Signals and Systems Fall 2003 Lecture #2 9 September 2003 Some examples of systems 2) System properties and examples a) Causality b) linearity c)Time invariance
Signals and Systems Fall 2003 Lecture #2 9 September 2003 1) Some examples of systems 2) System properties and examples a) Causality b) Linearity c) Time invariance
SYSTEM EXAMPLES CT System DT SyStem Ex #1 RlC circuit R L x(t) i(t) C y(t) R i(t)+ a+∥) (+)=C LC dt2+rc ay(t) 9(
SYSTEM EXAMPLES x ( t) CT System y ( t) x[n] D T System y[n] Ex. #1 RLC circuit
Ex.#2 Mechanical system K x(t)-applied force K -spring constant D -damping constant y(t)-displacement from rest y(t) Force Balance (t) 2 dt may(t)+d dt +Ky(t)=a(t) Observation: Very different physical systems may be modeled mathematically in very similar ways
Force Balance: Observation: Very different physical systems may be modeled mathematically in very similar ways. Ex. #2 Mechanical system
Ex. #3 Thermal system Cooling fin in Steady state Temperature y(t) rlt t distance along rod y(t)=Fin temperature as function of position x(t)= surrounding temperature along the fin
Ex. #3 Thermal system Cooling Fin in Steady State
Ex. #3(Continued) kla(t).(t) 9(10 0 dt Observations Independent variable can be something other than time. such as space Such systems may, more naturally, have boundary conditions rather than"initial conditions
Ex. #3 (Continued) Observations • Independent variable can be something other than time, such as space. • Such systems may, more naturally, have boundary conditions, rather than “initial” conditions
Ex.#4 Financial system Fluctuations in the price of zero-coupon bonds t=0 Time of purchase at price yo t=t Time of maturity at value yr y(t=values of bond at time t t=Influence of external factors on fluctuations in bond price dt x1(t),x2(),,xmN(t),t (①)=ym Observation: Even if the independent variable is time. there are interesting and important systems which have boundary conditions
Ex. #4 Financial system Observation: Even if the independent variable is time, there are interesting and important systems which have boundary conditions. Fluctuations in the price of zero-coupon bonds t = 0 Time of purchase at price y0 t = T Time of maturity at value yT y(t) = Values of bond at time t x(t) = Influence of external factors on fluctuations in bond price
Ex.#5 a rudimentary edge detector n+1]-2x{n]+x{n-1 econd difference,(e This system detects changes in signal slope (a)x{m]=m→9m]=0 (b)[m]=m[m] 0123 0
• A rudimentary “edge” detector • This system detects changes in signal slope Ex. #5 0 1 2 3
Observations 1)A very rich class of systems(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 system 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
Observations 1) A very rich class of systems (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 system 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
SYSTEM PROPERTIES (Causality, Linearity, Time-invariance, etc. WHY Important practical/physical implications They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply
SYSTEM PROPERTIES (Causality, Linearity, Time-invariance, etc.) • Important practical/physical implications • They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply. WHY ?
CAUSALITY A system is causal if the output does not anticipate future values of the input, 1. 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 tomorrows 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
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
