Approved by

AICTE, UGC Under Section 2f, Recognized by Govt. of Karnataka.

Faculty of Engineering &Technology (Co-Ed)

DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING
QUESTION BANK ACADEMIC YEAR 2023-24

Faculty Name: Dr. Rangayya and Harshavardhan Reddy.

Subject Name: Signals & Systems [22EC44]
Sem: IV th Semmester

Module-1

1. Define Signal & System. Explain the basic elementary continuous time signals.
2. Define Signal & System. Explain the basic elementary discrete time signals.
3. Sketch the following signals
a) x(t) = u(t+1)-2u(t)+u(t+1)
b) x(𝑡) = − 𝑢(𝑡 + 3) + 2𝑢(𝑡 + 1) − 2𝑢(𝑡 − 1) + 𝑢(𝑡 − 3)
4. A continuous-time signal 𝑥(𝑡) is shown in figure. Draw the signal,
𝑦(𝑡) = [𝑥(𝑡) + 𝑥(2 − 𝑡)]𝑢(1 − 𝑡)

5. For the continuous-time signal𝑥(𝑡) shown in figure. Obtain𝑦(𝑡) = 𝑥(3𝑡) + 𝑥(3𝑡 + 2).

6. Two continuous-time signals x(𝑡) and 𝑦(𝑡) are given in figure and draw
i. z(𝑡) = 𝑥(2𝑡)𝑦(2𝑡 + 1) ii 𝑧(𝑡) = 𝑥(𝑡)𝑦(− 1 − 𝑡)
7. The discrete time signals x(n) and y(n) are as shown in figure. Sketch
the signal z(n)=x(2n) y(n-4).

8. Determine whether the following signals are periodic or not periodic; if periodic find its
fundamental period T.
a. x(t)= cos8πt
𝜋𝑛 𝜋𝑛
b. x(n)= cos( ) sin(( )
5 3
c. x(t) = sin(20πt)+cos(30t)
𝜋𝑛 𝜋𝑛
d. x(n)= cos( 2 )+ sin( 4 )

9. Determine whether the continuous time signal y(t)=y1(t)+y2(t)+y3(t) is periodic,

where y1(t),y2(t) and y3(t) have periods of 1.08, 3.6 and 2.025 sec. respectively..

10. Determine and sketch the even and odd component of the signal 𝑥(𝑡) shown in
figure below

11. Determine and sketch the even and odd component of the signal 𝑥(n)
 12. What is the total energy of the rectangular pulse shown in figure.

13. Determine whether the given signal

x(t) is energy or power signal & find the same.

Module-2

System Classification and Properties Time- Domain Representation of LTI Systems

1. For the following system determine whether the system is a) Linear b)Time
invariance c) Memory d) Causal e) Stable
i. H{x(n)} = g(n).x(n)
ii. H{x(n)} = x(-n)
iii. H{x(t)} = e x(t)
iv. H{x(t} = x(t/2)
v. y(𝑡) = 𝑐𝑜𝑠( 𝑥(𝑡))
2. Evaluate the discrete-time convolution sum given. y(𝑛) = 𝑢(𝑛) * 𝑢(𝑛 − 3)
3. Consider an input x(𝑛) and an unit impulse response h(𝑛) given by,
x(𝑛) = α 𝑛𝑢(𝑛) ; 0 < α < 1 h(𝑛) = 𝑢(𝑛), evaluate the discrete-time convolution sum
4. Evaluate the discrete-time convolution sum given by y(n)=(1/2)n u(n-2)*u(n)
5. Evaluate the discrete-time convolution sum given by y(n)=βn u(n)*u(n-3) ; |β| <1
6. Consider a continuous-time LTI system with unit impulse response,h(𝑡) = 𝑢(𝑡) and

input 𝑥(𝑡) = 𝑒−𝑎𝑡 u(t) ; 𝑎> 0. Find the output y(𝑡) of the system.

