Chapter 1: Classification of Signal and System

Transcription

1 Chapter 1: Classification of Signal and System Houshou Chen Dept. of Electrical Engineering, National Chung Hsing University

2 H.S. Chen Chapter1: Classification of signals and systems 1 Siganls: 1. What is a signal 2. Classification of signals 3. Basic operations on signals 4. Elementary signals Systems: 1. What is a system 2. Classification of systems 3. LTI systems: circuit example 4. More example and motivation

3 H.S. Chen Chapter1: Classification of signals and systems 2 Signals Signal: a continuous-time signal x(t) (discrete-time signal x[n]) is a function of an independent continuous variable t (discrete variable n). Elementary continuous-time signals: 1. x(t) =e s 0t,s 0 = σ 0 + jω 0 (complex exponential) 2. x(t) =e jω 0t,s 0 = jω 0 (periodic complex exponential) 3. x(t) =e σ 0t,s 0 = σ 0 (real exponential) 4. x(t) =cosω 0 t =Re{e jω 0t } (sinusoidal signals) 5. impulse function: δ(t) 6. unit function: u(t) 7. ramp function: r(t)

4 H.S. Chen Chapter1: Classification of signals and systems 3 Elementary discrete-time signals: 1. x[n] =z n 0,z 0 = r 0 e jω 0 (complex exponential) 2. x[n] =e jω 0n,z 0 = e jω 0 (periodic complex exponential) 3. x[n] =r n 0,z 0 = r 0 (real exponential) 4. x[n] =cosω 0 n =Re{e jω 0n } (sinusoidal signals) 5. impulse function: δ[n] 6. unit function: u[n] 7. ramp function: r[n] We will treat continuous-time and discrete-time signals separately but in parallel.

5 H.S. Chen Chapter1: Classification of signals and systems 4 Classification of signals 1. continuous-time x(t) vs. discrete-time x[n] Usually a discrete-time signal x[n] is obtained from a continuous time signal x(t) by sampling: x[n] =x(nt ), n =0, ±1, ±2... forsomefixedt. 2. even vs. odd signals even (real): x( t) = x(t) odd (real): x( t) = x(t) symmetric (complex): x( t) =x (t) anti-symmetric (complex): x( t) = x (t)

6 H.S. Chen Chapter1: Classification of signals and systems 5 Any signal x(t) can be decompose into the even part x e (t) and the odd part x o (t) by: x(t) = 1 2 [x(t)+x( t)] + 1 [x(t) x( t)], 2 where x e (t) = 1 2 [x(t)+x( t)] and x o(t) = 1 [x(t) x( t)] 2 It is easy to check that x e (t) =x e ( t),x o (t) = x o (t).

7 H.S. Chen Chapter1: Classification of signals and systems 6 3. periodic vs. aperiodic signals A signal x(t) (x[n]) is called a periodic signal if there exist real number T (integer N) such that: x(t + T )=x(t) (x[n + N] =x[n]). The smallest T 0 (N 0 ) such that : x(t + T 0 )=x(t) (x[n + N 0 )=x[n]) is called the (fundamental) period of x(t) (x[n]). 2π T 0 ( 2π N 0 ) is called the fundamental frequency ( rad sec )ofx(t) (x[n]). x(t) (x[n]) is called aperiodic if it is not periodic.

8 H.S. Chen Chapter1: Classification of signals and systems 7 4. deterministic vs. random deterministic signal x(t) x(t 0 ) is a number, no uncertainity random signal x(t) x(t 0 )is a random variable (with some probability specification) x(t) = random signal = random process = stochastic process 5. energy signal vs. power signal for a continuous signal x(t): E = x2 (t)dt : energy 1 P=lim T x 2 (t)dt : power = 1 T T 2 T 2 T T 2 T 2 x 2 (t)dt if x(t) is periodic with period T

9 H.S. Chen Chapter1: Classification of signals and systems 8 for a discrete signal x[n] E = n= x[n]: energy 1 P = lim n = 1 N 2N N 1 n= N x2 [n]: power N 1 n=0 x2 [n] periodic with period N x(t)(x[n]) is an energy signal if 0 <E< or is a power signal if 0 <P < A signal x(t) (x[n]) can not be an energy signal and a power signal simultaneously.

10 H.S. Chen Chapter1: Classification of signals and systems 9 Difference between x(t) and x[n] There are many similarities between x(t) andx[n], but there is one important difference. For a continuous time x(t) =e jw 0t we have: 1. e jw 1t e jw 2t if w 1 w 2, i.e., any two signals with two different frequencies are distinct. 2. w 1 >w 2 e jw 1t oscillates faster than e jw 2t. 3. e jw 0t is periodic for any w 0, T 0 = 2π w 0.

