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Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
Dr. N.G.P. INSTITUTE OF TECHNOLOGY Coimbatore-641048 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
TWO MARKS WITH ANSWERS
EC6303-SIGNALS AND SYSTEMS
Prepared By
Approved By
Dr.K.Gayathri Devi
Dr.S.Sureshkumar
Prof/ECE
Hod-ECE
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering Syllabus EC6303
SIGNALS AND SYSTEMS
LTPC 3 1 04
OBJECTIVES: To understand the basic properties of signal & systems and the various methods of classification To learn Laplace Transform &Fourier transform and their properties To know Z transform & DTFT and their properties To characterize LTI systems in the Time domain and various Transform domains UNIT I - CLASSIFICATION OF SIGNALS AND SYSTEMS
9
Continuous time signals (CT signals) - Discrete time signals (DT signals) - Step, Ramp, Pulse, Impulse, Sinusoidal, Exponential, Classification of CT and DT signals - Periodic & Aperiodic signals, Deterministic & Random signals, Energy & Power signals - CT systems and DT systems- Classification of systems – Static & Dynamic, Linear & Nonlinear, Time-variant & Time-invariant, Causal &Non causal, Stable & Unstable. UNIT II- ANALYSIS OF CONTINUOUS TIME SIGNALS
9
Fourier series analysis-spectrum of Continuous Time (CT) signals- Fourier and Laplace Transforms in CT Signal Analysis - Properties. UNIT III- LINEAR TIME INVARIANT- CONTINUOUS TIME SYSTEMS
9
Differential Equation-Block diagram representation-impulse response, convolution integrals-Fourier and Laplace transforms in Analysis of CT systems UNIT IV- ANALYSIS OF DISCRETE TIME SIGNALS
9
Baseband Sampling - DTFT – Properties of DTFT - Z Transform – Properties of Z Transform UNIT V- LINEAR TIME INVARIANT-DISCRETE TIME SYSTEMS
9
Difference Equations-Block diagram representation-Impulse response - Convolution sum- Discrete Fourier and Z Transform Analysis of Recursive & Non-Recursive systems TOTAL (L:45+T:15): 60 PERIODS
OUTCOMES:
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering Upon the completion of the course, students will be able to: o
Analyze the properties of signals & systems
o
Apply Laplace transform, Fourier transform, Z transform and DTFT in signal analysis
o
Analyze continuous time LTI systems using Fourier and Laplace Transforms Analyze discrete time LTI systems using Z transform and DTFT
TEXT BOOK: 1. Allan V.Oppenheim, S.Wilsky and S.H.Nawab, “Signals and Systems”, Pearson, 2007. REFERENCES: 1.
B. P. Lathi, “Principles of Linear Systems and Signals”, Second Edition, Oxford, 2009.
2.
R.E.Zeimer, W.H.Tranter and R.D.Fannin, “Signals & Systems - Continuous and
Discrete”, Pearson, 2007. 3.
John Alan Stuller, “An Introduction to Signals and Systems”, Thomson, 2007.
4.
M.J.Roberts, “Signals & Systems Analysis using Transform Methods & MATLAB”,
Tata McGraw Hill, 2007.
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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
E
2
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
1 P lim x(t) T T T 2
2
dt
A Signal x(t) is called a power signal if the average power is finite and Energy is infinite.
-t 3. whether x(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
E=
x(t)
dt
-
2
= [Ae ] dt - t
Since u(t) = 1 for 0 t
0
= A 2 e-2t dt 0
e 2 t =A 2 0 2
=
A2 2
The energy is finite and non zero; hence the signal is energy signal
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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:
y(to ) = f [ x(t) ; t t o ] Stability: A system is said to be stable,if it produces bounded output for all bounded inputs Example: y(t) = x(t) sin(100t) 5. Show that the Complex exponential signal 𝒙(𝒕) = 𝒆𝒋𝝎𝒐𝒕 is periodic and that 𝟐𝝅
fundamental period is𝝎
𝟎
For a periodic signal x(t) =x(T + t),If
x(t) = e
jo t
Therefore x(t + T) = e
x(t + T) = e
jo (t + T)
jot joT e
This condition is satisfied x(t) = x(T + t) only if e o T 2 or T =
joT
=1 from which we have
2 o
6. Check whether the signal x(t ) 2 cos 3 t 7 cos 9t is periodic or not.
2 2 2 ; T2 3 3 9 2 T1 18 3 T 3 23 ; Therefore 1 T2 6 T2 9
T1
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.
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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 𝜋(t) is obtained from unit step signals 𝜋 (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. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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 Noncausal if the output of the system at any time t depends on the future inputs or outputs
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
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.
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 24. Sketch the following signals:𝒓𝒆𝒄𝒕 ( 𝑟𝑒𝑐𝑡 (
𝑡+1 4
𝒕+𝟏 𝟒
) ; 𝟓 𝒓𝒂𝒎𝒑(𝟎. 𝟏𝒕) (May/Jun 2016)
)
Shift the 𝜋(𝑡) by 0.25 units to the left and then do amplitude scaling
𝟓 𝒓𝒂𝒎𝒑(𝟎. 𝟏𝒕)
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 𝑟(𝑡) = ∫ 𝑢(𝑡) 𝑑𝑡 = ∫ 1 𝑑𝑡 = 𝑡 Relationship between Ramp and Step function 𝛿 (𝑡 ) = 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛
δ(t) →
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛
𝑢 (𝑡) →
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛
parabolic →
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛
𝑟(𝑡) →
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛
𝑟(𝑡) →
𝑑 𝑢(𝑡 ) 𝑑𝑡
𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑖𝑐 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛
𝑢(𝑡) →
δ(t)
𝟏, 𝟐, 𝟑, −𝟒, 𝟔 } .Plot the signal x(n-1)(Nov/Dec 2015) 26. Given 𝒙[𝒏] = { ↑
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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. ∫−∞ 𝑥(𝑡)𝛿 (𝑡)𝑑𝑡 = 𝑥 (0) 2. 𝑥(𝑡)𝛿(𝑡 − 𝑡𝑜 ) = 𝑥(𝑡0 )𝛿(𝑡 − 𝑡𝑜 ) 30. Draw the following signals: (Nov/Dec 2014) 𝒂) 𝒖(𝒕) − 𝒖(𝒕 − 𝟏𝟎) 𝒏 𝒃)(𝟏⁄𝟐) 𝒖(𝒏 − 𝟏) a)
b)
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
UNIT II - ANALYSIS OF CONTINUOUS TIME SIGNALS 1. Define Fourier series. Continuous Time Fourier series is defined as
a
x(n)
k
P
k
x[k]
e jk0 n 2
(Synthesis Equation)
(Analysis Equation)
k N
x(t) dt <
T0
Discrete Time Fourier series is defined as
x(n)
a
k
ak
k
e jk0n
1 a k e jk0n N n N
(Synthesis Equation) (Analysis Equation)
2. State Dirichlet condition for Fourier series.(Nov/Dec 2014,2013) i)The function x(t) should be within the interval T 0 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 T 0 iv ) The function x(t) should be absolutely integrable
x(t) dt <
T0
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
P
x (k)
