Unit 1: Introduction to Signals - Subjective Questions

ECE220 — Signal And Systems • Practice Questions with Detailed Answers

20 questions

Define a signal and a system. Classify signals as continuous-time (CT) and discrete-time (DT) signals with suitable examples.

Distinguish between energy signals and power signals. Give the mathematical conditions for each.

Energy of a signal:

Continuous: $E = \int_{-\infty}^{\infty} |x(t)|^2 \, dt$
Discrete: $E = \sum_{n=-\infty}^{\infty} |x[n]|^2$

Power of a signal:

Continuous: $P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 \, dt$
Discrete: $P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$

Property	Energy Signal	Power Signal
Energy $E$	Finite ( $0 < E < \infty$ )	Infinite
Power $P$	Zero	Finite ( $0 < P < \infty$ )
Nature	Non-periodic, aperiodic	Usually periodic
Example	Decaying exponential, single pulse	Sinusoidal signal, periodic signals

Important notes:

A signal cannot be both an energy and a power signal simultaneously.
Some signals (e.g., $x(t) = t$ ) are neither energy nor power signals.

Determine whether the signal $x(t) = e^{-2t}u(t)$ is an energy or a power signal by computing its energy and power.

Explain the various transformations of the independent variable in signals with neat diagrams: time shifting, time scaling, and time reversal.

Define a periodic signal. Derive the condition for periodicity for continuous-time and discrete-time signals, and determine the fundamental period of $x(t) = \cos(3\pi t) + \sin(4\pi t)$ .

Explain why a discrete-time sinusoid $x[n] = \cos(\omega_0 n)$ is not always periodic. State the condition for periodicity.

Define even and odd signals. Show how any arbitrary signal can be decomposed into its even and odd parts. Find the even and odd components of $x(t) = e^{jt}$ .

Describe the continuous-time real exponential signal $x(t) = Ce^{at}$ for different values of the parameter $a$ .

Explain the continuous-time complex exponential signal and show its relationship with sinusoidal signals using Euler's formula.

Define the continuous-time unit impulse function $\delta(t)$ . State and explain its important properties.

Define the continuous-time and discrete-time unit step functions. Explain the relationship between the unit step and the unit impulse functions.

Define the discrete-time unit impulse $\delta[n]$ and explain its sifting property with an example.

Explain the various operations performed on the amplitude of signals: amplitude scaling, addition, and multiplication with suitable examples.

A discrete-time signal is given as $x[n] = \{1, 2, 3, 4, 5\}$ where the arrow (origin $n=0$ ) is at the value 3. Sketch/describe: (i) $x[n-2]$ , (ii) $x[-n]$ , (iii) $x[2n]$ .

Compare continuous-time signals and discrete-time signals on the basis of definition, representation, periodicity, and examples.

Basis	Continuous-Time (CT) Signal	Discrete-Time (DT) Signal
Definition	Defined for every instant of time over a continuous interval	Defined only at discrete (integer) instants
Notation	$x(t)$ , $t \in \mathbb{R}$	$x[n]$ , $n \in \mathbb{Z}$
Independent variable	Continuous real number $t$	Integer $n$
Origin	Obtained naturally / analog sources	Obtained by sampling CT signals
Sinusoid periodicity	Always periodic	Periodic only if $\frac{2\pi}{\omega_0}$ is rational
Graphical form	Smooth continuous curve	Sequence of discrete points (stem plot)
Example	$x(t)=\sin(2\pi t)$ , speech signal	$x[n]=\{1,2,3\}$ , digital data
Processing	Analog systems	Digital systems / computers

Summary: CT signals model analog real-world phenomena, while DT signals are used in digital signal processing and are obtained by sampling CT signals at regular intervals.

Determine whether the following signals are energy signals, power signals, or neither: (i) $x(t) = A\cos(\omega_0 t)$ , (ii) $x(t) = u(t)$ .

Explain the significance of software simulation of elementary signals. Write pseudo-code/MATLAB commands to generate a unit step, unit impulse, and a sinusoidal signal.

Significance of Software Simulation:

Allows visualization of signals and their transformations without hardware.
Helps in verifying theoretical results and analyzing complex operations.
Provides a platform for DSP algorithm development and testing.
Tools commonly used: MATLAB, Python (NumPy/Matplotlib), Scilab, Octave.

