FUNDAMENTAL OF SIGNALS AND SYSTEMS

WAN HAMIDAH BINTI WAN ABAS

First Published 2021

© Politeknik Kuala Terengganu

e ISBN 978-967-2240-25-9

All rights reserved. No part of this book may be reproduced or

transmitted in any form or by any means, electronic, including

photocopying, recording or by any information storage or retrieval

system, without prior written permission from the Politeknik Kuala

Terengganu

Author :

Wan Hamidah Binti Wan Abas

Published by :

Politeknik Kuala Terengganu,

Jalan Sultan Ismail,

20200 Kuala Terengganu, Terengganu.

09-6204100

Fundamental of Signals And Systems was written for Electrical

Engineering students based on polytechnic syllabus. It also can be used

by engineering students in other technical institution The material in this

book are presented in a simple manner to assist student in understanding

the fundamental concepts and theory of signals and systems .

PREFACE

ACKNOWLEGDEMENTS

Special thanks to my family especially to my sweetie daughter, and

colleagues for their support and encouragement throughout the

preparation for this book. My hoping is that this book would helping

students to gain better understanding of the signals and systems

concepts.

CONTENT

INTRODUCTION TO SIGNALS

AND SYSTEMS

CLASSIFICATION OF SIGNALS2

1

3

4

5

CLASSIFICATION OF SYSTEMS

SIGNAL OPERATIONS

QUESTION AND ANSWER

1

7

20

28

6 REFERENCE

36

48

Introduction To Signals and

Systems

1

After completion this chapter, student

should be able to:

Understand the definition of signals

and systems.

Know the concept and theory of

Signals and Systems.

Learning Outcome

2

Chapter 1

INTRODUCTION TO SIGNALS AND SYSTEMS

Introduction To Signals And Systems

Student GuideStudent GuideDefinition of Signal

A function representing a physical quantity or variable, and typically it

contains information about the behavior or nature of the phenomenon.

Human voice

Radio wave signal Optical signal

Example of Signal

Electrical Current

Body Temperature signal

3

Introduction To Signals And Systems

Definition of System

A respond to an input signal by producing an output signal.

Example of System

SYSTEMOutput

Signals

Input

Signals

SYSTEMOutput

Signal

Input

Signal

Single input, single output

Multiple inputs, multiple outputs

Traffic light controller

Radio transmitter and receiver

4

Example of System

Human Voice System

Remote Sensing System

Auditory System

Introduction To Signals And Systems

5

Application of Signals and Systems

Acoustics

Restore speech in a noisy environment such as cockpit

Biometrics

Fingerprint identification,

speaker recognition, iris recognition

Communications

Transmission in mobile phones, GPS, radar

Biodemical

Extract information from biological signals:

-Electrocardogram (ECG) electrical signals generated

by heart -Electroencephalogram (EEG) electrical signals

generated by brain

Multimedia

Compress signals to store data such as

CDs, DVDs

Control System

Introduction To Signals And Systems

6

2Classification

Of Signals

After completion this chapter, student

should be able to:

Understand the types of signals

Understand the basic signals of

Continuous-Time Signals

Understand the basic signals of

Discrete-Time Signals

Learning Outcome

Classification Of Signals

8

A. Continuous-Time And Discrete-Time Signals

B. Analog And Digital Signals

C. Real And Complex Signals

D. Deterministic And Random Signals

E. Periodic And Non-Periodic Signals

F. Even And Odd Signals

CLASIFICATION OF SIGNAL

9

Chapter 2

CLASSIFICATION OF SIGNALS

Classification Of Signals

1. Continuous-Time And Discrete-Time Signals

Continuous-time signal x(t) Discrete-time signal x[n]

The identification is based on Horizontal: X-Axis or TIME

Sampling

A discrete-time signal x[n] can obtained by sampling a continuous-time

signal x(t) :

𝑥 𝑡0 , 𝑥 𝑡1 , … . , 𝑥 𝑡𝑛 ,…. or

𝑥 0 , 𝑥 1 , ……, 𝑥 𝑛 , ..… or

𝑥0, 𝑥1, …. , 𝑥𝑛 , …..

