5. Response of Linear Time-Invariant Systems to Random Inputs

Transcription

1 Sysem: 5. Response of inear ime-invarian Sysems o Random Inpus 5.. Discree-ime linear ime-invarian (IV) sysems 5... Discree-ime IV sysem IV sysem xn ( ) yn ( ) [ xn ( )] Inpu Signal Sysem S Oupu Signal We look a a sysem as a black box which generaes an oupu signal depending on he inpu signal and possibly some iniial condiions. inear: x ( n) x ( n) a x ( n) + a x ( n) IV sysem y ( n) y ( n) a y ( n) + a y ( n) We consider wo ypes of signals: Discree-ime signals or sequences xn ( ) ime-invarian: xn ( ) xn ( n ) IV sysem yn ( ) yn ( n ) 5... Seady-sae descripion of a IV sysem 3 4 n Impulse response: he impulse response (IR) hn ( ) of is he response of o he uni pulse Coninuous-ime signals x () δ( n) ; n, ; n namely hn ( ) [ δ( n) ] xn ( ) δ( n) hn ( ) [ δ( n) ] Discree-ime signals are obained by sampling coninuous-ime signals. 5- n 3 4 n 5-

2 Sable IV sysem: A IV sysem is sable if is response o a bounded signal is bounded. I can be shown [o his end we need (5.)] ha a IV sysem is sable if, and if, m hm ( ) < (Discree) Fourier ransform: Here, zn ( ) denoes an arbirary sequence. Z( f ) F { zn ( )} zn ( ) exp( jπnf ) ( f < ) n zn ( ) F { Z( f )} Z( f ) exp( jπnf ) d f Useful propery exensively used in he sequel: Causal IV sysem: xn ( ) δ( n) hn ( ) for n < hn ( ) [ δ( n) ] zn ( n ) exp( jπn f )Z( f ) n 3 4 n Inpu-oupu (I-O) relaionship of a IV sysem (ime domain): yn ( ) hm ( )xn ( m) m hn ( )*x( n) he symbol * denoes he discree convoluion operaion. ( 5.) (Frequency) ransfer funcion of a IV sysem: H( f ) F { hn ( )} hn ( ) exp( jπnf ) n I-O relaionship of a IV sysem (frequency domain): Y( f ) H( f )X( f )

3 Summary: I-O relaionship of a IV sysem: ime domain yn ( ) hn ( )*x( n) xn ( ) IV sysem yn ( ) hn ( ) - Auocorrelaion funcion: R YY ( n, n ) hm ( )hm ( )R XX ( n m, n m ) m m X( f ) H( f ) Y( f ) Frequency domain Y( f ) H( f ) X( f ) Firs- and second-order characerizaion of a VI sysem Random inpu and oupu sequences: If X( n) is a random sequence (or process), so is Y( n). IV sysem X( n) Y( n) [ X( n) ] Second-order characerizaion of random sequences: Here, Z( n) denoes an arbirary random sequence. - Expecaion: - Auocorrelaion funcion: µ Z ( n) E[ Z( n) ] R ZZ ( n, n ) E[ Z( n )Z( n )] Second-order properies of he oupu sequence Y( n) : - Expecaion: µ Y ( n) hn ( )*µ X ( n) Wide-sense-saionary (WSS) processes Definiion: A random sequence Z( n) is WSS if he following condiions are saisfied: - Expecaion: µ Z ( n) E[ Z( n) ] µ Z - Auocorrelaion funcion: R ZZ ( n, n + k) E[ Z( n )Z( n + k) ] R ZZ ( k) Whie process: Z( n) is a whie process if i saisfies he following condiions: - Z( n) is a random process - µ Z ( n) E[ Z( n) ]

