LTI Systems (1A) Young Won Lim 3/21/15

LTI Systems (1A)

Copyright (c) 214 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.

An Improper Integration F (s) = f (t)e s t dt Complex Number Real Number Real Number s = σ + i ω t Integration Variable R{s} real part I {s} imag part The improper integral converges if the limit defining it exists. LTI Systems (1A) 3

An Integration Function For a given function f(t) F (s) = f (t)e st dt Complex Number Real Number s = σ + i ω t Real Number Integration Variable s 1 s 2 s 3 F (s 1 ) = F (s 2 ) = F (s 3 ) = f (t )e s 1t f (t )e s 2 t f (t)e s 3 t dt dt dt F (s 1 ) F (s 2 ) F (s 3 ) Complex Numbers Complex Numbers LTI Systems (1A) 4

Complex Function Plot I F I R{F (s 1 )} F (s 1 ) R R I s 1 = σ 1 + iω 1 F (s 1 ) = R{F (s 1 )} + i I{F (s 1 )} R s 1 I F I I{F (s 1 )} arg {F (s 1 )} R R I s 1 s 1 = σ 1 + iω 1 F (s 1 ) = R{F (s 1 )} + i I{F (s 1 )} R LTI Systems (1A) 5

Two Functions: f(t) & F(s) For a given function f(t) there exits a unique F(s) s-domain function F(s) R{F (s 1 )} F (s 1 ) f (t) F (s) s 1 I{s} R{s} I{F (s 1 )} arg {F (s 1 )} t-domain function f(t) f (t) I{s} t R{s} s 1 Real number domain function f(t) Complex number domain function F(s) LTI Systems (1A) 6

Laplace Transform f a (x) L F A (s) 1 f 1 (t)e st dt = F 1 (s) 1 s f b (t ) L F B (s) e a t f 2 (t)e st dt = F 2 (s) 1 s+a f c (t) L F C (s) cos(k t) f 3 (t)e st dt = F 3 (s) s s 2 +k 2 f a (x) Real-valued Function F A (s) Complex Function f (t) I I t 1 s 1 = σ 1 + iω 1 f (t 1 ) ω 1 R{F (s 1 )} t 1 t σ 1 R R I{F (s 1 )} f (t 1 ) F (s 1 ) = R{F (s 1 )} + i I{F (s 1 )} LTI Systems (1A) 7

Some Laplace Transform Pairs 1 1 s e at 1 s+a t 1 s 2 sin(ωt) ω s 2 +ω 2 t 2 2 s 3 t 3 6 s 4 t n n! s n+1 cos(ωt ) sinh(ω t) cosh(ω t) s s 2 +ω 2 ω s 2 ω 2 s s 2 ω 2 LTI Systems (1A) 8

Partial Fraction Methods 1 (a x+b) + A (a x+b) + 1 (a x+b) k + A 1 (a x+b) + A 2 (a x+b) 2 + + A k (a x+b) k + 1 (a x 2 +b x+c) + A x+b (a x 2 +b x+c) + 1 (a x 2 +b x+c) k + A 1 x+ B 1 (a x 2 +b x+c) + A 2 x+b 2 (a x 2 +b x+c) 2 + + A k x+b k (a x 2 +b x+c) k + LTI Systems (1A) 9

Cover-up Method (1) 2s 1 (s+3)(s 2) = A (s+3) + B (s 2) 2s 1 (s+3)(s 2) (s+3) A = 2 s 1 (s 2) s= 3 = 6 1 3 2 = + 7 5 2s 1 (s+3)(s 2) (s 2) B = 2 s 1 (s+3) s=2 = 4 1 2+3 = +3 5 LTI Systems (1A) 1

Examples 2 s+5 (s 3) 2 = A 1 (s 3) + A 2 (s 3) 2 2 s+5 (s 3) 2 (s 3)2 = A 1 (s 3) (s 3)2 + A 2 (s 3) 2 (s 3)2 = A 1 (s 3) + A 2 2 s+5 (s 3) 2 (s 3)2 = (2s+5) s=3 = 11 = A 2 s=3 d ( 2s+5 (s 3)2) = d ds (s 3) 2 ds (A (s 3) + A ) = A 1 2 1 d ( 2 s+5 (s 3)2) ds (s 3) 2 s=3 = 2 = A 1 LTI Systems (1A) 11

