Background LTI Systems (4A) Young Won Lim 4/20/15

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1 Background LTI Systems (4A)

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.

3 Convolution Integrals LTI Systems (BGND.4A) 3

4 Convolution: delayed response of h (1) Impulse response δ(t ) h t y t 1 1 t 1 delayed response by 1 y t = h t y t =? 1 1 LTI Systems (BGND.4A) 4

5 Convolution: delayed response of h (2) Time Invariant h t 1 1 Time Invariant h t Time Invariant h t LTI Systems (BGND.4A) 5

6 Convolution: delayed response of h (3) δ(t ) Linear System h t y t 1 1 δ(t v) x(v) δ(t v) h(t v) x(v) h(t v) x v h t v v 1 input value at time v x(v) delayed impulse response x(v) h(t - v) v 1 x(t ) = x (v)δ(t v) dv y t = x v h t v dv 1 1 LTI Systems (BGND.4A) 6

7 Convolution: Commutative Law x h t y(t ) LTI x(v) Flip Shift h(v) h( v) h(t v) x(v)h(t v) dv = y h(v) x(t v) dv = y Flip Shift h(v) x(v) x( v) x(t v) h x y(t ) LTI LTI Systems (BGND.4A) 7

8 Superposition of Derivatives of x LTI Systems (BGND.4A) 8

9 ODE's and Causal LTI Systems d N y (t ) d N 1 y d y(t ) a N +a d t N N 1 + +a d t N 1 d t +a 0 y (t ) = x(t ) output y h 0 x input a N d N y d t N + a N 1 d N 1 y d t N + + a 1 d y d t d M x + a 0 y = b M d t + b M M 1 d M 1 x(t ) d t M 1 d x + + b 1 + b d t 0 x output y h(t ) =? x input LTI Systems (BGND.4A) 9

10 Derivative of inputs y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = x base system x h 0 y notation: y ( N) = dn y dt N = dn dt N y(t ) x ' h 0 y ' x ' '(t ) y ' '(t ) h 0 x (3) h 0 y (3) LTI Systems (BGND.4A) 10

11 Superposition of derivatives of an input y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 x ( N ) + b 1 x ( N 1) + + b N 1 x (1) + b N x y 0 (N) + a 1 y 0 (N 1) + + a N 1 y 0 (1) + a N y 0 = b N x y 0 b N x y 1 (N) + a 1 y 1 (N 1) + + a N 1 y 1 (1) + a N y 1 = b N 1 x (1) y 1 b N 1 x (1) y N ( N) + a 1 y N ( N 1) + + a N 1 y N (1) + a N y N = b 0 x ( N ) y N b 0 x (N ) y = y 0 + y y N LTI Systems (BGND.4A) 11

12 Superposition of derivatives of an delta function h (N ) + a 1 h (N 1) + + a N 1 h (1) + a N h = b 0 δ (N ) + b 1 δ ( N 1) + + b N 1 δ (1) + b N δ h 0 (N ) + a 1 h 0 ( N 1) + + a N 1 h 0 (1) + a N h 0 = b N δ h 0 b N δ h 1 ( N ) + a 1 h 1 (N 1) + + a N 1 h 1 (1) + a N h 1 = b N 1 δ (1) h 1 b N 1 δ (1) h N ( N ) + a 1 h N (N 1) + + a N 1 h N (1) + a N h N = b 0 δ ( N ) h N b 0 δ (N ) h = h 0 + h h N LTI Systems (BGND.4A) 12

13 Base System & Derived System y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = x base system x h 0 y notation: y ( N) = dn y dt N = dn dt N y(t ) y (N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 x ( N ) + b 1 x ( N 1) + + b N 1 x (1) + b N x derived system x h(t ) y notation: x ( N ) = d N x dt N = dn dt N x(t ) LTI Systems (BGND.4A) 13

14 Superposition of derivatives of a delta function y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = x base system x h 0 y notation: y ( N) = dn y dt N = dn dt N y(t ) b N δ b N h 0 h 0 h 0 (N) + a 1 h 0 ( N 1) + + a N h 0 = b N δ b N 1 δ (1) h 0 b N 1 h 0 (1) h 0 (N+1) + a 1 h 0 ( N 1) + + a N h 0 (1) = b N 1 δ (1) b 0 δ (N ) b 0 h 0 (N ) h 0 h 0 (2 N) + a 1 h 0 (2 N 1) + + a N h 0 (N) = b 0 δ (N) LTI Systems (BGND.4A) 14

