Signal and Systems Chapter 1: Signals and Systems Signals 1) Systems 2) Some examples of systems 3) System properties and examples 4) Causality a) Linearity b) Time invariance c) Reformatted version of open course notes from MIT opencourseware http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-003-signals-and-systems-spring-2010/lecture- notes /
Book Chapter#: Section# Signals Signals are functions of independent variables that carry information. For example: Electrical signals voltages and currents in a circuit Acoustic signals audio or speech signals(analog or digital) Video signals intensity variations in an image Biological signals sequence of bases in a gene Department of Computer Engineering, Signal and Systems 2
Book Chapter#: Section# The Independent Variables Can be continuous Trajectory of a space shuttle Mass density in a cross-section of a brain Can be discrete DNA base sequence Digital image pixels Can be 1-D, 2-D, ••• N-D For this course: Focus on a single (1-D) independent variable which we call "time” Continuous-Time (CT) signals: x(t), t continuous values integer values only Discrete-Time (DT) signals: x[n] , n Computer Engineering Department, Signal and Systems 3
Book Chapter#: Section# CT Signals Most of the signals in the physical world are CT signals — E.g. voltage & current, pressure, temperature, velocity, etc. Computer Engineering Department, Signal and Systems 4
Book Chapter#: Section# DT Signals x [ n ], n — integer, time varies discretely Examples of DT signals in nature: DNA based sequence Population of the nth generation of certain species Computer Engineering Department, Signal and Systems 5
Book Chapter#: Section# Many human-made DT Signals Ex.#2 digital image Ex.#1 Weekly Dow-Jones industrial average Why DT? — Can be processed by modern digital computers and digital signal processors (DSPs). Computer Engineering Department, Signal and Systems 6
Book Chapter#: Section# SYSTEMS For 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 or more output signals Computer Engineering Department, Signal and Systems 7
Book Chapter#: Section# EXAMPLES OF SYSTEMS An RLC circuit Dynamics of an aircraft or space vehicle An algorithm for analyzing financial and economic factors to predict bond prices An algorithm for post-flight analysis of a space launch An edge detection algorithm for medical images Computer Engineering Department, Signal and Systems 8
Book Chapter#: Section# SYSTEM INTERCONNECTIOINS An important concept is that of interconnecting systems To build more complex systems by interconnecting simpler subsystems To modify response of a system Cascade Parallel Feedback Computer Engineering Department, Signal and Systems 9
Book Chapter#: Section# SYSTEM EXAMPLES Example. #1 RLC circuit di ( t ) Ri ( t ) L y ( t ) x ( t ) dt dy ( t ) i ( t ) C dt 2 d y ( t ) dy ( t ) LC RC y ( t ) x ( t ) 2 dt dt Computer Engineering Department, Signal and Systems 10
Example. #2 Mechanical system Force Balance: 2 d y ( t ) dy ( t ) M x ( t ) Ky ( t ) D 2 dt dt 2 d y ( t ) dy ( t ) M D Ky ( t ) x ( t ) 2 dt dt Observation: Very different physical systems may be modeled mathematically in very similar ways. 11
Example. #3 Thermal system Cooling Fin in Steady State t = distance along rod y(t) = Fin temperature as function of position x(t) = Surrounding temperature along the fin 12
Example. #3 Thermal system (Continued) 2 d y ( t ) K [ y ( t ) x ( t )] 2 dt y ( T ) y 0 0 dy ( T ) 0 1 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. 13
Example. #4 Financial system Fluctuations in the price of zero-coupon bonds t = 0 Time of purchase at price y 0 t = T Time of maturity at value y t y(t) = Values of bond at time t x(t) = Influence of external factors on fluctuations in bond price 2 d y ( t ) dy ( t ) f y ( t ), , x ( t ), x ( t ),..., x ( t ), t 1 2 N 2 dt dt y ( 0 ) y , y ( T ) yT . 0 Observation: Even if the independent variable is time, there are interesting and important systems which have boundary conditions. 14
Example. #5 A Rudimentary “Edge” Detector y[n]=x[n+1]-2x[n]+x[n-1] ={x[n+1]-x[n]}-{x[n]-x[n-1]} = “Second difference” This system detects changes in signal slope x[n]=n → y[n]=0 a) x[n]=nu[n] → y[n] b) 15
Book Chapter#: Section# 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. Computer Engineering Department, Signal and Systems 16
Book Chapter#: Section# 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. Computer Engineering Department, Signal and Systems 17
Book Chapter#: Section# 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. Computer Engineering Department, Signal and Systems 18
Book Chapter#: Section# CAUSALITY (continued) Mathematically (in CT): A system x ( t ) → y ( t ) is causal iff when x 1 ( t ) → y 1 ( t ) x 2 ( t ) → y 2 ( t ) and x 1 ( t ) = x 2 ( t ) for all t ≤ t o Then y 1 ( t ) = y 2 ( t ) for all t ≤ t o Computer Engineering Department, Signal and Systems 19
Book Chapter#: Section# CAUSAL OR NONCAUSAL 𝑧(𝑢ሻ = 𝑦 2 (𝑢 − 1 ሻ depends on x(4) … causal ሻ 𝐹. . 𝑧(5 ሻ 𝑧(𝑢ሻ = 𝑦(𝑢 + 1 𝑧(5ሻ = 𝑦(6ሻ, y depends on future noncausal 𝐹. . ሿ 𝑧[𝑜ሿ = 𝑦[−𝑜 𝑧[5ሿ = 𝑦[−5 ok, but ሿ 𝐹. . 𝑧[−5ሿ = 𝑦[5 y depends on future noncausal ሿ 𝑜+1 1 𝑦 3 [𝑜 − 1 ] depends on x(4) … causal 𝑧[𝑜ሿ = 2 Computer Engineering Department, Signal and Systems 20
Book Chapter#: Section# TIME-INVARIANCE (TI) Informally, a system is time-invariant (TI) if its behavior does not depend on what time it is. Mathematically (in DT): A system x[n] → y[n] is TI iff for any input x[n] and any time shift n 0 , If x[n] → y[n] then x[n -n 0 ] → y[n -n 0 ] Similarly for a CT time-invariant system, If x(t) → y(t) then x(t – t 0 ) → y(t – t 0 ) . Computer Engineering Department, Signal and Systems 21
Book Chapter#: Section# TIME-INVARIANT OR TIME- VARYING? 𝑧(𝑢ሻ = 𝑦 2 (𝑢 + 1 is TI ሻ 𝑜+1 1 𝑦 3 [𝑜 − 1 ] is TV (NOT time-invariant) 𝑧[𝑜ሿ = 2 Computer Engineering Department, Signal and Systems 22
Book Chapter#: Section# NOW 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 These are the the same output, same input! i.e., y ( t ) = y ( t + T ). Computer Engineering Department, Signal and Systems 23
Book Chapter#: Section# LINEAR AND NONLINEAR SYSTEMS Many systems are nonlinear. For example: many circuit elements (e.g., diodes), dynamics of aircraft, econometric models,… However, in this course we focus exclusively on linear systems. Why? Linear 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 insight Computer Engineering Department, Signal and Systems 24
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