7. Consider a LTI system with unit impulse response h(𝑡) = 𝑒−t 𝑢(𝑡) . If the input
to the system is x(𝑡) = 𝑒−3t[𝑢(𝑡) − 𝑢(𝑡 − 2)]. Find the output y(𝑡) of the system.
8.Evaluate the continuous-time convolution integral given below.
y(t)=e-2tu(t)*u(t+2)
9. Evaluate the continuous-time convolution integral given below.
y(t)= u(t+1)*u(t-2)

Module-3

Differential & Difference Equation Representation of LTI Systems

1. Find the natural response for the system described by the differential equation,
ⅆ𝑦(𝑡)
5 ⅆ𝑡 + 10𝑦(𝑡) = 2𝑥(𝑡); y(0)=3
2. Find the zero-input response of the system described by the differential equation
ⅆ 2 𝑦(𝑡) ⅆ𝑦(𝑡) ⅆ𝑥(𝑡) ⅆ𝑦(𝑡)
+3 + 2𝑦(𝑡) = 𝑥(𝑡) with y(0)=1; | =1
ⅆ𝑡 2 ⅆ𝑡 ⅆ𝑡 ⅆ𝑡 𝑡=0

3. Find the natural response for the system described by the differential equation
ⅆ 2 𝑦(𝑡) ⅆ𝑦(𝑡)
+2 + 𝑦(𝑡) = 0 with y(0)=1; y’ (0)=1
ⅆ𝑡 2 ⅆ𝑡

4. Find the natural response for the system described by the differential equation
ⅆ 2 𝑦(𝑡) ⅆ𝑦(𝑡) ⅆ𝑥(𝑡) ⅆ𝑦(𝑡)
+2 + 2𝑦(𝑡) = with y(0)=1; | =0
ⅆ𝑡 2 ⅆ𝑡 ⅆ𝑡 ⅆ𝑡 𝑡=0
5. Determine the forced response for the system given by,
ⅆ𝑦(𝑡)
5 + 10𝑦(𝑡) = 2𝑥(𝑡) with input x(t)=2 u(t).
ⅆ𝑡
6. Determine the forced response for the system given by,
ⅆ𝑦(𝑡)
5 + 10𝑦(𝑡) = 2𝑥(𝑡) with input x(t)=e-t u(t).
ⅆ𝑡
7. Determine the complete response of the system described by the differential
equation.
ⅆ 2 𝑦(𝑡) ⅆ𝑦(𝑡) ⅆ𝑥(𝑡) ⅆ𝑦(𝑡)
8. +5 + 4𝑦(𝑡) = with y(0)=0; | = 1 for input x(t)=e-2t u(t).
ⅆ𝑡 2 ⅆ𝑡 ⅆ𝑡 ⅆ𝑡 𝑡=0

9. Find the natural response of the system described by the difference equation,

y(n)-1/4 y(n-1)-1/8 y(n-2) = x(n) + x(n-1) with y(-1) = 0 and y(-2) = 1

10. Find the zero-input response for the system described by the difference
equation y(n)+9/16 y(n-2) = x(n-1) with initial conditions y(-1) = 1 and y(-2) = -1
11. Find the forced response for the system described by the difference equation,

y(n)-1/4 y(n-1)-1/8 y(n-2) = x(n) + x(n-1) with input x(n) = (1/8)n u(n)
12. Find the zero-state response for the system described by the difference
equation y(n)-5/6 y(n-1) + 1/6 y(n-2) = x(n) with forcing conditions x(n)= 2n ; n≥0
and zero elsewhere
13. Find the response of the system described by the difference equation,
y(n)-1/9 y(n-2) = x(n-1) with y(-1) = 1, y(-2) = 0 and x(n)=u(n)
 14. Define the following properties of CTFS:
a. Linearity
b. Time shift
c. Frequency shift
d. Scaling
e. Time-differentiation
f. Convolution
g. Modulation
h. Parseval’s theorem
i. Symmetry
15. For the signal (𝑡) = 𝑠inω0𝑡, find the Fourier series and draw its spectrum.
16. Evaluate the fourier series representation for the signal,
x(𝑡) = 𝑠𝑖𝑛(2π𝑡) + 𝑐𝑜𝑠(3π𝑡).
17. Determine the fourier series representation for the signal x(𝑡) = 𝑐𝑜𝑠4𝑡 + 𝑠𝑖𝑛8𝑡 and
draw the Spectrum.
18. Find the Fourier series coefficients for the periodic signal (𝑡) with period T=2
given by, (𝑡) = 𝑒−𝑡; for − 1 <𝑡< 1.
Module 4
Fourier Representation of Aperiodic Signals