MATLAB code to generate elementary signals:

MATLAB

% Discrete-Time Unit Impulse
n = -5:5;
imp = (n == 0);
stem(n, imp); title('Unit Impulse');

% Discrete-Time Unit Step
u = (n >= 0);
stem(n, u); title('Unit Step');

% Continuous-Time Sinusoidal Signal
t = 0:0.001:1;
f = 5;               % frequency in Hz
x = sin(2*pi*f*t);
plot(t, x); title('Sinusoidal Signal');
xlabel('Time'); ylabel('Amplitude');

Python equivalent (using NumPy):

PYTHON

import numpy as np
import matplotlib.pyplot as plt

n = np.arange(-5, 6)
impulse = (n == 0).astype(int)
step = (n >= 0).astype(int)
plt.stem(n, impulse)
plt.show()

Simulation makes abstract concepts tangible and speeds up learning.

Sketch and describe the operation of obtaining $y(t) = x(-2t + 3)$ from a given signal $x(t)$ . Clearly state the sequence of transformations.

Explain the properties of a discrete-time complex exponential and sinusoidal signal, highlighting the differences from the continuous-time case.

Check whether the following signals are even, odd, or neither: (i) $x(t) = \sin(t) + \cos(t)$ , (ii) $x(t) = t^3$ , (iii) $x[n] = \cos(2n)$ .

Question 1

Define a signal and a system. Classify signals as continuous-time (CT) and discrete-time (DT) signals with suitable examples.

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Answer 1

Signal: A signal is a function of one or more independent variables that carries information about the behaviour or nature of a physical phenomenon. Mathematically it is represented as $x(t)$ for continuous time and $x[n]$ for discrete time.

System: A system is an entity that processes an input signal to produce an output signal, represented as $y(t) = T\{x(t)\}$ .

Classification:

Continuous-Time (CT) Signal: Defined for every value of time over a continuous interval. Denoted .
- Example: $x(t) = \sin(2\pi t)$ , human speech, temperature variation.
Discrete-Time (DT) Signal: Defined only at discrete instants of time. Denoted where is an integer.
- Example: $x[n] = \{1, 2, 3, 4\}$ , monthly rainfall data, digital audio samples.

Key difference: A CT signal has a value at every instant, whereas a DT signal exists only at integer indices and is often obtained by sampling a CT signal.

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Question 2

Distinguish between energy signals and power signals. Give the mathematical conditions for each.

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Answer 2

Energy of a signal:

Continuous: $E = \int_{-\infty}^{\infty} |x(t)|^2 \, dt$
Discrete: $E = \sum_{n=-\infty}^{\infty} |x[n]|^2$

Power of a signal:

Continuous: $P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 \, dt$
Discrete: $P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$

Property	Energy Signal	Power Signal
Energy $E$	Finite ( $0 < E < \infty$ )	Infinite
Power $P$	Zero	Finite ( $0 < P < \infty$ )
Nature	Non-periodic, aperiodic	Usually periodic
Example	Decaying exponential, single pulse	Sinusoidal signal, periodic signals

Important notes:

A signal cannot be both an energy and a power signal simultaneously.
Some signals (e.g., $x(t) = t$ ) are neither energy nor power signals.

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Question 4

Explain the various transformations of the independent variable in signals with neat diagrams: time shifting, time scaling, and time reversal.

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Answer 4

Transformations modify the independent variable ( $t$ or $n$ ) to produce a new signal.

1. Time Shifting:

Represented as $x(t - t_0)$ .
If $t_0 > 0$ : signal is delayed (shifted right).
If $t_0 < 0$ : signal is advanced (shifted left).

2. Time Scaling:

Represented as $x(at)$ .
If $|a| > 1$ : signal is compressed in time.
If $|a| < 1$ : signal is expanded/stretched in time.

3. Time Reversal (Folding):

Represented as $x(-t)$ .
The signal is mirrored/reflected about the vertical axis ( $t = 0$ ).

Combined transformation $x(at + b)$ is performed by:

First shift by $b$ , then scale by $a$ OR first scale then shift by $b/a$ .
Care must be taken with the order of operations.

Example: For $x(2t - 4)$ , shift $x(t)$ right by 4 to get $x(t-4)$ , then compress by factor 2 to get $x(2t-4)$ .

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Question 6

Explain why a discrete-time sinusoid $x[n] = \cos(\omega_0 n)$ is not always periodic. State the condition for periodicity.