𝑥𝑛 = samples

sampling interval = the time interval between samples (𝑥𝑛)

𝑥𝑛= x[n] = x(𝑡𝑛)

10

2. Analog And Digital Signals

The identification is based on Vertical: Y-Axis or AMPLITUDE

𝑥 𝑛 = 𝑥𝑛 = ൞1

2

𝑛

𝑛 ≥ 0

0 𝑛 < 0

or

𝑥 𝑛 = 1,1

2,1

4,… . . ,

1

2

𝑛

, …

Sampling intervals are equal (uniform sampling)

𝑥𝑛= x[n] = x(𝑛𝑇𝑠)

Example of calculating the nth value of the sequence.

Write the sequences of discrete signal.

{𝑥𝑛} = { ... , 0, 0, 1, 2, 2, 1, 0, 1, 0, 2, 0, 0, ... }

{𝑥𝑛} = {1, 2, 2, 1, 0, 1, 0, 2}

Arrow show the origin of the signal

“0” before and after significant signal are neglect

or

signal whose amplitude can take on any value in a continuous range

signal whose amplitude can take only a finite number of values (involve binary 1 and/or 0)

Classification Of Signals

11

Classification Of Signals

3. Real And Complex Signals

The identification is based on Vertical and Horizontal: Y-Axis and X-Axis

or AMPLITUDE and TIME

4. Deterministic And Random Signals

The identification is based on PATTERN of the signal

Deterministic Signals

Random Signals

• The signal is fixed and can be determined

• Can be represented in mathematic form since the present, past and future values can be predicted based on the equation

• A signal take random values• Cannot be put in

mathematic form

X(t) is a complex quantity and has :• A real and imaginary part or

equivalently• A magnitude and a phase angle

Important class of signals is complex exponentials:• CT signals of the form 𝑥(𝑡) = 𝑒𝑠𝑡

• DT signals of the form 𝑥 𝑛 = 𝑧𝑛

Where z and s are complex numbers.

12

Classification Of Signals

5. Periodic And Non-Periodic Signals

The identification is based on REPETITIVE of the signal

CT

DT

Periodic Signals Non-Periodic Signals

A signal which repeats itself after a specific interval of time

A signal which does not repeatitself after a specific interval of time

A signal that repeats it patternover a period

A signal that does not repeats its pattern over a period

They can be represented by a mathematical equation

They cannot be represented by any mathematical equation

Their value can be determined at any point of time

Their value cannot be determined with certainty at any given point of time

They are deterministic signals They are random signals

Example: sine cosine square sawtooth etc

Example: sound signals from radio, all types of noise signals

Differences between Periodic and Non-Periodic Signal

Non-Periodic Signal also known as Aperiodic Signal.

13

Where

𝑥𝑒 𝑡 =1

2{𝑥 𝑡 + 𝑥 −𝑡 } even part of 𝑥 𝑡

𝑥𝑜 𝑡 =1

2{𝑥 𝑡 − 𝑥 −𝑡 } odd part of 𝑥 𝑡

𝑥𝑒[𝑛] =1

2{𝑥[𝑛] + 𝑥[−𝑛]} even part of 𝑥[𝑛]

𝑥𝑜[𝑛] =1

2{𝑥[𝑛] − 𝑥[−𝑛]} odd part of 𝑥[𝑛]

Classification Of Signals

Any signal x(t) or x[n] can be expressed as a

sum of two signals, one of which is even and

one of which is odd. That is,

𝑥 𝑡 = 𝑥𝑒 𝑡 + 𝑥𝑜 𝑡

𝑥[𝑛] = 𝑥𝑒[𝑛] + 𝑥𝑜[𝑛]

Note:

Sum product of two signal 𝑥𝑒 𝑡 × 𝑥𝑒 𝑡 = 𝑥𝑒 𝑡𝑥𝑜 𝑡 × 𝑥𝑜 𝑡 = 𝑥𝑒 𝑡𝑥𝑒 𝑡 × 𝑥𝑜 𝑡 = 𝑥𝑜 𝑡𝑥𝑜 𝑡 × 𝑥𝑒 𝑡 = 𝑥𝑜 𝑡

6. Even And Odd Signals

The identification is based on SYMMETRICAL of the signal

Even Signals Odd Signals

CT

DT

Symmetrical at vertical axis

Double symmetrical operations:1. First at horizontal (or vertical)2. Then at vertical (or horizontal)

