 Cos2t 1 1 s 2
X(s) L{x(t)} L
2 2 s s 4 s(s 4)
   
       
    
j t j t
X( ) x(t) e dt = (t) e dt =1
    
 
   
19. Give the relation between Fourier and Laplace transform. (Nov/Dec 2015, 2014)
st
j t
X(s) x(t) e dt Sub s= +j
If s=j and =0 then it becomes fourier transform X(j ) x(t) e dt


 

  
   


20.What is ( − ) ∗ ( − )? where * represents convolution.(Nov/Dec 2015)
( ) =
0 ; < −1
( ); ≥ −1
n
Amplitude| ( )|

21.Find the ROC of the Laplace transform of x(t)=u(t). (Nov/Dec 2014)
ROC : Re{s}>0
22.Draw the block diagram of the LTI system described by (Nov/Dec 2014)
dy(t)
y(t) = 0.1x(t)
dt

23. Define Gibb’s phenomenon?
The truncated Fourier series approaches x(t) as n increases.ie the error between x(t) and xn(t)
decreases as n increases. However at the discontinuity,xn(t) exhibits an oscillatory behavior and
has ripples on both sides. As n increases the frequency of the ripple increases, but the amplitude
of ripples is about remain roughly same. This behavior is known as Gibb’s phenomenon.
24. State Rayleigh’s energy theorem?
Rayleigh‟s energy theorem states that the energy of the signal may be written in frequency
domain as superposition of energies due to individual spectral frequencies of the signal.
25. State the properties of ROC of Laplace transform ? [May/Jun 2014]
1. The ROC of X(S) consists of strips parallel to the jΩ axis in the s-plane.
2. The ROC does not contain any poles
3. If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s-
plane
4. It x(t) is a right-sided signal ,i.e. x(t) =0 for t< < ∞ then the ROC is of the form
Re(s) > max. here max equals the maximum real part of any of the poles of X(S).
5. It x(t) is a left-sided signal ,i.e. x(t) =0 for t> 1 > −∞ then the ROC is of the form
Re(s) < min.
-4 0 1 σ
s-plane
1
jΩ
1/s
-1
X(s)
0.1 Y(s)

6. It x(t) is a two-sided signal ,then the ROC is of the form 1 <Re(s) < 2.
26. What are the methods for evaluating inverse Laplace transform?
The two methods for evaluating inverse Laplace transform are
(i). By Partial fraction expansion method.
(ii).By convolution integral.
27. Find the Laplace transform of x(t) = sin2
t
2 2
1 Cos2t 1 1 s 2
X(s) L{x(t)} L
2 2 s s 4 s(s 4)
   
       
    
28. State Initial value theorem
The initial value theorem states that if x(t) and their derivatives are Laplace transformable then
(0) = →
( ) = →
( )
29. State the time scaling property of Laplace transform?(May/Jun 2013)
The time scaling property of Laplace transform says that,
If L(x(t)}=X(s) then
{ ( )} = | |
( )
30.What is the Fourier transform of a DC signal of amplitude 1? (May/June 2013)
[ ( )] =
1
2
( )
= ∫ ( ) = ( ) = 0
[ ( )] =
1
2
Thus [1] = [2 ( )]

UNIT III - LINEAR TIME INVARIANT- CONTINUOUS TIME SYSTEMS
1. List the blocks used for block diagram representation.
The block diagrams are implemented with the help of scalar multipliers, adders and multipliers.
2. State the significance of block diagram representation.
The LTI systems are represented with the help of block diagrams. The block diagrams are more
effective way of system description. Block Diagrams indicate how individual calculations are
performed. Various blocks are used for block diagram representation.
3. State the convolution integral.(Apr/May 2015)
The convolution of two signals is given by y(t)= x(t)*h(t)
Where    x t *h t x( )h(t ) d


     . This is known as convolution integral.
4. What are the properties of convolution?
1. Commutative
2. Associative.
3. Distributive
5. List the properties of convolution Integral?(Nov/Dec 2014)
Commutative property of Convolution is
x (t)*h(t) = h(t)*x(t)
Associative Property of convolution is
[x (t)*h1(t)]*h2(t) = x(t)*[h1(t)*h2(t)]
The Distributive Property of convolution is
{x (t)*[h1(t)+ h2(t)]} = x(t)*h1(t) + x(t)*h2(t)
6. Define natural response.
Natural response is the response of the system with zero input. It depends on the initial state of
the system. It is denoted by yn(t)
7. Define forced response.
Forced response is the response of the system due to input alone when the initial state of the
system is zero. It is denoted by yf (t).
8. Define complete response.
The complete response of a LTI-CT system is obtained by adding the natural response and
forced response. y(t)= yn(t)+ yf (t).

