Chapr 3: Fourir Rprsnaion of Signals and LTI Sysms Chih-Wi Liu
Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Fourir Sris Coninuous-im Priodic Signals Discr-im Nonpriodic Signals Fourir Transform Coninuous-im Nonpriodic Signals Propris of Fourir rprsnaions Linariy and Symmry Propris Convoluion Propry 2 Lc 3 - cwliu@wins..ncu.du.w
Oulin Diffrniaion and Ingraion Propris Tim- and Frquncy-Shif Propris Finding Invrs Fourir Transforms Muliplicaion Propry Scaling Propris Parsval Rlaionships Tim-Bandwidh Produc Dualiy 3 Lc 3 - cwliu@wins..ncu.du.w
Inroducion In his chapr, w rprsn a signal as a wighd suprposiion of complx sinusoids. AKA Fourir analysis Th wigh associad wih a sinusoid of a givn frquncy rprsns h conribuion of ha sinusoid o h ovrall signal. Four disinc Fourir rprsnaions: 4 Lc 3 - cwliu@wins..ncu.du.w
( ( ( ( ( ( ( H d h d h h x y Frquncy Rspons of LTI Sysm 5 Th rspons of h LTI sysm o a sinusoidal inpu : H{x(= }= H( For discr-im cas, h rspons of h LTI sysm o a sinusoidal inpu n is H{xn= n }= n H( LTI Sysm h( x ( ( ( H h d consan Dpndn on, bu indpndn on h H ( ( ( n n n H h h n h n x n y Dpndn on, bu indpndn on n
Frquncy Rspons of LTI Sysm Frquncy rspons of a coninuous-im LTI sysm x( or xn LTI Sysm y( or yn H( h( d H ( h Frquncy rspons of h LTI sysm can also b rprsnd by H( H( arg{ H( } Magniud rspons Phas rspons H ( arg{ H ( } 6
Exampl 3. RC Circui Sysm Th impuls rspons of h RC circui sysm is drivd in Exampl.2 as / RC Find an xprssion for h frquncy rspons, and plo h h ( u ( RC magniud and phas rspons. <Sol.> Frquncy rspons: H RC u d RC 7 RC RC RC RC RC RC RC Magniud rspons: H RC 2 RC 2 Phas rspons: arg d H arcanrc Lc 3 - cwliu@wins..ncu.du.w H Low-pass filr
Anohr Maning for Frquncy Rspons Th ignfuncion of h LTI sysm (: x( ( LTI Sysm h( x n n Th ign-rprsnaion of h LTI sysm H ( H y( ( ignvalu Indpndn of y n n n n H H 8 By rprsning arbirary signals as wighd suprposiion of ignfuncion, hn M M x a y( H{ x( } a H ( h wighs dscrib h signal as a funcion of frquncy. (frquncy-domain rprsnaion Muliplicaion in frquncy domain, c.f. convoluion in im-domain
Fourir Analysis Non-priodic signals hav (coninuous Fourir ransform rprsnaions, whil priodic signals hav (discr Fourir sris rprsnaions. Why Fourir sris rprsnaions for Priodic signals Priodic signal can b considrd as a wighd suprposiion of (priodic complx sinusoids (using priodic signals o consruc a priodic signal Rcall ha h priodic signal has a (fundamnal priod, his implis ha h priod (or frquncy of ach componn sinusoid mus b an ingr mulipl of h signal s fundamnal priod (or frquncy in frquncy-domain analysis, h wighd complx sinusoids loo li a discr sris of wighd frquncy impuls Fourir sris rprsnaion Qusion: Can any a priodic signal b rprsnd or consrucd by a wighd suprposiion of complx sinusoids? 9 Lc 3 - cwliu@wins..ncu.du.w
Approximad Priodic Signals n Suppos h signal xˆ n A is approximad o a discr-im priodic signal xn wih fundamnal priod N, whr = 2/N. ( N n Nn n 2n n n Sinc, hr ar only N n disinc sinusoids of h form :.g. =,,, N- Accordingly, w may rwri h signal as xˆ n N A n DTFS ˆ( A x For coninuous-im cas, w hn hav, whr = 2/T is h fundamnal frquncy of priodic signal x( Alhough is priodic, is disinc for disinc Hnc, an infini numbr of disinc rms, i.. xˆ( A FS Lc 3 - cwliu@wins..ncu.du.w