2
k N
4. State the time shifting property of Fourier series.
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering If the Fourier series co-efficients of x(t ) are cn then the Fourier co-efficient of the shifted signal
x(t t0 ) are
FS[ x(t t0 )] e
jn (
2 ) 0 T
cn
5. Find the Fourier transform of () Inverse Fourier transform is given as
x(t )
1 2
X ( ) e
dt
1 x(t ) 2 x(t )
jt
jt ( ) e dt
1 : = 0 0 : 0
( )
1 (0)e0 2 1 2
6. Find the Fourier Transform of e X ( )
x(t ) e
jt
at
u (-t)
dt
0
e
at
e jt dt
1 = sa
7. Find the Laplace Transform of a unit step function.(Nov/Dec 2015) st
X(s)
x(t) e
= 0
st
dt
1 for t 0 u(t) 0 for t 0
e st e dt s 0
=
1 s
8. Determine the Laplace transform of x(t) = e-at sin(ωt) u(t)
1 L e a j t e a jt t 2j
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
1 1 1 2j s (a j) s (a j)
, ROC : Re(s) >-a (s a) 2 2
9. Find the Laplace transform of x(t) t e-at u(t), where a > 0. L e-at u(t)
1 ROC : Re (s) > -a sa'
Differentiation in s-domain property gives, L -t x(t)
d X(s) ds
L t e-at u(t)
L t e-at u(t)
d 1 ds s a
1 ROC : Re (s) > -a (s a) 2'
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)
x(t) et 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)
x( j)
x(t )e
jt
dt for all and
x(t )
1 2
x( j)e
jt
d for all t.
12. State the frequency shifting property of Laplace transform. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
If L[ x(t )] X ( s) Then L[e at x(t )] X (s a)
13.Define Laplace transform.
X (s)
The Laplace transform is given by
x(t )e
st
dt
14. Find the Laplace transform of 𝒙(𝒕) = 𝒆−𝒂𝒕 𝒖(𝒕). (Nov/Dec 2016) st
X(s)
x(t) e
1 for t 0 u(t) 0 for t 0
dt
= 0
e (s a )t e .e dt (s a) 0 at
st
=
1 s+a
15. What is the inverse Fourier transform of 𝒊)𝒆−𝒋𝟐𝝅𝒇𝒕𝒐 𝒊𝒊)𝜹(𝒇 − 𝒇𝒐 ) ? May/Jun 2016 𝑖)𝐹 −1 [𝑒 −𝑗2𝜋𝑓𝑡𝑜 ] = 𝛿 (𝑡 − 𝑡𝑜 ) 𝑖𝑖)𝐹 −1 [𝛿(𝑓 − 𝑓𝑜 )] =
1 𝑗2𝜋𝑓 𝑡 𝑜 𝑒 2𝜋
16. Give the Laplace transform of 𝒙(𝒕) = 𝟑𝒆−𝟐𝒕 𝒖(𝒕) − 𝟐𝒆−𝒕 𝒖(𝒕) with ROC.(May/Jun 2016)
X(s) =
-2 1 2 There are two poles Re(s) <-4 and Re(s) >1 The overall ROC Re(s) >1 5 s 4 5(s 1) jΩ s-plane
-4
0
Question Bank- Two Marks With Answer
1
1
σ
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 17.Draw the spectrum of CT rectangular pulse.(Apr/May 2015)
Amplitude|𝑋(𝜔)|
n 18. Given 𝒙(𝒕) = 𝜹(𝒕).Find 𝑿(𝒔)𝒂𝒏𝒅 𝑿(𝝎) .(Apr/May 2015) A delta function is defined as
1 :t 0 (t) 0 :t 0 s 2 1 Cos2t 1 1 X(s) L{x(t)} L 2 2 2 2 s s 4 s(s 4)
X()
jt
x(t) e
dt = (t) e
jt
dt =1
19. Give the relation between Fourier and Laplace transform. (Nov/Dec 2015, 2014)
X(s)
x(t) e
st
dt Sub s=+j
If s=j and =0 then it becomes fourier transform X( j)
jt
x(t) e
dt
20.What is 𝒖(𝒕 − 𝟐) ∗ 𝒇(𝒕 − 𝟏)? where * represents convolution.(Nov/Dec 2015) 0 ; 𝑡 < −1 𝑦 (𝑡 ) = { ( ) 𝑓 𝑡 ; 𝑡 ≥ −1
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 21.Find the ROC of the Laplace transform of x(t)=u(t). (Nov/Dec 2014) ROC : Re{s}>0 jΩ
s-plane
-4
0
σ
11
22.Draw the block diagram of the LTI system described by (Nov/Dec 2014) dy(t) y(t) = 0.1x(t) dt X(s) 1/s -1
0.1
Y(s)
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 splane 4. It x(t) is a right-sided signal ,i.e. x(t) =0 for t< 𝑡𝑜 < ∞ then the ROC is of the form Re(s) > 𝛼 max. here 𝛼max 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> 𝑡1 > −∞ then the ROC is of the form Re(s) < 𝛼min. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 6. It x(t) is a two-sided signal ,then the ROC is of the form 𝛼1
< 𝛼2.
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) = sin2t s 2 1 Cos2t 1 1 X(s) L{x(t)} L 2 2 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 𝐿𝑡 𝐿𝑡 𝑥(0) = 𝑡→0 𝑥 (𝑡) = 𝑠→∞ 𝑠𝑋(𝑠) 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 1 𝑠 𝐿{𝑥(𝑎𝑡)} = |𝑎| 𝑋( 𝑎 )
30.What is the Fourier transform of a DC signal of amplitude 1? (May/June 2013) 𝐹
−1 [
𝛿(𝜔)] =
1 +∞ ∫ 𝑥(𝑡)𝑒 𝑗𝜔𝑡 𝑑𝑡 2𝜋 −∞ 1
+∞
1
= 2𝜋 ∫−∞ 𝛿(𝜔)𝑒 𝑗𝜔𝑡 𝑑𝑡 = 2𝜋 𝛿(𝜔)𝑒 𝑗𝜔𝑡 | 𝜔 = 0 𝐹 −1 [𝛿(𝜔)] =
1 Thus 𝐹𝑇[1] = [2𝜋𝛿(𝜔)] 2𝜋
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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). Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 9. Define Eigen function and Eigen value. Let the input to LTI-CT system be a complex exponential e st .Then using the convolution theorem the output becomes, y(t) = H(s).est H(s) is called Eigen value and e st 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) x t*h t
where
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?
p1 0.3 j 0.4, p2 0.3 j 0.4 z 0.3 H ( s)
(s z) ( s 0.3) ( s p1 )( s p2 ) ( s 0.3 j 0.41 )( s 0.3 j 0.4)
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 Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering h(t ) L1 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?