1. State and Prove the following properties related to continuous-time fourier transform:
a. Linearity
b. Time shift
c. Frequency shift
d. Time Differentiation
e. Parseval’s theorem
2. Find the DTFT of the signal, x(𝑛) = α 𝑛𝑢(𝑛) ; |α| < 1 Draw the magnitude spectrum
3. Find the DTFT of the signal, x(𝑛) = (− 1)n u(n) .
4. Find the DTFT of the signal x(𝑛) = 𝑢(𝑛) − 𝑢(𝑛 − 6)
5. Find the DTFT of the signal, x(𝑛) = 2 nu(-n)
6. Evaluate the DTFT of the signal, x(𝑛) = ( 1/4 ) n𝑢(𝑛 + 4)
7. Compute the DTFT of the following signal,
x(𝑛) = (1/ 2 ) n{𝑢(n+3)-u(n-2)}
8. Obtain the Fourier Transform of the signal and draw its spectrum:
x(𝑡) = 𝑒−𝑎𝑡𝑢(𝑡) ; 𝑎> 0
9. Evaluate the Fourier transform for the signal, x(t)=e-3t u(t-1). Find the
expression for magnitude and phase spectra.
10. Find the Fourier transform for the signal, x(t)=te-2t u(t). Obtain the expression
for the magnitude and phase spectra.
 Module 5
The Z-Transforms
1. What is Z-transform? Mention properties of Region of Convergence (ROC)
2. Find the Z-transform of the signal (𝑛) = α 𝑛(𝑛) and Draw ROC.
3. Find (𝑧) if (𝑛) = − α 𝑛𝑢(− 𝑛 − 1) and find the ROC.
4. For the signal 𝑥 (𝑛) = 7 ( 1/3 ) 𝑛𝑢(𝑛) – 6( 1 / 2 ) 𝑛𝑢(𝑛) find the Z-transform
and ROC.
5. Determine the Z-transform of, 𝑥 (𝑛) = − u(− 𝑛 − 1) + (1 / 2 ) n𝑢(𝑛) Find the
ROC and pole-zero location of (𝑧) in the z-plane.
1
6. Find the Z-transform for the signal x(n)=( 2)nu(n-2)and plot its ROC and
poles and zeros.
7. Find the Z-transform for the signal x(n)=2n u(-n-1) and plot its ROC and
poles and zeros.
8. State and Prove properties of Z-transforms:
a. Linearity
b. Time Shifting
c. Scaling in theZ-domain
d. Timer eversal
e. Convolution
f. Differentiation in the Z-domain
9. Find the Z-transform of the signal x(n)=3(2)nu(-n) using appropriate properties.
10. Determine the Z-transform of the following signals and sketch the ROC.
a) x1(n)=(1/3)n ; n≥0
=(1/2)-n ; n≤0
b) x2(n) =x1(n+4)
11. Find the discrete-time sequence x(n) which has Z-transform,

− 1 + 5 z −1
X ( z) = with ROC; |𝑧|>1
3 −1 1 − 2
1− z + z
2 2

12. Find the inverse Z-transform of the following using partial fraction expansion
method.
1 + 2 z −1 + z −2
X ( z) = with ROC; |𝑧|>1
3 −1 1 − 2
1− z + z
2 2
13. Find the inverse Z-transform of
z 2 − 3z
X ( z) = ROC: 1/ 2 < |𝑧| < 2
3
z + z −1
2

2
14. Find the inverse Z-transform of the following using partial fraction expansion
1 − 1 / 2 z −1
X ( z) = with ROC; |𝑧|>1/2
3 −1 1 −2
1+ z + z
4 8