SIGNALS AND SYSTEMS 402067

COURSE OBJECTIVES By the end of this course, student will be able to • understand and use math models to represent basic signals and systems • understand the relationship between time an

TON DUC THANG UNIVERSITY

FACULTY OF ELECTRICAL & ELECTRONICS

ENGINEERING

SIGNALS AND SYSTEMS

402067

Syllabus

ACKNOWLEDGEMENT

The picture content of this slide is from Charles L Phillips, [2014],Signals, Systems, and Transforms, 5e Pearson

COURSE OBJECTIVES

By the end of this course, student will be able to

• understand and use math models to represent basic signals and systems

• understand the relationship between time and frequency domain of basic systems’ math model

• transform signals and system models from time domain to frequency domain and vice versa

• understand the relationship between time and discrete-time models of system

continuous-COURSE CONTENTS

• Introduce the mathematical tools for analysis signals and systems

• Provide a basis for applying the above techniques

in control and communication engineering

COURSE CONTENTS

This course will cover the following topics:

• Continuous time signals and systems

• Fourier transform and its applications

• Linear time invariant systems

• Discrete time signals and systems

Simulation software: MATLAB

REFERENCES

Textbook:

Charles L Phillips

Signals, Systems, and Transforms

5e Pearson Prentice Hall

Other references please refer to the detail syllabus found in the library

PREREQUISITE

ENGINEERING ANALYSIS - 402064

GRADES

• 10%: writing test

• 20%: homework, quizzes, project

• 20%: midterm exam (writing test)

• 50%: final exam

(50% writing test + 50% multiple choices)

MOODLE

 ELEARNING  COURSE Course key: piytg2ie

CLASS POLICY

• MISSING 20% including classes, homework, quizzes, test will be banned from the final exam

TON DUC THANG UNIVERSITY

FACULTY OF ELECTRICAL & ELECTRONICS

ACKNOWLEDGEMENT

The picture content of this slide is from Charles L Phillips, [2014],Signals, Systems, and Transforms, 5e Pearson

OUTLINE

1 Information vs signals

2 Mathematical representation of signals

3 Continuous time signals vs discrete time signals

4 Energy signals vs power signals

5 Linear systems

OBJECTIVES

In this chapter, you will learn:

• the different between information and signals

• how to mathematical represent signals

• how to distinguish between continuous/discrete time signals, energy/power signals

• the definition of linear system, continuous/discrete time systems

1 WHAT IS A SIGNAL?

"A detectable physical quantity or impulse (as a voltage, current, or magnetic field strength) by which

messages or information can be transmitted."

"A signal is a function of independent variables that

carry some information."

"A signal is a physical quantity that varies with time, space or any other independent variable by which

information can be conveyed“

* Definitions of signal from Merrian-Webster dictionary

1 WHAT IS A SIGNAL?

2 REPRESENTING A SIGNAL

• Using mathematical model

• Signal = function of independent variables

Function of one independent variable: time

3 CONTINUOUS vs DISCRETE

TIME SIGNALS

Continuous time signals:

• Most signals in real world

Discrete time signals:

• Some signals like pixel…

• Digital signals

(sampled signals)

4 ENERGY vs POWER SIGNALS

• Energy of a signal x(t) is 𝐸𝑥 = |𝑥 𝑡 |−∞∞ 2𝑑𝑡

Signal x(t)

Energy of x(t)

Finite energy Infinite energy

x(t) is energy signal Finite power

x(t) is power signal

ENERGY vs POWER SIGNALS

• Power of a signal x(t) is 𝑃𝑥 = lim

𝑇→∞

1 2𝑇 |𝑥 𝑡 |−𝑇𝑇 2𝑑𝑡

 For periodic signal: 𝑃𝑥 = 1

𝑇 |𝑥 𝑡 |0𝑇 2𝑑𝑡

A signal can be a power signal, or an energy signal, or neither, but not both

RECOGNIZING ENERGY vs POWER SIGNALS

Energy signals:

• Aperiodic and truncated

RECOGNIZING ENERGY vs POWER SIGNALS

Energy signals:

• Approach zero asymptotically as t  ∞

RECOGNIZING ENERGY vs POWER SIGNALS

-1

input (IP) and output (OP)



WHAT IS A LINEAR SYSTEM?