11 H.S. Chen Chapter1: Classification of signals and systems 10 The above three properties are not true for a discrete-time signal x[n] =e jω 0n. 1. For a discrete-time signal, we have x[n] =e j(ω 0+2π)n = e jω 0n e j2πn = e jω on i.e., the signal x[n] atfrequency(ω 0 +2π) is the same as that at frequency Ω 0, that is unlike the continuous case: e jw 1t e jw 2t if w 1 w 2 2. I.e., for continuous-time signal, e jw 0t are all distinct for distinct w 0. On the other hand, in discrete-time, the signal x[n] =e jω 0n = e j(ω 0+2mπ)n for any m Z. = we only need to consider a frequency interval of length 2π, usually π Ω <πor 0 Ω < 2π.

12 H.S. Chen Chapter1: Classification of signals and systems Ω 0 is larger e jω 0n oscillate faster is not true in discrete-time case In discrete-time, since we only need to consider a frequency interval of length 2π, say π Ω <πor 0 Ω < 2π. We have: frequencies close to 0, 2π are termed as low frequencies and frequencies close to π, or π are termed as high frequencies. I.e., As Ω 0, 2π, e jω 0n oscillates slower, and as Ω π, π, e jω 0n oscillates faster. cos(0n) =1,cos( πn 8 ),cos( πn 4 π, e jωn oscillates slower to faster 3πn ),cos( 4 e jωn oscillates faster to slower cos( 3πn 2 ),cos( πn 2 πn ),cos( 1 ), Ω from 0 to 8πn ),cos( 7 )cos(2πn) =1,Ωfromπ to 2π,

13 H.S. Chen Chapter1: Classification of signals and systems The period of discrete-time signal e jω 0n e jω 0(n+N) = e jω 0n e jω 0N = e jω 0n ( need e jω 0N =1) Ω 0 N =2πm Ω 0 2π = m N i.e., a discrete-time signal e jω 0n is not necessary periodic for any Ω 0. For a periodic e jω 0n,wemusthaveΩ 0 = s2π, where s Q. e j nπ 4 (Ω o = π 4 = 1 82π, N = 8) periodic e j3n (Ω o =3 m N 2π) not periodic

14 H.S. Chen Chapter1: Classification of signals and systems 13 Figure 1: ( 1) n

15 H.S. Chen Chapter1: Classification of signals and systems 14 Operations on signals operation on t axis x axis of x(t) On dependent variable x(t) i.e., given x(t), = want to find y(t) =Ax(t)+B y 1 (t) =Ax(t) scaling first y 2 (t) =y 1 (t)+b shift next y 2 (t) =y(t) =Ax(t)+B.

16 H.S. Chen Chapter1: Classification of signals and systems 15 y(t) =Ax(t)+B A > 1 expand(a < 0 reverse) A < 1 compress Remark: B>0 shift up B<0 shift down

17 H.S. Chen Chapter1: Classification of signals and systems 16 y(t) =3x(t)+4 Figure 2:

18 H.S. Chen Chapter1: Classification of signals and systems 17 If we do y 1 (t) =x(t)+b shift next y 2 (t) =Ay 1 (t) scaling first = y 2 (t) =A(x(t)+B) Conclusion: y(t) = Ax(t)+ B then A first B next y(t) = A(x(t)+ B) then B first A next

19 H.S. Chen Chapter1: Classification of signals and systems 18 On independent variable t i.e., given x(t) y(t) =x(at + b) y 1 (t) =x(t + b) shift first y 2 (t) =y 1 (at) scaling next y 2 (t) =y(t) =y 1 (at + b) Remark: a > 1 compress(a < 0 reverse) a < 1 expand b>0 shift left (advance version) b<0 shift right (delayed version)

20 H.S. Chen Chapter1: Classification of signals and systems 19 Figure 3:

21 H.S. Chen Chapter1: Classification of signals and systems 20 y 1 (t) =x(at) scaling first If we do y 2 (t) =y 1 (t + b) shift next y 2 (t) =y 1 (t + b) =x(a(t + b)) = x(at + ab) Conclusion: y(t) =x(at + b) y(t) =x(a(t + b)) b first a next a first b next