4 - R ZZ ( nn, + k) E[ Z( n)z( n + k) ] R ZZ ( k) σ Z δ( k) Auocorrelaion funcion of he impulse response: R ZZ ( k) σ Z n R hh ( k) hm ( )hm ( + k) m Second-order I-O relaionship of a VI sysem (ime domain): - Expecaion: hk ( )*h( k) µ Y hm ( ) µ X H ( )µ X m Power specrum of a WSS process: he power specrum of he WSS process Z( n) wih he auocorrelaion funcion R ZZ ( k) is defined o be Noice ha from he inverse Fourier ransformaion -- we conclude ha S ZZ ( f ) F { R ZZ ( k) } R ZZ ( k) exp( jπkf ) n Specrum of a whie process: If Z( n) is a whie process: R ZZ ( k) S ZZ ( f ) exp( jπkf ) d f E[ Z( n) ] R ZZ ( ) S ZZ ( f ) d f S ZZ ( f ) σ Z -- - Auocorrelaion funcion: R YY ( k) R hh ( k)*r ( k) XX S ZZ ( f ) σ Z.5.5 f

5 Second-order I-O relaionship of a VI sysem (frequency domain): S YY ( f ) H( f ) S XX ( f ) Example: Firs order recursive filer Block diagram and recursive equaion: yn ( ) xn ( ) + Uni delay yn ( ) φy( n ) φ Summary: Second-order I-O relaionship of a IV sysem: ime domain µ Y H ( )µ X R YY ( k) R hh ( k)*r ( k) XX µ IV sysem X, R XX ( k) hn ( ) R µ Y, R YY ( k) hh ( k) S XX ( f ) H( f ) H( f ) S YY ( f ) yn ( ) xn ( ) + φyn ( ) ( yn ( ) n < ) Impulse response: xn ( ) δ( n) hn ( ) φ φ φ 3 φ 4 n 3 4 n Frequency domain S YY ( f ) H( f ) S XX ( f ) hn ( ) ; n < φ n ; n 5-9 Sabiliy condiion: n hn ( ) φ n φ N lim N < φ φ < n N a n a N + -- a n 5-

6 ransfer funcion:.5 Anoher more direc way o compue he ransfer funcion: Random inpu and oupu: H( f ) F { hn ( )} φ n exp( jπnf ) n [ φexp( jπf )] n n φexp( jπf ) H( f ) -- φ -- + φ.5 f yn ( ) xn ( ) + φyn ( ) Y( f ) X( f ) + φexp( jπf ) Y( f ) Second-order I-O relaionship: - ime domain: µ Y H ( )µ X --µ φ X R YY ( k) R hh ( k)*r ( k) XX φ k -- φ *R XX ( k) R hh ( k) φ m φ m+ k φ k φ m φ k -- m m φ - Frequency domain: S YY ( f ) H( f ) S XX ( f ) Special case: AR() process (see Secion 6.): If X( n) is a whie Gaussian process, S XX ( f ) φexp( jπf ) µ Y Y( n) Xn ( ) + Uni delay Y( n ) R YY ( k) φ k -- φ R YY ( k) φy( n ) φ φ - φ φ - φ - φ φ - φ φ - φ Yn ( ) Xn ( ) + φy( n ) k 5-5-

7 S YY ( f ) φexp( jπf ) 5.. Coninuous-ime IV sysems 5... Coninuous-ime IV sysem.5 S YY ( f ) - φ -- + φ.5 f inear: ime-invarian: IV sysem x () y () [ x ()] x () x () a x () + a x () IV sysem y () y () a y () + a y () One realizaion of Y( n) : x () x ( ) IV sysem y () y ( ) 5... Seady-sae descripion of a IV sysem Impulse response: he impulse response h () of is he response of o he Dirac impulse yn ( ) namely δ() ; ; h () [ δ() ] δ() d, x () δ () h () [ δ() ]