Examples X (s) = P(s) (s+ p)(s+r) k = K (s+ p) + A (s+r) k + A 1 (s+r) k 1 + + A k 1 (s+r) 1 A = X (s)(s+r) k s= r A 1 = d ds [ X (s)(s+r)k ] s= r A 2 = 1 2! d 2 ds 2 [ X (s)(s+r)k ] s= r A m = 1 m! d m ds m [ X (s)(s+r)k ] s= r LTI Systems (1A) 12

Differentiation in the s-domain f (t) F (s) F(s) = f (t) e s t dt t f (t) F '(s) d d s F (s) = [ s f ] (t) e st dt = ( t) f (t) e st dt +t 2 f (t ) F ' '(s) d 2 d s 2 F (s) = 2 [f (t) e s t ] dt = s 2 ( t) 2 f (t) e s t dt t 3 f (t ) F (3) (s) d 3 d s 3 F (s) = 3 [f (t ) e s t ] dt = s 3 ( t ) 3 f (t) e st dt t n f (t) ( 1) n d n ds n F (s) LTI Systems (1A) 13

Differentiation in the t-domain (1) f (t) F (s) F(s) = f (t) e s t dt f '(t) s F (s) f () f ' (t) e s t dt = [f (t) e s t ] = f () + s ( s) f (t) e st dt f (t) e s t dt = s F (s) f () f ' '(t ) s(s F (s) f ()) f '() f (3) (t ) s(s(s F (s) f ()) f '()) f ' ' () LTI Systems (1A) 14

Differentiation in the t-domain (2) f (t ) F (s) f '(t) s F (s) f () f ' '(t ) s 2 F (s) s f () f '() f (3) (t ) s 3 F (s) s 2 f () s f '() f ' ' () f (n) (t) s n F(s) s (n 1) f () s (n 2) f '() f (n 1) () LTI Systems (1A) 15

Integration in the t-domain f (t) F (s) g(t ) G(s) t f (τ)d τ F (s) s t f (τ) d τ? f (t) F (s) f (t ) = d dt { t f (τ)d τ} = d dt g (t) g '(t) sg(s) g() g(t ) = t f (τ)d τ g() = f (τ)d τ = F (s) = F(s) s LTI Systems (1A) 16

Translation in the s-domain f (t) F (s) F(s) = f (t) e s t dt e +a t f (t ) F (s a) F(s a) = f (t) e (s a )t dt = [e +a t f (t)]e s t dt e ±at f (t) F (s a) LTI Systems (1A) 17

Translation in the t-domain f (t) F (s) F(s) = f (t) e s t dt f (t a)u(t a) f (t a)u(t a) e a s F(s) e as F (s) shift right : always o.k. shift left: only when no information is lost during improper integration by the left shift a = = a = = e a s f (t a)u(t a) e st dt f (t a)u(t a) e s t dt + a f (t a)u(t a) e s t dt f (t a) e s t dt ν = t a d ν = d t f (ν) e s (ν+a) d ν = e a s F (s) f (ν) e s ν d ν = a a LTI Systems (1A) 18

Initial Value Theorem lim t + f (t ) = lim s s F (s) F (s) = s F (s) f ( ) = f (t ) e s t dt f ' (t) e s t dt lim s s F(s) = f ( ) + lim s f ' (t ) e s t dt = f ( ) + f ( + ) f ( ) = f ( + ) LTI Systems (1A) 19

Final Value Theorem lim s lim t f (t ) = lim s F (s) = s F (s) f ( ) = lim s [ s F (s) f ( )] = lim s s F (s) s F (s) = f ( ) + lim s = f ( ) + f () f ( ) = f () f (t ) e s t dt f ' (t) e s t dt f ' (t ) e st dt f ' (t ) e st dt LTI Systems (1A) 2

Laplace Transform and ODE's y ' ' + 3 y ' + 2 y = e 3 x y() = k 1, y ' () = k 2 y (x) y ' (x) y ' '(x) Y (s) sy (s) y() s 2 Y (s) s y () y ' () e 3 x 1 (s+3) [ s 2 Y (s) s y() y ' ()] + 3 [s Y (s) y() ] + 2 [Y (s)] = 1 s+3 (s 2 + 3s + 2)Y (s) = k 1 s+k 2 +3 k 1 + 1 s+3 LTI Systems (1A) 21