22 H.S. Chen Chapter1: Classification of signals and systems 21 Why need this: convolutional sum, integral x(t) Ax(t)+B A first,b next x(t) x(at + b) b>0 shift left b first, a next b<0 shift right or equivalent x(t) x(at b) b>0 shift right x(t) x(at b) b<0 shift left by changing variable h(t τ)x(τ)dτ t τ = λ τ = t λ dτ = dλ = h(λ)x(t λ)( dλ) = x(λ)h(t λ)(dλ) =x(t) h(t) x[n] h[n] = k= h[n k]x[k] = m= h[m]x[n m] =x[n] h[n]

23 H.S. Chen Chapter1: Classification of signals and systems 22 Recall h(τ) h(t τ) =h( τ + t)(h( τ ( t)) = h( (τ t))) 1. y(t) = h(t τ)x(τ) =h(t) x(t) 2. y[n] = k h[n k]x[k] =h[n] x[n] 3. X(D)h(D) Figure 4:

24 H.S. Chen Chapter1: Classification of signals and systems 23 Figure 5:

25 H.S. Chen Chapter1: Classification of signals and systems 24 Other Elementary signals 1. ramp function: 0 t 0 r(t) = t t 0 0 n 0 r[n] = n n 0 Figure 6:

26 H.S. Chen Chapter1: Classification of signals and systems unit function 0 t 0 u(t) = 1 t 1 step function 0 n = 1, 2,... u[n] = 1 n =0, 1,... Figure 7:

27 H.S. Chen Chapter1: Classification of signals and systems 26 Remark: Many functions x(t) can be written in term of step function. This will be very useful since we can deal with the transform of x(t) by the transform of u(t), e.g., r(t) = tu(t).

28 H.S. Chen Chapter1: Classification of signals and systems 27 u(t) u(t 1) u(t a) u(t b) Figure 8:

29 H.S. Chen Chapter1: Classification of signals and systems 28 t (u(t) u(t 1)) t (u(t) u(t 1)) + (u(t 1) u(t 2)) Figure 9:

30 H.S. Chen Chapter1: Classification of signals and systems 29 In general, if we have x(t) in the form as follows. Figure 10:

31 H.S. Chen Chapter1: Classification of signals and systems 30 We can always partition x(t) into: x(t) =g 1 (t)[u(t a 1 ) u(t a 2 )] + g 2 (t)[u(t a 2 ) u(t a 3 )] +. + g n (t)[u(t a n ) u(t a n+1 )] as follows. Figure 11:

32 H.S. Chen Chapter1: Classification of signals and systems impulse function δ(t) = δ[n] = 0 t 0 1 t = 0 impulse function 0 n 0 1 n = 0 delta function In general, δ(t) is not a function, it is a generalized function. (but δ[n] is a function).

33 H.S. Chen Chapter1: Classification of signals and systems 32 For example, δ(t) can be defined as the limit of some function. We can think of the continuous-time impulse function with the property and δ(t) = 0 (t 0) (t =0) δ(t)dt =1 In other words, continuous-time impulse δ(t) has the property: δ(t) = 0 for all t except at t = 0 and the total area under δ(t) is1.

34 H.S. Chen Chapter1: Classification of signals and systems 33 Figure 12:

35 H.S. Chen Chapter1: Classification of signals and systems 34 Properties of impulse function There are many property of δ(t) 1. sampling property: x(t) δ(t t 0 )=x(t 0 ) δ(t t 0 ) 2. sifting property: x(t)δ(t t 0)dt = x(t 0 ) b a x(t)δ(t t 0)dt = x(t 0 ) if t 0 [a, b] 0 else

36 H.S. Chen Chapter1: Classification of signals and systems 35 sampling and sifting property Figure 13:

37 H.S. Chen Chapter1: Classification of signals and systems δ(at) = 1 a δ(t) Figure 14:

38 H.S. Chen Chapter1: Classification of signals and systems δ(at + b) =δ(a(t + b a )) = 1 a δ(t + b a ) Figure 15:

39 H.S. Chen Chapter1: Classification of signals and systems 38 All of these properties can be proved by thinking δ(t) asa generalized function. From the above properties, we have x(t 0 ) = x(t)δ(t t 0)dt (by1) = x(τ)δ(τ t 0)dτ ( replace t by τ) = x(τ)δ(t 0 τ)dτ (by3) Since this is true for t 0 (, ), we can replace t 0 by t. Finally, we have x(t) =x(t) δ(t) x(τ)δ(t τ)dτ = x(t), t

40 H.S. Chen Chapter1: Classification of signals and systems 39 From this property, δ(t) (or δ[n]) is the identity of convolutional integral (convolutional sum) x(t) = x(τ)δ(t τ)dτ or = x(t τ)δ(τ)dτ (continuous-time) x[n] =Σ k= x[k]δ[n k] or = Σ k= δ[k]x[n k] (discret-time) We see that any signal x(t) (x[n]) can be written as the linear combination of δ(t) (δ[n]) and it s shift version δ(t τ) (δ[n k]), i.e., the linear integral for continuous-time, and linear sum for discrete-time.