8 Sable IV sysem: A IV sysem is sable if is response o a bounded signal is bounded. I can be shown [o his end we need (5.)] ha a IV sysem is sable if, and if, Causal IV sysem: Inpu-oupu relaionship of a IV sysem (ime domain): Here, he symbol * denoes he coninuous convoluion operaion. (Coninuous) Fourier ransform: h () d < (Frequency) ransfer funcion of a IV sysem I-O relaion ship of a IV sysem (frequency domain): h () < y () h( τ)x ( τ) dτ h ()*x() Z( f ) F { z ()} z () exp( jπf) dτ ( 5.) z () F { Z( f )} Z( f ) exp( jπf) d f H( f ) F { h ()} h () exp( jπf) d Y( f ) H( f )X( f ) Summary: I-O relaion ship of a IV sysem: ime domain Frequency domain Second-order characerizaion of IV Random inpu and oupu processes: Second-order characerizaion of random processes: Here, Z() denoes an arbirary random process. - Expecaion: - Auocorrelaion funcion: Second-order properies of he oupu process Y() : - Expecaion: - Auocorrelaion funcion: y () h ()*x() x () IV sysem h () y () X( f ) H( f ) Y( f ) Y( f ) H( f ) X( f ) IV X() Y() [ X() ] µ Z () E[ Z() ] R ZZ (, ) E[ Z( )Z( )] µ Y () h ()*µ X () R YY (, ) hu ( )hu ( )R XX ( u, u ) du du

9 5..4. Wide-sense-saionary (WSS) processes Wide-sense saionariy: A random process Z() is WSS if he following condiions are saisfied: - Expecaion: - Auocorrelaion funcion: µ Z () µ Z R ZZ (, + τ) R ZZ ( τ) Whie process: Z() is a whie process if i saisfies he following condiions: - Z() is a random process - µ Z () E[ Z() ] - R ZZ (, + τ) E[ Z()Z ( + τ) ] R ZZ ( τ) σ Z δτ ( ) R ZZ ( τ) σ Z δτ ( ) - Auocorrelaion funcion: Power specrum of a WSS process: Z() is WSS wih he auocorrelaion funcion R ZZ ( τ). Noice he ideniies Specrum of a whie process: If Z() is a whie process: R YY ( τ) R hh ( τ)*r ( τ) XX S ZZ ( f ) F { R ZZ ( τ) } E[ Z() ] R ZZ ( ) S ZZ ( f ) d f S ZZ ( f ) σ Z S ZZ ( f ) σ Z τ Auocorrelaion funcion of he impulse response: R hh ( τ) hu ( )hu ( + τ) du h( τ)*h( τ) Second-momen I-O relaionship of a VI sysem (ime domain): - Expecaion: µ Y hu ( ) u µ X H ( )µ X Summary: Second order I-O relaionship of a IV sysem: ime domain µ X, R XX ( τ) Frequency domain S XX ( f ) µ Y H ( )µ X R YY ( τ) R hh ( τ)*r ( τ) XX IV sysem h () R hh ( τ) H( f ) H( f ) S YY ( f ) H( f ) S XX ( f ) f µ Y, R YY ( τ) S YY ( f )

10 5..5. Example: Ideal inegraor Block diagram and inpu-oupu relaionship: Impulse response: x () x( α) dα y () x () δ () y () x( α) dα h () [ δ() ] Sabiliy condiion: h ransfer funcion: H( f ) F { h ()} h () exp( jπf) d sin( πf) exp( jπf) d exp( jπf) πf exp( jπf) sinc( f) H( f ) h () δα ( ) dα ; < < ; elsewhere ; (, ) ; elsewhere f δα ( ) α

11 Random inpu and oupu: X () x( α) dα Y() Y () Second-order I-O relaionship: - ime domain: X ( α) dα µ Y H ( )µ X µ X - Frequency domain: S YY ( f ) H( f ) S XX ( f ) sinc( f) S XX ( f ) wih R YY ( τ) R hh ( τ)*r ( τ) XX R hh ( τ) τ - ; < τ < ; elsewhere Special case: X() is a whie process, µ Y R YY ( τ) - τ - ; < τ < ; elsewhere R hh ( τ) R YY ( τ) - τ τ 5-5-

12 S YY ( f ) sinc( f) S YY ( f ) f 5-3