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1、1. Some examples of systems2. System properties and examples a) Causality b) Linearity c) Time invarianceSignals and Systems 2#1. Some examples of systemsA system is a set of elements (often called components instead) and relationships which are different from relationships of the set or its element
2、s to other elements or sets.系统系统：为实现规定功能以达到某一目标而构成的相互关：为实现规定功能以达到某一目标而构成的相互关联的一个集合体或装置联的一个集合体或装置(部件部件)。 1. Some examples of systemsFor the most part, our view of systems will be from an input-output perspective:A system responds to applied input signals, and its response is described in terms of one
3、 or more output signals1. Some examples of systemsInterconnection of two systems:Series (Cascade) InterconnectionParallel InterconnectionSeries- Parallel InterconnectionBlock Diagrams1. Some examples of systemsAn important concept is that of interconnecting systems To build more complex systems by i
4、nterconnecting simpler subsystems To modify response of a systemSignal flow (Block) diagramFeedback InterconnectionI1. Some examples of systemsEx. #1: RLC circuit1. Some examples of systemsEx. #2: Mechanical systemForce Balance:1. Some examples of systemsObservation: 1). A very rich class of systems
5、 (but by no means all systems of interest to us) are described by differential and difference equations.2). Such an equation, by itself, does not completely describe the input-output behavior of a system: we need auxiliary conditions (initial conditions, boundary conditions).3). In some cases the sy
6、stem of interest has time as the natural independent variable and is causal. However, that is not always the case. 4). Very different physical systems may have very similar mathematical descriptions.1. Some examples of systems2. System properties and examples a) Causality b) Linearity c) Time invari
7、anceSignals and Systems 2#2. System properties and examplesCAUSALITY(因果因果)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. 因果系统是指当且仅当输入信号激励系统时，才会出因果系统是指当且仅当输入信号激励系统时，才会出现输出（响应）的系统。也就是
8、说，因果系统的（响现输出（响应）的系统。也就是说，因果系统的（响应）不会出现在输入信号激励系统的以前时刻。应）不会出现在输入信号激励系统的以前时刻。2. System properties and examplesAll 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 tomorrows stock price.)Caus
9、ality 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.2. System properties and examplesCAUSALITY(因果因果)Mathematically (in CT): A system x(t) y(t) is
10、causal if when x1(t) y1(t) x2(t) y2(t) and x1(t) = x2(t) for all t toThen y1(t) = y2(t) for all t to2. System properties and examplesCAUSAL OR NONCAUSAL2. System properties and examplesTIME-INVARIANCE (TI)若系统的特征行为不随时间而变，则为时不变系统。若系统的特征行为不随时间而变，则为时不变系统。2. System properties and examplesTIME-INVARIANT O
11、R TIME-VARYING ?2. System properties and examples2. System properties and examplesNOW WE CAN DEDUCE SOMETHING!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
12、the same output, i.e., y(t) = y(t + T). These are the same input!2. System properties and examplesLINEAR AND NONLINEAR SYSTEMSMany systems are nonlinear. For example: many circuit elements (e.g., diodes), dynamics of aircraft, econometric models,However, we focus exclusively on linear systems.Why?Li
13、near models represent accurate representations of behavior of many systems (e.g., linear resistors, capacitors, other examples given previously,)Can often linearize models to examine “small signal” perturbations around “operating points”Linear systems are analytically tractable, providing basis for important tools and considerable insight2. System properties and examplesLINEARITY2. System properties and examplesPROPERTIES OF LINEAR SYSTEMS2. System properties and examples2. System properties and examples2. System properties and examplesHomework1.211.31