15 General System: Impulse Response y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 x ( N ) + b 1 x (N 1) + + b N 1 x (1) + b N x δ(t ) h(t ) h(t ) h = b 0 h 0 ( N ) + b 1 h 0 ( N 1) + + b N 1 h 0 (1) + b N h 0 = P( D)h 0 (t ) Q( D) = ( D N +a 1 D N 1 + +a N 1 D+a N ) P( D) = (b 0 D N +b 1 D N 1 + +b N 1 D+b N ) Q( D)h 0 = δ(t ) Q( D) P( D)h 0 = P( D)δ δ(t ) h 0 h 0 P( D)h 0 (t ) h(t ) LTI Systems (BGND.4A) 15

16 General System: Output y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 x ( N ) + b 1 x (N 1) + + b N 1 x (1) + b N x x h(t ) y y = b 0 w ( N ) + b 1 w (N 1) + + b N 1 w (1) + b N w = P( D)w (t ) Q( D) = ( D N +a 1 D N 1 + +a N 1 D+a N ) P( D) = (b 0 D N +b 1 D N 1 + +b N 1 D+b N ) x h 0 w (t ) Q( D) w = x Q(D) P( D)w = P ( D) x P( D)w (t ) y LTI Systems (BGND.4A) 16

17 ODE's and Causal LTI Systems a N d N y d t N + a N 1 d N 1 y d t N + + a 1 d y d t d M x + a 0 y = b M d t + b M M 1 d M 1 x(t ) d t M 1 d x + + b 1 + b d t 0 x N: the highest order of derivatives of the output y (LHS) M: the highest order of derivatives of the input x (RHS) N < M : (M-N) differentiator magnify high frequency components of noise (seldom used) N > M : (N-M) Integrator x h(t ) y LTI Systems (BGND.4A) 17

18 System Response (a) Zero Input Response (b) Zero State Response (c) Natural Response (d) Forced Response LTI Systems (BGND.4A) 18

19 Interval of Interest Total = ZSR + ZIR together with Laplace transform, the time interval [0, ) is assumed x y(t ) 0 < t < causal x x * h x non-causal x y(t ) No restriction in the time interval But if the same time interval [0, ) can be assumed, then the two total responses will be the same < t < LTI Systems (BGND.4A) 19

20 Responses after t = 0 x y(t ) y(t ) = t x(τ) h(t τ)d τ xu assume ( y h + y p )u ZSR t y x+ (t ) = 0 x(τ)h(t τ)d τ xu( t) assume y h ZIR 0 y + (t ) = ( x(τ)h(t τ)d τ) u(t ) xu y(t )u ZSR + ZIR Natural + Forced LTI Systems (BGND.4A) 20

21 ODE Solutions and System Responses y h = i c i e λ Characteristic Mode Terms < t < + y p (t ) x Similar form as the input x < t < + y n y p (t ) y zi y zs Natural response with all coefficients determined Forced response Zero input response Zero state response y zi + y zs (t ) y n + y p (t ) Responses after t = 0 (t > 0) to the applied input x (t 0) Causal Input Causal System Causal Output LTI Systems (BGND.4A) 21

22 ZIR & ZSR in terms of y h & y p ZIR ZSR x assume y zi = x assume y zs (t ) = y h (t ) ( y h + y p ) u(t ) causal x original x The effect of x (t ) (t < 0) is summarized as a ZIR with initial conditions at t=0- original y(t ) y zi + y zs = y(t )u causal x y zi = i c i e λ i t h(t ) x = ( b 0 δ(t ) + i u(t ) ( y h + y p ) = u(t ) ( i d i e λ it ) x(t ) k i e λ i t + y p (t ) ) LTI Systems (BGND.4A) 22

23 Types of System Responses Zero Input Response State only Natural Response Homogeneous 0 h(t ) y 0 y 0 (1) y 0 (N 1) y zi d n y d t + a d n 1 y n 1 d t + + a y = 0 n 1 n Zero State Response Input only Forced Response Particular x h(t ) y zs d n y d t + a d n 1 y n 1 d t + + a n 1 n y(t ) = b 0 d n x d t n + b 1 d n 1 x d t + + b n x LTI Systems (BGND.4A) 23

