Sgnals and Sysems Texbooks: Fundamenals of sgnals and sysems usng he web and Malab Auhors: Edward W. Kamen and Bonne S. Heck Prence-Hall Inernaonal, I

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1 教育部改善師資經費補助 正修科技大學 14 年度教師編纂教材成果報告 ************************** 信號與系統 ************************** 團體 個人 單 位 : 電機系 單位主管 : ( 簽章 ) 職 姓 稱 : 副教授 名 : 廖炳松 執行期間 :14 年 3 月 1 日至 14 年 6 月 3 日 教評會審查期間 : 學年度第次 ( 年月日 ) 13- 信號與系統 1

2 Sgnals and Sysems Texbooks: Fundamenals of sgnals and sysems usng he web and Malab Auhors: Edward W. Kamen and Bonne S. Heck Prence-Hall Inernaonal, Inc. Web se: hp://users.ece.gaech.edu/~bonne/book/ Domesc dealer: Chwa books Corp. Lecure noe compler: Png-Sung Lao Chaper 1 Fundamenal Conceps 1.1 Sgnals and Sysems Sgnals: x() s a real-valued, or scalar-valued, funcon of he me varable. For nsance, f ( ) = sn(w) The represenaon of a sgnal may be descrbed by eher a connuous-me sgnal or a se of sample values. For easly undersandng he composon of a sgnal, he sgnal may be n erms of he frequency specrum hrough Fourer ransform. Sgnal Processng Sgnal processng plays an mporan role n eher he exracon of he nformaon carred n a sgnal or he reconsrucon of a sgnal whch has been corruped by spurous sgnals of nose. Reconsruc he x() from m() (esmaon/flerng) by cancelng he nose n() m( ) = x( ) + n( ) Sysems A sysem s an nerconnecon of componens wh ermnals or access pors hrough whch energy and nformaon can be appled or exraced. A mahemacal model of a sysem s usually an dealzed represenaon of he sysem. There are wo ypes basc of mahemacal models: one s npu/oupu represenaon; he oher s sae model. Four ypes of npu/oupu represenaons are suded here, 1. The npu/oupu dfferenal equaon or dfference equaon, 13- 信號與系統

3 . The convoluon model, 3. The Fourer ransform represenaon, 4. The ransfer funcon represenaon. 1. Connuous-Tme Sgnals A sgnal x() s sad o be a connuous-me sgnal or analog sgnal when me varable akes s values from he se of he real number. Sep funcon The un-sep funcon of u() s descrbed mahemacally by 1, u () =, < The magnude of un-sep funcon u() s equal o 1 for all. Ramp funcon The un-ramp funcon of r() s descrbed mahemacally by, r () =, < Noe ha for, he slope f r() s 1. Un-mpulse funcon δ() The un mpulse funcon, also called he dela funcon or he Drac funcon, s defned n generalzed form by δ () =, ε ε δ ( ) = 1, for any real number ε > Perodc sgnals A connuous-me sgnal x() s perodc wh perod T f x( + T) = x( ), for all. Noe ha he fundamenal perod s he smalles posve number T whch sasfes he foregong defnon of perodc sgnals. Tme-Shf sgnals Gven a connuous-me sgnal, x(), he shfed verson of x() usually s denoed as x(- 1 ) or x(+ 1 ) where 1 >. The sgnal of x(- 1 ) s shfed o he rgh by 1 seconds and he sgnal of x(+ 1 ) s shfed o he lef by 1 seconds. 13- 信號與系統 3

4 Connuous and Pecewse-Connuous Sgnals + x() s dsconnuous a 1 f x( 1 ) x( 1 ). + x() s connuous a 1 f x( 1 ) = x( 1 ). x() s connuous sgnal f s connuous a all pons. Remark: connuous >> connuous-me sgnal connuous >> connuous-me sgnal s connuous as a funcon of. A connuous sgnal x() s sad o be pecewse connuous f s connuous a all pons excep a a fne or counably nfne collecon of pons, = 1,, 3,... Dervave of a connuous-me sgnal A connuous-me sgnal x() s sad o be dfferenable a pon 1 f s ordnary dervave dx d () = 1 x( 1+ h) x( 1) = lm h h has a lm as h->. Generalzed dervave of x() (f x() dsconnuous a 1 ) Pecewse connuous sgnal x() may have a dervave n he generalzed sense such as dx() + + [ x( 1 ) x( 1) 1] δ ( 1 ) d Dscree-Tme Sgnals A dscree-me sgnal s a sgnal ha s a funcon of he dscree-me varable n ; n oher words, a dscree-me sgnal has values only a he dscree-me pons = n, where n s neger number. The waveform of x[n] s usually depced by sem plo. Samplng One of he mos common ways n whch dscree-me sgnals arse s n samplng connuous-me sgnals. x[ n] = x( ) = x( nt) where T s he samplng perod. = nt Noe ha nonunform samplng s somemes ulzed n praccal applcaons bu s no consdered here. Dscree-me un sep uncon 13- 信號與系統 4