For parallel connection h(t ) h1 (t ) h2 (t ) For series connection h(t ) h1 (t )* h2 (t ) 16. State the properties needed for connecting interconnecting LTI Systems. The Two LTI Systems with Impulse Responses h1 (t ) and h2 (t ) can be interconnected in the following ways
For parallel connection h(t ) h1 (t ) h2 (t ) For series connection h(t ) h1 (t )* h2 (t ) 17. Find the unit step of the system given by h(t )
1 t RC e u (t ) RC
The step response can be obtained from the impulse response as,
y (t )
h( ) d
t
1 t RC 1 t RC -t = e u ( ) d = e d = 1-e RC for t 0 RC RC 0 18. What is the impulse response of the system y(t)=x(t-to) Taking the Laplace transform of the given equation
H ( s ) e st0 X ( s ) Y ( s) H (s) e st0 X ( s)
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering Taking Inverse Laplace transform h(t ) (t t0 ) 19. The impulse response of the LTI-Ct system is given by h(t ) et u (t ) .Determine the transfer function and check whether the system is causal and stable.
h(t ) et u (t ) Taking Laplace transform
H ( s)
1 ( s 1)
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 ii) The transfer function of ideal delay of T s
Seconds is e sT 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?
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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 𝜹(𝒕 − 𝟏).(Nov/Dec 2016) 𝟏∶𝒕=𝟏 𝒖(𝒕 − 𝟏) ∗ 𝜹(𝒕 − 𝟏) = { 𝟎∶𝒕≠𝟏 28. Given
𝒔
𝑯(𝒔) = 𝒔𝟐 +𝟐𝒔+𝟏.Find
the
differential
equation
representation
of
the
system.(Nov/Dec 2016) 𝑌(𝑠) 𝑋(𝑠)
𝑠
= 𝑠 2+2𝑠+1
𝑠𝑋(𝑠) = 𝑠 2 𝑌(𝑠) + 2𝑠𝑌 (𝑠) + 𝑌(𝑠)
d 2 y(t) dy(t) dx(t) 2 y(t) = 2 dt dt dt 29. Given the differential equation representation of a system 2 d y(t) dy(t) 2 3y(t) =2 x(t) .Find the Frequency response H(jω).(Nov/Dec 2015) 2 dt dt 2𝑋(𝑠) = 𝑠 2 𝑌(𝑠) + 2𝑠𝑌 (𝑠) − 3𝑌(𝑠) 𝑠
H(j𝜔)= 𝑠 2+2𝑠−3 30. Determine the unit step response h(t)=3t.u(t). 𝝀=𝒕
𝒔(𝒕) = ∫
of the system whose impulse response are 𝝀=𝒕
𝒉(𝝀)𝒅𝝀 = ∫
𝝀=−∞
Question Bank- Two Marks With Answer
𝝀=𝟎
𝒕
𝝀𝟐 𝟑 𝟑𝝀 𝒅𝝀 = 𝟑 [ ] = 𝒕𝟐 𝟐 𝟎 𝟐
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
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 2fm samples per second. 2. 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 filter. This effect is known as aliasing effect. 3. 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 Fs 2fm using the sampling theorem, the frequency components outside m will appear as low-frequency signals s due to the aliasing effect and lead to loss of information. To 2 avoid aliasing, we use an analog low filter before sampler to reshape the frequency spectrum of the signal so that the frequency for s is negligible. This filter is known as 2 anti-aliasing filter.
of frequencies between 0 and
4.
Define the Z- Transform.(Nov/Dec 2015)
The z-transform of a discrete signal x(n) is defined as x(Z )
x(n)z
-n
n -
j Where z is a complex variable, In polar form Z can be expressed as z re , where r is the radius of the circle.
5. 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. 6. 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. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 3. The ROC of an LTI stable system contains the unit circle. 4. The ROC must be a connected region. 7. Explain the convolution property of the z-transform. Zx1 (n) X1(z) and Zx2(n) X 2(z) , then
If
Z{ x1 (n)* x2(n)} X1(z)X 2(z) 8. State Parseval’s relation in z-transform.
If x1 (n) and x 2 (n) are complex valued sequences ,then
1
x ( n ) x ( n) 2 j X (v ) X
x
* 2
1
1
* 2
c
1 1 * v dv v
9. State the initial value theorem and the final value theorem. Initial value theorem: If x(n) is casual, then X(0) = limX(z) 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( ) lim(z 1)x(z) z1
10. Determine z-transform and ROC of the finite-duration signal X(n)={2,4, 5 ,7,0,1}
X(z) =
x(n)z
n
n
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 z2 4z 5 7z1 z3 ROC: entire z-plane except z = 0 and z = 11. 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 0, where x(n) is 1, we find
X(z) =
z n 0
n
1 1 z1
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
12. 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
13. 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 14. 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). 15. Prove initial value theorem. If X (z) z{x(n)}, then x(0) LimX (z) z
Proof:
X (z) x(n)z n As z , all the vanish except x(0), which proves the n 0
theorem i.e.; LimX (z) Lim x(n)z n x(0) z
z
n 0
16. Define DTFT pair. The Fourier transform pair of discrete time signal is
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering x ( n)
1 2
X (e
j
) e j n d
2
x ( n )e
X ( e j )
j n
n
17. 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
x ( n)
n