41 H.S. Chen Chapter1: Classification of signals and systems 40 Remark: r (t) =u(t) u (t) =δ(t) t δ(τ)dτ = u(t) t u(τ)dτ = r(t)

42 H.S. Chen Chapter1: Classification of signals and systems 41 Also r[n] r[n 1] = u[n] u[n] u[n 1] = δ[n] Σ n k= δ[k] =u[n] Σ n k= u[k] =r[n]

43 H.S. Chen Chapter1: Classification of signals and systems 42 The relationship between u[n] and δ[n] From the identity of convolutional sum, we have u(t) = u(t τ)δ(τ)dτ Similarly, we have u(t) = = t δ(τ)dτ u(τ)δ(t τ)dτ = 0 δ(t τ)dτ

44 H.S. Chen Chapter1: Classification of signals and systems 43 The relationship between u[n] and δ[n] From the identity of convolutional sum, we have u[n] = k= u[n k]δ[k] = n k= δ[k] Similarly, we have u[n] = = δ[n k]u[k] k= δ[n k] k=0

45 H.S. Chen Chapter1: Classification of signals and systems 44 System A continuous-time (discrete-time) system H is an operator that transfer the input x(t) (x[n]) into the output y(t) (y[n]). We denote the process by Figure 16:

46 H.S. Chen Chapter1: Classification of signals and systems 45 Example: the RLC circuit Figure 17: How to describe the relationship between the input v i (t) andthe output v 0 (t)?

47 H.S. Chen Chapter1: Classification of signals and systems 46 Classification of system 1. linear vs. nonlinear H is called linear if H has the superposition property: H{x 1 (t)+x 2 (t)} = H{x 1 (t)} + H{x 2 (t)} H{cx(t)} = ch{x(t)} H{c 1 x 1 (t)+c 2 x 2 (t)} = c 1 H{x 1 (t)} + c 2 H{x 2 (t)} H{ n i=1 c ix i (t)} = n i=1 c ih{x i (t)}

48 H.S. Chen Chapter1: Classification of signals and systems 47 Figure 18:

49 H.S. Chen Chapter1: Classification of signals and systems time-invariant vs. time-variant H is called time-invariant if the following is true H{x(t)} = y(t) = H{x(t t 0 )} = y(t t 0 ) I.e., a time-shift to in the input x(t) results in an identical time-shift to in the output

50 H.S. Chen Chapter1: Classification of signals and systems 49 Figure 19:

51 H.S. Chen Chapter1: Classification of signals and systems memory vs. memoryless A system H is memoryless if the value y(t 0 ) (i.e.,y(t = t 0 )) only depends on the value x(t 0 ) for any t 0. example: y(t) =x 2 (t) is memoryless since y(t 0 )=x 2 (t 0 )for t 0. example: y(t) =x(t 1) is a system with memory since y(t 0 )=x(t 0 1), e.g., y(0) = x( 1). y(t 0 ) depends on x(t) at t = t 0 1, not at t 0. In other words, output y(t) at current time t = t 0 is only affected by input x(t) at current time t = t 0

52 H.S. Chen Chapter1: Classification of signals and systems causal vs. noncausal A system H is causal if the value y(t 0 ) only depends on {x(t) :t t 0 }. I.e., current output is produce by current input and past input, not future input. the system y[n] =x[n 1] is causal (y[0] = x[ 1]) the system y[n] =x[n + 1] is noncausal (y[0] = x[1]) the system y(t) =x(t + a) iscausalifa 0 and is noncausal if a>0

53 H.S. Chen Chapter1: Classification of signals and systems stable vs. nonstable H is stable if x(t) M x < t then y(t) M y < t I.e., bounded input x(t) produces bounded output y(t)

54 H.S. Chen Chapter1: Classification of signals and systems 53 We will focus on a linear time-invariant system (LTI system) H. If H is a LTI system, x(t) and y(t) are usually described by impulse response h(t) transfer function H(s) differential equation block diagram