24 Comparison of System Responses (1) Zero Input Response State only Natural Response Homogeneous 0 h(t ) y 0 y 0 (1) y 0 (N 1) y zi d n y d t + a d n 1 y n 1 d t + + a n 1 n y = 0 Response of a system when the input x is zero (no input) Solution due to characteristic modes only Zero State Response Input only Forced Response Particular x h(t ) y zs d n y d t + a d n 1 y n 1 d t + + a y(t ) = n 1 n b 0 d n x d t n + b 1 d n 1 x d t + + b n x Response of a system when system is at rest initially Solution excluding the effect of characteristic modes LTI Systems (BGND.4A) 24

25 Comparison of System Responses (2) Zero Input Response State only Natural Response Homogeneous 0 h(t ) y 0 y 0 (1) y 0 (N 1) y zi d n y d t + a d n 1 y n 1 d t + + a n 1 n y = 0 response to the initial conditions only all characteristic modes response Zero State Response Input only Forced Response Particular x h(t ) y zs d n y d t + a d n 1 y n 1 d t + + a y = n 1 n b 0 d n x d t n + b 1 d n 1 x d t + + b n x response to the input only non-characteristic mode response LTI Systems (BGND.4A) 25

26 Comparison of System Responses (3) Zero Input Response State only Natural Response Homogeneous response to the initial conditions only all characteristic modes response {y ( N 1) (0 ),, y (1) (0 ), y(0 )} y n (t ) = i K i e λ i t Zero State Response Input only Forced Response Particular response to the input only non-characteristic mode response h(t ) x = ( b 0 δ(t ) + i causal x d i e λ it ) x(t ) y p (t ) LTI Systems (BGND.4A) 26

27 Forms of System Responses Zero Input Response State only Natural Response Homogeneous y zi = i c i e λ i t {y (N 1) (0 ),, y (1) (0 ), y (0 )} y n (t ) = i K i e λ i t the coefficients K i 's are determined by the initial conditions. determines the coefficients y n (t ) + y p (t ) {y (N 1) (0 + ),, y (1) (0 + ), y(0 + )} determines the coefficients Zero State Response Input only Forced Response Particular convolution form y zs (t ) = x ( i d i e λ i t + b 0 δ(t ) ) Impulse matching y p (t ) = βe ζ t or (t r + β r 1 t r β 1 t + β 0 ) step function form y zs (t ) = u ( i k i e λ i t + y p ) balancing singularities y p (t ) similar to the input, with the coefficients determined by equating the similar terms y p is similar to the input x LTI Systems (BGND.4A) 27

28 Valid Intervals of System Responses ZIR + ZSR y(t ) non-causal x x * h y(t ) 0 < t < < t < General Assumption General Assumption x = 0 (t < 0) h = 0 (t < 0) Causal Input Causal System h = 0 (t < 0) Causal System y(t ) = 0 (t < 0) Causal Output LTI Systems (BGND.4A) 28

29 Causal and Everlasting Exponential Inputs ZIR + ZSR y(t ) x * h non-causal x y(t ) 0 < t < < t < Suitable for causal exponential inputs Suitable for everlasting exponential inputs LTI Systems (BGND.4A) 29

30 The effect of an input for t < 0 ZIR + ZSR x * h causal x x = 0 (t < 0) y zs (t ) non-causal x y(t ) h = 0 (t < 0) h = 0 (t < 0) y(t ) = 0 (t < 0) y zi y(t ) The effect of x (t < 0) is summarized as a ZIR with initial conditions at t=0- LTI Systems (BGND.4A) 30

31 Interval of Validity x(t ) y t 0 h(t ) t > 0 an impulse is excluded an impulse can be present in x and h at t = 0 δ discontinuity at t = 0 creates initial conditions x (N 1) (0 ) y (N 1) (0 ) y (N 1) (0) y (N 1) (0 + ) x (N 2) (0 ) y (N 2) (0 ) y (N 2) (0) y (N 2) (0 + ) x (1) (0 ) x (0 ) The initial conditions in the term Zero State refers this initial conditions y (1) (0 ) y (0 ) y (1) (0) y (0 ) y (1) (0 + ) y (0 + ) Usually known IC Needed IC LTI Systems (BGND.4A) 31