5 1, n=, 1,... un [ ] =, n=-1, -,... Dscree-me Ramp funcon n, n=, 1,... rn [ ] =, n=-1, -,... Dscree-me un-pulse funcon δ[n] 1, n= δ[ n] =, n Dscree-Tme Perodc Sgnals x[n + r] = x[n], for all neger n and r s called he perod (fundamenal perod). For nsance, x[n] = Acos(Ωn +θ ) The sgnal s perodc f Acos[Ω(n+r) +θ ]= Acos(Ωn +θ ). In oher words, Ωr=πq for some neger q. 1.4 Examples of Sysems Four examples of connuous-me sysems RC crcu dy() 1 C + y () = () = x () d R Assumpon : nal value y( ), x() = u() 1 λ C + (/ RC)( λ ) y () = e d ( / RC) = y ( ) R[1 e ], Car on a level surface 13- 信號與系統 5

6 d y() M + k () () f y = x d Le v() = dy()/ d Thus, dv() M + kv f () = x () d Mass-Sprng-Damper sysem d y() M + Dy() + Ky() = x() d Smple pendulum d θ () I + MgLsn θ ( ) = Lx( ) d If he magnude of he angle θ () s small, so ha sn θ () s approxmaely equal o θ () and he foregong equaon can be approxmaed by d θ () I + MgLθ () = Lx() d 13- 信號與系統 6

7 Example of dscree-me sysem Repaymen of he a bank loan The npu x[n] s he amoun of he loan paymen n he nh monh, and he oupu y[n] s he balance of he loan afer he nh monh, and I s he yearly neres rae. I yn [ ] 1 + yn [ 1] = xn [ ], n=, 1,, Basc Sysem Properes Causaly A sysem s sad o be causal f for any me 1, he oupu response y( 1 ) resulng from he npu sgnal x() s only dependen on values of he npu x() for > 1. Memoryless sysem and memory sysem A causal sysem s memoryless or sac sysem f for any me 1, he value of he oupu a me 1 s depends only he value of he npu a me 1. A causal sysem ha s no memoryless s sad o have memory. Lneary A sysem s sad lnear f s boh addve and homogeneous. x () y () 1 1 x () y () addve: x1() + x() y1() + y() homogeneous: ax () ay (), where a s a scalar 1 1 lneary: a x () + a x () a a y () + a y (), where a and a are any real scalar A sysem ha s no lnear s sad o be nonlnear. Noe ha he sysem analyss on more wdely used sysems are based on lnearzaon mehod. Tme-nvarance The sysem s sad o be me-nvaran f for any npu x() and any me varable 1, he response o he shfed npu x(- 1 ) s equal o y(- 1 ), where y() s he response o x() wh zero nal energy. A sysem s me varyng or me varan f s no me nvaran. 13- 信號與系統 7