Approximaion Error Man-squar rror (MSE prformanc: N 2 MSE xn xn d N n T 2 MSE x ( x ( d T W s h wighs or cofficins A such ha h MSE is minimum Th DTFS and FS cofficins (Fourir analysis achiv h minimum MSE (MMSE prformanc. Lc 3 - cwliu@wins..ncu.du.w
Fourir Analysis Why Fourir ransform rprsnaions for Non-priodic signals 2 Using priodic sinusoids (h sam approach o consruc a nonpriodic signal, hr ar no rsricions on h priod (or frquncy of h componn sinusoids hr ar gnrally having a coninuum of frquncis in frquncy-domain analysis Fourir ransform rprsnaion Fourir ransform: Coninuous-im cas xˆ( ( 2 Discr-im cas d FT n xˆ n ( d 2 DTFT Frquncis sparad by an ingr mulipl of 2 ar idnical xˆ n xˆ( A N A n FS DTFS
Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Coninuous-im Priodic Signals Discr-im Nonpriodic Signals Fourir Transform Coninuous-im Nonpriodic Signals Propris of Fourir rprsnaions Linariy and Symmry Propris Convoluion Propry 3 Lc 3 - cwliu@wins..ncu.du.w
Discr-Tim Fourir Sris (DTFS Th DTFS-pair of a priodic signal xn wih fundamnal priod N and fundamnal frquncy =2/N is xn DTFS; N xn N x n N n n n Th DTFS cofficins ar calld h frquncy-domain rprsnaion for xn Th valu drmins h frquncy of h sinusoid associad wih Th DTFS is xac. (Any priodic discr-im signal can b dscribd in rms of DTFS cofficins xacly Th DTFS is h only on of Fourir analysis ha can b valuad and manipulad in compur for a fini s of N numbrs. 4 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.2 DTFS Cofficins Find h frquncy domain rprsnaion of h signal dpicd in Fig. 3.5. <Sol.>. Priod: N = 5 o = 2/5 2. Odd symmry W choos n = 2 o n = 2 3. Fourir cofficin: 5 2 x n 2 n/5 n2 4 / 5 2 / 5 2 2 / 5 2 x x x x x 4 / 5 5 2 2 2 /5 2 /5 { } 5 5 { sin( 2 / 5} 5 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.2 (coni. If w calcula using n = o n = 4: 8 / 2 / 5 4 2 / 5 3 6 x x x x / 5 x 4 5 5 2 2 / 5 8 / 5 8 /5 2 2 /5 2 /5 5 2 2 { sin( 2 / 5} 5 2 2 /5 2 /5 { } sinc 5 Th sam xprssion for h DTFS cofficins!!! 6 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.2 (coni. Evn funcion Magniud spcrum of x n arg Odd funcion arg Phas spcrum of xn 7 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.3 Compuaion by Inspcion Drmin h DTFS cofficins of xn = cos (n/3 +, using h mhod of inspcion. <Sol.>. Priod: N = 6 o = 2/6 = /3 2. Using Eulr s formula, xn can b xprssd as ( n ( n 3 3 n n 3 3 xn (3.3 2 2 2 3. Compar Eq. (3.3 wih h DTFS of Eq. (3. wih o = /3, wrin by summing from = 2 o = 3: xn 3 n /3 2 2 2 n/3 n/3 n/3 2 2 n/3 3 n 8 DTFS; 3 xn /2, /2,, ohrwis on 2 3 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.4 Find h DTFS cofficins of h N-priodic impuls rain xn nln. l <Sol.>. Priod: N. 2. By (3., w hav N n2 / N N n n N 9 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.6 Find h DTFS cofficins for h N-priodic squar wav givn by <Sol.>. Priod = N, hnc o = 2/N 2. I is convnin o valua DTFS cofficins ovr h inrval n = M o n = NM. 2 x n,, M n M M n N M o o 3. For =, N, 2N,, w hav M 2M,, N, 2N, N N nm For, N, 2N,, w hav N M nm N N M M n n x n nm M (2M n Lc 3 - cwliu@wins..ncu.du.w N,, N nm N, 2N,
Exampl 3.6 (coni. M M (2M,, N, 2 N,... N 2M /2 2M 2M /2 2M /2 /2 /2 /2 N N sin 2M / 2 Th numraor and dnominaor of, abov Eq. ar dividd by 2 N sin / 2, N, 2 N, Evn symmry 2 Th DTFS cofficins for h squar wav, assuming a priod N = 5: (a M = 4. (b M = 2.