32 Zero Input Response LTI Systems (BGND.4A) 32

33 Zero Input Response : y zi (D N +a 1 D N 1 + +a N 1 D+a N ) y = (b 0 D N +b 1 D N 1 + +b N 1 D+b N ) x(t ) 0 h(t ) y 0 y 0 (1) y 0 (N 1) y zi d 2 y d t 2 + a 1 d y d t + a 2 y = 0 x = 0 x = 0 Q( D) y zi (t ) = 0 (D N +a 1 D N 1 + +a N 1 D+a N ) y zi = 0 linear combination of {y zi and its derivatives} = 0 ce λ t only this form can be the solution of y zi Q(λ) = 0 (λ N + a 1 λ N a N 1 λ+ a N ) ce λ t = 0 = 0 0 LTI Systems (BGND.4A) 33

34 Characteristic Modes (D N +a 1 D N 1 + +a N 1 D+a N ) y = (b 0 D N +b 1 D N 1 + +b N 1 D+b N ) x(t ) Q(λ) = (λ N + a 1 λ N a N 1 λ+ a N ) = 0 Q(λ) = (λ λ 1 )(λ λ 2 ) (λ λ N ) = 0 λ i characteristic roots y zi = c 1 e λ t 1 + c 2 e λ t 2 + c N e λ Nt = i c i e λ i t e λ i t characteristic modes ZIR a linear combination of the characteristic modes of the system (1, a 1,, a N 1, a N ) the initial condition before t=0 is used {y ( N 1) (0 ),, y (1) (0 ), y(0 )} 0 x = 0 h(t ) (b 0, b 1,, b N 1, b N ) y 0 y 0 (1) y 0 (N ) y zi = i c i e λ i t {y ( N 1) (0 + ),, y (1) (0 + ), y(0 + )} any input is applied at time t=0, but in the ZIR: the initial condition does not change before and after time t=0 since no input is applied LTI Systems (BGND.4A) 34

35 Zero Input Response IVP d N y d N 1 y(t ) + a d t N 1 d t N 1 d y + +a N 1 +a d t N y(t ) = 0 (D N +a 1 D N 1 + +a N 1 D+a N ) y (t ) = 0 y (N 1) (0 ) y (1) (0 ) y (0 ) = = = y (N 1) (0) y (1) (0) y (0) = = = y (N 1) (0 + ) y (1) (0 + ) y (0 + ) = k N 1 y (N 2) (0 ) = y (N 2) (0) = y (N 2) (0 + ) = k N 2 = k 1 = k 0 ZIR Initial Value Problem (IVP) x = 0 y zi Zero Input Response h(t ) input is zero only initial conditions drives the system t=0 t=0 t=0 + y zi (0 ) = y zi (0) = y zi (0 + ) y zi (0 ) = y zi (0 ) = y zi (0) = y zi (0) = y zi (0 + ) y zi (0 + ) LTI Systems (BGND.4A) 35

36 Zero State Response LTI Systems (BGND.4A) 36

37 Zero State Response y Zero State Response + y(t ) = x (t ) h = x( τ)h(t τ ) d τ Convolution Impulse response h(t ) causal system h: response cannot begin before the input h(t τ) = 0 t τ < 0 causal input x: the input starts at t=0 x( τ) = 0 τ < 0 Causality t y(t ) = 0 x(τ)h(t τ ) d τ, t 0 LTI Systems (BGND.4A) 37

38 Delayed Impulse Response (D N +a 1 D N 1 + +a N 1 D+a N ) y = (b 0 D N +b 1 D N 1 + +b N 1 D+b N ) x(t ) All initial conditions are zero y ( N 1) (0 ) = = y (1) (0 ) = y (0) (0 ) = 0 superposition of inputs delayed impulse (1, a 1,, a N 1, a N ) the sum of delayed impulse responses x h(t ) y = h(t ) x (b 0, b 1,, b N 1, b N ) + = x(τ)h(t τ) d τ scaling delayed impulse response LTI Systems (BGND.4A) 38