55 H.S. Chen Chapter1: Classification of signals and systems 54 Preview and Review: t and s domain 1. t-domain: impulse response h(t) x(t) = x(τ)δ(t τ)dτ y(t) =H{x(t)} = H{ x(τ)δ(t τ)dτ} = x(τ)h{δ(t τ)}dτ = x(τ)h(t τ)dτ 2. s-domain: transfer function H(s) x(t) = X(w)ejwt dw y(t) =H{x(t)} = X(w)H{ejwt }dw = X(w)H(w)ejwt dw

56 H.S. Chen Chapter1: Classification of signals and systems 55 e st is an eigenfunction of a continuous-time LTL system y(t) = h(τ)x(t τ)dτ = h(τ)es(t τ) dτ =( h(τ)e sτ dτ)e st = H(s)e st (= H(s)x(t)) z n is an eigenfunction of a discrete-time LTL system y[n] = h[k]x[n k] = h[k]zn k =( h[k]z k )z n = H(z)z n (= H(z)x[n])

57 H.S. Chen Chapter1: Classification of signals and systems 56 Change of basis: two domains A vector x in terms of one basis {e 1,e 2,e n } x =(x 1,x 2, x n )=x 1 (100 0) + x 2 (010 0) + + x n (000 1) = x 1 e 1 + x 2 e 2 + x n e n ( e 1,e 2 e n ) The same vector x in terms of another basis {v 1,v 2 v n } x = x 1 e 1 + x 2 e 2 + x n e n = x 1v 1 + x 2v x nv n = x A vector x has two representations in terms of two bases x =(x 1,x 2,,x n )=(x 1,x 2,,x n) We can change from {e i } n i=1 to {v i} n i=1 and vice versa; if {v i} n i=1 are eigenvectors, we can simplify operation y = Ax in {e i } n i=1 domain to y = Dx in {v i } n i=1 domain.

58 H.S. Chen Chapter1: Classification of signals and systems 57 The reason is as follows. In {e i } n i=1 domain, we have y = Ax. If Av i = λv i for all i {v 1,v 2 v n } = eigenvectors with eigenvalues {λ 1,λ 2 λ n } x = x 1 e 1 + x 2 e 2 + x n e n = x 1v 1 + x 2v x nv n = x x =(x 1x 2 x n)=x 1v 1 + x 2v x nv n y = Ax = A(x 1v 1 + x 2v x nv n ) = x 1λ 1 v 1 + x 2λ 2 v x nλ n v n =y 1v 1 + y nv n where y i = λ ix i

59 H.S. Chen Chapter1: Classification of signals and systems 58 Or equivalently, A(v 1 v 2 v n )=(Av 1,Av 2, Av n )=(λ 1 v 1,λ 2 v 2, λ n v n ) =(v 1 v 2 v n ) λ AV = VD A = VDV λ n y = VDV 1 x y = Dx V 1 y = DV 1 x

60 H.S. Chen Chapter1: Classification of signals and systems 59 Motivation of LTI system Motivation I: O.D.E and Circuit signal and system A RLC circuit Figure 20:

61 H.S. Chen Chapter1: Classification of signals and systems 60 Or block diagram Figure 21:

62 H.S. Chen Chapter1: Classification of signals and systems 61 From the circuit theory, we have V R (t) =R i(t) V L (t) =L di(t) dt i(t) =C dv C(t) dt di(t) dt = C d2 V C (t) dt 2 Therefore, by KVL, we have :V C (t)+v L (t)+v R (t) =V s (t) V R (t) =R i(t) =RC dv C(t) dt V L (t) =L C d2 V C (t) dt 2

63 H.S. Chen Chapter1: Classification of signals and systems 62 Finally, we have L C d2 V C (t) dt 2 + RC dv C(t) dt + V C (t) =V s (t) V c (t)+ R L V c (t)+ 1 Lc V c(t) = 1 LC V s(t) Input signal: x(t) =V s (t) output signal: y(t) =V c (t) The differential equation describing the relationship between input x(t) & output y(t) is as follows. y (t)+ R L y (t)+ 1 LC y(t) = 1 LC x(t) This is a 2nd order constant coefficient linear ODE.

64 H.S. Chen Chapter1: Classification of signals and systems 63 A complete solution y(t) is given by: y(t) =y h (t)+y p (t) (O.D.E.) = y Z.I.R (t)+y Z.S.R. (t) (circuit) = y natural (t)+y forced (t) (circuit) In general, y(t) for t t 0 depends on both the initial state s(t 0 )and the input function x(τ), t t 0 we write: y(t) =F (s(t 0 ); x(τ),τ t 0 ) then ZIR(t) =f(s(t 0 ); 0); ZSR(t) =f(0; x(τ),τ t 0 ) 1. For a linear-system, Complete system response=zir+zsr 2. We will assume s(t 0 ) = 0 from now on and turn attention to ZIR when we discuss the Laplace Transform.