8 Fne dmensonaly Gven a connuous sysem wh npu x() and oupu y(), he sysem s fne dmensonal f for some posve neger N and nonnegave M, can be wren n he form ( N) (1) ( N 1) (1) ( M) y = f y y y x x x ( ) ( ( ), ( ),..., ( ), ( ), ( ),... ( ), ), where N 1, M > Gven a dscree-me sysem wh npu x[n] and oupu y[n], he sysem s fne dmensonal f for some posve neger N and nonnegave M, can be wren n he form yn [ ] = f( yn [ 1], yn [ ],..., yn [ N], xn [ ], xn [ 1],..., xn [ M], n), w here N 1, M > A connuous-me sysem wh memory s nfne dmensonal f s no fne dmensonal. The examples of nfne dmensonal sysem are ofen proved by he law of conradcon. For nsance, sysem wh me delay as gven below, Infne dmensonal sysem-1 dy() + ay( 1) = x( ) d In pracce, s seldom possble o express he soluon of a delay-dfferenal equaon n analyc form. Infne dmensonal sysem- Consder he dscree-me sysem wh he gven npu/oupu relaonshp, n 1 1 = yn [ ] = x [ ] n yn [ ] = x[] + ( ) x[1] xn [ 1] + xn [ ] n n = x[ n] + x[ n 1] ( ) x[ n ( n 1)] + x[ n n] 1 n 1 n Because he las erm n he rgh-sde canno be specfed by x[n-m] n whch M s a deermnsc non-negave neger, hs dscree-me dfference equaon s nfne dmensonal. 13- 信號與系統 8

9 Lnear fne-dmensonal sysem ( N) (1) ( N 1) (1) ( M) y ( ) = f( y( ), y ( ),..., y ( ), x( ), x ( ),... x ( ), ), where N 1, M > N-1 M () () a( ) y ( ) b( ) y ( ) = = = + y[ n] = f( yn [ 1], yn [ ],..., yn [ N], xn [ ], xn [ 1],..., xn [ M], n), N 1 = a ( n) y[ n ] + b( n) x[ n ] = 1 = where N 1, M > M Lnear me-nvaran fne-dmensonal sysem ( N) (1) ( N 1) (1) ( M) y ( ) = f( y( ), y ( ),..., y ( ), x( ), x ( ),... x ( ), ), N-1 M () () a( ) y ( ) b( ) x ( ) = =, a ( ) and b ( ) are consans, and N 1, M > = + y[ n] = f( yn [ 1], yn [ ],..., yn [ N], xn [ ], xn [ 1],..., xn [ M], n), N 1 = a ( n) y[ n ] + b( n) x[ n ] M = 1 =, a ( n) and b ( n) are consans, and N 1, M > 13- 信號與系統 9

10 Sgnals and Sysems Texbooks: Fundamenals of sgnals and sysems usng he web and Malab Auhors: Edward W. Kamen and Bonne S. Heck Prence-Hall Inernaonal, Inc. Web se: hp://users.ece.gaech.edu/~bonne/book/ Domesc dealer: Chwa books Corp. Chaper Sysem defned by Dfferenal or dfference equaons.1 Lnear Inpu/Oupu Dfferenal Equaons wh Consan Coeffcens N-1 M ( N) () () + = = = y () a () y () b() x () Here s assumed ha, a ( ) and b ( ) are real consans, and N 1, M > In oher words, N-1 M ( N) () () + = = = y () a y () bx () Here s assumed ha, a and b are real consans, and N 1, M > Inal condon To solve he foregong equaon, s necessary o specfy he N nal condons, eher (1) ( N 1) (1) ( N 1) y(), y (),... y () for mos cases or y( ), y ( ),... y ( ) f he Mh dervave of he npu x() conans an mpulse kδ() or a dervave of an mpulse. Frs-order case Canoncal formaon dy() + ay() = bx() (.4) d Is oupu response y() for he nal condon y() and npu x() s gven by ( a ) a( λ ) y () y() e e bx( λ) dλ, (.5) or = + ( a) a( λ ) y () y() e e bx( λ) dλ, (.6) = 信號與系統 1

11 READING SKILL dy() dx() + ay() = b1 + bx() (.7) d d d[ y( ) bx 1 ( )] + ay() = b x(), d d[ y( ) bx 1 ( )] + ay [ () bx 1 ()] = bx () abx 1 (), d dq() + aq() = ( b ab1) x() d where q() = y() b x() 1 (.11) Ths gves ha ( a) a( λ ) q () = q() e + e ( b ab1) x( λ) dλ, (.1) Thus, y() can be yelds by subsung y() = q() + bx 1 () no (.1) 1 1 ( a) a( λ ) ( b ab1) λ λ (.13) y() = bx () + [ y() bx()] e + e x( ) d, Noe ha f he npu x() s he un-sep funcon u(), he response of oupu y() n (.13) s expressed as ( a) a( λ ) 1 1 ( a ) a( λ ) 1 ( b ab1) + y() = bx() + [ y() bx()] e + e ( b ab) x( λ) dλ, ( a ) a a( λ ) = bu 1 ( ) + y( ) e + ( b ab1) e ( 1/ a)[1 e ] dλ, =. Sysem Modelng In buldng he npu/oupu equaon of a sysem, one mus be famlar wh he laws of physcs and he mehodologes of mahemacs. Elecrcal crcus 1 = bu( ) + y( ) e + e dλ, Ressor: v ( ) = R ( ) dv() 1 Capacor : ( ) = C or v( ) = ( λ) dλ d C - d() 1 Inducor : v() = L or () = v( λ) dλ d L 信號與系統 11