Partitioning y ' ' + 3 y ' + 2 y = e 3 x y() = k 1, y ' () = k 2 [ s 2 Y (s) s y() y ' ()] + 3 [s Y (s) y() ] + 2 [Y (s)] = 1 s+3 (s 2 + 3 s + 2)Y (s) = (k 1 s+k 2 +3 k 1 ) + 1 s+3 depends only on initial conditions k 1, k 2 depends only on input e -3x LTI Systems (1A) 22

Decomposed Y(s) output input (s 2 + 3 s + 2)Y zi (s) = (k 1 s+k 2 +3 k 1 ) depends only on initial conditions k 1, k 2 No Input output input (s 2 + 3 s + 2)Y zs (s) = 1 s+3 depends only on input e -3x No State output input (s 2 + 3 s + 2)Y (s) = (k 1 s+k 2 +3 k 1 ) + 1 s+3 depends only on initial conditions k 1, k 2 depends only on input e -3x LTI Systems (1A) 23

ZIR & ZSR y zi (x) Y zi (s) = k 1 s+k 2 +3 k 1 (s+1)(s+2) Zero Input Response y zs (x) Y zs (s) = 1 (s+1)(s+2)(s+3) Zero State Response y (x) Y (s) = k 1s+k 2 +3 k 1 (s+1)(s+2) + 1 (s+1)(s+2)(s+3) y (x) Y (s) = Y zi (s) + Y zs (s) LTI Systems (1A) 24

Laplace Transform and IVP's y ' ' + 3 y ' + 2 y = y() = k 1, y ' () = k 2 ZIR IVP y zi (x) Y zi (s) = k 1 s+k 2 +3 k 1 (s+1)(s+2) y ' ' + 3 y ' + 2 y = e 3 x y() =, y ' () = ZSR IVP y zs (x) Y zs (s) = 1 (s+1)(s+2)(s+3) y ' ' + 3 y ' + 2 y = e 3 x y() = k 1, y ' () = k 2 LTI Systems (1A) 25

To include impulse inputs y ' ' + 3 y ' + 2 y = e 3 x y() = k 1, y ' () = k 2 y (x) y ' (x) y ' '(x) Y (s) sy (s) y( ) s 2 Y (s) s y ( ) y '( ) e 3 x 1 (s+3) [ s 2 Y (s) s y( ) y ' ( )] + 3 [s Y (s) y( )] + 2 [Y (s) ] = 1 s+3 LTI Systems (1A) 26

ODEs with an input g(x) y ' ' + 3 y ' + 2 y = e 3 x y() = k 1, y ' () = k 2 usually known i.c. y( ) = k 1, y ' ( ) = k 2 i.c. to be calculated y( + ) = m 1, y ' ( + ) = m 2 solution to be found y(t) (t > ) LTI Systems (1A) 27

ZIR & ZSR y ' ' + 3 y ' + 2 y = x (t ) y () = k 1, y ' () = k 2 (s 2 + 3 s + 2)Y (s) = k 1 s+k 2 +3 k 1 + X (s) s 2 Y (s) s y() y ' ()+3(s Y (s) y())+2 Y (s)=x (s) Y (s) = k 1 s+k 2 +3 k 1 (s+1)(s+2) + X (s) (s+1)(s+2) Zero Input Response x (t) = Zero State Response y () =, y ' () = Y (s) = s+5 (s+1)(s+2) Y (s) = X (s) (s+1)(s+2) x (t) = e + t 1 = +4 (s+1) 3 1 (s+2) = + 1 3 1 (s+2) 1 2 1 (s+1) + 1 6 1 (s 1) y = 4 e t 3 e 2t y = 1 2 e t + 1 3 e 2 t + 1 6 e+t LTI Systems (1A) 28

Transfer Function y ' ' + 3 y ' + 2 y = x (t ) y () = k 1, y ' () = k 2 (s 2 + 3 s + 2)Y (s) = k 1 s+k 2 +3 k 1 + X (s) s 2 Y (s) s y() y ' ()+3(s Y (s) y())+2 Y (s)=x (s) Y (s) = k 1 s+k 2 +3 k 1 (s+1)(s+2) + X (s) (s+1)(s+2) Zero State Response y () =, y ' () = Y (s) = X (s) (s+1)(s+2) Y (s) X (s) = 1 (s+1)(s+2) Transfer Function LTI Systems (1A) 29