18. State the relationship between z-transform and DTFT?(Nov/Dec 2015) The z-transform of x(n) is given by
X(z)=
x(n)z
n
--------------(1)
n
j Where z = re
---------------(2)
Substituting Eq. (2) in Eq. (1) we get X(re j )
x(n)r
n jn
e
---------------(3)
n
The Fourier transform of x(n) is given by X(e
j
)
x(n)e
jn
---------------(4)
n
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 equal to H(z) evaluated along the unit circle,or X(e
j
j
) is
) X(z) zej
For X(e j ) to exist, the ROC of X(z) must include the unit circle 19. 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 . 20. Find the Fourier transform of a sequence x(n) = 1 for -2 n 2 0 otherwise
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
X (e j )
x ( n)e
j n
n
a ne jn n 0
n
2
a a a j 1 j j ..... e e n 0 e 21. Determine z-transform and ROC of the finite-duration signal X(n)={2,-1, 3 ,0,2}
(Nov/Dec 2016)
X(z) =
x(n)z
n
n
x(0) =3; x(-1) =-1; x(-2) =2; x(1) =0; x(2) = 2 Substituting the sequence values we get X(z)= 2 z2 1z 3 2z2 ROC: entire z-plane except z = 0 and z = 22. Determine the Nyquist rate of sampling for 𝒙(𝒕) = 𝑺𝒊𝒏(𝟐𝟎𝟎𝝅𝒕) − 𝑪𝒐𝒔(𝟏𝟎𝟎𝝅𝒕). (Nov/Dec 2016) 𝑥(𝑡) = 𝑆𝑖𝑛(200𝜋𝑡) − 𝐶𝑜𝑠(100𝜋𝑡) = 𝑆𝑖𝑛(2𝜋𝑓1 𝑡) − 𝐶𝑜𝑠(2𝜋𝑓2 𝑡). 2𝜋𝑓1 = 200𝜋 ℎ𝑒𝑛𝑐𝑒 𝑓1 = 100 𝐻𝑧 2𝜋𝑓2 = 100𝜋 ℎ𝑒𝑛𝑐𝑒 𝑓1 = 50 𝐻𝑧 The maximum analog frequency is 100 Hz .Hence 𝐹𝑠 ≥ 2𝐹𝑚𝑎𝑥 = 2𝑋100 𝐻𝑧 = 200 𝐻𝑧 23. Determine the Nyquist rate 𝟐( 𝟑 𝑺𝒊𝒏 𝟏𝟎𝟎𝝅𝒕).(Apr/May 2015)
of
sampling
𝒙(𝒕) = 𝑺𝒊𝒏(𝟐𝟎𝟎𝝅𝒕) +
for
𝑥(𝑡) = 𝑆𝑖𝑛(200𝜋𝑡) + 3 𝑆𝑖𝑛2 (100𝜋𝑡) == 𝑆𝑖𝑛(2𝜋𝑓1 𝑡) − 𝐶𝑜𝑠(2𝜋𝑓2 𝑡) 2𝜋𝑓1 = 200𝜋 ℎ𝑒𝑛𝑐𝑒 𝑓1 = 100 𝐻𝑧 2𝜋𝑓2 = 100𝜋 ℎ𝑒𝑛𝑐𝑒 𝑓1 = 50 𝐻𝑧 The maximum analog frequency is 100 Hz .Hence 𝐹𝑠 ≥ 2𝐹𝑚𝑎𝑥 = 2𝑋100 𝐻𝑧 = 200 𝐻𝑧 24. State the need for Sampling.(Nov/Dec 2015) Sampling is the process of converting a continuous time signal into discrete time signal 25. Find the z-transform and its associated ROC for x[n]={1,-1, 2, 3 ,4}(Nov/Dec 2015)
X(z) =
x(n)z
n
n
x(0) =5; x(-1) = 2; x(-2) =-1; x(-3) =1; x(1) =4
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering Substituting the sequence values we get X(z)= 2 z3 z2 2z 3 4z1 ROC: entire z-plane except z = 0 and z = 26. Find the DTFT of 𝒙[𝒏] = 𝜹(𝒏) + 𝜹(𝒏 − 𝟏).(Nov/Dec 2014) 𝟏∶𝒏=𝟎 𝜹(𝒏) = { 𝟎: 𝒏 ≠ 𝟎 𝟏∶𝒏=𝟏 𝜹( 𝒏 − 𝟏) = { 𝟎: 𝒏 ≠ 𝟏
X (e j )
n
n 0
n 0
x(n)e jn (n)e jn (n 1)e jn
Sub n=0 and n=1 in the first and
1 e j Second 27. 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) =
x(n)zn Z{x[−n]} =
n
Let –n = p Z{x[−n]} =
x(n)z
n
n
x(p)(z
1 p
) X(z1 )
p
𝟏
28. Find the inverse Z transform for 𝒁+𝟎.𝟏(Nov/Dec 2016) 𝑧 𝑥(𝑛) = (−0.1)𝑛 𝑢(𝑛) = 𝑧 + 0.1 1 𝑥(𝑛 − 1) = (−0.1)𝑛−1 𝑢(𝑛 − 1) = 𝑧 + 0.1 29. Find the Z transform of a) Impulse b)Unit step.(Nov/Dec 2016)
Impulse
x(n)zn
X(z) =
n
Unit Step X(z) =
(n)z
n
1
n
n
n 0
u(n)zn 1.zn
z z 1
30. Define Unilateral and Bilateral Z transform.(Nov/Dec 2013) Bilateral Z transform X(z)
n
n 0
x(n)zn Unilateral Z transform X(z) x(n)zn
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
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 of output.
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, n
m
k 1
k 0
y (n) ak y (n k ) bk 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
1.
Output of the system with zero input
Question Bank- Two Marks With Answer
Forced Response Output of the system with given input. EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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 band of the filter to preserve the shape of a given signal within the 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. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering The LTI system is causal and stable if its ROC is outside of circle of radius r<1 Thus z > r <1 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: x(n) h 1 (n) h 2 (n) x(n) h 1 (n) h 2 (n) 3. Distributive Property: x(n) h 1 (n) h 2 (n) x(n) h 1 (n) x(n) h 2 (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 z{x(n) * h(n)} X ( z ) H ( z ) X ( z ) (1 2 z 1 z 2 ) H ( z ) (1 z 1 z 2 ) X ( z ) H ( z ) 1 3 z 1 4 z 2 3 z 3 z 4 x(n) * h(n) { 1, 3, 4, 3, 1 }
15. Convolve the following sequence x(n) = { 1 ,2,3} h(n) = { 1 ,1,2}.(Nov/Dec 2016)
We know z{x(n) * h(n)} X ( z ) H ( z ) X ( z ) (1 2 z 1 3 z 2 ) H ( z ) (1 z 1 2 z 2 ) X ( z ) H ( z ) 1 3 z 1 7 z 2 7 z 3 6 z 4 x(n) * h(n) { 1, 3, 7, 7, 6 }
16. Given the system function H ( z ) 2 3z 1 4 z 3 5 z 4 .Determine the impulse response h[n].(Nov/Dec 2016) Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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) 𝑵
𝑴
𝒚(𝒏) + ∑ 𝒂𝒌 𝒚(𝒏 − 𝒌) = ∑ 𝒃𝒌 𝒙(𝒏 − 𝒌) 𝒌=𝟏
𝒌=𝟎
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 z{x(n) * h(n)} X ( z ) H ( z ) X ( z ) (1 z 1 3 z 2 ) H ( z ) (1 4 z 1 1z 2 ) X ( z ) H ( z ) 1 5 z 1 6 z 2 11z 3 3z 4 x(n) * h(n) { 1, 5, 6, 11, 3 }
