1 “Figures and images used in these lecture notes by permission, copyright 1997 by Alan V. Oppenheim and Alan S. Willsky” Signals and Systems Spring 2003 Lecture #2 Jacob White (Slides thanks to A. Willsky, T. Weiss, Q. Hu, and D. Boning)

2 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

3 Hierarchical Design Robot Car

4 Robot Car Block Diagram Top Level of Abstraction

5 Wheel Position Controller Block Diagram 2 nd Level of the Hierarchy

6 Motor Dynamics Differential Equations 3 nd Level of the Hierarchy

7 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

8 Compositional Design Mechanics - Sum Element Forces

9 Circuit - Sum Element Currents

10 Differential Equation representation –Mechanical and Electrical Systems Dynamically Analogous –Can reason about the system using either physical representation. System Representation

11 Integrator-Adder-Gain Block Diagram

12 Four Representations for the same dynamic behavior Pick the representation that makes it easiest to solve the problem

13 Discrete-Time Example - Blurred Mandril Blurrer (system model) Deblurrer System Original Image Blurred Image Deblurred Image

14 How do we get ? Difference Equation Representation of the model of a Blurring System Deblurring System Difference Equation Representation Note Typo on handouts

15 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!

16 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. System Properties

17 Causal and Non-causal Systems

18 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. Observations on Causality

19 Linearity

20 Superposition If Then Key Property of Linear Systems

21 a)Suppose system is causal. Show that (*) holds. b)Suppose (*) holds. Show that the system is causal. A linear system is causal if and only if it satisfies the conditions of initial rest: “Proof” Linearity and Causality

22 Mathematically (in DT): A system x[n]  y[n] is TI if for any input x[n] and any time shift n 0, Similarly for CT time-invariant system, If x[n]  y[n] then x[n - n 0 ]  y[n - n 0 ]. If x(t)  y(t) then x(t - t o )  y(t - t o ). Time-Invariance

23 These are the same input! Fact: If the input to a TI System is periodic, then the output is periodic with the same period. “Proof”: Suppose x(t + T) = x(t) and x(t)  y(t) Then by TI x(t + T)  y(t + T).   So these must be the same output, i.e., y(t) = y(t + T). Interesting Observation

24 Example - Multiplier

25 Multiplier Linearity

26 Multiplier – Time Varying

27 Example – Constant Addition

28

29 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 Linear Time-Invariant (LTI) Systems

30 Example – DT LTI System

31 Conclusions