39 Zero State Response IVP d N y d N 1 y(t ) +a d t N 1 d t N 1 d y d M x(t ) + +a N 1 +a d t N y(t ) = b 0 d t +b M 1 d M 1 x(t ) d t M 1 d x + +b N 1 +b d t N x(t ) (D N +a 1 D N 1 + +a N 1 D+a N ) y (t ) = (b 0 D M +b 1 D M 1 + +b N 1 D+b N ) x all initial conditions are zero y(0 ) = y (1) (0 ) = y (2) (0 ) = y ( N 2) (0 ) = y ( N 1) (0 ) = 0 ZSR Initial Value Problem (IVP) x h(t ) y * = t=0 t=0 t=0 + t=0 t=0 t=0 + * an impulse in x & h at t = 0 creates non-zero initial conditions y(0 + ) = k 0, y (1) (0 + ) = k 1, y ( N 2) (0 + ) = k N 2, y ( N 1) (0 + ) = k N 1 LTI Systems (BGND.4A) 39

40 Total Response LTI Systems (BGND.4A) 40

41 Total Response y = N c k e λ kt k =1 Zero Input Response + x h Zero State Response y = y n + y p (t ) Natural Response Forced Response Classical Approach LTI Systems (BGND.4A) 41

42 Total Response = ZIR + ZSR (1) y = N c k e λ kt k =1 + x h ZIR y zi ZSR y zs (t ) y(t ) = y zi because the input has not started yet y zi (0 ) = y zi (0 + ) y zi (0 ) = y zi (0 + ) t 0 possible discontinuity at t = 0 in general, the total response y zs (0 ) = y zs (0 + ) y zs (0 ) = y zs (0 + ) x y h(t ) LTI Systems (BGND.4A) 42

43 Total Response = ZIR + ZSR (2) initial conditions y(t ) homogeneous solution y h (t ) ZIR y zi ZSR x h no impulse y zs (t ) * ZSR x h impulse at t=0 y zs (t ) * LTI Systems (BGND.4A) 43

44 Total Response = y h + y p (1) y = N c k e λ kt k =1 Zero Input Response + x h Zero State Response x ( i u ( i d i e λ i t + b 0 δ(t ) ) k i e λ i t + y p (t ) ) convolution form step function form y = y n + y p (t ) Classical Approach Natural Response Forced Response y p is similar to the input x LTI Systems (BGND.4A) 44

45 Total Response = y h + y p (2) y = y p (t ) + y n Forced Response Natural Response x = δ y p (t ) = 0 y h (t ) x = k y p = β y h (t ) x = t u y p = β 1 t+β 0 y h (t ) x(t ) = e ζ t ζ λ i y p = βe ζ t y h (t ) LTI Systems (BGND.4A) 45

46 Total Response ZIR ZSR ZIR ZSR i i c i e λ i t k i e λ i t y p y p y zi = i c i e λ it y zs = h x(t ) Natural Response Forced Response h = b 0 δ(t ) + i d i e λ i t y p (t ) = 0 x = δ y n (t ) = i K i e λ i t y p y p = β y p = β 1 t+β 0 x = k x = t u y p = βe ζ t x = e ζ t ζ λi LTI Systems (BGND.4A) 46

47 Characteristic Mode Coefficients the initial condition after t=0 is used the same initial condition before t=0 since no input is applied N c k e λ k t k=1 ZIR + x h ZSR characteristic mode terms only c i e λ i t characteristic mode terms may exist k i e λ i t ZSR includes the particular solution y p the initial condition after t=0 is used So the effects of the char. modes of ZSR are included. y n + y p (t ) Natural Response lumped all the characteristic mode terms K i e λ i t Forced Response characteristic mode terms t r e λ i t possible LTI Systems (BGND.4A) 47

48 Total Response and Initial Conditions [, 0 ] [ 0 +, + ] zero input response + zero state response y(t ) = y zi t 0 because the input has not started yet continuous at t = 0 y(0 ) = y zi (0 ) = y zi (0 + ) ẏ(0 ) = y zi (0 ) = y zi (0 + ) y(t ) = y zi + y zs y(0 + ) y (0 ) possible discontinuity at t = 0 y(0 + ) = y zi (0 + )+ y zs (0 + ) ẏ(0 + ) = y zi (0 + )+ y zs (0 + ) natural response + forced response y h (0 ) y zi (0 ) ẏ h (0 ) y zi (0 ) y p (0 ) y zi (0 ) y p (0 ) y zi (0 ) [ 0 +, + ] y(t ) = y h (t ) + y p (t ) y(0 + ) = y zi (0 + )+ y zs (0 + ) ẏ(0 + ) = y zi (0 + )+ y zs (0 + ) y(0 + ) = y h (0 + )+ y p (0 + ) ẏ(0 + ) = y h (0 + )+ y p (0 + ) Interval of validity t > 0 LTI Systems (BGND.4A) 48

