Signals and Systems Linear Time-Invariant (LTI) Systems

Transcription

1 Signals and Sysems Linear Time-Invarian (LTI) Sysems Chang-Su Kim

2 Discree-Time LTI Sysems

3 Represening Signals in Terms of Impulses Sifing propery 0 x[ n] x[ k] [ n k] k x[ 2] [ n 2] x[ 1] [ n1] x[0] [ n] x[1] [ n1] x[2] [ n2] xn [] x[ 2] [ n2] + x[ 1] [ n1] + x[0] [ n] + x[1] [ n1] + x[2] [ n2]

4 Impulse Response The response of a sysem H o he uni impulse [n] is called he impulse response, which is denoed by h[n] h[n] = H[[n]] [n] Sysem H h[n]

5 Convoluion Sum Le h[n] be he impulse response of an LTI sysem. Given h[n], we can compue he response y[n] of he sysem o any inpu signal x[n]. [n] h[n]

6 Convoluion Sum Le h[n] be he impulse response of an LTI sysem. Given h[n], we can compue he response y[n] of he sysem o any inpu signal x[n].

7 Convoluion Sum Le h[n] be he impulse response of an LTI sysem. Given h[n], we can compue he response y[n] of he sysem o any inpu signal x[n]. k x [ n] x[ k] [ n k] y[ n] H[ x[ n]] H x[ k] [ n k] k k k k H x[ k] [ n k] x[ k] H [ n k] x[ k] h[ n k]

8 Convoluion Sum Noaion for convoluion sum * y[ n] x[ n] h[ n] x[ k] h[ n k] k The characerisic of an LTI sysem is compleely deermined by is impulse response. [n] x[n] LTI sysem h[n] x[n] * h[n]

9 Convoluion Sum To compue he convoluion sum y[ n] x[ n]* h[ n] x[ k] h[ n k] k Sep 1 Plo x and h vs k since he convoluion sum is on k. Sep 2 Flip h[k] around he verical axis o obain h[-k]. Sep 3 Shif h[-k] by n o obain h[n-k]. Sep 4 Muliply o obain x[k]h[n-k]. Sep 5 Sum on k o compue x[k]h[n-k]. Sep 6 Change n and repea Seps 3-6.

10 Example Consider an LTI sysem ha has an impulse response h[n] = u[n] Wha is he response when an inpu signal is given by x[n] = a n u[n] where 0<a <1? For n0, Therefore, a a a 1 1 ] [ 1 0 n n k k n y ] [ 1 1 ] [ 1 n u n y n a a

11 Convoluion Sum Demonsraion

12 Coninuous-Time LTI Sysems

13 Impulse Response The response of a sysem H o he uni impulse () is called he impulse response, which is denoed by h() h() = H(()) () D () Sysem H As D 0, D () () and h D () h() h() h D () Recall he definiion of approximaed impulse funcion D 1, 0 D () D 0, oherwise D () 1/D D

14 Saircase Approximaion of x() xd( ) x( kd) D( kd) D k x() D 0 D 2D kd

15 Convoluion Inegral D () h D ()

16 Convoluion Inegral

17 Convoluion Inegral The derivaion shows ha a saircase approximaion o he inpu xd( ) x( kd) D( kd) D k yields an approximaion o he oupu yd( ) x( kd) hd( kd) D k Now we ake he limi. As 0, () δ(), h () h(), x () x(), and y () y(). Also, he sums approach he inegrals x( ) x( ) ( ) d y( ) x( ) h( ) d Sifing propery Convoluion inegral

18 Anoher Inerpreaion of Sifing Propery To see he meaning of he sifing propery x( ) x( ) ( ) d we approximae he impulse wih a all, narrow pulse δ (-)

19 Convoluion Inegral To compue he convoluion inegral y( ) x( )* h( ) x( ) h( ) d Sep 1 Plo x and h vs τ since he convoluion inegral is on τ. Sep 2 Flip h(τ ) around he verical axis o obain h(-τ ). Sep 3 Shif h(-τ ) by o obain h( -τ ). Sep 4 Muliply o obain x(τ ) h( -τ ). Sep 5 Inegrae on τ o compue x(τ ) h( -τ ) dτ. Sep 6 Increase and repea Seps 3-6.

20 Example 1 Le x() be he inpu o a LTI sysem wih uni impulse response h() For >0 We can compue y() for >0 So for all ) ( ) ( 0 ) ( ) ( u h a u e x a oherwise 0 0 ) ( ) ( e h x a a a a a a e e d e y 1 ) ( ) ( 1 ) ( 1 u e y a a In his example a=1

21 Example 2 Calculae he convoluion of he following signals x( ) h( ) e 2 u( u( ) 3) For <3, he convoluion inegral becomes y( ) 3 e 2 d 1 2 e 2( 3) For -30, he produc x()h(-) is non-zero for -< <0, so he convoluion inegral becomes e d 2 y( )

22 Properies of LTI Sysems

23 Properies of Convoluion x[ n]* y[ n] x[ k] y[ n k] k x( )* y( ) x( ) y( ) d Commuaive x[n]*y[n]=y[n]*x[n] x()*y()=y()*x() Disribuive x[n]*(y 1 [n] + y 2 [n])=x[n]*y 1 [n] + x[n]*y 2 [n] x()*(y 1 () + y 2 ())=x()*y 1 () + x()*y 2 () Associaive x[n]*(y 1 [n]*y 2 [n])=(x[n]*y 1 [n])*y 2 [n] x()*(y 1 ()*y 2 ())=(x()*y 1 ())*y 2 ()

24 Causaliy of LTI Sysems A sysem is causal if is oupu depends only on he pas and he presen values of he inpu signal. Consider he following for a causal DT LTI sysem: k y [ n] x[ k] h[ n k] Because of causaliy h[n-k] mus be zero for k>n. In oher words, h[n]=0 for n<0. Similarly for a CT LTI sysem o be causal h() = 0 for <0.

25 Causaliy of LTI Sysems So he convoluion sum for a causal LTI sysem becomes Similarly, he convoluion inegral for a causal LTI sysem becomes So, if a given sysem is causal, one can infer ha is impulse response is zero for negaive ime values, and use he above simpler convoluion formulas. 0 ] [ ] [ ] [ ] [ ] [ k n k k n x k h k n h k x n y 0 ) ( ) ( ) ( ) ( ] [ d x h d h x n y

26 Sabiliy of LTI Sysems A sysem is sable if a bounded inpu yields a bounded oupu (BIBO). In oher words, if x[n] < k 1 hen y[n] < k 2. Noe ha y[ n] x[ n k] h[ k] x[ n k] h[ k] k h[ k] 1 k k k Therefore, a DT sysem is sable if k hk [ ] Similarly, a CT sysem is sable if h() d