12 where v()and () are he ermnal volage across he componen such as R, L, C. Mechancal sysems There are hree ypes of forces ha ress he ranslaon moon. d y() Inera force x I( ): xi( ) = M d Dampng force x (): x () = k d d d d y() d Sprng force x (): x () = k y() s d s D alember s prncple Any fxed me he sum of all exernal forces appled o a body n a gven drecon and all he forces ressng he moon n ha drecon mus be equal o zero. Roaonal mechancal sysems In analogy wh he hree ypes of forces ressng ranslaon moon, here are hree ypes of forces ressng roaonal moon. d θ () Inera orque x I( ): xi( ) = I d 13- 信號與系統 1

13 Dampng orque x ( ): x ( ) = k d d d d θ () d Sprng orque x ( ): x ( ) = kθ ( ) s d s.3 Lnear Inpu/Oupu Dfference Equaons wh Consan Coeffcens N 1 yn [ ] + ayn [ ] = b( nxn ) [ ] M = 1 =, a and b are real consans, and N 1, M > (.35) The foregong dfference equaon s lnear, me-nvaran, causal and fne dmensonal. The soluon of (.35) can be compued recursvely as follows. N 1 M y[ n] = a y[ n ] + bx[ n ], n=,1,,... = 1 = For an nfne dmensonal dfference equaon s soluon canno be compued recursvely as above. Example: By consderng he frs-order lnear dfference equaon yn [ ] = ayn [ 1] + bxn [ ] wh nal condon y[]. Frs, seng n=1, n=, n=3 n (.39) gves y[1] = ay[] + bx[1] y[] = ay[1] + bx[] = a( ay[] + bx[1]) + bx[] = a y[] abx[1] + bx[] y[3] = ay[] + bx[3] = a( a y[] abx[1] + bx[]) + bx[3] 3 = a y[] + a bx[1] abx[] + bx[3] Fnally, can be seen ha n 1 n n yn [ ] = ( a) y[] + ( a) bxn [ ] n = 信號與系統 13

14 .4 Dscrezaon n Tme of Dfferenal Equaons The dscrezaon n me acually resuls n a dscree-me represenaon of he connuous-me sysem governed by he gven npu/oupu dfferenal equaon. Frs-order case dy() + ay() = bx() (.46) d The dervave of y() can be approxmaed by dy() y[( n + 1) T] y[ nt ] = (.48) d T = nt Inserng he approxmaon (.46) no (.45) gves y[( n+ 1) T] y[ nt] = ay[ nt ] + bx[ nt ] (.49) T Le xn [ ] = x ( ) and yn [ ] = y ( ) = nt = nt In erms of hs noaon, (.49) becomes y[( n+ 1)] y[ n] = ay[ n] + bx[ n] T Be careful, he samplng perod T n he denomnaor canno be omed here. Fnally, he dscree approxmaon o (.46) can be expressed as yn [ ] yn [ 1] = atyn [ 1] + btxn [ 1] or yn [ ] = (1 at) yn [ 1] + btxn [ 1] (.51) Second-order case Consder a lnear me-nvaran connuous-me sysem wh he second-order npu/oupu dfferenal equaon d y () () () + a dy 1 + a dx y = 1 + b () b x() (.6) d d d Dealng wh he above equaon, he followng approxmaons are needed o approxmae he frs-order and he second-order dervaves. 13- 信號與系統 14