9. Define Eigen function and Eigen value.
Let the input to LTI-CT system be a complex exponential est
.Then using the convolution
theorem the output becomes,
y(t) = H(s).est
H(s) is called Eigen value and est
is called Eigen function.
10. List the four steps to compute the convolution integral.
The convolution of two signals is given by y(t)= x(t)*h(t)
where
   x t *h t x( )h(t ) d


    
1. Folding-One of the signal h(t) is folded about t=0.
2. Shifting-The folded signal is shifted right or left depending upon the time at which the output
is calculated to get h(t ) 
3. Multiplication-The shifted signal is multiplied by input signal x(t) to evaluate the range of
integration.
4. Integration-The multiplied signals are integrated to get the convolution output with the
integrating range in the overlapping interval.
11. Define Transfer function of an LTI system with example.(Nov/Dec 2016)
Let x(t) and y(t) is the input and output sequences of an LTI system with impulse response h(t).
Then the system function of the LTI system is defined as the ratio of Y(s) and X(s), i.e.,
H(s)=
( )
( )
Y s
X s
Where Y(s) is the Laplace transform of the output signal y(n) and X(s) is the Laplace transform
of the input signal x(n).
12. What is the transfer function of a system whose poles are at -0.2 ± j0.7 and a zero at
0.3?
1 2
1 2 1
0.3 0.4, 0.3 0.4
0.3
( ) ( 0.3)
( )
( )( ) ( 0.3 0.4 )( 0.3 0.4)
p j p j
z
s z s
H s
s p s p s j s j
     

 
 
     
13. Define impulse response of a continuous system
The impulse response is the output produced by the system when unit impulse is applied as
input. The impulse response is denoted by h(t) .It can be obtained from the transfer function also

 
 
1
( ) ( )
( ) ( )
h t L H s
h t IFT H f



14. Write the conditions for the LTI system to be causal and stable.
The LTI system is stable if if the impulse response of the system is absolutely integrable
( )h 


 
An LTI continuous time system is causal if and only if its impulse response is zero for negative
values of t.
15. What is the overall impulse response h(t) when two systems with impulse responses
h1(t) and h2 (t) are in parallel and in series?
1 2
1 2
( ) ( ) ( )
( ) ( )* ( )
For parallel connection h t h t h t
For series connection h t h t h t
 

16. State the properties needed for connecting interconnecting LTI Systems.
The Two LTI Systems with Impulse Responses 1 2( ) and ( )h t h t can be interconnected in the
following ways
1 2
1 2
( ) ( ) ( )
( ) ( )* ( )
For parallel connection h t h t h t
For series connection h t h t h t
 

17. Find the unit step of the system given by
1
( ) ( )


t
RC
h t e u t
RC
The step response can be obtained from the impulse response as,
t- RC
0
( ) ( )
1 1
= ( ) = = 1-e 0



 




 
t
t t
RC RC
y t h d
e u d e d for t
RC RC
 
  
18. What is the impulse response of the system y(t)=x(t-to)
Taking the Laplace transform of the given equation
0
0
( ) ( )
( )
( )
( )
st
st
H s e X s
Y s
H s e
X s



 

Taking Inverse Laplace transform 0( ) ( ) h t t t
19. The impulse response of the LTI-Ct system is given by ( ) ( )
 t
h t e u t .Determine the
transfer function and check whether the system is causal and stable.
( ) ( )t
h t e u t

Taking Laplace transform
1
( )
( 1)
H s
s


Here pole at s=-1 is located in left half of S Plane. Hence this system is causal and stable.
20. What are the different types of realization?
1. Direct Form I Realization
2. Direct Form II Realization
3. Cascade Realization
4. Parallel Realization
21. What is the condition for stability of a system? (Nov/Dec 2016)
For a system to be stable, the poles of the transfer must be in the left half of S Plane.
22. What is the transfer function of the following?
i) An Ideal Integrator ii) An ideal delay of T seconds
i)The transfer function of ideal integrator is
1
s
ii) The transfer function of ideal delay of T
Seconds is sT
e
23. Define the poles and Zeros of a transfer function.
The transfer function of a system is the ratio of two polynomials. The roots of the numerator
polynomial are called the zeros of the transfer function and the roots of the denominator
polynomial are called poles of the transfer function
24. When the LTI-CT system is said to be dynamic?
In LTI CT system, the system is said to be dynamic if the present output depends only on the
present input.
25. When the LTI-CT system is said to be causal?

