LTI Systems-Digital Signal Processing-Lecture Slides, Slides of Digital Signal Processing

This lecture is part of lecture series delivered by Dr Muhammad Fasih Uddin Butt for Digital Signal Processing course at COMSATS Institute of Information Technology. Its main points are: Linear, Time-Invariant, Systems, Impulse, Response, Convolution, Properties, Commutative, Cascade, Connection

Typology: Slides

2011/2012

Uploaded on 07/06/2012

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Linear Time-Invariant Systems
Quote of the Day
The longer mathematics lives the more abstract
and therefore, possibly also the more practical it
becomes.
Eric Temple Bell
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck,
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pf4
pf5
pf8
pf9
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Linear Time-Invariant Systems

Quote of the Day The longer mathematics lives the more abstract – and therefore, possibly also the more practical – it becomes.

Eric Temple Bell

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck,

Linear-Time Invariant System

  • Special importance for their mathematical tractability
  • Most signal processing applications involve LTI systems
  • LTI system can be completely characterized by their impulse

response

  • Represent any input
  • From time invariance we arrive at convolution

[n-k] T{.} hk[n]

  (^)     



k

xn xk n k

  (^)      (^)    ^  ^     







k

k k k

yn T xk n k xk T n k xkh n

y  n x k hn k x k h k k

 (^)    



Convolution Demo

Joy of Convolution Demo from John Hopkins University

Properties of LTI Systems

  • Convolution is commutative
  • Convolution is distributive

x  k h k x  khn k h  kxn k h k x k k k

        





x  k h 1  k h 2  k   x k h 1  k  x k h 2  k

x[n] h[n] y[n] h[n] x[n] y[n]

h 1 [n]

x[n] y[n]

h 2 [n]

+ x[n] h 1 [n]+ h 2 [n] y[n]

Stable and Causal LTI Systems

  • An LTI system is (BIBO) stable if and only if
    • Impulse response is absolute summable
    • Let’s write the output of the system as
    • If the input is bounded
    • Then the output is bounded by
    • The output is bounded if the absolute sum is finite
  • An LTI system is causal if and only if

  ^  

k

h k

  (^)      (^)     





k k

yn hkxn k hk xn k

x[ n] Bx

  (^)   



k

yn Bx hk

h  k  0 fork  0

Linear Constant-Coefficient Difference Equations

  • An important class of LTI systems of the form
  • The output is not uniquely specified for a given input
    • The initial conditions are required
    • Linearity, time invariance, and causality depend on the initial conditions
    • If initial conditions are assumed to be zero system is linear, time invariant, and causal
  • Example
    • Moving Average
    • Difference Equation Representation

 ^ ^  ^   

M

k 0

k

N

k 0

ak yn k b xn k

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

a yn k b xn k where ak bk 1

3

k 0

k

0

k 0

 (^) k ^       

Eigenfunction Demo

LTI System Demo

From FernÜniversität, Hagen, Germany