𝑥 −𝑡 = 𝑥(𝑡)𝑥 −𝑡 = −𝑥(𝑡)

𝑥[−𝑛] = 𝑥[𝑛] 𝑥 −𝑛 = −𝑥[𝑛]

14

A. Continuous-Time Signals

i. Unit Step Function

ii. Unit Impulse Function

iii. Complex Exponential Signals

iv. Sinusoidal Signal

v. Arbitrary Signals

B. Discrete-Time Signals

i. Unit Step Sequence

ii. Unit Impulse Sequence

iii. Complex Exponential Signals

iv. Sinusoidal Signal

v. Arbitrary Signals

TYPES OF SIGNAL

15

Continuous-Time Signal

Unit Step Function

The unit step function 𝑢(𝑡) , alsoknown as the Heaviside unit

function

The shifted unit step function𝑢(𝑡 − 𝑡0)

Discrete-Time Signal

Unit Step Sequence

The unit step sequence 𝑢[𝑛],

The shifted unit step sequence 𝑢[𝑛 − 𝑘]

𝑢[𝑛] = ቊ1 𝑛 ≥ 00 𝑛 < 0

𝑢 𝑡 = ቊ1 𝑡 > 00 𝑡 < 0

𝑢 𝑡 − 𝑡0 = ቊ1 𝑡 > 𝑡00 𝑡 < 𝑡0

𝑢[𝑛 − 𝑘] = ቊ1 𝑛 ≥ 𝑘0 𝑛 < 𝑘

Classification Of Signals

16

Continuous-Time Signal

Unit Impulse Function

The unit impulse function (𝑡), also known as the Dirac delta function

The delayed delta function (𝑡 – 𝑡0)

Discrete-Time Signal

Unit Impulse Sequence

The unit impulse sequence (or unit sample) [𝑛]

The shifted unit impulse sequence [𝑛 − 𝑘]

𝛿 𝑡 = ቊ0 𝑡 ≠ 0∞ 𝑡 = 0

න−∞

∞

𝛿 𝑡 𝑑𝑡 = 1

𝛿 𝑛 = ቊ1 𝑛 = 00 𝑛 ≠ 0

𝛿 𝑛 − 𝑘 = ቊ1 𝑛 = 𝑘0 𝑛 ≠ 𝑘

𝑥 𝑛 =

𝑘=−∞

∞

𝑥 𝑘 𝛿[𝑛 − 𝑘]න−∞

∞

∅(𝑡)𝛿 𝑡 − 𝑡0 𝑑𝑡 = ∅(𝑡0)

Classification Of Signals

17

Continuous-Time Signal

Complex Exponential Signal

Euler’s Formula

Discrete-Time Signal

Complex Exponential Signal

A signal x(t) is a real signal if its value is a real number

A signal x(t) is a complex signal if its value is a complex number.

General complex signal x(t):

where 𝑥1(𝑡) and 𝑥2(𝑡) are real

signals and 𝑗 = −1t represents either a continuous or a discrete variable.

𝑥 𝑡 = 𝑥1 𝑡 + 𝑗𝑥2(𝑡)

𝑥 𝑛 = 𝑒𝑗0𝑛 = 𝑐𝑜𝑠0𝑛 + 𝑗𝑠𝑖𝑛0𝑛

Euler’s Formula

𝑥 𝑛 = 𝑒𝑗0𝑛𝑥(𝑡) = 𝑒𝑗𝜔0𝑡

𝑥(𝑡) = 𝑒𝑗𝜔0𝑡 = 𝑐𝑜𝑠𝜔0𝑡 + 𝑗𝑠𝑖𝑛𝜔0𝑡

Increasing signal

Decreasing signal

𝛼 > 1

1 > 𝛼 > 0

0 > 𝛼 > −1

𝛼 < −1

Classification Of Signals

18

Continuous-Time Signal

Sinusoidal Signal

Discrete-Time Signal

Sinusoidal Sequences Signal

where Also can expressed as

𝑥 𝑛 = 𝐴 𝑐𝑜𝑠 0𝑛 + 𝜃

𝐴 𝑐𝑜𝑠 0𝑛 + 𝜃 = 𝐴𝑅𝑒 𝑒𝑗0𝑛+𝜃

𝑥(𝑡) = 𝐴 𝑐𝑜𝑠 𝜔0𝑡 + 𝜃

A = amplitude (real)𝜔0 = radian frequency in radian

per seconds𝜃 = phase angle in radians

Fundamental period :