An LTI continuous time system is causal if and only if its impulse response is zero for negative
values of t.
26. Mention the advantages of direct form II structure over direct form I structure.
No. of integrators are reduced to half.
27. Convolve the following signals u(t-1) and ( − ).(Nov/Dec 2016)
( − ) ∗ ( − ) =
∶ =
∶ ≠
28. Given ( ) = .Find the differential equation representation of the
system.(Nov/Dec 2016)
( )
( )
=
( ) = ( ) + 2 ( ) + ( )
2
2
d y(t) dy(t) dx(t)
2 y(t) =
dt dt dt
 
29. Given the differential equation representation of a system
2
2
d y(t) dy(t)
2 3y(t) =2 x(t)
dt dt
  .Find the Frequency response H(jω).(Nov/Dec 2015)
2 ( ) = ( ) + 2 ( ) − 3 ( )
H(j )=
30. Determine the unit step response of the system whose impulse response are
h(t)=3t.u(t).
( ) = ( ) = = =

UNIT IV- ANALYSIS OF DISCRETE TIME SIGNALS
1. State sampling theorem.(Nov/Dec 2016)
A band limited continuous time signal, with higher frequency m
 Hertz, can be uniquely
recovered from its samples provided that the sampling rate F m
2f samples per second.
1. What is aliasing effect?(Nov/Dec 2014)
Let us consider a band limited signal x(t) having no frequency component for m
   . If
we sample the signal x(t) with a sampling frequency F<2fm, the periodic continuation of
X(j  ) results in spectral overlap. In this case, the spectrum X(j  ) cannot be recovered
using a low pass filter. This effect is known as aliasing effect.
2. What is an anti-aliasing filter?
The frequency spectra of real signals do not confined to a band limit m
 .There are almost
always frequency components outside m
 . If we select sampling frequency s m
F 2f using the
sampling theorem, the frequency components outside m
 will appear as low-frequency signals
of frequencies between 0 and s
2

due to the aliasing effect and lead to loss of information. To
avoid aliasing, we use an analog low pass filter before sampler to reshape the frequency
spectrum of the signal so that the frequency for s
2

  is negligible. This filter is known as
anti-aliasing filter.
3. Define the Z- Transform.(Nov/Dec 2015)
The z-transform of a discrete signal x(n) is defined as




-n
n-
x(n)z)(Zx
Where z is a complex variable, In polar form Z can be expressed as z
j
re , where r is the
radius of the circle.
4. Write a note on ROC( Region of convergence)?(Nov/Dec 2014)
The region of convergence (ROC) of X(z) is the set of all values of z for which X(z) attains a
finite value.
5. What are the properties of region of convergence?
1. The ROC is a ring or disk in the z-plane centered at the origin.
2. The ROC cannot contain any poles.

3. The ROC of an LTI stable system contains the unit circle.
4. The ROC must be a connected region.
6. Explain the convolution property of the z-transform.
If  1 1
Z x (n) X (z) and  2 2
Z x (n) X (z) , then
Z{ 1 2 1 2
x (n)* x (n)} X (z)X (z)
7. State Parseval’s relation in z-transform.
1 2
* * 1
1 2 1 2 *
( ) and x ( ) are complex valued sequences ,then
1 1
( ) ( ) ( )
2x c
If x n n
x n x n X v X v dv
j v



 
  
  
 
8. State the initial value theorem and the final value theorem.
Initial value theorem: If x(n) is casual, then X(0) =
z
limX(z)

Final value theorem: If x(n) is casual, Z[x(n)] = X(z), where the ROC for x(z) includes, but is
not necessarily confined to z >1 and (z-1)X(z) has no poles on or outside the unit circle, then
x(
z 1
) lim(z 1)x(z)

