Frequency Response. Response of an LTI System to Eigenfunction
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- Piers Banks
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1 Frquncy Rsons Las m w Rvsd formal dfnons of lnary and m-nvaranc Found an gnfuncon for lnar m-nvaran sysms Found h frquncy rsons of a lnar sysm o gnfuncon nu Found h frquncy rsons for cascad, fdbac, dffrnc quaon, and dffrnal quaon sysms Today w wll Exnd h rsuls o accommoda snusodal nu, and hn any nu va Fourr srs rrsnaon Wr h Fourr srs n rms of comlx xonnals Provd a mhod o calcula Fourr srs coffcns Drmn rors of hs coffcns Rsons of an LTI Sysm o Egnfuncon Las m, w rovd ha for an nu sgnal x gvn by x( for all œ Rals h corrsondng ouu y of an LTI sysm can b xrssd as y( H( for all œ Rals whr H( s calld h frquncy rsons of h sysm. Th comlx xonnal s calld an gnfuncon of h sysm, bcaus cras an ouu wh h sam form, only dffrng by a scalng facor. Th sam s ru for a dscr nu, x( n for all n œ Ingrs lads o y(n H( n for all n œ Ingrs
2 Cosns as Comlx Exonnals Rcall ha cosns can b xrssd as comlx xonnals: cos( If w l x ( and x ( -, w xrss h cosn as cos( ½(x ( x ( If w aly h cosn as nu o an LTI sysm S, w fnd S(½(x x ½ (S(x S(x and snc x and x ar gnfuncons, w can wr y( ½ (S(x S(x ( ½ (H( H(- - So w can us frquncy rsons o xrss h ouu for snusodal nu. Conjuga Symmry So for h nu x( cos(, w oban h ouu y( ½ (H( H(- -. Ralsc sysms wll roduc urly ral ouu (no magnary comonn for a urly ral nu l cos(. Ths mans ha h magnary ars of H( and H(- - mus cancl ou; hy mus b oos n sgn. Ths s h sam as sayng ha on s h conjuga of h ohr: H( (H(- - * H(-* For sysms ha roduc ral ouu for ral nu, s ru ha H( H(-*
3 Imlcaons: Scald and Shfd Snusods Sysms ha roduc urly ral ouu for a urly ral nu ar calld conjuga symmrc. L s loo agan a h ouu for our cas x( cos(, y( ½ (H( H(- - Usng h fac ha z z* R{z}, y( ½ ( R{H( } R{H( } If w xrss H( n olar form, H( H( H(, y( R{ H( H( } R{ H( ( H( } H( cos( H( Comung Snusodal Rsons So, gvn h sysm rsons o an gnfuncon, H(, w can comu h magnud rsons H( and h has rsons H(. Ths form h scalng facor and has shf n h ouu, rscvly. Th frquncy of h ouu snusod wll b h sam as h frquncy of h nu snusod n any LTI sysm. Th LTI sysm scals and shfs snusods. Ths rsuls hold ru for boh connuous and dscr sgnals and sysms.
4 Examl R Consdr our RC crcu from las m, whr w found x( C y( _ H( _ RC To comu h volag ovr h caacor, y(, for a snusodal nu volag x(, I smly nd o fnd h magnud and has of H( and lug n: H( RC ( RC H( ( RC an RC Rsons o Fourr Srs Inu In Char 7, w mnond ha any rodc sgnal can b rrsnd by a Fourr srs: x( cos( Snc w ar dalng wh LTI sysms, whr w can ull ou consans and dsrbu ovr sums, w can g h sysm ouu for any nu by scalng and summng h ouu for h ndvdual snusods n h Fourr srs.
5 lrna Fourr Srs Rrsnaon Rmmbrng ha w may wr and also and lng w oban cos( ( ( ( x( x( < > f f f x( smly or x( lrna Fourr Srs Rrsnaon: Dscr For a dscr rodc sgnal, wh h nw noaon Th roof s gvn n h x on ag 33. > < f f cos( f f n x(n
6 Rsons o Fourr Srs Inu Now l s aly a connuous nu x( o an LTI sysm wh frquncy rsons H( and fnd h ouu y(: x( Du o lnary, w can dsrbu ovr h sum and ull ou h consans. Th rsul s a scald sum of h ouu gnrad by ach ndvdual comlx xonnal. Snc ach has corrsondng ouu H(, y( H( Drmnng Fourr Srs Coffcns W now gv formula for h Fourr srs coffcns for a rodc sgnal of rod : m For connuous sgnals m x( d x(, œ Rals: For dscr sgnals x(n x(n, n œ Ingrs: m mn x(n n Th xboo rovds a valdaon of hs formula on ag 36, bu hr drvaon wll b nuv onc w hav covrd Fourr ransforms.
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