49 Requirements of an Impulse Response LTI Systems (BGND.4A) 49

50 Requirements of h at t 0 (1) d N h(t ) d N 1 h d h(t ) +a d t N 1 + +a d t N 1 N 1 d t +a N h(t ) = b 0 d M δ(t ) d t M +b 1 d M 1 δ d t M 1 d δ + +b M 1 +b d t M δ requirements at t 0 h (N ) (t ) + a 1 h (N 1) + a N h = 0 (t 0) The linear combination of all the derivatives of h must add to zero for any time t 0 all the derivatives of δ exists only t=0. It is zero for any time t 0 LTI Systems (BGND.4A) 50

51 Requirements of h at t 0 (2) d N h(t ) d N 1 h d h(t ) +a d t N 1 + +a d t N 1 N 1 d t +a N h(t ) = b 0 d M δ(t ) d t M +b 1 d M 1 δ d t M 1 d δ + +b M 1 +b d t M δ u=0 : step function the linear combination of all the derivatives of y p results to zero : homogeneous solution u = 0 y h (N ) + a 1 y h (N 1) + + a N y h = 0 for t<0, u = 0 for t>0, y h (N ) + a 1 y h (N 1) + + a N y h = 0 derivatives of {y h u} produce derivatives of δ when t =0 y h (t )u when t 0 A possible candidate of h LTI Systems (BGND.4A) 51

52 Requirements of h at t=0 (1) d N h(t ) d N 1 h d h(t ) +a d t N 1 + +a d t N 1 N 1 d t +a N h(t ) = b 0 d M δ(t ) d t M +b 1 d M 1 δ d t M 1 d δ + +b M 1 +b d t M δ requirements at t = 0 All the derivatives of h up to N must match the corresponding derivatives of an impulse δ up to M at time t=0 Need to add a δ and its derivatives in case that (N M) Need to integrate y h u several times in case that (N > M) LTI Systems (BGND.4A) 52

53 Derivatives of y h u h = y h u u (i) = δ (i 1) (t ) f (t )δ = f (0)δ(t ) h = y h u h (1) = y (1) h u + y h u (1) y h (0)δ(t ) h (2) = y (2) h u + 2 y (1) h u (1) + y h u (2) 2 y h (1) (0)δ + y h (0)δ (1) (t ) h (3) = y h (3) u + 3 y h (2) u (1) + 3 y h (1) u (2) + y h u (3) 3 y h (2) (0)δ(t ) + 3 y h (1) (0)δ (1) + y h (0) δ (2) (t ) h = y h u All the derivatives of h up to N incurs the derivatives of an impulse δ up to N-1 h (N ) (t ) = d N dt N {y h u} K 1 δ (N 1) + K 2 δ (N 2) + + K N 1 δ (1) + K N δ(t ) LTI Systems (BGND.4A) 53

54 h can have at most a δ for most systems y (t ) h(t ) x y (N) + a 1 y (N 1) + + a N 1 y (1) + a N y = b 0 x (N) + b 1 x (N 1) + + b N 1 x (1) + b N x h(t ) h(t ) δ(t ) h (N) + a 1 h ( N 1) + + a N 1 h (1) + a N h = b 0 δ ( N) + b 1 δ ( N 1) + + b N 1 δ (1) + b N δ if h contain δ C δ (N) b 0 δ (N) the highest order derivatives of δ cannot be matched if h contain δ (1) C δ (N+1) b 0 δ (N) the highest order derivatives of δ cannot be matched if h contain δ (2) C δ (N+2) b 0 δ (N) the highest order derivatives of δ t=0 h can have at most an impulse b 0 δ(t ) no derivatives of δ possible at all in most systems N M LTI Systems (BGND.4A) 54