15 dy() y[( n + 1) T ] y[ nt ] = d T = nt d y y n+ T y n+ T + y nt ( ) [( ) ] [( 1) ] [ ] = d T = nt Seng =nt n (.6) and usng he approxmaons gven above resul n he followng me dscrezaon of (.6). yn [ + ] yn [ + 1] + yn [ ] yn [ + 1] yn [ ] xn [ + 1] xn [ ] [ ] + b x[ n] + a 1 + ay n = b1 T T T Afer summarzng, yn at yn at at yn [ ] + ( 1 ) [ 1] + (1 1 + ) [ ] = btx[ n 1] + ( bt bt) x[ n ] Sysems Defned by Tme-Varyng or Nonlnear Equaons Frs-order me-varyng sysem dy() + a () y () = bx () () d The me-varyng propery of capacance C() s a resul of changng he poson of he delecrc. The charge-volage relaonshp of capacor s gven by q () = Cv () () c Thus, akng he dervave of boh sdes of he prevous equaon 13- 信號與系統 15

16 dq() vc () c() = C () Cv () c() d = d + The soluon of a lnear npu/oupu dfferenal equaon wh me-varyng coeffcens canno be obaned exacly by analyc form n general, bu can be obaned by numercal-soluon echnques. In general, due o he nonlneary f s no possble o derve an analyc expresson for he oupu response y() of a nonlnear sysem wh an nal condon y() and he npu x(). Smlarly, he response y() of a nonlnear sysem s ofen compued usng a numercal soluon echnques as he lnear npu/oupu dfferenal equaon wh me-varyng coeffcens does. The sysem behavor around he nomnal condons y norm () for a nonlnear sysem can be lnerzed no he lnearzed equaon wh respec o he nomnal funcons y norm () and x norm (). Tme-varyng dscree-me sysem y[ n] = f( yn [ 1], yn [ ],..., yn [ N], xn [ ], xn [ 1],..., xn [ M], n), N 1 = a ( n) y[ n ] + b( n) x[ n ] M = 1 =, a ( n) and b ( n) are consans, and N 1, M > For nsance, yn [ ] + an ( ) yn [ 1] = b( nxn ) [ ] + b( nxn ) [ 1] 1 Load-paymen process In ( ) yn [ ] 1 + yn [ 1] = xn [ ] 1 Nonlnear npu/oupu dfference equaons For nsance, yn [ ] = f( yn [ 1], xn [ ],[ n 1]) If he explc expresson of funcon f(.) s gven, he soluon can be obaned as he recurson process (program) for he lnear sysem does. 13- 信號與系統 16

17 Sgnals and Sysems Texbooks: Fundamenals of sgnals and sysems usng he web and Malab Auhors: Edward W. Kamen and Bonne S. Heck Prence-Hall Inernaonal, Inc. Web se: hp://users.ece.gaech.edu/~bonne/book/ Domesc dealer: Chwa books Corp. Chaper 3 Convoluon Represenaon 3.1 Convoluon Represenaon of Lnear Tme-Invaran Dscree-Tme Sysems Throughou hs secon, here s no nal energy n he dscussed lnear me-nvaran dscree sysem whch s also causal, bu no necessarly fne dmensonal. Le h[n] s he oupu response when a un pulse (mpulse) δ[n] s appled o he sysem wh no nal energy a me n=. If he npu sgnal s shf o rgh wh delay me =, he shfed un-pulse response wh respec o he same sysem shown n Fg. 3.1 s gven below. In general case, he npu sgnal x[n] can be expressed n he form 13- 信號與系統 17

18 x[ n] = x[] δ[ n] + x[1] δ[ n 1] + x[] δ[ n ] +... xn [ ] = x [ ] δ[ n ] = The response o x[] δ[ n ] s gven by xn [ ] = x[] δ[ n] + x[1] δ[ n 1] + x[] δ[ n ] +... y[ n] = x[ ] δ[ n ] h[ n ] = xhn [ ] [ ] (3.3) (3.4) By addve propery, he response o he sum gven by (3.3) mus be equal o he sum of he ndvdual sum y [n]. Thus he response o x[n] s yn [ ] = y[ n] = = xhn [ ] [ ], n (3.5) = 3. Convoluon of Dscree-Tme Sgnals In hs secon, he convoluon operaon s defned for arbrary dscree-me sgnals x[n] and v[n] ha are no necessarly zero for n<. The convoluon of x[n] and v[n] s defned by x[ n]* v[ n] = x[ ] v[ n ] = = vxn [ ] [ ] = (3.7) If x[n] s zero for n<, hs convoluon operaon s gven by, n < xn [ ]* vn [ ] = xvn [ ] [ ], n = Smlarly, f v[n] s zero for n<, he convoluon operaon s gven by 13- 信號與系統 18