Fundamental frequency :

Fundamental angular frequency :

Euler’s Formula for sinusoidal signal: Real part:

Imaginary part:

𝑇0 =2𝜋

𝜔0

𝑓0 =1

𝑇0

𝜔0 = 2𝜋𝑓0

𝐴 𝑐𝑜𝑠 𝜔0𝑡 + 𝜃 = 𝐴 𝑅𝑒 𝑒𝑗𝜔0𝑡+𝜃

𝐴 𝑠𝑖𝑛 𝜔0𝑡 + 𝜃 = 𝐴 𝐼𝑚 𝑒𝑗𝜔0𝑡+𝜃

Periodic Signal

Non-Periodic Signal

Classification Of Signals

19

3Classification Of Systems

After completion this chapter, student

should be able to:

Understand the classification of the

systems

Learning Outcome

Classification Of Systems

21

Chapter 3

CLASSIFICATION OF SYSTEMS

System is a mathematical model of a physical process that relates the input

(or excitation) signal to the output (or response) signal.

Where T is a transformation (or mapping) of x into y.

Continuous-time system :

If the input and output signals x and y are continuous-time signals

Discrete-time system :

If the input and output signals are discrete-time signal or sequences

Classification Of Systems

System

T Output Signal

x(t)

System

T Output Signal

x[n]

Input Signal

y(t)

Input Signal

y[n]

𝑦 = 𝐓𝑥

System classification refers to how the system interacts with the input signal.

The interaction can be linear or non linear, time-varying or time-invariant, with

memory or memoryless and causal or non-causal.

22

Example 2

Example 1

Systems With Memory And Without Memory

A Memoryless

System

A Memory

System

The output at any

time depends on the

past or current

The output at any time

depends on only the input at

that same time.

A resistor R with the input 𝒙(𝒕) taken

as the current flow through the resistorand the voltage taken as the output𝒚(𝒕) is a potential different across theresistor. The input-output relationship

(Ohm's law) of a resistor is

𝑦 𝑡 = 𝑅𝑥(𝑡)

𝑣 𝑡 = 𝑅𝑖(𝑡)

A capacitor C with the current as the

input 𝒙(𝒕) flow across the capacitor andthe voltage as the output 𝒚(𝒕) is a

potential different through the capacitor.

𝑦 𝑡 =1

𝐶න−∞

𝑡

𝑥 𝜏 𝑑𝜏

𝑣 𝑡 =1

𝐶න−∞

∞

𝑖 𝑡 𝑑𝑡

Example 3

A discrete-time system whose input and output sequences are related by 𝑦 𝑛 =

𝑘=−∞

𝑛

𝑥[𝑘]

Classification Of Systems

23

Causal And Non-Causal Systems

Non-Causal System Causal System

The output at any time depends

on the past or current only

The output at any time

depends on future input.

i.e. the output only depends on the

input for values of 𝑡 ≤ 𝑡0

The output only exists after an

input was applied to the system

Most of the practical systems are causal

Causality is a property

that is very similar to

memory

Linear System And Non-Linear Systems

Linear System

• Output is proportional to an input

• 3 properties of linear system

i. Additivity

ii. Scaling

iii. Superposition

Classification Of Systems

24

Linear System And Non-Linear Systems

1. Additivity property

If 𝑥1 𝑡 𝑦1(𝑡)

and 𝑥2 𝑡 𝑦2(𝑡)

therefore 𝑥1 𝑡 + 𝑥2 𝑡 𝑦1 𝑡 + 𝑦2(𝑡)

2. Scaling property

If 𝑥1 𝑡 𝑦1(𝑡)

therefore 𝑘𝑥1 𝑡 𝑘𝑦1 𝑡

3. Superposition property

If 𝑥1 𝑡 𝑦1(𝑡)

and 𝑥2 𝑡 𝑦2(𝑡)

therefore 𝑘1𝑥1 𝑡 + 𝑘2𝑥2 𝑡 𝑘1𝑦1 𝑡 + 𝑘2𝑦2(𝑡)

Non-Linear System

Any system that does not satisfy properties of Linear System is

classified as a nonlinear system.

