Signals and Systems Spring 2003 Lecture 2 Jacob

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 2 nd Level of the Hierarchy 5

Motor Dynamics Differential Equations 3 nd 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 n 0, • If then • x[n] y[n] x[n - n 0] y[n - n 0]. 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) 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

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Linear Time-Invariant (LTI) Systems • Focus of most of this course - Practical importance (Eg. #1 -3 earlier this lecture 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