Signals and Systems
Spring 2003
Lecture #3
Jacob White
(Slides thanks to A. Willsky, T. Weiss,
Q. Hu, and D. Boning)
“Figures and images used in these lecture notes by permission,
copyright 1997 by Alan V. Oppenheim and Alan S. Willsky”
1
Amazing Property of LTI Systems
2
Outline
• Superposition Sum for DT Systems
– Representing Inputs as sums of unit samples
– Using the Unit Sample Response
• Superposition Integral for CT System
– Use limit of tall narrow pulse
• Unit Sample/Impulse Response and Systems
– Causality, Memory, Stability
3
Representing DT Signals with Sums of Unit Samples
4
Written Analytically
Coefficients
Basic Signals
Note the Sifting Property of the Unit Sample
5
The Superposition Sum for DT Systems
Graphic View of Superposition Sum
6
Derivation of Superposition Sum
7
Convolution Sum
8
Convolution Notation
Notation is confusing, should not have [n]
takes two sequences and produces a third sequence
makes more sense
Learn to live with it.
9
Convolution Computation Mechanics
10
DT Convolution Properties
Commutative Property
11
Associative Property
12
Distributive Property
+
13
Delay Accumulation
14
Superposition Integral for CT Systems
Graphic View of Staircase Approximation
15
Tall Narrow Pulse
16
Derivation of Staircase Approximation of
Superposition Integral
17
The Superposition Integral
18
Sifting Property of Unit Impulse
19
CT Convolution Mechanics
20
CT Convolution Properties
21
Computing Unit Sample/Impulse Responses
Circuit Example
22
Narrow pulse approach
23
Narrow pulse response
24
Narrow pulse response cont’d
25
Convergence of Narrow pulse
response
26
Alternative Approach – Use
Differentiation
27
Alternative Approach – Use
Differentiation cont’d
28
How to measure Impulse Responses
29
Unit Sample/Impulse Responses of
Different Classes of Systems
30
Bounded-Input Bounded-Output Stability
31
Conclusions
32