• Linear system: mathematical model of the system is based on linear operators

• Which operator is linear?

is linear if with

CONTINOUS-TIME SYSTEM

A continuous time system is one in which no sampled signals appear

CONTINOUS-TIME SYSTEM

PROPERTIES

• Memory: OP y(t0) IP values other than x(t0)

• Invertibility: distinct IPs result in distinct OPs

Inverse of a system T is Ti

• Causality: OP at any time t0 IP only for t t0

 All physical systems are causal



DISCRETE-TIME SYSTEM

A discrete-time system operates on a time signal (IP signal) to produce another discrete-time signal (OP signal or response)

SUMMARY

In this chapter, you have learned:

• the fundamentals of signal and how to mathematical represent signals

• the difference between continuous/discrete time signals, energy/power signals

• the definition of linear system, continuous/discrete time systems and their properties

MATLAB TUTORIAL 1

MATLAB computing:

• Signals described in Math form

HOMEWORK

• N/A

PREP FOR NEXT TIME

[1] Chapter 2: Continuous-time signals and systems Section 2.1 to 2.7

Refer to the syllabus for more reading on references

TON DUC THANG UNIVERSITY

FACULTY OF ELECTRICAL & ELECTRONICS

ENGINEERING

SIGNALS AND SYSTEMS

402067

Continuous-Time Signals & Systems

ACKNOWLEDGEMENT

The picture content of this slide is from Charles L Phillips, [2014],Signals, Systems, and Transforms, 5e Pearson

OUTLINE

1 Signal characteristics

2 Basic signal operations

3 Common signals in engineering

4 Continuous-time systems

OBJECTIVES

In this chapter, you will learn:

• characteristics of signal

• basic operations on signal

• common signals that are used in engineering

• continuous-time systems and their properties

SIGNAL CHARACTERISTICS

INTEGRAL

• Finite duration signal 𝑥(𝑡)

• Infinite duration signal 𝑥(𝑡)

SIGNAL CHARACTERISTICS

AVERAGE

• Finite duration signal 𝑥(𝑡)

• Infinite duration signal 𝑥(𝑡)

SIGNAL CHARACTERISTICS

ENERGY

• Finite duration signal 𝑥(𝑡)

• Infinite duration signal 𝑥(𝑡)

Recall: Energy signal:

SIGNAL CHARACTERISTICS

AVERAGE POWER

• Finite duration signal 𝑥(𝑡)

• Infinite duration signal 𝑥(𝑡)

2 BASIC SIGNAL OPERATIONS

BASIC SIGNAL OPERATIONS

AMPLITUDE SCALING

𝐴𝑥(𝑡): affecting the amplitude of the signal

• 0<A<1: Vertically squeezed

• A>1: Vertically expanded

• A<0: Vertically squeezed or expanded & inverted

BASIC SIGNAL OPERATIONS

AMPLITUDE SHIFTING

𝐴 + 𝑥(𝑡): adding DC component to the signal

• A>0: Vertically move the signal up A units

• A<0: Vertically move the signal down A units

DIY: Sketch the following signals:

3 sin(1000 )

3 sin(1000 )

t t



 

BASIC SIGNAL OPERATIONS

TIME SCALING

𝑥(A𝑡): time scaling by a factor of A

• If 0<A<1: signal is wider

• A>1: signal is squeezed

• A<0: squeezed or expanded and reflected

x(0.5t) x(2t)

-2 2 -1 1 -4 4

x(t)

x(-0.5t) x(t)

BASIC SIGNAL OPERATIONS

TIME SHIFTING

𝑥(𝑡 - 𝑡0): shifting signal 𝑥(𝑡) by 𝑡0 to the right

Why to the right?

