Discrete Time Sigals Samples of a CT sigal: EE123 Digital Sigal Processig x[] =X a (T ) =1, 2, x[0] x[2] x[1] X a (t) T 2T 3T t Lecture 2 Or, iheretly discrete (Examples?) 1 2 Basic Sequeces Uit Impulse [] = 1 =0 0 6= 0 Basic Sequeces Expoetial x[] = A 0 0 <0 0 < < 1 > 1 Uit Step U[] = 1 0 0 <0 1 < < 0 Bouded ubouded < 1 3 4
Discrete Siusoids x[] = A cos(! 0 + ) x[] = Ae j! 0+j or, Discrete Siusoids x[] = A cos(! 0 + ) x[] = Ae j! 0+j or, Q: Periodic or ot? x[ + ] =x[] for iteger Q: Periodic or ot? x[ + ] =x[] for iteger A: If! 0 / is ratioal (Differet tha C.T.!) To fid fudametal period Fid smallest itegers K, :! 0 / =2 K 5 6 Discrete Siusoids Example: cos(5/7 ) = 14 (K = 5) cos( /5) = 10 (K = 1) Discrete Siusoids Aother Differece: Q: Which oe is a higher frequecy?! 0 = or! 0 = 3 2 cos(5/7 ) + cos( /5) ) = S.C.M{10, 14} = 70 7 8
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 = A: cos( ) cos(3 /2) = cos( /2) =4 9 10 Discrete Siusoids - 3 2!0 = =2 r or!0 = Discrete Siusoids Recall the periodicity of DTFT!0 = 0!0 = /8 cos(w0 )!0 = /4!0 = 11 12
Discrete Siusoids Discrete Time Systems cos(w 0 )! 0 =2! 0 = 15 8! 0 = 7 4 x[] What properties? Causality: T { } y[] y[ 0 ] depeds oly o x[] for 0! 0 = 13 14 Properties of D.T. Systems Cot. Memoryless: y[] depeds oly o x[] Example: y[] = x[] 2 Liearity: Superpositio: T {x 1 []+x 2 []} = T {x 1 []} + T {x 2 []} = y 1 []+y 2 [] Properties of D.T. Systems Cot. Time Ivariace: y[] = T {x[]} If: The: BIBO Stability x[] apple B x < 1 If: The: y[ 0 ] = T {x[ 0 ]} y[] apple B y < 1 8 8 Homogeeity: T {ax 1 []} = at {x[]} = ay[] 15 16
Example: Examples Time Shift y[] = x[ d ] Accumulator y[] = X x[k] Causal L TI memoryless if d>=0 Why the compressor is OT Time Ivariat? Suppose M=2, if d=0 BIBO stable x[ Compressor y[] = x[m ], - 0 Q 1 tft, 6J 6= y[ 1] 17 18 Example: Removig Shot oise From PR This America Life, ep.203 The Greatest Phoe Message i the World /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 * j0 cl] o-liear system: Media Filter 1] M >1 Examples y[] x[] = cos[ /2 ] 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.. '}CxJ :> corrupted message -i - -1 s?- e ee }'lhl 19 20
Spectrum of Speech Speech Low Pass Filterig LP-Filter Spectrum Corrupted Speech 21 22 Low-Pass Filterig of Shot oise Corrupted Low-Pass Filterig of Shot oise Corrupted LP-filtered Med-filter 23 24
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... 25 Discrete-Time LTI Systems The impulse respose h[] completely characterizes a LTI system DA of LTI [] x[] LTI LTI y[] = 1X = 1 h[] y[] =h[] x[] h[m]x[ m] discrete covolutio Sum of weighted, delayed impulse resposes! 26 BIBO Stability of LTI Systems BIBO Stability of LTI Systems A LTI system is BIBO stable iff h[] is absolutely summable 1X h[k] < 1 Proof: if y[] = apple 1X h[k]x[ k] 1X h[k] x[ k] apple B x apple B x 1 X h[k] < 1 27 28
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