22. In 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?
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering S.No 1 2 3 4 5
direct form -I Separate delay for input and output M+N+1 Multiplications are involved M+N delays are involved M+N Memory locations are required Noncanonical structure
direct form -II Same delay for input and output M+N+1 Multiplications are are involved N delays are involved N Memory locations are required 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) X(z)
Y(z) H1(Z)
Question Bank- Two Marks With Answer
H2(Z)
EC6303/ Signals and Systems
Dr. N.G.P. INSTITUTE OF TECHNOLOGY Coimbatore-641048 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
TWO MARKS WITH ANSWERS
EC6303-SIGNALS AND SYSTEMS
Prepared By
Approved By
Dr.K.Gayathri Devi
Dr.S.Sureshkumar
Prof/ECE
Hod-ECE
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering Syllabus EC6303
SIGNALS AND SYSTEMS
LTPC 3 1 04
OBJECTIVES: To understand the basic properties of signal & systems and the various methods of classification To learn Laplace Transform &Fourier transform and their properties To know Z transform & DTFT and their properties To characterize LTI systems in the Time domain and various Transform domains UNIT I - CLASSIFICATION OF SIGNALS AND SYSTEMS
9
Continuous time signals (CT signals) - Discrete time signals (DT signals) - Step, Ramp, Pulse, Impulse, Sinusoidal, Exponential, Classification of CT and DT signals - Periodic & Aperiodic signals, Deterministic & Random signals, Energy & Power signals - CT systems and DT systems- Classification of systems – Static & Dynamic, Linear & Nonlinear, Time-variant & Time-invariant, Causal &Non causal, Stable & Unstable. UNIT II- ANALYSIS OF CONTINUOUS TIME SIGNALS
9
Fourier series analysis-spectrum of Continuous Time (CT) signals- Fourier and Laplace Transforms in CT Signal Analysis - Properties. UNIT III- LINEAR TIME INVARIANT- CONTINUOUS TIME SYSTEMS
9
Differential Equation-Block diagram representation-impulse response, convolution integrals-Fourier and Laplace transforms in Analysis of CT systems UNIT IV- ANALYSIS OF DISCRETE TIME SIGNALS
9
Baseband Sampling - DTFT – Properties of DTFT - Z Transform – Properties of Z Transform UNIT V- LINEAR TIME INVARIANT-DISCRETE TIME SYSTEMS
9
Difference Equations-Block diagram representation-Impulse response - Convolution sum- Discrete Fourier and Z Transform Analysis of Recursive & Non-Recursive systems TOTAL (L:45+T:15): 60 PERIODS
OUTCOMES:
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering Upon the completion of the course, students will be able to: o
Analyze the properties of signals & systems
o
Apply Laplace transform, Fourier transform, Z transform and DTFT in signal analysis
o
Analyze continuous time LTI systems using Fourier and Laplace Transforms Analyze discrete time LTI systems using Z transform and DTFT
TEXT BOOK: 1. Allan V.Oppenheim, S.Wilsky and S.H.Nawab, “Signals and Systems”, Pearson, 2007. REFERENCES: 1.
B. P. Lathi, “Principles of Linear Systems and Signals”, Second Edition, Oxford, 2009.
2.
R.E.Zeimer, W.H.Tranter and R.D.Fannin, “Signals & Systems - Continuous and
Discrete”, Pearson, 2007. 3.
John Alan Stuller, “An Introduction to Signals and Systems”, Thomson, 2007.
4.
M.J.Roberts, “Signals & Systems Analysis using Transform Methods & MATLAB”,
Tata McGraw Hill, 2007.
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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
E
2
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
1 P lim x(t) T T T 2
2
dt
A Signal x(t) is called a power signal if the average power is finite and Energy is infinite.
-t 3. whether x(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
E=
x(t)
dt
-
2
= [Ae ] dt - t
Since u(t) = 1 for 0 t
0
= A 2 e-2t dt 0
e 2 t =A 2 0 2
=
A2 2
The energy is finite and non zero; hence the signal is energy signal
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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:
y(to ) = f [ x(t) ; t t o ] Stability: A system is said to be stable,if it produces bounded output for all bounded inputs Example: y(t) = x(t) sin(100t) 5. Show that the Complex exponential signal 𝒙(𝒕) = 𝒆𝒋𝝎𝒐𝒕 is periodic and that 𝟐𝝅
fundamental period is𝝎
𝟎
For a periodic signal x(t) =x(T + t),If
x(t) = e
jo t
Therefore x(t + T) = e
x(t + T) = e
jo (t + T)
jot joT e
This condition is satisfied x(t) = x(T + t) only if e o T 2 or T =
joT
=1 from which we have
2 o
6. Check whether the signal x(t ) 2 cos 3 t 7 cos 9t is periodic or not.
2 2 2 ; T2 3 3 9 2 T1 18 3 T 3 23 ; Therefore 1 T2 6 T2 9
T1
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.
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EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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
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EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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 𝜋(t) is obtained from unit step signals 𝜋 (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. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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 Noncausal if the output of the system at any time t depends on the future inputs or outputs
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
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.
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EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 24. Sketch the following signals:𝒓𝒆𝒄𝒕 ( 𝑟𝑒𝑐𝑡 (
𝑡+1 4
𝒕+𝟏 𝟒
) ; 𝟓 𝒓𝒂𝒎𝒑(𝟎. 𝟏𝒕) (May/Jun 2016)
)
Shift the 𝜋(𝑡) by 0.25 units to the left and then do amplitude scaling
𝟓 𝒓𝒂𝒎𝒑(𝟎. 𝟏𝒕)
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 𝑟(𝑡) = ∫ 𝑢(𝑡) 𝑑𝑡 = ∫ 1 𝑑𝑡 = 𝑡 Relationship between Ramp and Step function 𝛿 (𝑡 ) = 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛
δ(t) →
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛
𝑢 (𝑡) →
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛
parabolic →
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛
𝑟(𝑡) →
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛
𝑟(𝑡) →
𝑑 𝑢(𝑡 ) 𝑑𝑡
𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑖𝑐 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛
𝑢(𝑡) →
δ(t)
𝟏, 𝟐, 𝟑, −𝟒, 𝟔 } .Plot the signal x(n-1)(Nov/Dec 2015) 26. Given 𝒙[𝒏] = { ↑
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EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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. ∫−∞ 𝑥(𝑡)𝛿 (𝑡)𝑑𝑡 = 𝑥 (0) 2. 𝑥(𝑡)𝛿(𝑡 − 𝑡𝑜 ) = 𝑥(𝑡0 )𝛿(𝑡 − 𝑡𝑜 ) 30. Draw the following signals: (Nov/Dec 2014) 𝒂) 𝒖(𝒕) − 𝒖(𝒕 − 𝟏𝟎) 𝒏 𝒃)(𝟏⁄𝟐) 𝒖(𝒏 − 𝟏) a)
b)
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EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
UNIT II - ANALYSIS OF CONTINUOUS TIME SIGNALS 1. Define Fourier series. Continuous Time Fourier series is defined as
a
x(n)
k
P
k
x[k]
e jk0 n 2
(Synthesis Equation)
(Analysis Equation)
k N
x(t) dt <
T0
Discrete Time Fourier series is defined as
x(n)
a
k
ak
k
e jk0n
1 a k e jk0n N n N
(Synthesis Equation) (Analysis Equation)
2. State Dirichlet condition for Fourier series.(Nov/Dec 2014,2013) i)The function x(t) should be within the interval T 0 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 T 0 iv ) The function x(t) should be absolutely integrable
x(t) dt <
T0
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
P
x (k)
