EE123 Digital Signal Processing

Transcription

1 Today EE123 Digital Sigal Processig Lecture 2 Last time: Admiistratio Overview Today: Aother demo Ch. 2 - Discrete-Time Sigals ad Systems 1 2 Discrete Time Sigals Samples of a CT sigal: x[] =X a (T ) =1, 2, Basic Sequeces Uit Impulse [] = 1 =0 0 6= 0 x[0] x[2] x[1] X a (t) T 2T 3T t Uit Step U[] = <0 Or, iheretly discrete (Examples?) 3 4

2 Basic Sequeces Expoetial x[] = A 0 0 <0 0 < < 1 > 1 Discrete Siusoids x[] = A cos(! 0 + ) x[] = Ae j! 0+j or, Q: Periodic or ot? x[ + ] =x[] for iteger 1 < < 0 < 1 Bouded ubouded 5 6 Discrete Siusoids x[] = A cos(! 0 + ) Q: Periodic or ot? x[] = Ae j! 0+j x[ + ] =x[] for iteger A: If! 0 / is ratioal (Differet tha C.T.!) To fid fudametal period Fid smallest itegers K, : or,! 0 =2 K 7 Discrete Siusoids Example: cos(5/7 ) = 14 (K = 5) cos( /5) = 10 (K = 1) cos(5/7 ) + cos( /5) ) = S.C.M{10, 14} = 70 8

3 Discrete Siusoids Discrete Siusoids Aother Differece: Aother Differece: Q: Which oe is a higher frequecy?!0 = Q: Which oe is a higher frequecy? or!0 = 3 2!0 = cos( ) or!0 = 3 2 cos(3 /2) = cos( /2) 9 10 Discrete Siusoids Discrete Siusoids Aother Differece: r - Recall the periodicity of DTFT Q: Which oe is a higher frequecy?!0 = A: or!0 = 3 2!0 = =2 =

4 Discrete Siusoids Discrete Siusoids! 0 =0! 0 =2 cos(w 0 )! 0 = /8 cos(w 0 )! 0 = 15 8! 0 = /4! 0 = 7 4! 0 =! 0 = Discrete Time Systems Properties of D.T. Systems Cot. x[] T { } y[] Memoryless: y[] depeds oly o x[] What properties? Causality: y[ 0 ] depeds oly o x[] for 0 Example: y[] = x[] 2 Liearity: Superpositio: T {x 1 []+x 2 []} = T {x 1 []} + T {x 2 []} = y 1 []+y 2 [] Homogeeity: T {ax 1 []} = at {x[]} = ay[] 15 16

5 Properties of D.T. Systems Cot. Example: Time Ivariace: If: The: y[ y[] = 0 ] = T {x[]} T {x[ Time Shift y[] = x[ 0 ]} d ] Accumulator BIBO Stability x[] Bx < 1 8 If: The: y[] By < 1 8 y[] = X x[k] BIBO stable Causal L TI memoryless if d>=0 if d=0 Compressor y[] = x[m ] M > Examples Examples * j0 cl] Why the compressor is OT Time Ivariat? o-liear system: Media Filter Suppose M=2, /MIL y[]a_= tj»-l.f).)f?_ S'(<;r M.!.Mf/2!/VI ffltf-f( MED{x[ k],, x[ + k]} =- m o[ r[jj-k.'],...1 :X{+-tJ1 y[] x[] = cos[ /2 ] x[, - 1] 0 Q 1 tft, 6J '}CxJ :> -i = y[ s?- e ee }'lhl 1] 19 20

6 Example: Removig Shot oise From PR This America Life, ep.203 The Greatest Phoe Message i the World Spectrum of Speech Speech em.. (giggle) There comes a time i life, whe.. eh... whe we hear the greatest phoe mail message of all times ad... well here it is... eh.. you have to hear it to believe it.. Corrupted Speech corrupted message Low Pass Filterig LP-Filter Spectrum Low-Pass Filterig of Shot oise Corrupted LP-filtered 23 24

7 Low-Pass Filterig of Shot oise Corrupted Med-filter The Greatest Message of All Times... I thought you ll get a kick out of a message from my mother... Hi Fred, you ad the little mermaid ca go blip yourself. I told you to stay ear the phoe... I ca t fid those books.. you have other books here.. it must be i Le hoya.. call me back... I m ot goig to stay up all ight for you Bye Bye Discrete-Time LTI Systems The impulse respose h[] completely characterizes a LTI system DA of LTI BIBO Stability of LTI Systems A LTI system is BIBO stable iff h[] is absolutely summable [] LTI h[] discrete covolutio 1X h[k] < 1 x[] LTI y[] =h[] x[] 1X y[] = h[m]x[ m] = 1 Sum of weighted, delayed impulse resposes! 27 28

8 BIBO Stability of LTI Systems Proof: if 1X y[] = h[k]x[ k] apple 1X h[k] x[ k] apple B x X 1 apple B x h[k] < 1 BIBO Stability of LTI Systems Proof: oly if P 1 h[k] = 1 suppose show that there exists bouded x[] that gives ubouded y[] Let: 29 30