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Answer 6

For a discrete-time sinusoid $x[n] = \cos(\omega_0 n)$ to be periodic, there must exist a positive integer $N$ such that:

$x[n + N] = x[n]$

$\cos(\omega_0(n+N)) = \cos(\omega_0 n)$

This requires $\omega_0 N = 2\pi k$ , where $k$ is an integer. Therefore:

$N = \frac{2\pi k}{\omega_0}$

Condition for periodicity: The ratio $\dfrac{2\pi}{\omega_0} = \dfrac{N}{k}$ must be a rational number.

Explanation:

Unlike continuous-time sinusoids (which are always periodic), a DT sinusoid is periodic only if $\frac{\omega_0}{2\pi}$ is rational.
If $\frac{\omega_0}{2\pi}$ is irrational, no integer $N$ satisfies the periodicity condition, so the signal is aperiodic.

Example:

$x[n] = \cos\left(\frac{\pi}{4}n\right)$ : $\frac{2\pi}{\pi/4} = 8$ (rational) $\Rightarrow$ periodic with $N = 8$ .
$x[n] = \cos(n)$ : $\frac{2\pi}{1} = 2\pi$ (irrational) $\Rightarrow$ aperiodic.

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Question 8

Describe the continuous-time real exponential signal $x(t) = Ce^{at}$ for different values of the parameter $a$ .

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Answer 8

The continuous-time real exponential signal is given by:

$x(t) = C e^{at}$

where $C$ and $a$ are real constants.

Cases based on the value of $a$ :

$a > 0$ (Positive): The signal is a growing exponential. Its amplitude increases with time. Represents unstable systems.
$a < 0$ (Negative): The signal is a decaying exponential. Its amplitude decreases with time and approaches zero. Represents stable/damped systems (e.g., RC circuit discharge).
$a = 0$ : The signal becomes a constant (DC signal) $x(t) = C$ .

Role of $C$ :

$C$ is the value of the signal at $t = 0$ (i.e., $x(0) = C$ ).

Applications: Modelling radioactive decay, capacitor charging/discharging, population growth, and RC/RL transient responses.

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Question 13

Explain the various operations performed on the amplitude of signals: amplitude scaling, addition, and multiplication with suitable examples.

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Answer 13

Operations on the dependent variable (amplitude) modify the signal's value at each instant.

1. Amplitude Scaling:

Multiplying a signal by a constant $A$ : $y(t) = A\,x(t)$ .
If $A > 1$ : amplification; if $0 < A < 1$ : attenuation; if $A < 0$ : inversion + scaling.

2. Signal Addition:

Two signals are added instant by instant: $y(t) = x_1(t) + x_2(t)$ .
For DT: $y[n] = x_1[n] + x_2[n]$ .
Example: $x_1 = \{1,2,3\}$ , $x_2 = \{4,1,2\}$ $\Rightarrow y = \{5,3,5\}$ .

3. Signal Multiplication:

Two signals are multiplied point by point: $y(t) = x_1(t)\cdot x_2(t)$ .
For DT: $y[n] = x_1[n]\cdot x_2[n]$ .
Example: $x_1 = \{1,2,3\}$ , $x_2 = \{4,1,2\}$ $\Rightarrow y = \{4,2,6\}$ .

Note: Addition and multiplication require the signals to be defined over the same time index; otherwise, they are extended with zeros.

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Question 15

Compare continuous-time signals and discrete-time signals on the basis of definition, representation, periodicity, and examples.

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Answer 15

Basis	Continuous-Time (CT) Signal	Discrete-Time (DT) Signal
Definition	Defined for every instant of time over a continuous interval	Defined only at discrete (integer) instants
Notation	$x(t)$ , $t \in \mathbb{R}$	$x[n]$ , $n \in \mathbb{Z}$
Independent variable	Continuous real number $t$	Integer $n$
Origin	Obtained naturally / analog sources	Obtained by sampling CT signals
Sinusoid periodicity	Always periodic	Periodic only if $\frac{2\pi}{\omega_0}$ is rational
Graphical form	Smooth continuous curve	Sequence of discrete points (stem plot)
Example	$x(t)=\sin(2\pi t)$ , speech signal	$x[n]=\{1,2,3\}$ , digital data
Processing	Analog systems	Digital systems / computers

Summary: CT signals model analog real-world phenomena, while DT signals are used in digital signal processing and are obtained by sampling CT signals at regular intervals.