2
k N
4. State the time shifting property of Fourier series.
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering If the Fourier series co-efficients of x(t ) are cn then the Fourier co-efficient of the shifted signal
x(t t0 ) are
FS[ x(t t0 )] e
jn (
2 ) 0 T
cn
5. Find the Fourier transform of () Inverse Fourier transform is given as
x(t )
1 2
X ( ) e
dt
1 x(t ) 2 x(t )
jt
jt ( ) e dt
1 : = 0 0 : 0
( )
1 (0)e0 2 1 2
6. Find the Fourier Transform of e X ( )
x(t ) e
jt
at
u (-t)
dt
0
e
at
e jt dt
1 = sa
7. Find the Laplace Transform of a unit step function.(Nov/Dec 2015) st
X(s)
x(t) e
= 0
st
dt
1 for t 0 u(t) 0 for t 0
e st e dt s 0
=
1 s
8. Determine the Laplace transform of x(t) = e-at sin(ωt) u(t)
1 L e a j t e a jt t 2j
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
1 1 1 2j s (a j) s (a j)
, ROC : Re(s) >-a (s a) 2 2
9. Find the Laplace transform of x(t) t e-at u(t), where a > 0. L e-at u(t)
1 ROC : Re (s) > -a sa'
Differentiation in s-domain property gives, L -t x(t)
d X(s) ds
L t e-at u(t)
L t e-at u(t)
d 1 ds s a
1 ROC : Re (s) > -a (s a) 2'
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)
x(t) et 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)
x( j)
x(t )e
jt
dt for all and
x(t )
1 2
x( j)e
jt
d for all t.
12. State the frequency shifting property of Laplace transform. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
If L[ x(t )] X ( s) Then L[e at x(t )] X (s a)
13.Define Laplace transform.
X (s)
The Laplace transform is given by
x(t )e
st
dt
14. Find the Laplace transform of 𝒙(𝒕) = 𝒆−𝒂𝒕 𝒖(𝒕). (Nov/Dec 2016) st
X(s)
x(t) e
1 for t 0 u(t) 0 for t 0
dt
= 0
e (s a )t e .e dt (s a) 0 at
st
=
1 s+a
15. What is the inverse Fourier transform of 𝒊)𝒆−𝒋𝟐𝝅𝒇𝒕𝒐 𝒊𝒊)𝜹(𝒇 − 𝒇𝒐 ) ? May/Jun 2016 𝑖)𝐹 −1 [𝑒 −𝑗2𝜋𝑓𝑡𝑜 ] = 𝛿 (𝑡 − 𝑡𝑜 ) 𝑖𝑖)𝐹 −1 [𝛿(𝑓 − 𝑓𝑜 )] =
1 𝑗2𝜋𝑓 𝑡 𝑜 𝑒 2𝜋
16. Give the Laplace transform of 𝒙(𝒕) = 𝟑𝒆−𝟐𝒕 𝒖(𝒕) − 𝟐𝒆−𝒕 𝒖(𝒕) with ROC.(May/Jun 2016)
X(s) =
-2 1 2 There are two poles Re(s) <-4 and Re(s) >1 The overall ROC Re(s) >1 5 s 4 5(s 1) jΩ s-plane
-4
0
Question Bank- Two Marks With Answer
1
1
σ
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 17.Draw the spectrum of CT rectangular pulse.(Apr/May 2015)
Amplitude|𝑋(𝜔)|
n 18. Given 𝒙(𝒕) = 𝜹(𝒕).Find 𝑿(𝒔)𝒂𝒏𝒅 𝑿(𝝎) .(Apr/May 2015) A delta function is defined as
1 :t 0 (t) 0 :t 0 s 2 1 Cos2t 1 1 X(s) L{x(t)} L 2 2 2 2 s s 4 s(s 4)
X()
jt
x(t) e
dt = (t) e
jt
dt =1
19. Give the relation between Fourier and Laplace transform. (Nov/Dec 2015, 2014)
X(s)
x(t) e
st
dt Sub s=+j
If s=j and =0 then it becomes fourier transform X( j)
jt
x(t) e
dt
20.What is 𝒖(𝒕 − 𝟐) ∗ 𝒇(𝒕 − 𝟏)? where * represents convolution.(Nov/Dec 2015) 0 ; 𝑡 < −1 𝑦 (𝑡 ) = { ( ) 𝑓 𝑡 ; 𝑡 ≥ −1
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Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 21.Find the ROC of the Laplace transform of x(t)=u(t). (Nov/Dec 2014) ROC : Re{s}>0 jΩ
s-plane
-4
0
σ
11
22.Draw the block diagram of the LTI system described by (Nov/Dec 2014) dy(t) y(t) = 0.1x(t) dt X(s) 1/s -1
0.1
Y(s)
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 splane 4. It x(t) is a right-sided signal ,i.e. x(t) =0 for t< 𝑡𝑜 < ∞ then the ROC is of the form Re(s) > 𝛼 max. here 𝛼max 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> 𝑡1 > −∞ then the ROC is of the form Re(s) < 𝛼min. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 6. It x(t) is a two-sided signal ,then the ROC is of the form 𝛼1
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) = sin2t s 2 1 Cos2t 1 1 X(s) L{x(t)} L 2 2 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 𝐿𝑡 𝐿𝑡 𝑥(0) = 𝑡→0 𝑥 (𝑡) = 𝑠→∞ 𝑠𝑋(𝑠) 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 1 𝑠 𝐿{𝑥(𝑎𝑡)} = |𝑎| 𝑋( 𝑎 )
30.What is the Fourier transform of a DC signal of amplitude 1? (May/June 2013) 𝐹
−1 [
𝛿(𝜔)] =
1 +∞ ∫ 𝑥(𝑡)𝑒 𝑗𝜔𝑡 𝑑𝑡 2𝜋 −∞ 1
+∞
1
= 2𝜋 ∫−∞ 𝛿(𝜔)𝑒 𝑗𝜔𝑡 𝑑𝑡 = 2𝜋 𝛿(𝜔)𝑒 𝑗𝜔𝑡 | 𝜔 = 0 𝐹 −1 [𝛿(𝜔)] =
1 Thus 𝐹𝑇[1] = [2𝜋𝛿(𝜔)] 2𝜋
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EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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). Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 9. Define Eigen function and Eigen value. Let the input to LTI-CT system be a complex exponential e st .Then using the convolution theorem the output becomes, y(t) = H(s).est H(s) is called Eigen value and e st 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) x t*h t
where
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?
p1 0.3 j 0.4, p2 0.3 j 0.4 z 0.3 H ( s)
(s z) ( s 0.3) ( s p1 )( s p2 ) ( s 0.3 j 0.41 )( s 0.3 j 0.4)
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 Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering h(t ) L1 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?