  
9. Determine z-transform and ROC of the finite-duration signal X(n)={2,4, 5

,7,0,1}
X(z) = n
n
x(n)z




x(0) =5; x(-1) =4; x(-2) =2; x(1) =7; x(2) =0;x(3) =1
Substituting the sequence values we get
X(z)= 2 2 1 3
z 4z 5 7z z 
   
ROC: entire z-plane except z = 0 and z = 
10. Find the z-transform of (a) A digital impulse (b) a digital step.
(a) Since x(N) is zero except for n=0,where x(n) is 1, we find X(z)=1
(b) Since x(n) is zero except for n findwe1,isx(n)where,0
X(z) = n
1
n 0
1
z
1 z








11. The z- transform of a sequence x(n) is X(z), what is the z-transform of nx(n)?
If Z{x(n)} = X(z)
Z{n x(n)}= -z
d
X(z)
dz
12. List the different methods used for finding inverse z-transform?(Apr/May 2015)
The inverse z-transform can be evaluated using several methods
Long division method
Partial function expansion method
Residue Method
Convolution method
13. Define system function.
Let x(n) and y(n) is the input and output sequences of an LTI system with impulse response
h(n). Then the system function of the LTI system is defined as the ratio of Y(z) and X(z), i.e.,
H(z) =
Y(z)
X(z)
Where Y(z) is the z-transform of the output signal y(n) and X(z) is the z-transform of the input
signal x(n).
14. Prove initial value theorem.
If
z
X (z) z{x(n)}, then x(0) LimX (z) 
 
Proof: n
n 0
X (z) x(n)z




  As z ,  all the terms vanish except x(0), which proves the
theorem i.e.; n
z z
n 0
LimX (z) Lim x(n)z x(0)


 

 
15. Define DTFT pair.
The Fourier transform pair of discrete time signal is








n
njj
njj
enxeX
deeXnx






)()(
)(
2
1
)(
2
16. Write the condition for the existence of DTFT?(May/Jun 2016)
The sufficient condition for the existence of DTFT for a sequence x(n) is


n
nx )(
17. State the relationship between z-transform and DTFT?(Nov/Dec 2015)
The z-transform of x(n) is given by
X(z)= n
n
x(n)z



 --------------(1)
Where z =
j
re ---------------(2)
Substituting Eq. (2) in Eq. (1) we get
X(re j n j n
n
) x(n)r e

   

  ---------------(3)
The Fourier transform of x(n) is given by
X(e j j n
n
) x(n)e

  

  ---------------(4)
Eq. (3) and Eq (4) are identical, when r =1
In the z-plane this corresponds to the locus of points on the unit circle z =1. Hence X(e j
)
is
equal to H(z) evaluated along the unit circle,or X(e j
j
z e
) X(z) 



For X(e j
) to exist, the ROC of X(z) must include the unit circle
18. What are the properties of Fourier spectrum of a discrete-time aperiodic sequence?
The Fourier spectrum of an aperiodic sequence is continuous and periodic with period 2 .
19. Find the Fourier transform of a sequence x(n) = 1 for -2 n 2 
0 otherwise

0
2
0
( ) ( )
1 .....
j j n n j n
n n
n
j j j
n
X e x n e a e
a a a
e e e
  
  
 
 
 


 
   
       
   
 

20. Determine z-transform and ROC of the finite-duration signal X(n)={2,-1, 3

,0,2}
(Nov/Dec 2016)
X(z) = n
n
x(n)z




x(0) =3; x(-1) =-1; x(-2) =2; x(1) =0; x(2) = 2
X(z)= 2 2 2
z 1z 3 2z
   
21. Determine the Nyquist rate of sampling for ( ) = ( ) − ( ).
(Nov/Dec 2016)
( ) = (200 ) − (100 ) = (2 ) − (2 ).
2 = 200 ℎ = 100
2 = 100 ℎ = 50
The maximum analog frequency is 100 Hz .Hence ≥ 2 = 2 100 = 200
22. Determine the Nyquist rate of sampling for ( ) = ( ) +
( ).(Apr/May 2015)
( ) = (200 ) + 3 (100 ) == (2 ) − (2 )
2 = 200 ℎ = 100
2 = 100 ℎ = 50
The maximum analog frequency is 100 Hz .Hence ≥ 2 = 2 100 = 200
23. State the need for Sampling.(Nov/Dec 2015)
Sampling is the process of converting a continuous time signal into discrete time signal
24. Find the z-transform and its associated ROC for x[n]={1,-1, 2,3