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Question 17

Explain the significance of software simulation of elementary signals. Write pseudo-code/MATLAB commands to generate a unit step, unit impulse, and a sinusoidal signal.

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Answer 17

Significance of Software Simulation:

Allows visualization of signals and their transformations without hardware.
Helps in verifying theoretical results and analyzing complex operations.
Provides a platform for DSP algorithm development and testing.
Tools commonly used: MATLAB, Python (NumPy/Matplotlib), Scilab, Octave.

MATLAB code to generate elementary signals:

MATLAB

% Discrete-Time Unit Impulse
n = -5:5;
imp = (n == 0);
stem(n, imp); title('Unit Impulse');

% Discrete-Time Unit Step
u = (n >= 0);
stem(n, u); title('Unit Step');

% Continuous-Time Sinusoidal Signal
t = 0:0.001:1;
f = 5;               % frequency in Hz
x = sin(2*pi*f*t);
plot(t, x); title('Sinusoidal Signal');
xlabel('Time'); ylabel('Amplitude');

Python equivalent (using NumPy):

PYTHON

import numpy as np
import matplotlib.pyplot as plt

n = np.arange(-5, 6)
impulse = (n == 0).astype(int)
step = (n >= 0).astype(int)
plt.stem(n, impulse)
plt.show()

Simulation makes abstract concepts tangible and speeds up learning.

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Question 18

Sketch and describe the operation of obtaining $y(t) = x(-2t + 3)$ from a given signal $x(t)$ . Clearly state the sequence of transformations.

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Answer 18

To obtain $y(t) = x(-2t + 3)$ from $x(t)$ , we apply three transformations in a proper sequence.

Method (Shift first, then scale/fold):

Step 1 — Time Shift: Replace $t$ with $t+3$ to get $x(t + 3)$ . This advances (shifts left) the signal by 3 units.

Step 2 — Time Scaling: Replace $t$ with $2t$ in $x(t+3)$ to get $x(2t + 3)$ . This compresses the signal by a factor of 2.

Step 3 — Time Reversal (Folding): Replace $t$ with $-t$ to get $x(-2t + 3)$ . This reflects the signal about the vertical axis.

Verification using key points: For a point where the argument $= 0$ :

$-2t + 3 = 0 \Rightarrow t = 1.5$

So the feature that was at origin in $x$ now appears at $t = 1.5$ in $y(t)$ .

Alternative method (Scale/fold first, then shift):

$x(t) \to x(-2t)$ (fold + compress) $\to$ shift right by $\frac{3}{2}$ to get $x(-2(t - 1.5)) = x(-2t + 3)$ .

Both methods yield the same result; care with the order avoids errors.

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Question 19

Explain the properties of a discrete-time complex exponential and sinusoidal signal, highlighting the differences from the continuous-time case.

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Answer 19

Discrete-Time Complex Exponential:

$x[n] = C\alpha^n \quad \text{or} \quad x[n] = e^{j\omega_0 n}$

Key Properties and Differences from CT:

1. Periodicity:

CT signal $e^{j\omega_0 t}$ is always periodic.
DT signal $e^{j\omega_0 n}$ is periodic only if $\frac{\omega_0}{2\pi}$ is rational, i.e., $\frac{2\pi}{\omega_0} = \frac{N}{k}$ .

2. Distinct frequencies:

In CT, distinct values of $\omega_0$ give distinct signals.
In DT, frequencies separated by $2\pi$ are identical: $e^{j(\omega_0 + 2\pi)n} = e^{j\omega_0 n}$
Hence DT frequencies are considered only over an interval of length $2\pi$ (e.g., $-\pi$ to $\pi$ ).

3. Rate of oscillation:

In DT, highest rate of oscillation occurs at $\omega_0 = \pi$ (or odd multiples), and lowest near $\omega_0 = 0$ (or multiples of $2\pi$ ).

General DT exponential $x[n] = C\alpha^n$ with $\alpha = |\alpha|e^{j\omega_0}$ :

$|\alpha| > 1$ : growing envelope
$|\alpha| < 1$ : decaying envelope
$|\alpha| = 1$ : constant amplitude sinusoid

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Unit 1: Introduction to Signals - Subjective Questions

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