Transfer Function & Impulse Response y ' ' + 3 y ' + 2 y = x (t ) y () = k 1, y ' () = k 2 (s 2 + 3 s + 2)Y (s) = k 1 s+k 2 +3 k 1 + X (s) s 2 Y (s) s y() y ' ()+3(s Y (s) y())+2 Y (s)=x (s) Y (s) = k 1 s+k 2 +3 k 1 (s+1)(s+2) + X (s) (s+1)(s+2) Transfer Function Y (s) X (s) = H (s) Zero State Response y () =, y ' () = Y (s) = H (s) X (s) y (t) = h(t) x (t) Y (s) = X (s) (s+1)(s+2) H (s) Transfer Function h(t) Impulse Response Y (s) X (s) = 1 (s+1)(s+2) Transfer Function LTI Systems (1A) 3

Everlasting & Causal Exponential Function exponential function e st = e σ t + i ω t s = σ + i ω sinusoid function e st = e i ω t s = i ω everlasting exponential function everlasting sinusoid function applied at t = applied at t = e st e i ωt causal exponential function causal sinusoid function applied at t = applied at t = e st u(t) e i ωt u(t) LTI Systems (1A) 31

Sinusoidal Functions and Initial Conditions everlasting sinusoid function causal sinusoid function applied at t = applied at t = e i ωt u(t) zero state (no initial conditions) non-zero state at t = + sin(ω t) sin(ω t)u (t) cos(ωt ) cos(ωt )u(t) creates zero conditions at time t = non-zero conditions at time t = + LTI Systems (1A) 32

ZSR to an everlasting exponential input input applied at t = - δ(t ) e s t zero initial conditions (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) h(t ) H (s)e s t y (t) = h(t ) e s t + = h(τ)e s(t τ) d τ + = e s t h(τ)e s τ d τ h(t ) = (t < ) = e s t H (s) + H (s) = h(τ)e s τ d τ LTI Systems (1A) 33

Forced Response to a causal exponential input input applied at t = δ(t) e ζ t u(t) (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) y y (1) y (N 1) ZIR i i natural ZSR c i e λ i t + h(t) K i e λ i t + βe ζ t forced initial conditions initial conditions at time t = at time t = + y n (t ) = y n (t ) + y p (t ) K 1 e λ 1 t + K 2 e λ 2t + K N e λ N t + {y ( N 1) ( + ),, y (1) ( + ), y( + )} = i K i K i e λ i t LTI Systems (1A) 34

Everlasting Exponential Total Response input applied at t = - δ(t ) e s t zero initial conditions (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) h(t ) H (s)e s t ZSR Laplace Transform of h(t) + H (s) = h(τ)e s τ d τ Polynomials of Differential Equation for a given s = ζ ZSR y (t) = H (ζ)e ζ t < t < + H (s) = P(s) Q(s) Transfer function (t-domain) H (s) = [ y (t ) x(t ) ]x(t )=e st y (t) = H (ζ)x (t) Y (s) = H (s) X (s) x(t ) = e ζ t Transfer function (s-domain) H (s) = Y (s) X (s) LTI Systems (1A) 35

Transfer Function & Frequency Response input applied at t = - δ(t ) e s t zero initial conditions (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) h(t ) H (s)e s t input applied at t = - δ(t ) e j ω t zero initial conditions (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) h(t ) H ( j ω)e j ω t y (t) = h(t ) e s t s = σ+ j ω y (t) = h(t ) e j ω t s = j ω + = h(τ)e s(t τ) d τ + = h(τ)e j ω(t τ) d τ + = e s t h(τ)e s τ d τ h(t ) = (t < ) + = e j ω t h(τ)e j ω τ d τ h(t ) = (t < ) = e s t H (s) = e j ω t H ( j ω) + H (s) = h(τ)e s τ d τ Transfer function + H ( j ω) = h(τ)e j ω τ d τ Frequency response LTI Systems (1A) 36

Frequency Response Laplace Transform of h(t) Fourier Transform of h(t) + H (s) = h(τ)e s τ d τ s = σ+ j ω + H ( j ω) = h(τ)e j ω τ d τ s = j ω Polynomials of Differential Equation Polynomials of Differential Equation H (s) = P(s) Q(s) s = σ+ j ω H ( j ω) = P( j ω) Q( j ω) s = j ω Transfer function (t-domain) Frequency response (t-domain) H (s) = [ y (t ) x(t ) ]x(t )=e st s = σ+ j ω H ( j ω) = [ y (t ) x(t ) ]x (t )=e jω t s = j ω Transfer function (s-domain) Frequency response (ω-domain) H (s) = Y (s) X (s) s = σ+ j ω H ( j ω) = Y ( j ω) X ( j ω) s = j ω LTI Systems (1A) 37