Classification Of Systems

25

Time-Invariant System And Time-Varying Systems

Time-Invariant System Time-Varying System

Shifted output signal will

produce if the input signal

had a time shifted

If 𝑥1 𝑡 𝑦1(𝑡)

therefore 𝑥 𝑡 − 𝑡0 𝑦 𝑡 − 𝑡0

Also known as Fixed Parameter System

A system which does not satisfy

either continuous-time system or

discrete-time system

Linear Time-Invariant Systems

If the system is linear and also time-invariant, then it is called a linear

time-invariant (LTI) system.

There are many well developed techniques for dealing with the response

of linear time invariant systems, such as Laplace and Fourier transforms

Its input output characteristics do not

change with time.

A system in which certain

quantities governing the system's

behavior change with time, so

that the system will respond

differently to the same input at

different times

Classification Of Systems

26

Feedback Systems

The output signal is fed back and added to the input to the system.

Stable Systems

A system is said to be bounded-input, bounded-output (BIBO) stable if

and only if every bounded input results in a bounded output, otherwise it

is said to be unstable.

Classification Of Systems

27

4Basic Signal Operations

After completion this chapter, student

should be able to:

Understand the transformation on

dependent variable

Understand the transformation on

independent variable

Apply the problem for

transformation on dependent and

independent variable

Learning Outcome

Basic Signal Operations

29

Chapter 4

BASIC SIGNAL OPERATIONS

An important concept in signal and system analysis is the transformation of a

signal. A signal either continuous time or discrete time can be manipulated

by modifying or transforming its dependent (amplitude) or independent (time)

variable. Multiple transformations can be applied to a signal in a certain

sequence to manipulate it in a particular way. The sequence of time

transformations is significant.

Types of Transformation Signals

Dependent Variable

(Amplitude)

Independent Variable

(Time)

a. Amplitude Scaling

b. Addition

c. Multiplication

d. Differentiation

e. Integration

a. Time Scaling

b. Time Inversion (Reflection)

c. Time Shifting

1. time transformation time parameter

2. amplitude transformation amplitude.

Remember !!!

affects only

affects only

Basic Signal Operations

30

Transformation on Dependent variable (amplitude)

1. Amplitude Scaling

𝑌(𝑡) = 𝑋(𝑡)

.

2. Addition

𝑌(𝑡) = 𝑋1(𝑡) + 𝑋2(𝑡)

x(t) y1(t) y2(t)

t t t

t = 0.5t = 1.5

1

-1 -1 -1

-1.5

1

0.51

1.5

00 0

(a) (b) (c)

(a) (b) (c)

<1 signal is attenuated >1 signal is amplified.

The identification is based on Vertical: Y-Axis or AMPLITUDE

The identification is based on ADDITION AMPLITUDE of two signal

Basic Signal Operations

31

3. Multiplication

𝑌(𝑡) = 𝑋1(𝑡)𝑋2(𝑡)

4. Differentiation

(a) (b) (c)

(a) (b) (c) (d)

Y 𝑡 =𝑑

𝑑𝑡𝑋(𝑡)

The identification is based on DIFFERENTIATION AMPLITUDE of signal

The identification is based on MULTIPLICATION AMPLITUDE of two signal

Basic Signal Operations

32

Transformation on Independent variable (time)

5. Integration

𝑌 𝑡 = න−∞

𝑡

𝑋(𝑡)dt

(a) (b) (c) (d)

The identification is based on INTEGRATION AMPLITUDE of signal

The periodicity of the signal is varied by modifying the horizontal axis values,

while the amplitude or the strength remains constant.

1. Time Scaling

(a) (b) (c)

𝑌(𝑡) = 𝑋(𝛽𝑡)

The identification is based on DIVIDE X-Axis with a CONSTANT

Basic Signal Operations

33

2. Time Inversion or Reflection

(a) (b)

The identification is based on MIRROR at Vertical: Y-Axis or AMPLITUDE

Y(t) = X(-t) or Y(-t) = X(t)

Basic Signal Operations

3. Time Shifting

(a) (b) (c)

X(t) Y1(t) Y2(t)

Y(t) = X(t - t0)

The identification is based on MOVING the WHOLE SIGNAL along TIME-axis by a CONSTANT

A time delay or advances the signal in time by a time interval +t0 or -t0

without changing its shape.

i. If t0 is positive, the signal of y(t) is obtained by shifting x(t) toward

the relative to the x-axis (Delay)

ii. If t0 is negative, x(t) is shifted to the left (Advances)

𝑌1(𝑡) = 𝑋(𝑡 − 3) = 𝑋(𝑡 – (+3))

𝑌2(𝑡) = 𝑋(𝑡 + 4) = 𝑋(𝑡 – (−4))

34

Basic Signal Operations

Region of Signals

Remember !!!III

III IV

shifted

shifted

shifted

shifted

35

5Question & Answers

Question & Answers

Example 1

Sketch each of the following continuous-time signals.