Symmry Propry of DTFS Cofficins If = -, i is insruciv o considr h conribuion of ach rm in N xn n of priod N Assum ha N is vn, so ha N/2 is ingr. o = 2/N Rwri h DTFS cofficins by ling rang from N/2 + o N/2, i.. x n N / 2 N / 2 Dfin nw s of cofficins B n N /2 mn mn n xn N/2 2m m 2 N /2 N n m m n / 2cos 2 cos,, N /2 2,, 2,, N /2 m N /2 xn B cos( n 22 Lc 3 - cwliu@wins..ncu.du.w A similar xprssion may b drivd for N odd.
Exampl 3.7 Th conribuion of ach rm in DTFS sris o h squar wav may b illusrad by J dfining h parial-sum approximaion o xn as x J n B cos( n whr J N/2. This approximaion conains h firs 2J + rms cnrd on = in h squar wav abov. Assum a squar wav has priod N = 5 and M = 2. Evalua on priod of h Jh rm and h 2J + rm approximaion for J =, 3, 5, 23, and 25 <Sol.> J = J = 3 23 Lc 3 - cwliu@wins..ncu.du.w
J = 5 J = 23 J = 25 Th cofficins B associad wih valus of nar zro rprsn h lowfrquncy or slowly varying faurs in h signal, whil h cofficins associad wih valus of nar N/2 rprsn h high frquncy or rapidly varying faurs in h signal. 24 Lc 3 - cwliu@wins..ncu.du.w
Fourir Sris (FS Th DT-pair of a priodic signal x( wih fundamnal priod T and fundamnal frquncy =2/T is 25 x ( FS; x T x( d T a on priod of x( Th FS cofficins ar calld h frquncy-domain rprsnaion for x( Th valu drmins h frquncy of h sinusoid associad wih Th infini sris in x( is no guarand o convrg for all possibl signals. Suppos w dfin xˆ( approach o? x( Lc 3 - cwliu@wins..ncu.du.w If x( is squar ingrabl, hn T T x x 2 d a zro powr in hir diffrncs.
