Signals and Systems
Spring 2003
Lecture #2
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
Outline - Systems
• How do we construct complex systems
– Using Hierarchy
– Composing simpler elements
• System Representations
– Physical, differential/difference Equations, etc.
• System Properties
– Causality, Linearity and Time-Invariance
2
Hierarchical Design
Robot Car
3
Robot Car Block Diagram
Top Level of Abstraction
4
Wheel Position Controller Block Diagram
2nd Level of the Hierarchy
5
Motor Dynamics Differential Equations
3nd Level of the Hierarchy
6
Observations
• If we “flatten” the hierarchy, the
system becomes very complex
• Human designed systems are often
created hierarchically.
• Block input/output relations provide
communication mechanisms for team
projects
7
Compositional Design
Mechanics - Sum Element Forces
8
Circuit - Sum Element Currents
9
System Representation
Differential Equation representation
– Mechanical and Electrical Systems
Dynamically Analogous
– Can reason about the system using either
physical representation.
10
Integrator-Adder-Gain Block Diagram
11
Four Representations for the same
dynamic behavior
Pick the representation that makes it
easiest to solve the problem
12
Discrete-Time Example - Blurred Mandril
Blurrer (system
model)
Blurred
Image
Original
Image
Deblurred
Image
Deblurrer System
13
Difference Equation Representation
• Difference Equation Representation of the
model of a Blurring System
• Deblurring System
•Note Typo on handouts
How do we get
?
14
Observations
• CT System representations include circuit and
mechanical analogies, differential equations, and
Integrator-Adder-Gain block diagram.
• Discrete-Time Systems can be represented by
difference equations.
• The Difference Equation representation does not
help us design the mandril deblurring
• New representations and tools for manipulating
are needed!
15
System Properties
•
•
•
Important practical/physical implications
Help us select appropriate representations
They provide us with insight and
structure that we can exploit both to
analyze and understand systems more
deeply.
16
Causal and Non-causal Systems
17
Observations on Causality
•
A system is causal if the output does not anticipate future
values of the input, i.e., if the output at any time depends
only on values of the input up to that time.
•
All real-time physical systems are causal, because time only
moves forward. Effect occurs after cause. (Imagine if you
own a noncausal system whose output depends on
tomorrow’s stock price.)
•
Causality does not apply to spatially varying signals. (We
can move both left and right, up and down.)
•
Causality does not apply to systems processing recorded
signals, e.g. taped sports games vs. live broadcast.
18
Linearity
19
Key Property of Linear Systems
• Superposition
If
Then
20
Linearity and Causality
• A linear system is causal if and only if it satisfies the
conditions of initial rest:
“Proof”
a) Suppose system is causal. Show that (*) holds.
b) Suppose (*) holds. Show that the system is causal.
21
Time-Invariance
Mathematically (in DT): A system x[n]  y[n] is TI if for
any input x[n] and any time shift n0,
•
If
then
•
x[n]  y[n]
x[n - n0]  y[n - n0] .
Similarly for CT time-invariant system,
If
then
x(t)  y(t)
x(t - to)  y(t - to) .
22
Interesting Observation
Fact: If the input to a TI System is periodic, then the output is
periodic with the same period.
“Proof”:
Suppose
and
x(t + T) = x(t)
x(t)

y(t)
Then by TI
x(t + T)  y(t + T).

These are the
same input!
23

So these must be
the same output,
i.e., y(t) = y(t + T).
Example - Multiplier
24
Multiplier Linearity
25
Multiplier – Time Varying
26
Example – Constant Addition
27
28
Linear Time-Invariant (LTI) Systems
•
Focus of most of this course
- Practical importance (Eg. #1-3 earlier this lecture
are all LTI systems.)
- The powerful analysis tools associated
with LTI systems
•
A basic fact: If we know the response of an LTI system
to some inputs, we actually know the response to many
inputs
29
Example – DT LTI System
30
Conclusions
31