65 H.S. Chen Chapter1: Classification of signals and systems Solving 2nd oreder O.D.E.: y h (t): solving λ 2 + R L λ + 1 LC = 0 (two roots λ 1 & λ 2 ) y h (t) =c 1 e λ 1t + c 2 e λ 2t (λ 1 λ 2 distinct roots ) or y h (t) =c 1 e λ 1t + c 2 te λ 2t (λ 1 = λ 2 repeat roots ) or y n (t) =e α 1t (c 1 cos β 1 t + c 2 sin β 1 t) where λ 1 = α 1 + ıβ 1,λ 2 = λ 1 = α 1 ıβ 1

66 H.S. Chen Chapter1: Classification of signals and systems Solving 1st order differential system: states:i(t) &v C (t) V C (t) = 1 c i(t) i (t) = 1 L V L(t)= 1 L (V s(t) V C (t) V R (t)) = 1 L ( V C(t) Ri(t)+V s (t)) V C(t) i(t) x 1(t) x 2 (t) = = A 0 1 C 1 L x 1(t) x 2 (t) R L V C(t) i(t) + F (t) L V s(t)

67 H.S. Chen Chapter1: Classification of signals and systems 66 in general,we have X (t) =Ax(t)+F (t) and we have the solution of the first order differential system: X(t) =X h (t)+x p (t) where X h (t) is obtained by diagonalizing the matrix A [ A V 1 V 2 ] = A = VDV 1 [ λ 1 V 1 λ 2 V 2 ] = [ ] V 1 V 2 λ λ 2 x (t) =VDV 1 x(t) V 1 x 1 (t) }{{} Y (t) = DVx(t) }{{} Y (t)

68 H.S. Chen Chapter1: Classification of signals and systems 67 y 1 (t) =c 1 e λ 1t y 2 (t) =c 2 e λ 2t x 1(t) x 2 (t) x 1(t) x 2 (t) = = [ [ ] v 1 v 2 y 1(t) y 2 (t) v 1 (t) v 2 (t) ] c 1e λ 1t c 2 e λ 2t = c 1 e λ 1t v 1 + c 2 e λ 2t v 2

69 H.S. Chen Chapter1: Classification of signals and systems 68 Summary From the above example, we can see that there are several ways to describe the relationship between the input x(t) and the output y(t) for a LIT sytem x(t) y(t). These are: 1. Block diagram Figure 22:

70 H.S. Chen Chapter1: Classification of signals and systems differential equation (SISO system) y + R L y (t)+ 1 LC y(t) = 1 LC x(t) (λ2 + R L λ + 1 LC = 0 two roots) y(t) =y h (t)+y p (t) =y ZIR (t)+y ZSR (t) where y h (t) = c 1 e λt + c 2 e λt (λ 1 λ 2 real) c 1 e λt + c 2 te λt (λ 1 = λ 2 real) y h (t) =c 1 e α1t (cos βt +sinβt) (λ 1 = λ 2 = α + ıβ)

71 H.S. Chen Chapter1: Classification of signals and systems differential system (MIMO system) y (t) = y 1(t) = A y 1(t) + F (t) y 2 (t) y 2 (t) y(t) =y h (t)+y p (t) [ ] & y h (t) = v 1 v 2 c 1e λ 1t c 2 e λ 2t where Av 1 = λ 1 v 1, Av 2 = λ 2 v 2 (λ 1 λ 2 )

72 H.S. Chen Chapter1: Classification of signals and systems our focus time domain h(t): impulse response frequency domain H(s): transfer function We can find the transfer function H(s), or the frequency response H(jw) (H(e jω )) directly from the circuit diagram or from the differential equation (system). After that, we can get the impulse response h(t) from H(s). The idea is connecting with phasors in circuit theory.

73 H.S. Chen Chapter1: Classification of signals and systems 72 Phasors R: V R (t) =Ri(t) i(t) =e jwt V R (t) = L: V L (t) =L di(t) dt i(t) =e jwt V L (t) = C: i(t) =C dv C(t) dt V C (t) =e jwt i(t) = Z R {}}{ R e jwt Z R = R( independce) Z L {}}{ L jw e jwt Z L = jwl (SL) Z C {}}{ jwc e jwt Z C = 1 jwc ( 1 sc )

74 H.S. Chen Chapter1: Classification of signals and systems 73 Figure 23: We can replace R by R, C by 1/jwC, andl by jwl; thenbykvl or KCL we can solve the transfer function H(s) directly from the circuit diagram.