55 ODE's and Causal LTI Systems d N h(t ) d N 1 h d h(t ) +a d t N 1 + +a d t N 1 N 1 d t +a N h(t ) = b 0 d M δ(t ) d t M +b 1 d M 1 δ d t M 1 d δ + +b M 1 +b d t M δ N > M : (N-M) integrator N < M : (M-N) differentiator magnify high frequency components of noise (seldom used) x h(t ) y N M in most systems h = h = y h (t )u y h (t )u + m 0 δ(t ) (N >M ) (N =M ) LTI Systems (BGND.4A) 55

56 An Impulse Response and System Responses LTI Systems (BGND.4A) 56

57 h as a ZIR (t > 0) d N h(t ) d N 1 h d h(t ) +a d t N 1 + +a d t N 1 N 1 d t + a N h(t ) = b 0 d M δ(t ) d t M +b 1 d M 1 δ d t M 1 d δ + +b M 1 + b d t M δ Interval of Validity : (t > 0) h(t ) = y h u(t ) The solution of the IVP with the following I.C. Zero Input (no input for t > 0 ) δ=0 for t > 0 creates nonzero i. c h (N 1) (0 + ) h (N 2) (0 + ) = k N 1 = k N 2 h (1) (0 + ) = k 1 h(0 + ) = k 0 δ an impulse may be present h No impulse is assumed y t 0 t 0 t > 0 LTI Systems (BGND.4A) 57

58 Impulse Response h δ h(t ) δ(0 ) = 0 δ(0 + ) = 0 h(t ) t=0 t=0 t=0 + t=0 t=0 t=0 + * general y cannot be a ZIR for a general input x (generally x 0 for t > 0) * impulse input vanishes (x = δ = 0 for t > 0 ) (N M ) h : ZIR with the newly created I.C. (t > 0) h (N 1) (0 + ) h (N 2) (0 + ) h (1) (0 + ) h(0 + ) = k N 1 = k N 2 = k 1 = k 0 t 0 + (t 0) t=0 h = char mode terms h = b 0 δ at most an impulse non-zero i. c. i, k i 0 t 0 h = b 0 δ + char mode terms h : ZSR to an impulse input LTI Systems (BGND.4A) 58

59 Assumed Finite Jumps δ h(t ) δ(0 ) = 0 δ(0 + ) = 0 h(t ) t=0 t=0 t=0 + t=0 + generates energy storage creates nonzero i. c. at t=0 + h (N 1) (0 initially ) = 0 h (N ) (0) = f h (N 1) (0 + ) at rest N 1 (k i, δ (i) ) h (N 2) (0 ) = 0 h (N 1) (0) = f h (N 2) (0 + N 2 (k i,δ (i) ) ) h (1) (0 ) h(0 ) = 0 = 0 h (2) (0) h (1) (0) = f 1 (k i,δ (i) ) = f 0 (k i,δ (i) ) Impulse matching h (1) (0 + ) h(0 + ) = k non-zero initial N 1 conditions = k N 2 i, k i 0 = k 1 = k 0 Assumed finite jumps LTI Systems (BGND.4A) 59

60 Impulse Matching Method Summary d N h(t ) d N 1 h d h(t ) +a d t N 1 + +a d t N 1 N 1 d t +a N h(t ) = b 0 d M δ(t ) d t M +b 1 d M 1 δ d t M 1 d δ + +b M 1 +b d t M δ h(0 ) = 0 h(0 + ) h (1) (0 ) = 0 h (1) (0 + ) h (N 2) (0 ) = 0 h (N 2) (0 + ) h (N 1) (0 ) = 0 h (N 1) (0 + ) = k 0 = k 1 = k N 2 = k N 1 h (1) h (2) h (N 1) h (N ) (t ) = f 0 (k i,δ (i) ) = f 1 (k i,δ (i) ) = f N 2 (k i,δ (i) ) = f N 1 (k i, δ (i) ) initially at rest assumed finite jumps h (N ) (0) + a 1 h ( N 1) (0) + + a N 1 h (1) (0) + a N h(0) = b 0 δ ( N ) (t ) + b 1 δ (N 1) + + b N 1 δ (1) (t ) + b N δ(t ) Impulse matching h = (c 0 e λ 0 t + c 1 e λ 1t + + c N 2 e λ N 2t + c N 1 e λ N 1 t ) u LTI Systems (BGND.4A) 60

61 References [1] [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 [3] M. J. Roberts, Fundamentals of Signals and Systems [4] S. J. Orfanidis, Introduction to Signal Processing [5] B. P. Lathi, Signals and Systems