Rmars A zro MSE dos no imply ha h wo signals ar qual poinwis. Dirichl s condiions:. x( is boundd 2. x( has a fini numbr of maximum and minima in on priod 3. x( has a fini numbr of disconinuiis in on priod Poinwis convrgnc of ˆx and x( is guarand a all xcp hos corrsponding o disconinuiis saisfying Dirichl s condiions. If x( saisfis Dirichl s condiions and is no coninuous, hn ˆx convrgs o h midpoin of h lf ad righ limis of x( a ach disconinuiy. 26 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.9 Drmin h FS cofficins for h signal x(. <Sol.>. Th priod of x( is T = 2, so o =2/2 =. 2. Ta on priod of x(: x( = 2, 2. Thn 2 2 4 2 2 2 2 d d 2 2 4 2 2 2 4 2 4 2 Th Magniud of h magniud spcrum of x( 27 Lc 3 - cwliu@wins..ncu.du.w Th phas of h phas spcrum of x(
Exampl 3. Drmin h FS cofficins for h signal x( dfind by x 4l <Sol.>. Fundamnal priod of x( is T = 4, ach priod conains an impuls. 2. By ingraing ovr a priod ha is symmric abou h origin, 2 < 2, o obain : 2 /2 4 2 4 d 3. Th magniud spcrum is consan and h phas spcrum is zro. l 28 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3. Compuaion by Inspcion Drmin h FS rprsnaion of h signal x 3cos / 2 / 4 <Sol.>. Fundamnal frquncy of x( is o = 2/4= /2, so T = 4. 2. Rwri h x( as /2 /4 /2 /4 3 3 x 3 2 2 2 3. Compar wih 3 /4, 2 /2 x ( 3 /4, 2, ohrwis /4 /2 /4 /2 /4 29 Lc 3 - cwliu@wins..ncu.du.w /4
Exampl 3.2 Invrs FS Find h (im-domain signal x( corrsponding o h FS cofficins Assum ha h fundamnal priod is T=2. <Sol.>. Fundamnal frquncy: o = 2/T=. Thn /2 /2 x ( x /2 /2 x / 2 / 2 / 2 l l l / 2 l / 2 / 2 / 2 / 2 / 2 / 2 3 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.3 Drmin h FS rprsnaion of h squar wav: <Sol.>. Th priod is T, so h fundamnal frquncy o = 2/T. 2. W considr h inrval T/2 T/2 o obain h FS cofficins. Thn ( For, w hav (2 For =, w hav T /2 T x d d T 2T TT /2 TT d T T T T, T 3 2 T 2sinT T T 2 T T,, By mans of L Hôpial s rul 2sinT 2T lim T T 2sin T T
Exampl 3.3 (coni. Figur 3.22 Th FS cofficins,, 5 5, for hr squar wavs. (a T o /T = /4. (b T o /T = /6. (c T o /T = /64. 32 Lc 3 - cwliu@wins..ncu.du.w
Sinc Funcion sinc( u sin( u u Maximum of sinc(u is uniy a u =, h zro crossing occur a ingr valus of u, and h ampliud dis off as /u. Th porion of sinc(u bwn h zro crossings a u = is nown as h mainlob of h sinc funcion. Th smallr rippls ousid h mainlob ar rmd sidlobs 33 Lc 3 - cwliu@wins..ncu.du.w
Mor on h FS Pairs 34 Th original FS pairs ar dscribd in xponnial form: L s considr h rigonomric form For ral-valud signal x(:, and ( x arg arg ( ( ( x arg arg arg ( arg ( sin( sin(arg cos( cos(arg 2 arg cos( 2 ( ( x ( cos( sin( x B B A Rwri h signal as w hav
Trigonomric FS Pair for Ral Signals x ( B B cos( A sin( whr B, B 2 cos(arg 2 R{ } A 2 sin(arg 2 Im{ } Or, if w us rigonomric FS rprsnaion for a ral-valud priodic signal x( wih priod T, hn T 2 T 2 T B xd ( T B x( cos( d T A x( sin( d T 35 Lc 3 - cwliu@wins..ncu.du.w
Exampl 3.5 L us find h FS rprsnaion for h oupu y( of h RC circui in rspons o h squar-wav inpu dpicd in Fig. 3.2, assuming ha T o /T = ¼, T = s, and RC =. s. <Sol.>. If h inpu o an LTI sysm is xprssd as a wighd sum of sinusoids (ignfuncions, hn h oupu is also a wighd sum of sinusoids. 2. Inpu: 3. Oupu: FS; 2sin T y x Y H T y H 4. Frquncy rspons of h RC circui: / RC H / RC 5. Subsiuing for H( o wih RC =. s and 36 o = 2, and T o /T = ¼ Y 2 sin / 2