Part 2-System Basics - Introduction to Systems Systems 2-1...

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Systems 2-1 Introduction to Systems
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Systems 2-2 Outline 2.1 Introduction 2.2 Classification of Systems 2.3 Convolution Sum for Discrete-Time LTI Systems 2.4 Convolution Integral for Continuous- Time LTI Systems 2.5 Properties and Characterizations of LTI Systems
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Systems 2-3 What is a system? System: any process that results in the transformation of a signal. A system is continuous-time if both input and output are continuous-time. A system is discrete-time if both input and output are discrete-time. x(t) y(t) Continuous-time x[n] y[n] Discrete-time
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Systems 2-4 Example 1: RC Circuit v s ( t ) + v c ( t ) 0 ) 0 ( ) ( 1 ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( c s c c c c s v t v RC t v RC dt t dv dt t dv C t i R t v t v t i Input-Output Relationship Initial Condition
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Systems 2-5 Example 2: Mass-Spring-Damper System Mass M is supported by a spring with stiffness constant K and a damper with damping constant D. An external force x(t) is applied and causes the mass to move with displacement y(t). M x ( t ) y ( t ) D K ) ( ) ( ) ( ) ( 2 2 t x t Ky dt t dy D dt t y d M
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Systems 2-6 Time-Domain Input-Output Representation Observation: A continuous-time system can be represented by an ordinary differential equation . What about a discrete-time system? Consider taking the samples of and approximate T nT v T n v dt t dv c c c ) ( ] ) 1 [( ) ( ) ( nT v c ) ( ) ( ) 1 ( ] ) 1 [( nT v RC T nT v RC T T n v s c c
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Systems 2-7 Difference Equation Observation: A discrete-time system can be represented by an difference equation . ) ( ) ( ) 1 ( ] ) 1 [( nT v RC T nT v RC T T n v s c c ) ( ) ( ) 1 ( )] 1 [( n v RC T n v RC T n v s c c
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Systems 2-8 Classification of Systems Memoryless Causality Stability Time-invariance Linearity
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Systems 2-9 (1). Memoryless A system is memoryless if the output at ANY given time depends ONLY on the input value at that (same) time. A system with memory is called a dynamic system.
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Systems 2-10 Examples 1. A resister is a memoryless component: (1) y ( t ) = R x ( t ) (2) y(t) = x(t-1), y(t) = x(t+1), y(t) = x(2t) (3) y(t) = sin[x(t)], y(t) = [x(t)]^2 (4) y(t) = 2. Inductors and capacitors are components having infinite memory. 3. The following system has finite memory: ] 2 [ ] 1 [ ] [ ] [ n x n x n x n y t d x ) (
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Systems 2-11 (3). Causality A system is causal if the output at ANY time depends ONLY on the input values at and prior to that time. Past, Present, and Future Examples Causal: Non-causal: ] 1 [ ] [ ] 1 [ 3 1 ] [ n x n x n x n y ] 1 [ ] [ ] [ n x n x n y
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Systems 2-12 Examples of Causal & Non-Causal Systems (1) y ( t ) = R x ( t ) (2) y(t) = x(t-1), y(t) = x(t+1), y(t) = x(2t) (3) y(t) = sin[x(t)], y(t) = [x(t)]^2 (4) y(t) = t d x ) ( Observation: Every memoryless system is causal.
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