For parallel connection h(t ) h1 (t ) h2 (t ) For series connection h(t ) h1 (t )* h2 (t ) 16. State the properties needed for connecting interconnecting LTI Systems. The Two LTI Systems with Impulse Responses h1 (t ) and h2 (t ) can be interconnected in the following ways
For parallel connection h(t ) h1 (t ) h2 (t ) For series connection h(t ) h1 (t )* h2 (t ) 17. Find the unit step of the system given by h(t )
1 t RC e u (t ) RC
The step response can be obtained from the impulse response as,
y (t )
h( ) d
t
1 t RC 1 t RC -t = e u ( ) d = e d = 1-e RC for t 0 RC RC 0 18. What is the impulse response of the system y(t)=x(t-to) Taking the Laplace transform of the given equation
H ( s ) e st0 X ( s ) Y ( s) H (s) e st0 X ( s)
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering Taking Inverse Laplace transform h(t ) (t t0 ) 19. The impulse response of the LTI-Ct system is given by h(t ) et u (t ) .Determine the transfer function and check whether the system is causal and stable.
h(t ) et u (t ) Taking Laplace transform
H ( s)
1 ( s 1)
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 ii) The transfer function of ideal delay of T s
Seconds is e sT 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?
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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 𝜹(𝒕 − 𝟏).(Nov/Dec 2016) 𝟏∶𝒕=𝟏 𝒖(𝒕 − 𝟏) ∗ 𝜹(𝒕 − 𝟏) = { 𝟎∶𝒕≠𝟏 28. Given
𝒔
𝑯(𝒔) = 𝒔𝟐 +𝟐𝒔+𝟏.Find
the
differential
equation
representation
of
the
system.(Nov/Dec 2016) 𝑌(𝑠) 𝑋(𝑠)
𝑠
= 𝑠 2+2𝑠+1
𝑠𝑋(𝑠) = 𝑠 2 𝑌(𝑠) + 2𝑠𝑌 (𝑠) + 𝑌(𝑠)
d 2 y(t) dy(t) dx(t) 2 y(t) = 2 dt dt dt 29. Given the differential equation representation of a system 2 d y(t) dy(t) 2 3y(t) =2 x(t) .Find the Frequency response H(jω).(Nov/Dec 2015) 2 dt dt 2𝑋(𝑠) = 𝑠 2 𝑌(𝑠) + 2𝑠𝑌 (𝑠) − 3𝑌(𝑠) 𝑠
H(j𝜔)= 𝑠 2+2𝑠−3 30. Determine the unit step response h(t)=3t.u(t). 𝝀=𝒕
𝒔(𝒕) = ∫
of the system whose impulse response are 𝝀=𝒕
𝒉(𝝀)𝒅𝝀 = ∫
𝝀=−∞
Question Bank- Two Marks With Answer
𝝀=𝟎
𝒕
𝝀𝟐 𝟑 𝟑𝝀 𝒅𝝀 = 𝟑 [ ] = 𝒕𝟐 𝟐 𝟎 𝟐
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
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 2fm samples per second. 2. 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 filter. This effect is known as aliasing effect. 3. 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 Fs 2fm using the sampling theorem, the frequency components outside m will appear as low-frequency signals s due to the aliasing effect and lead to loss of information. To 2 avoid aliasing, we use an analog low filter before sampler to reshape the frequency spectrum of the signal so that the frequency for s is negligible. This filter is known as 2 anti-aliasing filter.
of frequencies between 0 and
4.
Define the Z- Transform.(Nov/Dec 2015)
The z-transform of a discrete signal x(n) is defined as x(Z )
x(n)z
-n
n -
j Where z is a complex variable, In polar form Z can be expressed as z re , where r is the radius of the circle.
5. 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. 6. 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. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 3. The ROC of an LTI stable system contains the unit circle. 4. The ROC must be a connected region. 7. Explain the convolution property of the z-transform. Zx1 (n) X1(z) and Zx2(n) X 2(z) , then
If
Z{ x1 (n)* x2(n)} X1(z)X 2(z) 8. State Parseval’s relation in z-transform.
If x1 (n) and x 2 (n) are complex valued sequences ,then
1
x ( n ) x ( n) 2 j X (v ) X
x
* 2
1
1
* 2
c
1 1 * v dv v
9. State the initial value theorem and the final value theorem. Initial value theorem: If x(n) is casual, then X(0) = limX(z) 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( ) lim(z 1)x(z) z1
10. Determine z-transform and ROC of the finite-duration signal X(n)={2,4, 5 ,7,0,1}
X(z) =
x(n)z
n
n
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 z2 4z 5 7z1 z3 ROC: entire z-plane except z = 0 and z = 11. 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 0, where x(n) is 1, we find
X(z) =
z n 0
n
1 1 z1
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
12. 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
13. 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 14. 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). 15. Prove initial value theorem. If X (z) z{x(n)}, then x(0) LimX (z) z
Proof:
X (z) x(n)z n As z , all the vanish except x(0), which proves the n 0
theorem i.e.; LimX (z) Lim x(n)z n x(0) z
z
n 0
16. Define DTFT pair. The Fourier transform pair of discrete time signal is
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering x ( n)
1 2
X (e
j
) e j n d
2
x ( n )e
X ( e j )
j n
n
17. 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
x ( n)
n
18. State the relationship between z-transform and DTFT?(Nov/Dec 2015) The z-transform of x(n) is given by
X(z)=
x(n)z
n
--------------(1)
n
j Where z = re
---------------(2)
Substituting Eq. (2) in Eq. (1) we get X(re j )
x(n)r
n jn
e
---------------(3)
n
The Fourier transform of x(n) is given by X(e
j
)
x(n)e
jn
---------------(4)
n