ZSR to everlasting sinusoidal inputs e + j ω t h(t ) H (+ j ω)e + j ω t = H (+ j ω) e + j arg{h (+ j ω)} e + j ω t e j ω t h(t ) H ( j ω)e j ω t = H ( j ω) e + j arg{h ( j ω)} e j ω t e + j ω t + e j ω t h(t ) H (+ j ω) e [+ j ω t +arg{h (+ j ω)}] [ j ω t+arg{h ( j ω)}] + H ( j ω) e = H (+ j ω) {e [+ j ω t +arg {H (+ j ω)}] + e [ j ω t arg{h (+ j ω)}] } 2 cos(ω t) h(t ) H ( j ω) 2cos(ω t+arg {H ( j ω)}) A cos(ωt +α) h(t ) A H ( j ω) cos(ωt +α+arg{h ( j ω)}) A sin (ω t+α) h(t ) A H ( j ω) sin (ω t+α+arg {H ( j ω)}) LTI Systems (1A) 38

Sinusoidal Steady State Response (1) input applied at t = ZIR ZSR δ(t) e ζ t (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) y y (1) y (N 1) i i natural c i e λ i t + h(t) K i e λ i t + βe ζt forced input applied at t = - δ(t ) e s t zero initial conditions (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) h(t ) H (s)e s t total response steady state response natural forced forced y (t) = i K i e λ i t + H (ζ)e ζ t t y ss (t) = H (ζ)e ζ t t y (t) = i K i e λ i t + H (ζ)x(t) x(t) = e ζ t y ss (t) = H (ζ) x(t ) x (t) = e ζ t Y (s) = [ i K i (s λ i ) + H (s) ] X (s) Y ss (s) = H (s) X (s) LTI Systems (1A) 39

Sinusoidal Steady State Response (2) input applied at t = ZIR ZSR δ(t) e ζ t (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) y y (1) y (N 1) i i natural c i e λ i t + h(t) K i e λ i t + βe ζt forced steady state response x(t) = A h(t ) y ss (t ) = H () A e t (N 1) y ξ = = A H () y y (1) x(t) = A e j ωt ξ = j ω h(t ) y y (1) y (N 1) y ss (t ) = H ( j ω) A e jωt = A H ( j ω) e jω x(t) = A cos(ωt ) h(t ) y y (1) y ss (t ) = H ( j ω) A cos(ωt ) (N 1) ξ = j ω y = A H ( j ω) cos(ω t) LTI Systems (1A) 4

Sinusoidal Steady State Response (3) y ss (t ) = H () A e t = A H () x(t) = A ξ = input applied at t = ZIR ZSR δ(t) e ζ t (1, a 1,, a N 1, a N ) h(t ) (b, b 1,, b N 1, b N ) y y (1) y (N 1) i i natural c i e λ i t + h(t) K i e λ i t + βe ζ t forced steady state response steady state response Forced response Forced response steady state response y ss (t ) = H ( j ω) A cos(ωt ) x(t) = A cos(ωt ) = A H ( j ω) cos(ω t) ξ = j ω LTI Systems (1A) 41

Frequency Response y ss (t ) = H ( j ω) A cos(ωt ) x(t) = A cos(ωt ) = A H ( j ω) cos(ω t) ξ = j ω ω 1 ω 1 ω 2 ω 3 ω 1 ω 2 ω 2 ω 3 ω 3 http://en.wikipedia.org/wiki/file:bode_low-pass.png LTI Systems (1A) 42

Transient Response y ss (t ) = H () A e t = A H () x(t) = A ξ = Natural + Forced response transient response http://en.wikipedia.org/wiki/file:high_accuracy_settling_time_measurements_figure_1.png http://en.wikipedia.org/wiki/file:step_response_for_two-pole_feedback_amplifier.png LTI Systems (1A) 43

