INC 212 Signals and systems Lecture#1: Introduction to signals and - - PowerPoint PPT Presentation

INC 212 Signals and systems

Lecture#1: Introduction to signals and systems

• Assoc. Prof. Benjamas Panomruttanarug

benjamas.pan@kmutt.ac.th

Course details

• Textbook:
• Signals and Systems, Wiley, 2nd Edition, ISBN‐13: 978‐0471164746
• Signals and Systems, Pearson, ISBN: 978‐1‐29202‐590‐2
• Web site: http://inc.kmutt.ac.th/~yoodyui/courses/inc212/
• Assignment 10%, midterm exam 40%

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In Intr trod

• duction

tion

What is a signal?

• A signal is formally defined as a function of one or more variables that

conveys information on the nature of a physical phenomenon.

What is a system?

• A system is formally defined as an entity that manipulates one or more

signals to accomplish a function, thereby yielding new signals.

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Classific Classification tion of

• f Signals

Signals

Continuous‐time and discrete‐time signals Continuous‐time signals: x(t) Discrete‐time signals:

 

( ), 0, 1, 2, .......

s

x n x nT n    

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Classific Classification tion of

• f Signals

Signals

Periodic and nonperiodic signals (Continuous‐Time Case) Periodic signals:

( ) ( ) for all x t x t T t   Fundamental period T T  

1 f T 

Fundamental frequency: Angular frequency:

2 2 f T     

Fundamental period:

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Example of periodic and nonperiodic signals

(a) Square wave with amplitude A = 1 and period T = 0.2s. (b) Rectangular pulse of amplitude A and duration T1.

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‘square’ in Matlab

Basic Basic Oper Operations ns on

• n Signals

Signals

• Amplitude scaling:
• Addition:
• Multiplication:
• Differentiation:
• Integration:

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( ) ( ) y t cx t 

1 2

( ) ( ) ( ) y t x t x t  

1 2

( ) ( ) ( ) y t x t x t 

( ) ( ) d y t x t dt  ( ) ( )

t

y t x d  



 

Operations Performed on dependent Variables

Basi Basic Oper Operations ions on

• n Sign

Signals als

• Time scaling:

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( ) ( ) y t x at 

a >1  compressed 0 < a < 1  expanded

Operations Performed on independent Variables

‘sawtooth’ in Matlab

• Reflection:
• Time shifting:

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( ) ( ) y t x t t  

( ) ( ) y t x t  

Basi Basic Oper Operations ions on

• n Sign

Signals als

Operations Performed on independent Variables t0 > 0  shift toward right t0 < 0  shift toward left

Ex.

• Ex. 1‐5 Pre

Precedence Rule Rule fo for Con Continuous tinuous‐Tim Time Sign Signal al

Case 1: Shifting first, then scaling Case 2: Scaling first, then shifting

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El Elem emen entary Signals Signals

• 1. Exponential Signals

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( )

at

x t Be 

• 1. Decaying exponential, for which a < 0
• 2. Growing exponential, for which a > 0

El Elem emen entary Signals Signals

• 2. Sinusoidal Signals

Plot the following sinusoidal signals (a) Sinusoidal signal 5 cos( t + Φ) with phase Φ = +/2 radians. (b) Sinusoidal signal 5 sin ( t + Φ) with phase Φ = +/2 radians.

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( ) cos( ) x t A t    

2 T   

Re Relation Betw Between een Sinusoidal Sinusoidal and and Com Comple lex Exponen Exponential ial Signals Signals

• Euler’s identity:
• Complex exponential signal:

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cos sin

j

e j



   

j

B Ae  

   

j e e e e

j j j j

2 sin 2 cos

   

 

 

   

El Elem emen entary Signals Signals

• 3. Exponential Damped Sinusoidal Signals

Exponentially damped sinusoidal signal Ae at sin(t), with A = 60 and  = 6.

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( ) sin( ),

t

x t Ae t



  



  

El Elem emen entary Signals Signals

• 4. Step Function
• 5. Impulse Function

Properties of impulse function:

• 1. Even function:
• 2. Shifting property:
• 3. Time‐scaling property:

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

1, ( ) 0, t u t t   

( ) for t t   

( ) 1 t dt 

 





( ) ( ) t t    

( ) ( ) ( ) x t t t dt x t 

 

 



1 ( ) ( ), at t a a    

El Elem emen entary Signals Signals

• 6. Ramp Function

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, ( ) 0, t t r t t      

Systems

• Classification of systems

Continuous time vs. discrete time

• Stability

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

• nly if every bounded input results in a bounded output.
• Causality

A system is said to be causal if its present value of the output signal depends

• nly on the present or past values of the input signal.

A system is said to be noncausal if its output signal depends on one or more future values of the input signal.

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Consider the following systems

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1 [ ] ( [ ] [ 1] [ 2]) 3 y n x n x n x n     

Causal !

1 [ ] ( [ 1] [ ] [ 1]) 3 y n x n x n x n     

Noncausal !

[ ] [ ] [ ]

n n

y n r x n r x n   ．

With r > 1  The system is unstable.

LTI Systems

• Time Invariance

A system is said to be time invariance if a time delay or time advance

• f the input signal leads to an identical time shift in the output signal.

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• A time‐invariant system do not change with time.

y2(t) = y1(t‐t0) if H is time invariant

LTI Systems

• Linearity

A system is said to be linear in terms of the system input (excitation) x(t) and the system output (response) y(t) if it satisfies the following two properties of superposition and homogeneity:

• 1. Superposition:
• 2. Homogeneity:

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1

( ) ( ) x t x t 

1

( ) ( ) y t y t 

2

( ) ( ) x t x t 

2

( ) ( ) y t y t 

1 2

( ) ( ) ( ) x t x t x t  

1 2

( ) ( ) ( ) y t y t y t  

( ) x t ( ) y t ( ) ax t ( ) ay t