BASIC SIGNAL OPERATIONS

BASIC SIGNAL OPERATIONS

BASIC SIGNAL OPERATIONS

COMPONENT ANALYZING

• Even signal:

if for any time t, we have 𝑥(−𝑡) = 𝑥(𝑡)

(has mirror symmetry w.r.t the vertical axis)

• Odd signal:

if for any time t, we have 𝑥 −𝑡 = −𝑥(𝑡)

(has mirror symmetry w.r.t the origin)

Note: (even function) (even function) = even function

(even function) (odd function) = odd function (odd function) (odd function) = even function

BASIC SIGNAL OPERATIONS

• Follows will be some common signals used in engineering

3 COMMON SIGNALS IN

ENGINEERING

𝑥 𝑡 = 𝑎

Signal has constant value at all time

Practical example: DC voltage / current source

COMMON SIGNALS IN ENGINEERING

CONSTANT, DC SIGNAL

a

Unit impulse function is related to Unit step function

Also called Dirac delta function

Properties of unit impulse function:

COMMON SIGNALS IN ENGINEERING

CONSTANT, DC SIGNAL

Properties of unit impulse function:

COMMON SIGNALS IN ENGINEERING

CONSTANT, DC SIGNAL

COMMON SIGNALS IN ENGINEERING

SINUSOID

𝑥 𝑡 = 𝐴𝑐𝑜𝑠 𝑤0𝑡 + 𝜑 = 𝐴𝑐𝑜𝑠 2𝜋𝑓0𝑡 + 𝜑

A: amplitude or maximum value

𝑤0: mathematical frequency (rad/sec)

𝑓0: real or natural frequency (Hz)

𝜑: phase angle (rad)

T: period of the signal (sec)

AC signal is one-sided: V(t)= 𝐴𝑐𝑜𝑠 𝑤0𝑡 + 𝜑 𝑢 𝑡

𝑥 𝑡 = 𝐶𝑒𝑎𝑡

Case 1: C and a are real

a>0: growing function a<0: decaying function

C is the y-intercept

COMMON SIGNALS IN ENGINEERING

EXPONENTIAL SIGNAL

Case 2: C is complex and a is imaginary

Case 3: C and a are complex

C = A𝑒𝑗∅ and a = 𝜎0 + j𝑤0

𝑥 𝑡 = 𝐶 𝑒𝑎𝑡 = 𝐴𝑒𝑗∅𝑒(𝜎0 +𝑗𝑤0) 𝑡 = 𝐴𝑒𝜎0 𝑡 𝑒𝑗(∅+𝑤0 𝑡)

= 𝐴𝑒𝜎0 𝑡 𝑐𝑜𝑠 ∅ + 𝑤0𝑡 + 𝑗𝐴𝑒𝜎0 𝑡 𝑠𝑖𝑛(∅ + 𝑤0𝑡)

Plot of the real part:

COMMON SIGNALS IN ENGINEERING

EXPONENTIAL SIGNAL

𝜎0 < 0

𝜎0 > 0

Use rectangular function to represent a pulse:

𝑝 𝑡 = 𝐴 (𝑡 − 𝑐

𝑤 )

with c is center of the pulse

w is width of the pulse

A is height of the pulse

COMMON SIGNALS IN ENGINEERING

PULSE SIGNAL

c

w

A

𝑥 𝑡 = 𝐴 ∧ (𝑡−𝑐

𝑊/2)

where A: height of the triangle signal

c: center of the triangle signal

W: width of the base

COMMON SIGNALS IN ENGINEERING

TRIANGLE SIGNAL

c

A

4 CONTINUOUS TIME SYSTEM

Recall: A general system

( ) [ ( )]

y t  T x t

CONTINUOUS TIME SYSTEM

INTERCONNECTING SYSTEM

Block diagram elements

CONTINUOUS TIME SYSTEM

INTERCONNECTING SYSTEM

Basic connections of system

CONTINUOUS TIME SYSTEM

FEEDBACK SYSTEM

Feedback control system

CONTINUOUS TIME SYSTEM

Recall:

• Time invariant system: time shift in input signal

results in the same time shift in the output signal

Example: Test for time invariance

CONTINUOUS TIME SYSTEM

TIME INVARIANCE

y t  t  T x t  t

CONTINUOUS TIME SYSTEM

STABILITY

Recall:

• BIBO Stability: a system is stable if the OP remains bounded for any bounded IP

Def: a signal x(t) is bounded if there exists a number

M such that 𝑥 𝑡 ≤ 𝑀 for all t

SUMMARY

In this chapter, you have learned:

• the characteristics of signal

• the basic operations of signal

• the common signals in engineering

• the basic properties of continuous time systems

MATLAB TUTORIAL 2

MATLAB computing:

• Elementary signals

Pages 1-2 to 1-25

HOMEWORK

[1] page 102 to 113

2.1, 2.3, 2.4, 2.5, 2.6, 2.10, 2.11, 2.12, 2.14, 2,15, 2.16, 2.17, 2.18, 2.19, 2.20, 2.23, 2.24, 2.25, 2.26, 2.27, 2.30, 2.32, 2.34, 2.35

PREP FOR NEXT TIME

[1] Chapter 4: Fourier Series

TON DUC THANG UNIVERSITY

FACULTY OF ELECTRICAL & ELECTRONICS

ACKNOWLEDGEMENT

The picture content of this slide is from Charles L Phillips, [2014],Signals, Systems, and Transforms, 5e Pearson

OUTLINE

1 Fourier series

2 Convolution

3 Fourier transform of aperiodic signals

4 Fourier transform of periodic signals

OBJECTIVES

In this chapter, you will learn:

• how to approximate periodic functions

• Fourier series and frequency spectra

• Convolution

• Fourier transform of different types of signals

1 HISTORY OF FOURIER SERIES

To predict astronomical

events, the idea of using

trigonometric sums was

Euler studied vibrating

strings

Fourier showed that periodic signals can be represented as the integrals of sinusoids that are not all harmonically related

PERIODIC SIGNALS & FOURIER SERIES

• 𝑥(𝑡) is a periodic signal if 𝑥 𝑡 = 𝑥 𝑡 + 𝑇

Where T is the fundamental period

𝑤0 = 2𝜋𝑇 is the fundamental frequency

• Fourier series is of the form:

PERIODIC SIGNALS & FOURIER SERIES

• Fourier series is of the form:

• Forms of the Fourier series

PERIODIC SIGNALS & FOURIER SERIES

FOURIER SERIES & FREQ SPECTRA

Example: Find Fourier series of a square wave

A method of displaying frequency content of a periodic signal is plotting the Fourier coefficients C k

FOURIER SERIES & FREQ SPECTRA

Frequency content:

PROPERTIES OF FOURIER SERIES

Assume 𝑥𝑝(𝑡) ↔ *𝐶𝑛+ and 𝑦𝑝(𝑡) ↔ *𝐷𝑛+ then

• Linearity: 𝐴𝑥𝑝 𝑡 + 𝐵𝑦𝑝 𝑡 ↔ 𝐴𝐶𝑛 + 𝐵𝐷𝑛

• Multiplication: 𝑥𝑝 𝑡 𝑦𝑝 𝑡 ↔ * ∞𝑘=−∞ 𝐶𝑘𝐷𝑛−𝑘+

• Time shifting: 𝑥𝑝 𝑡 − 𝜏 ↔ *𝐶𝑛𝑒−𝑗2𝜋𝑛𝑓𝑜 𝜏+

PROPERTIES OF FOURIER SERIES

Assume 𝑥𝑝(𝑡) ↔ *𝐶𝑛+ and 𝑦𝑝(𝑡) ↔ *𝐷𝑛+ then

2 CONVOLUTION

• A function derived from two given functions by

integration

• Convolution expresses how the shape of one

function is modified by the other

• Convolution of functions 𝑥1 𝑡 and 𝑥2 𝑡 is 𝑥(𝑡)

defined as

𝑥 𝑡 ≝ 𝑥1 𝑡 ∗ 𝑥2 𝑡 = 𝑥∞ 1 𝑡 − 𝜏 𝑥2 𝜏 𝑑𝜏

−∞

CONVOLUTION

Wikipedia: Convolution Wikimedia Foundation, Inc 22 July 2015 Web 13 May 2016

<https://en.wikipedia.org/wiki/Convolution>

3 FOURIER TRANSFORM

Example:

Find the Fourier transform of 𝑥 𝑡 = 𝑒−𝑎𝑡𝑢 𝑡 with

𝑎 > 0 , the magnitude spectrum and the phase spectrum?

• Fourier transform (frequency spectrum) of 𝑥(𝑡)

• Magnitude spectrum:

• Phase spectrum:

0

1 ( )w a t j t w t

PROPERTIES OF F TRANSFORM

Assume then

• Frequency shifting:

• Multiplying by a sinusoid:

• Differentiation in frequency domain

• Differentiation in time domain

𝑥(𝑡) ↔ 𝑋(𝜔) and y(𝑡) ↔ Y(𝜔)

PROPERTIES OF F TRANSFORM

Example:

• Find the frequency spectrum of

• Find the time function of signal 𝑥(𝑡) that has

frequency spectrum as follow:

𝑥(𝑡) = 𝑡 2 𝛱(𝑡 − 1

4 )

X(w) A

PROPERTIES OF F TRANSFORM

Assume then

• Duality:

• Multiplication in time domain:

• Convolution in time domain:

𝑥 𝜏 𝑑𝜏 ↔ 1

𝑗𝜔

𝑡

𝑋(𝜔) + 𝜋𝑋(0)𝛿(𝜔)

4 FOURIER TRANSFORM OF

PERIODIC FUNCTIONS

Recall:

• Periodic function of time 𝑥(𝑡) can be represented

by its Fourier series

with

• Take Fourier transform on both sides, we have

𝑥(𝑡) = 𝐶𝑛𝑒𝑗𝑛𝜔0 𝑡

∞ 𝑛=−∞

= 2𝜋 𝐶𝑛𝛿(𝜔 − 𝑛𝜔0)

∞ 𝑛=−∞

• Periodic function can also be represented using

∞

𝛿 𝜔 − 𝑛𝜔0

4 FOURIER TRANSFORM OF

PERIODIC FUNCTIONS

Example: Find the frequency spectrum of 𝑓(𝑡)

4 FOURIER TRANSFORM OF

PERIODIC FUNCTIONS

APPLICATION OF F TRANSFORM

• Fourier transform are used in finding the frequency response of linear systems and the frequency

spectra of signals

• Finding the frequency response of linear systems

will be covered in the next chapters

ENERGY AND POWER DENSITY SPECTRA

• Energy/Power spectral density functions are used

to determine the energy/power distribution of an energy/power signal, in the frequency spectrum

• Energy density spectrum of signal 𝑥(𝑡):

Then energy of signal 𝑥(𝑡) is

+∞

−∞

• Power density spectrum of periodic signal 𝑥(𝑡):

with

Recall: spectrum of periodic signal x(t) is

Then power of signal 𝑥(𝑡) is

SUMMARY

In this chapter, you have learned:

• how to represent periodic signals using Fourier series

• Fourier transform and frequency spectrum of various types of signal

• properties of frequency spectrum of signals and how to manipulate signals in frequency domain

PREP FOR NEXT TIME

[1] Chapter 3: Continuous-Time Linear Time-Invariant Systems

Section 3.1 to 3.6

Refer to the syllabus for more reading on references

TON DUC THANG UNIVERSITY

FACULTY OF ELECTRICAL & ELECTRONICS

ACKNOWLEDGEMENT

The picture content of this slide is from Charles L Phillips, [2014],Signals, Systems, and Transforms, 5e Pearson

OUTLINE

1 Linear time-invariant (LTI) systems

2 Impulse response of LTI systems

3 Recall of Convolution

4 Frequency response of LTI systems