,4}(Nov/Dec 2015)
X(z) = n
n
x(n)z




x(0) =5; x(-1) = 2; x(-2) =-1; x(-3) =1; x(1) =4

X(z)= 2 3 2 1
z z 2z 3 4z
   
25. Find the DTFT of [ ] = ( ) + ( − ).(Nov/Dec 2014)
( ) =
∶ =
: ≠
( − ) =
∶ =
: ≠
0 0
( ) ( ) ( ) ( 1)
1
  
  
  

   
 
  j j n j n j n
n n n
j
X e x n e n e n e
e
   

 
Sub n=0 and n=1 in the first and
Second terms
26. State and prove the time folding property of Z transform.(Nov/Dec 2014)
Z{x[n]}=X(z) then Z{x[-n]}=X(z-1
)
Proof
X(z) = n
n
x(n)z



 Z{x[−n]} = n
n
x( n)z




Let –n = p Z{x[−n]} = 1 p 1
p
x(p)(z ) X(z )

  


27. Find the inverse Z transform for .
(Nov/Dec 2016)
( ) = (−0.1) ( ) =
+ 0.1
( − 1) = (−0.1) ( − 1) =
1
+ 0.1
28. Find the Z transform of a) Impulse b)Unit step.(Nov/Dec 2016)
Impulse X(z) = n n
n n
x(n)z (n)z 1
 
 
 
   
Unit Step X(z) = n n
n n 0
z
u(n)z 1.z
z 1
 
 
 
 

 
29. Define Unilateral and Bilateral Z transform.(Nov/Dec 2013)
Bilateral Z transform n
n
X(z) x(n)z



  Unilateral Z transform n
n 0
X(z) x(n)z



 

UNIT V-LINEAR TIME INVARIANT-DISCRETE TIME SYSTEMS
1. Distinguish between IIR and FIR system.
S.NO FIR SYSTEM IIR SYSTEM
1.
Length of impulse response is
limited.
Length of impulse response is
infinite.
2. There is no feedback of output. Feedback of output is taken.
3. These systems are non recursive. These systems are recursive.
2. Why direct form-II structure is called canonic structure?
The number of delay elements in the structure is equal to order of the difference
equation or order of the transfer function. Hence it is called canonic structure.
3. Write the general difference equation relating input and output of a system?
The generalized difference equation is given as,
1 0
( ) ( ) ( )
 
     
n m
k k
k k
y n a y n k b x n k
4. Realize the following system y(n)=2y(n-1)-x(n)+2x(n-1) in direct form-I method.
y(n)=2y(n-1)-x(n)+2x(n-1)
5. Differentiate natural response and forced response?
S.No Natural Response Forced Response
1. Output of the system with
zero input
Output of the system with given
input.

2. Obtained only with initial
conditions.
Obtained with zero initial
conditions.
6. How many number of additions, multiplications and locations are required
realize a IIR system with transfer function, ( )H z having ‘M’ zeros and ‘N’ poles in
direct form-1 and direct form-II realizations.
Operation Direct Form-1 Direct Form-2
Addition M+N M+N
Multiplication M+N+1 M+N+1
Memory
locations
M+N M or N whichever is high
7. What is meant by recursive systems?
When the output y(n) of the system depends upon the present and past inputs as
well as past outputs, then it is called recursive system. IIR filters are of this type.
8. What is meant by linear phase response?
For linear phase filter Ө(ω) is linearly proportional to ω. The linear phase filter
does not alter the shape of the original signal. If the phase response of the filter is non
linear the output signal may be distorted one. In many cases a linear characteristics is
required throughout the pass band of the filter to preserve the shape of a given signal
within the pass band. IIR filter cannot produce a linear phase. The FIR filter can give
linear phase, when the impulse response of the filter is symmetric about its midpoint.
9. What are the different types of filters based on impulse response?
Based on the impulse response the filters are of two types 1. IIR filter 2. FIR filter.
The IIR filters are of recursive type, whereby the present output sample depends
on the present input, past input samples and output samples.
The FIR filters are of non recursive type whereby the present output sample
depends on the present input sample and previous input samples.
10. What are the four steps to obtain Convolution?(Nov/Dec 2015)
1. Folding 2.Shifting 3.Multiplication 4.Summation
11. Write the condition for the LTI system to be causal and stable.