(a) 𝑦 𝑡 = 𝑢 𝑡 + 1 − 𝑢(𝑡)

(b) 𝑦 𝑡 = 𝑢 𝑡 + 4 − 𝑢(𝑡 − 4)

(c) 𝑦 𝑡 = 𝑡 − 1 + (𝑡 + 2)

(d) 𝑦 𝑡 = 𝑡 − 2 𝑡 − 1 + 3(𝑡 + 2)

Solution:

(a) 𝑦 𝑡 = 𝑢 𝑡 + 1 − 𝑢(𝑡)

(b) 𝑦 𝑡 = 𝑢 𝑡 + 4 − 𝑢(𝑡 − 4)

(c) 𝑦 𝑡 = 𝑡 − 1 + (𝑡 + 2)

37

(d) 𝑦 𝑡 = 𝑡 − 2 𝑡 − 1 + 3(𝑡 + 2)

Example 2

Sketch each of the following discrete-time signals.

(a) 𝑦[𝑛] = 𝑢 𝑛 + 3 − 0.5𝑢 𝑛 − 1

(b) 𝑦[𝑛] = 𝑢[𝑛 − 1] − 𝑢[𝑛 − 4]

(c) 𝑦 𝑛 = 𝑛 − 1 + [𝑛 + 2]

(d) 𝑦[𝑛] = 2𝛿 𝑛 + 4 − 𝑛 − 2 + 𝛿[𝑛 − 3]

(e) 𝑦[𝑛] = 𝛿 𝑛 − 2 𝑛 − 1 + 3[𝑛 + 2]

Solution:

(a) 𝑦[𝑛] = 𝑢 𝑛 + 3 − 0.5𝑢 𝑛 − 1

Question & Answers

38

Question & Answers

(b) 𝑦[𝑛] = 𝑢[𝑛 − 1] − 𝑢[𝑛 − 4]

(c) 𝑦 𝑛 = 𝑛 − 1 + [𝑛 + 2]

(d) 𝑦[𝑛] = 2𝛿 𝑛 + 4 − 𝑛 − 2 + 𝛿[𝑛 − 3]

(e) 𝑦[𝑛] = 𝛿 𝑛 − 2 𝑛 − 1 + 3[𝑛 + 2]

39

Sketch and label the even and odd component of the signal x(t) and x[n]

below.

Example 3

(a) (b)

(c) (d)

Solution:

(a)

(b)

Even odd Odd signal

Even odd Odd signal

Question & Answers

40

Question & Answers

(c)

(d)

Even odd Odd signal

Even odd Odd signal

Sketch and label each of the following signals for a continuous-time signal 𝑥(𝑡)

that shown in Fig. 1.

(a) 𝑥(𝑡 − 2) (b) 𝑥(𝑡 + 2)

(c) 𝑥(−𝑡) (d) 𝑥(3 − 𝑡)

(e) 𝑥(2𝑡) (f) 𝑥𝑡

2

(g) 2𝑥(𝑡)

Example 4

Fig.1

41

Question & Answers

Solution:

(a) 𝑥(𝑡 − 2) (b) 𝑥(𝑡 + 2)

(c) 𝑥(−𝑡) (d) 𝑥(3 − 𝑡)

𝑡 − 2 = 0For

𝑡 = 2

𝑡 − 2 = 4

𝑡 = 6

𝑡 + 2 = 0For

𝑡 = −2

𝑡 + 2 = 4

𝑡 = 2

−𝑡 = 0For

𝑡 = 0

−𝑡 = 4

𝑡 = −4

3 − 𝑡 = 0For

𝑡 = 3

3 − 𝑡 = 4

𝑡 = −1

For For

For For

Reflect signal Reflect signal

(e) 𝑥(2𝑡) (f) 𝑥𝑡

2

(g) 2𝑥(𝑡)

2𝑡 = 0For

𝑡 = 0

2𝑡 = 4

𝑡 = 2

𝑡

2= 0For

𝑡 = 0Not changing

at time

but only

changing on

amplitude

𝑡

2= 4

𝑡 = 8

For

For

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Example 5

Sketch and label each of the following signals for a discrete-time signal 𝑥[𝑛]that shown in Fig. 2.