75 H.S. Chen Chapter1: Classification of signals and systems 74 therefore by voltage divider, we have V c = 1 jwc 1 jwc + R + jwl }{{} H(jw) i.e. H(jw) = V s,( jw 1 L 1 LC (jw) 2 + R L jw+ 1 LC on top and bottom) or H(s) = 1 LC S 2 + R L S+ 1 LC Note: O.D.E. V c (t)+ R L V c (t) 1 LC + V c(t) = 1 LC V s(t) It seems that we can find H(s) aslo from the ODE.

76 H.S. Chen Chapter1: Classification of signals and systems 75 Figure 24: x(t) =e jwt (or in general x(t) =e st ) is an eigenfunction of a continuous-time LTI system. x[n] =e jωn (or in general x[n] =z n ) is an eigenfunction of a discrete-time LTI system. Let x(t) =e jwt then y(t) =H(jw)e jwt Let x[n] =e jωn then y[n] =H(e jωn )e jωn

77 H.S. Chen Chapter1: Classification of signals and systems 76 Figure 25: I.e., mathematically, for a LTI system H, we have 1. h(t) =H{δ(t)}, h[n] =H{δ[n]} 2. H(jw)= H{ejwt } e jwt, H(e jω )= H{ejΩn } e jωn

78 H.S. Chen Chapter1: Classification of signals and systems 77 e.g. the ODE for RLC circuit is: Let x(t) =e jwt,theny(t) =H(jw)e jwt y (t)+ R L y + 1 LC y(t) = 1 LC x(t) then y (t) =(jw)h(jw)e jwt, y (t) =(jw) 2 H(jw)e jwt ((jw) 2 + R L jw + 1 RL )H(jw)ejwt = 1 RL ejwt H(jw)= 1 LC (jw) 2 + R L jw+ 1 LC

79 H.S. Chen Chapter1: Classification of signals and systems 78 This is always true for any nth order linear constant coefficient ODE. That is, given a differential equation for a LTI system a n y (n) (t)+a n 1 y (n 1) (t)+ + a 1 y (t)+a 0 y(t) = b m x (m) (t)+b m 1 x (m 1) (t)+ + b 1 x 1 (t)+b 0 x(t) i.e., n i=1 a iy (i) (t) = m j=1 b jx (j) (t)

80 H.S. Chen Chapter1: Classification of signals and systems 79 Substitute: x(t) =e jwt & y(t) =H(jw)e jwt into the ODE & use the fact di dt i e jw =(jwt) i e jwt we have (a n (jw) n + a n 1 (jw) n a 1 (jw)+a 0 )H(jw)e jwt =(b m (jw) m + b m 1 (jw) m b 1 (jw)+b 0 )e jwt H(jw)= b m(jw) m + +b 1 (jw)+b 0 a n (jw) n + +a 1 (jw)+a 0 H(s) = b ms m + +b 1 s m +b 0 a n s n + +a 1 s+a 0

81 H.S. Chen Chapter1: Classification of signals and systems 80 Figure 26:

82 H.S. Chen Chapter1: Classification of signals and systems 81 Usually H(s) = N(s) D(s) = A 1 s+p 1 + A 2 s+p 2 (degd(s) =n) + + A n s+p n (assume D(s) has n distinct toots) (by P.E.F. partial fraction Expansion) h(t) =L 1 {H(s)} h(t) =A 1 e p 1t u(t)+a 2 e p 2t u(t)+ + A n e p nt u(t)

83 H.S. Chen Chapter1: Classification of signals and systems 82 In general, block diagram > differential system > O.D.E > h(t)(h(s)) where > means providing more information. In signal & system,we study the zero-state response Figure 27: in particular,the system H will be a L.I.T. system.(linear & time invariant)

84 H.S. Chen Chapter1: Classification of signals and systems 83 Motivation II: (linear algebra signal & system) A = a d c b b a c d c b a d d c b a (circulant matrix) How to find the eigenvectors and eigenvalues for the circulant matrix A? We can use the fact that A represents a discrete-time LTI system to find the eigenvectors and eigenvalues.

