module 2b linear systems

7/23/2019 Module 2B Linear Systems

http://slidepdf.com/reader/full/module-2b-linear-systems 1/8

Dr. Clinton L. Edwards Dr. M. Lee Edwards

Discrete Linear Time Invariant(LTI) Systems (Part 2)



!

Discrete Linear Systems

( )( ) y n L x n= ! "# $

( ) ( ) ( ) ( )1 1 2 2 1 1 2 2 L a x n a x n a L x n a L x n+ = +! " ! " ! "# $ # $ # $

( )1 x n L[!] ( )1

y n

Formal Definition of Linearity:

Superposition and Scalability



( )1 1

a x n L[!]

( )1 1

a y n

Scalability

( ) ( ) ( )1 1 1 1 1 1a y n a L x n L a x n= =

! " ! "# $ # $



Superposition

( )1 x n L[!] ( )1

y n

L[!] ( )2 y n( )2

x n

IF THEN ( ) ( )1 2 x n x n+ L[!] ( ) ( )

1 2 y n y n+

( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 1 2 L x n x n L x n L x n y n y n+ = + = +! " ! " ! "# $ # $ # $


Superposition and Scalability Imply Linearity



#$%&'()* +, -.*/)&*

0

( ) ( ) ( ) 2

y n L x n x n= =! " ! "# $ # $

Non-Linear System




[ ]0 x n n! L[!] [ ]0 y n n!

Time-Invariant

A shift in the inputs results in a shift in the outputs.

[ ] y n

0[ ] y n n!0

[ ] x n n!

[ ] x n



0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t, (s)

h ( t ) , i m p u l s e r e s p o n

s e

1

Impulse Response



0 5 10 150

0.5

x

( 0 ) h ( n )

0 5 10 150

0.5

x ( 1 ) h ( n - 1 )

0 5 10 150

0.5

x ( 2 )

h ( n - 2 )

0 5 10 150

0.5

x ( 3 ) h ( n - 3 )

0 5 10 150

0.5

y ( n

)

n

!"#$"%&'"# )&*

2

( ) ( )( )k

y n x k h n k !

="!

= "#

y n( ) ! x(n)!h n( )

+,- ."/#01

x(n)=[1 2 4 1]

For n=[0 1 2 3]

module 2b linear systems

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