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 equal to H(z) evaluated along the unit circle,or X(e
j
j
) is
) X(z) zej
For X(e j ) to exist, the ROC of X(z) must include the unit circle 19. 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 . 20. Find the Fourier transform of a sequence x(n) = 1 for -2 n 2 0 otherwise
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
X (e j )
x ( n)e
j n
n
a ne jn n 0
n
2
a a a j 1 j j ..... e e n 0 e 21. Determine z-transform and ROC of the finite-duration signal X(n)={2,-1, 3 ,0,2}
(Nov/Dec 2016)
X(z) =
x(n)z
n
n
x(0) =3; x(-1) =-1; x(-2) =2; x(1) =0; x(2) = 2 Substituting the sequence values we get X(z)= 2 z2 1z 3 2z2 ROC: entire z-plane except z = 0 and z = 22. Determine the Nyquist rate of sampling for 𝒙(𝒕) = 𝑺𝒊𝒏(𝟐𝟎𝟎𝝅𝒕) − 𝑪𝒐𝒔(𝟏𝟎𝟎𝝅𝒕). (Nov/Dec 2016) 𝑥(𝑡) = 𝑆𝑖𝑛(200𝜋𝑡) − 𝐶𝑜𝑠(100𝜋𝑡) = 𝑆𝑖𝑛(2𝜋𝑓1 𝑡) − 𝐶𝑜𝑠(2𝜋𝑓2 𝑡). 2𝜋𝑓1 = 200𝜋 ℎ𝑒𝑛𝑐𝑒 𝑓1 = 100 𝐻𝑧 2𝜋𝑓2 = 100𝜋 ℎ𝑒𝑛𝑐𝑒 𝑓1 = 50 𝐻𝑧 The maximum analog frequency is 100 Hz .Hence 𝐹𝑠 ≥ 2𝐹𝑚𝑎𝑥 = 2𝑋100 𝐻𝑧 = 200 𝐻𝑧 23. Determine the Nyquist rate 𝟐( 𝟑 𝑺𝒊𝒏 𝟏𝟎𝟎𝝅𝒕).(Apr/May 2015)
of
sampling
𝒙(𝒕) = 𝑺𝒊𝒏(𝟐𝟎𝟎𝝅𝒕) +
for
𝑥(𝑡) = 𝑆𝑖𝑛(200𝜋𝑡) + 3 𝑆𝑖𝑛2 (100𝜋𝑡) == 𝑆𝑖𝑛(2𝜋𝑓1 𝑡) − 𝐶𝑜𝑠(2𝜋𝑓2 𝑡) 2𝜋𝑓1 = 200𝜋 ℎ𝑒𝑛𝑐𝑒 𝑓1 = 100 𝐻𝑧 2𝜋𝑓2 = 100𝜋 ℎ𝑒𝑛𝑐𝑒 𝑓1 = 50 𝐻𝑧 The maximum analog frequency is 100 Hz .Hence 𝐹𝑠 ≥ 2𝐹𝑚𝑎𝑥 = 2𝑋100 𝐻𝑧 = 200 𝐻𝑧 24. State the need for Sampling.(Nov/Dec 2015) Sampling is the process of converting a continuous time signal into discrete time signal 25. Find the z-transform and its associated ROC for x[n]={1,-1, 2, 3 ,4}(Nov/Dec 2015)
X(z) =
x(n)z
n
n
x(0) =5; x(-1) = 2; x(-2) =-1; x(-3) =1; x(1) =4
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering Substituting the sequence values we get X(z)= 2 z3 z2 2z 3 4z1 ROC: entire z-plane except z = 0 and z = 26. Find the DTFT of 𝒙[𝒏] = 𝜹(𝒏) + 𝜹(𝒏 − 𝟏).(Nov/Dec 2014) 𝟏∶𝒏=𝟎 𝜹(𝒏) = { 𝟎: 𝒏 ≠ 𝟎 𝟏∶𝒏=𝟏 𝜹( 𝒏 − 𝟏) = { 𝟎: 𝒏 ≠ 𝟏
X (e j )
n
n 0
n 0
x(n)e jn (n)e jn (n 1)e jn
Sub n=0 and n=1 in the first and
1 e j Second 27. 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) =
x(n)zn Z{x[−n]} =
n
Let –n = p Z{x[−n]} =
x(n)z
n
n
x(p)(z
1 p
) X(z1 )
p
𝟏
28. Find the inverse Z transform for 𝒁+𝟎.𝟏(Nov/Dec 2016) 𝑧 𝑥(𝑛) = (−0.1)𝑛 𝑢(𝑛) = 𝑧 + 0.1 1 𝑥(𝑛 − 1) = (−0.1)𝑛−1 𝑢(𝑛 − 1) = 𝑧 + 0.1 29. Find the Z transform of a) Impulse b)Unit step.(Nov/Dec 2016)
Impulse
x(n)zn
X(z) =
n
Unit Step X(z) =
(n)z
n
1
n
n
n 0
u(n)zn 1.zn
z z 1
30. Define Unilateral and Bilateral Z transform.(Nov/Dec 2013) Bilateral Z transform X(z)
n
n 0
x(n)zn Unilateral Z transform X(z) x(n)zn
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering
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 of output.
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, n
m
k 1
k 0
y (n) ak y (n k ) bk 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
1.
Output of the system with zero input
Question Bank- Two Marks With Answer
Forced Response Output of the system with given input. EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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 band of the filter to preserve the shape of a given signal within the 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. Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering The LTI system is causal and stable if its ROC is outside of circle of radius r<1 Thus z > r <1 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: x(n) h 1 (n) h 2 (n) x(n) h 1 (n) h 2 (n) 3. Distributive Property: x(n) h 1 (n) h 2 (n) x(n) h 1 (n) x(n) h 2 (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 z{x(n) * h(n)} X ( z ) H ( z ) X ( z ) (1 2 z 1 z 2 ) H ( z ) (1 z 1 z 2 ) X ( z ) H ( z ) 1 3 z 1 4 z 2 3 z 3 z 4 x(n) * h(n) { 1, 3, 4, 3, 1 }
15. Convolve the following sequence x(n) = { 1 ,2,3} h(n) = { 1 ,1,2}.(Nov/Dec 2016)
We know z{x(n) * h(n)} X ( z ) H ( z ) X ( z ) (1 2 z 1 3 z 2 ) H ( z ) (1 z 1 2 z 2 ) X ( z ) H ( z ) 1 3 z 1 7 z 2 7 z 3 6 z 4 x(n) * h(n) { 1, 3, 7, 7, 6 }
16. Given the system function H ( z ) 2 3z 1 4 z 3 5 z 4 .Determine the impulse response h[n].(Nov/Dec 2016) Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering 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) 𝑵
𝑴
𝒚(𝒏) + ∑ 𝒂𝒌 𝒚(𝒏 − 𝒌) = ∑ 𝒃𝒌 𝒙(𝒏 − 𝒌) 𝒌=𝟏
𝒌=𝟎
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 z{x(n) * h(n)} X ( z ) H ( z ) X ( z ) (1 z 1 3 z 2 ) H ( z ) (1 4 z 1 1z 2 ) X ( z ) H ( z ) 1 5 z 1 6 z 2 11z 3 3z 4 x(n) * h(n) { 1, 5, 6, 11, 3 }
22. In 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?
Question Bank- Two Marks With Answer
EC6303/ Signals and Systems
Dr.N.G.P. Institute of Technology / Electronics And Communication Engineering S.No 1 2 3 4 5
direct form -I Separate delay for input and output M+N+1 Multiplications are involved M+N delays are involved M+N Memory locations are required Noncanonical structure
direct form -II Same delay for input and output M+N+1 Multiplications are are involved N delays are involved N Memory locations are required 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) X(z)
Y(z) H1(Z)
Question Bank- Two Marks With Answer
H2(Z)
EC6303/ Signals and Systems
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