85 H.S. Chen Chapter1: Classification of signals and systems 84 The matrix A represents a LTI system for a discrete-time with periodic input x[n]. That is, if x is a periodic input, then y = Ax is the periodic output with the fact that y[n] is obtained by the circular convolution between x[n] and h[n]: y[n] = N x[k]h[n k] k=1 In this example, we have h[n] =(a, b, c, d).

86 H.S. Chen Chapter1: Classification of signals and systems 85 Find the eigenvalues and eigenvectors for A. First,we can find eigenvalues of A by with v 0 Av = λv (λi A)v =0 Therefore we must have λi A is a singular matrix, i.e. det(λi A) = 0 (characteristic polynomial). This is a poly of degree n if A is a n n matrix. In general, it is not easy to find the eigenvalues for a given n n matrix A.

87 H.S. Chen Chapter1: Classification of signals and systems 86 For this circulant matrix,we can show that 1 1 v 1 = =(e j 2π 4 0 n ) n=0,1,2,3 =(i 0 ) n=0,1,2, i v 2 = =(e j 2π 4 1 n ) 0 n 3 =(i 1 n ) 0 n 3 1 i

88 H.S. Chen Chapter1: Classification of signals and systems v 3 = =(e j 2π 4 2 n ) 0 n 3 =(i 2 n ) 0 n i v 4 = =(e j 2π 4 3 n ) 0 n 3 =(i 3 n ) 0 n 3 1 i are eigenvectors of A.

89 H.S. Chen Chapter1: Classification of signals and systems 88 eigenvalue of v 1 = a + d + c + b eigenvalue of v 2 (a c)+i(d b) eigenvalue of v 3 (a + c) (d + b) eigenvalue of v 4 (a c) i(d b)

90 H.S. Chen Chapter1: Classification of signals and systems 89 Moveover, v 1,v 2,v 3,v 4 are orthogonal vectors, i.e.,. (v i,v j ) = 0 for any i j Let e i = 1 4 v i {e 1,e 2,e 3,e 4 } are orthonormal eigenvector for A. In other words,we have A [e 1,e 2,e 3,e 4 ] }{{} V λ 1 0 λ 2 where D =, λ 3 =[e 1,e 2,e 3,e 4 ] D }{{} V 0 λ 4 and λ i is an eigenvalue of e i.

91 H.S. Chen Chapter1: Classification of signals and systems 90 we can define a = h(0),b= h(1),c= h(2),d= h(3) then h(0) h(3) h(2) h(1) h(1) h(0) h(3) h(2) [ ] A = = h((n k)) h(2) h(1) h(0) h(3) 4 h(3) h(2) h(1) h(0) where h(3) 4 = h( 1), h(2) 4 = h( 2),

92 H.S. Chen Chapter1: Classification of signals and systems 91 Then Ax = 3 h[n k]x[k] = k=0 3 h[k]x[n k] k=0 This is just the discrete-time convolution sum. If we let x[n] =e j 2π 4 nk 0 (k 0 =0, 1, 2, 3) Ax = 3 k=0 h[k]ej 2π 4 (n k)k 0 3 = h[k]e j 2π 4 kk 0 e j 2π 4 nk 0 }{{}. k=0 } {{ } x[n] λ

93 H.S. Chen Chapter1: Classification of signals and systems 92 i.e., e j 2π 4 nk 0,(0 k 0 3) is an eigenvector of A with eigenvalue k h[k]e j 2π 4 kk 0. In matrix language, we have y = Ax -time domain y = VDV 1 x (since V 1 = V t ) = VDV T x V T y = DV T x y = Dx -frequency domain If V is an orthonormal matrix,then V T = V 1

94 H.S. Chen Chapter1: Classification of signals and systems 93 In general,we can show h(0) h(n 1) h(1) h(1) h(0) h(2) A = h(2) h(1) h(3).... h(n 1) h(n 2) h(0) always has eigenvectors 1 N (e j 2π N 0n ) 0 n N 1 = e 1 1 N (e j 2π N 1n ) 0 n N 1 = e N (e j 2π N (N 1)n ) 0 n N 1 = e N, (N eigenvectors)

95 H.S. Chen Chapter1: Classification of signals and systems 94 Such {e 1,...,e N } are orthonormal eigenvector for A and Ax = k h[n k]x[k] = k h[k]x[n k] Similarly, if we let x[n] =e j 2π N nk 0,0 k 0 N 1 Ax = k h[k]e j 2π N (n k)k 0 = h[k]e j 2π N kk 0 k }{{} λ e j 2π N nk 0 }{{} x[n] Also y = Ax = VDV 1 x V 1 y = DV 1 x y = DV T x y = Dx