(a) 𝑥[𝑛 − 2] (b) 𝑥[𝑛 + 2] (c) 𝑥[−𝑛]

(d) 𝑥[3 − 𝑛] (e) 𝑥[2𝑛] (f) 𝑥𝑛

2

(g) 2𝑥[𝑛]

Fig. 2

Solution:

(a) 𝑥[𝑛 −2] (b) 𝑥[𝑛 + 2]

(c) 𝑥[−𝑛] (d) 𝑥[3 − 𝑛]

𝑛 − 2 = 0For

𝑛 = 2

𝑛 − 2 = 4

𝑛 = 6

For

𝑛 + 2 = 0For

𝑛 = −2

𝑛 + 2 = 4

𝑛 = 2

For

−𝑛 = 0For

𝑛 = 0

−𝑛 = 4

𝑛 = −4

For

3 − 𝑛 = 0For

𝑛 = 3

3 − 𝑛 = 4

𝑛 = −1

For

Question & Answers

43

Question & Answers

(e) 𝑥[2𝑛] (f) 𝑥𝑛

2

(g) 2𝑥[𝑛]

2𝑛 = 0For

𝑛 = 0

2𝑛 = 4

𝑛 = 2

For

𝑛

2= 0

For

𝑛 = 0For 𝑛

2= 1

𝑛 = 2

Not changing

at time but

only changing

on amplitude𝑛

2= 2

𝑛 = 4

𝑛

2= 3

𝑛 = 6𝑛

2= 4

𝑛 = 8

Example 6

Fig. 3

A continuous-time signal 𝑥(𝑡) is shown in Fig. 3. Write the mathematical

equation in terms of Unit Step Function.

44

Solution:

)𝑦 = 2𝑢 𝑡 + 1 − 𝑢 𝑡 − 2𝑢 𝑡 − 1 + 𝑢(𝑡 − 2

Sketch individual signal as follows:

Mathematical equation in terms of Unit Step Function:

45

Example 7

A discrete-time signal x[n] and z[n] are shown in Fig. 4. Sketch and label the

signals of 𝑦1[𝑛] = 𝑥[𝑛] + 𝑧[𝑛], and 𝑦2[𝑛] = 𝑥[𝑛]𝑧[𝑛].

Fig. 4

Solution:

(a) 𝑦1[𝑛] = 𝑥[𝑛] + 𝑧[𝑛]

(b) 𝑦2[𝑛] = 𝑥[𝑛]𝑧[𝑛]

46

Example 8

A continuous-time signal 𝑥1(𝑡) and 𝑥2(𝑡) is shown in Fig. 5. Sketch and label

the signals of:

(a) 𝑦1(t) = 𝑥1(𝑡) + 𝑥2(𝑡)(b) 𝑦2(t) = 𝑥1(𝑡)𝑥2(𝑡)(c) 𝑦3(t) = 2𝑢 𝑡 𝑦1(𝑡)(d) 𝑦4(t) = 𝑥1(𝑡)𝛿(𝑡 − 2)

Solution:

(a) 𝑦1(t) = 𝑥1(𝑡) + 𝑥2(𝑡) (b) 𝑦2(t) = 𝑥1(𝑡)𝑥2(𝑡)

(c) 𝑦3(t) = 2𝑢 𝑡 𝑦1(𝑡) (d) 𝑦4(t) = 𝑥1(𝑡)𝛿(𝑡 − 2)

47

Fig. 5

1. Boulet, B. (2006). Fundamental Of Signals And Systems. Charles River Media.

2. Hsu, H. (2011). Schaum’s Outline: Signals And Systems, Second Edition. Mc Graw Hill.

3. Basic Signal Operations. Retrieved on 07092021 from https://www.electrical4u.com/basic-signal-operations/

References

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