Linear Time Invariant Systems

Transcription

1 1 Liear Time Ivaria Sysems Oulie We will show ha he oupu equals he covoluio bewee he ipu ad he ui impulse respose: sysem for a discree-ime, for a coiuous-ime sysdem, y x h y x h

2 2 Discree Time LTI Sysems h he ui impulse respose of he LTI sysem: ha is, h. x he ipu The he oupu sigal y is he covoluio sum: y x h Cosider a discree-ime liear ad ime ivaria (LTI) sysem. Suppose he LTI sysem produces he oupu h whe he ipu is, he ipu is x 2 3. Fid he sysem oupu y. By defiiio, h Time-Ivaria Propery: 3 3 h h Superposiio Propery: We ca see y h 2h 3 h

3 3 The oupu of a LTI sysem ca c be foud easily if i) x is represeed as a superposiio of shifed ui impulses, ad ii) he ui impulse respose his kow. Represe x as a superposiio of shifed ui impulses. Wrie he ipu discree-ime sigal x x k xk k. as Le h deoe he ui impulse respose of he LTI sysem: ha is, The he oupu sigal y y k Eq. d1 is refered o as he covoluio sum. y h. iss xkh k. x h d1

4 4 y k x 0 h xkh k x1 h1 0.5 h2h1

5 5 y k xkhh k. To fidd y 0, y To fidd y 1, To fidd y 2, y y k k k k k k 2.5 x k h 0 k x k h x k h 1 k k x k h k1 x k h 2 k x k h k 2

6 6 x h u u, 0 1 y k xkh k k u k uk k oig u k for 0 k for u 1 0 for 0

7 7 x h 1, 0 4, 0 6, 1. y k xkh k Some k 1 k Oher k 0 Divide he ime io iervals 1) 0 o overlap 2) 0 4 k 0 3) k 0 4) ) 4 k 6 11 o overlap

8 8 Coiuous Time LTI Sysems h x he ui impulse respose of hee LTI sysem: ha is, he ipu The he oupu sigal y y x h equals he covoluio iegral: h. Approximae he ipu sigal shifed boxcar fucios x as he superposiio of. xˆ k Noe x ˆ x(). Also lim 0 0 xk k lim xˆ x.,.

9 9 Le h deoe he respose of he LTI sysem o he ipu ˆx is The he oupu of he sysem for he ipu. k yˆ xk h k. Defie he coiuous-ime ui impulse respose h () limh. The y 0 lim yˆ 0 k lim x k h k 0 1 x h d c Eq.c1 is referred o as he covoluio iegral. y x h

10 10 x h e a u u, a 0 y? x e x a h h 1 d d 1) For 0, o overlap occurs. y 0. 2) For 0, y a e 0 d

11 11 x h 1, 0 T, 0 2T h y x h 2T? 1 x h d. d h 1) For 0, o overlap. 2T 2) 0 ad T, 0 x 3) T ad 2 T 0, 4) 2T 0 ad 2 T T, T 0 T 2T T 5) 2 T T, o overlap.

12 12 Properies of LTI Sysems Commuaive * x h h x oupu of he LTI sysem wih he ui impulse repose h ad he ipu x oupu of he LTI sysem wih he ui impulse repose x ad he ipu h Similarly, for discree-ime sysems, xhh x. x h h x Proof x h x h d x h d h h x d x leig

13 13 Disribuive x * h h x h xh 2 Similarly, for discree-ime sysems, x h h 1 2 x h 1 2 x h Proof x h h 1 2 k k xk x h h k h k x k h k 2 2 k x h x k h 2 k

14 14 Associaive x Similarly, for discree-ime sysems, 1 h 1 x h 2 2 * 1 h x h * 1 h x h h 2 h. 2 Combig he associaive propery wih he commuaive propery, 1 x h x h 2 h x h x h h h 1 h 2 1 The overall sysem respose does o deped o he order of hee sysems i a cascade of LTI sysems.

15 15 Prove he associaive propery Proof x h h x* h h x h h x h d h x h d h d chagig he order he double iegral, x h h d d defiig a ew variable,, x h h d d x h h x () h h d

16 16 Memoryless LTI Sysems Cosider a discree-ime LTI sysem. y x h xkh k. k I a memoryless sysem, y mus deped oly o x. Therefore, for he LTI sysem o be memoryless, hk should be 0 uless k, ha is, h K, 0. 0, 0 I a discree-ime memoryless LTI sysem, he impulse respose is a ui imulse sigal; hk. Similarly, i a coiuous-ime memoryless LTI sysem, he impulse respose is a ui imulse sigal; h K.

17 17 Iveribiliy of LTI Sysems s A sysem is iverible oly if a iverse sysem exiss. Le h The 1 deoe he ui impulse respose of he iverse sysem. h h 1 Similarly, for a discree-ime iverible i LTI sysem, 1 h h. should be. Cosider a LTI sysem whichh shifs he ipu sigal i ime y x 0. The ui impulse respose of he sysem is he respose of he sysemm o he ipu. Therefore he ui impulse respose of he ime shif sysem is h ad he impulse respose of he iverse sysem is h 0. Ideed we ca show hh 1 0, 1 0 * 0.

18 18 Problem Show *. 0 0 * lim 0 0 lim d However 0 lim r 0 lim r u, which implies ha r 0 0 is a sigal which behaves like a impulse: lim r r

19 19 Cosider a accumulaor, y xk k. k The ui impulse respose of he accumulaor is h k 1, u he ui sep sigal The oupu of he sysem wih he ui sep as he ui impulse respose y x u k k xkuk xk is ideed he accumulaor Now cosider he iverse sysem of he accumulaor, y x x The ui impulse respose of he iverse sysem is 1 h Show h h h h hkh k k k 1 u k k k u 1 u 1.

20 20 Sabiliy for LTI Sysems A discree-ime LTI sysem wih he ui impulse respose he ui impulse respose is absoluely summable: ha is, h is sable if ad oly if k hk Proof For a bouded ipu sigal Therefore y xkh k k k k k x, x B for all. x k h k B h k B h k So he absolue summabiliy is a sufficie codiio for sabiliy.

21 21 Show ha he absolue summabiliy of he ui impulse respose is also a ecessary codiio. Suppose hk. k * Cosider he followig ipu sigal x. where h 0 if h 0 x * h if h0 h is he complex cojugae., The ipu sigal x x h * h is bouded for all,sice 1 However he oupu sigal For 0, 0 y x k h k k k k k h h * h hk k k k h k y diverges for he bouded ipu sigal.

22 22 The ui impulse respose of a accumulaor is h u. k k h k u k o absoluely summable. So he accumulaor is usable. Similarly, a coiuous-ime LTI sysem wih he ui impulse respose oly if he ui impulse respose is absoluely iegrable: ha is, h is sable if ad h d The ui impulse respose of a iegraor is h d u. h d u d o absoluely iegrable. So he iegraor is usable.

23 23 The Ui Sep Respose The respose of a LTI sysem o he ui sep is referred o as he ui sep respose s or s. For a discree-ime sysem wih he ui impulse respose h, u h s Due o he commuaive propery, h u accumulaor s s k hk 1 h s s Similarly, for a coiuous-ime sysem wih he ui impulse respose h, u h s Due o he commuaive propery, h u iegraor s s h d h d s d