The LTI system is causal and stable if its ROC is outside of circle of radius r<1
Thus > r <1z
Thus ROC of causal and stable system includes unit circle. All the poles of causal and
stable system are inside unit circle.
12. State the properties needed for connecting interconnecting LTI Systems.
1. Commutative Property: ( ) ( ) ( ) ( )  x n h n h n x n
2. Associative Property:    1 2 1 2( ) ( ) ( ) ( ) ( ) ( )    x n h n h n x n h n h n
3. Distributive Property:  1 2 1 2( ) ( ) ( ) ( ) ( ) ( ) ( )     x n h n h n x n h n x n h n
13. List out the different ways of interconnecting any two systems.
1. Cascade Connection
2. Parallel connection
3. Commutative Connection
14. Find the convolution of the following sequences
X(n) = { 1

,2,1} h(n) = { 1

,1,1}
We know
1 2
1 2
1 2 3 4
{ ( )* ( )} ( ) ( )
( ) (1 2 )
( ) (1 )
( ) ( ) 1 3 4 3
( )* ( ) { 1, 3, 4, 3, 1 }
z x n h n X z H z
X z z z
H z z z
X z H z z z z z
x n h n
 
 
   


  
  
    

15. Convolve the following sequence x(n) = { 1

,2,3} h(n) = { 1

,1,2}.(Nov/Dec 2016)
We know
1 2
1 2
1 2 3 4
{ ( )* ( )} ( ) ( )
( ) (1 2 3 )
( ) (1 2 )
( ) ( ) 1 3 7 7 6
( )* ( ) { 1, 3, 7, 7, 6 }
 
 
   


  
  
    

z x n h n X z H z
X z z z
H z z z
X z H z z z z z
x n h n
16. Given the system function 1 3 4
( ) 2 3 4 5  
   H z z z z .Determine the impulse
response h[n].(Nov/Dec 2016)

Impulse Response h(n) = { 2

,3,0,4,-5}
17. Define the non-recursive system.(May/Jun 2016)
If the present output is dependent upon the present and past value of input and past
Value of output then the system is said to be non recursive system
18. Name the basic building block of LTIDT system block diagram.(Apr/May 2015)
 Multiplier
 Adder
 Unit Delay Element
19. Write the nth order difference equation.(Apr/May 2015)
( ) + ( − ) = ( − )
20. Distinguish recursive system and non-recursive system.(Nov/Dec 2015)
If the present output is dependent upon the present and past value of input then the
System is said to be recursive system
If the present output is dependent upon the present and past value of input and past
Value of output then the system is said to be non recursive system
21. following sequence x(n) = { 1

,1,3} h(n) = { 1

,4,-1}.( Nov/Dec 2015)
We know
1 2
1 2
1 2 3 4
{ ( )* ( )} ( ) ( )
( ) (1 3 )
( ) (1 4 1 )
( ) ( ) 1 5 6 11 3
( )* ( ) { 1, 5, 6, 11, 3 }
 
 
   


  
  
    
 
z x n h n X z H z
X z z z
H z z z
X z H z z z z z
x n h n
22. In terms of ROC,state the condition for an LTI discrete time system to be causal
and stable.
A causal LTI system is BIBO stable if and only if the poles of H(z) are inside the
unit circle.
23. Compare direct form –I and direct form -II?

S.No direct form -I direct form -II
1 Separate delay for input and
output
Same delay for input and output
2 M+N+1 Multiplications are
involved
M+N+1 Multiplications are are
involved
3 M+N delays are involved N delays are involved
4 M+N Memory locations are
required
N Memory locations are required
5 Noncanonical structure canonical structure
24. What are the different types of structure realization.
i. Direct form I
ii. Direct form II
iii. Cascade form
iv. Parallel Form.
25. How the Cascade realization structure obtained..
The given transfer function H(z) is splitted into two or more sub systems.
That is for eg. H(z) = H1(z). H2(z)
H1(Z) H2(Z)
X(z) Y(z)

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