Frequency Response in Control Theory (1) A 2 +B 2 cos(ωt θ) G(s) ( ) G( j ω) A 2 +B 2 cos(ω t θ+arg {G( j ω)}) A cos(ωt)+b sin(ωt ) A cos(ωt)+b sin(ωt ) = A 2 +B 2 [ A cos(ωt )+ B sin(ω A 2 + B 2 A 2 + B t)] 2 = A 2 +B 2 [cos(θ)cos(ωt)+sin(θ)sin(ωt )] = A 2 +B 2 cos(θ ωt ) = A 2 +B 2 cos(ωt θ) = A 2 + B 2 cos(ωt θ) cos(θ) = sin(θ) = A A 2 +B 2 B A 2 +B 2 LTI Systems (1A) 44

Frequency Response in Control Theory (2) A 2 +B 2 cos(ωt θ) G(s) G( j ω) A 2 +B 2 cos(ω( t θ+arg {G( j ω)}) ) A cos(ωt)+b sin(ωt ) = A 2 +B 2 cos(ωt θ) A s s 2 +ω + B ω 2 s 2 +ω 2 = A s+b ω s 2 +ω 2 Y (s) = A s+b ω s 2 +ω 2 G(s) = A s+b ω (s+ j ω)(s j ω) G(s) = K 1 s+ j ω + K 2 s j ω + F( s) Partial Fraction [ K 1 = A s+b ω [ (s+ j = A s+b ω s 2 +ω 2 (s j ω) ω)g(s)]s= j G(s) = ω ]s= j ω A jω+b ω 2 jω G ( j ω) = 1 ( A+ j B)G( j ω) 2 [ K 2 = A s+b ω [ (s j = A s+b ω s 2 +ω 2 (s+ j ω) ω)g(s)]s=+ G(s) = A j ω+b ω j ω ]s=+ j ω +2 j ω G(+ j ω) = 1 ( A j B)G(+ j ω) 2 LTI Systems (1A) 45

Frequency Response in Control Theory (3) A 2 +B 2 cos(ωt θ) G(s) G( j ω) A 2 +B 2 cos(ω( t θ+arg {G( j ω)}) ) Y (s) = K 1 s+ j ω + K 2 s j ω + F( s) K 1 = 1 2 ( A+ j B)G( j ω) K 2 = 1 ( A j B)G(+ j ω) 2 A± j B = A 2 + B 2[ A A 2 +B 2 ± j = A 2 +B 2 [cos θ ± j sin θ ] = A 2 + B 2 e ± j θ B ] A 2 +B 2 Stable System e p i e σ i lim t f (t ) = lim s system modes p i <, σ i < s F (s) = Ignore F (s) Y ss (s) = K 1 s+ j ω + K 2 s j ω K 1 = 1 2 A2 +B 2 e + jθ G( j ω) K 2 = 1 2 A 2 +B 2 e j θ G ( j ω) LTI Systems (1A) 46

Frequency Response in Control Theory (4) A 2 +B 2 cos(ωt θ) G(s) G( j ω) A 2 +B 2 cos(ω( t θ+arg {G( j ω)}) ) Y ss (s) = K 1 s+ j ω + K 2 s j ω = 1 2 ( A + j B)G( j ω) 1 s+ j ω + 1 2 ( A j B)G(+ j ω) 1 s j ω = 1 2 A 2 +B 2 e + jθ G( j ω) 1 s+ j ω + 1 A 2 +B 2 e jθ 1 G(+ j ω) 2 s j ω A 2 +B 2 y ss (t ) = [G( j ω)e jωt e + jθ 2 + G(+ j ω)e + jωt e jθ ] A 2 +B 2 = [G(+ j ω)e j(ωt θ) 2 + G(+ j ω)e + j( ωt θ) ] = A 2 +B 2 R{G(+ j ω) e + j(ωt θ) } = A 2 +B 2 G(+ j ω) cos(ωt θ+arg{g(+ j ω)}) LTI Systems (1A) 47

Impulse Response h(t) http://en.wikipedia.org/wiki/ LTI Systems (1A) 48

Impulse Response h(t) http://en.wikipedia.org/wiki/ LTI Systems (1A) 49

Impulse Response h(t) http://en.wikipedia.org/wiki/ LTI Systems (1A) 5

Impulse Response h(t) LTI Systems (1A) 51

References [1] http://en.wikipedia.org/ [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 23 [3] M. J. Roberts, Fundamentals of Signals and Systems [4] S. J. Orfanidis, Introduction to Signal Processing [5] B. P. Lathi, Signals and Systems