19 , n < xn [ ]* vn [ ] = xvn [ ] [ ], n = The graphcal llusraon of he convoluon operaon s gven below. 13- 信號與系統 19

20 Compue he convoluon by abulaon for specal cases. For nsance, suppose ha x[n]=u(n-n) for all n<n and v[n] = u(n-m) for all n< M, where N and M are posve or non-negave negers. 13- 信號與系統

21 The convoluon of x[n] and v[n] wren n he form, n< ( N + M) yn [ ] = xn [ ]* vn [ ] N M = x[ vn ] [ ], n ( N+ M) = N can be compued by he mulplcaon of wo rows of x[n] and v[n] usng array srucure. xn [ ] xn [ + 1] xn [ + ] xn [ + 3]... vm [ ] xm [ + 1] xm [ + ] xm [ + 3] xnvm [ ] [ ] xn [ + 1] vm [ ] xn [ + ] vm [ ] xn [ + 3] vm [ ]... xnvm [ ] [ + 1] xn [ + 1] vm [ + 1] xn [ + ] vm [ + 1] xn [ + 3] vm [ + 1]... xnvm [ ] [ + ] xn [ + 1] vm [ + ] xn [ + ] vm [ + ] ym [ + N] ym [ + N+ 1] ym [ + N+ ] ym [ + N+ 3] Properes of he convoluon operaon Assocavy: x[ n]*( v[ n]* w[ n]) = ( x[ n]* v[ n])* w[ n] Commuavy: x[ n]* v[ n] = v[ n]* x[ n] Dsrbuve wh addon: x[ n]*( v[ n] + w[ n]) = ( x[ n]* v[ n]) + ( x[ n]* w[ n]) Shf propery: 13- 信號與系統 1

22 w[ n q] = x [ n]* v[ n] = x[ n]* v [ n] q where x [ n] and v [ n] are q- sep rgh shfs ( q- sep lef shfs) q of xn [ ] andvn [ ] whenq> ( q< ). q Convoluon wh un pulse: x[ n]* δ [ n] = x[ n] q Convoluon wh he shfed un pulse: x[ n]* δ [ n] = x[ n q] Example: gven a causal lnear me-nvaran dscree-me sysem wh un-pulse response h[n] and s nal energy beng zero, when he npu x[n] wh x[n]= for n < s appled o hs sysem, as shown n (3.5), he oupu response y[n] wll be yn [ ] = xhn [ ] [ ], n (3.3) = Snce h[n]= for n < (by causaly), h[n-]= for n <. n yn [ ] = xhn [ ] [ ], n = Noe ha he sysem s causal and he npu sgnal s zero before n=. q Noncausal sysems: yn [ ] = xn [ ]* hn [ ] = xhn [ ] [ ] = Snce s b-nfne sum, he resul of convoluon sum canno be evaluaed n a fne number of compuaons. 3.3 Convoluon Represenaon of Lnear Tme-Invaran Connuous-Tme Sysems For a causal lnear me-nvaran connuous-me sysem, he mpulse response h() could be deermned expermenally by applyng a large-amplude shor-duraon pulse (as an approxmaon o δ()), bu n pracce s usually no possble o apply such an npu o he sysem. 13- 信號與系統

23 The convoluon of a sysem h() wh an npu x() s denoed by y() = x()* h() = x( λ) h( λ) dλ Consder a causal lnear me-nvaran connuous-me sysem h() wh npu x()= for <, y () = x ()* h () = x( λ) h ( λ) dλ = x( λ) h ( λ) dλ+ x( λ) h ( λ) dλ = x( λ) h( λ) dλ ( by causaly), < = x( λ) h( λ) dλ, 3.4 Convoluon of Connuous-Tme Sgnals Gven wo connuous-me sgnals x() and v(), he convoluon of x() and v() s defned by x()* v() = x( λ) v( λ) dλ 13- 信號與系統 3

24 If x()= and v()= for <, he foregong equaon becomes x ()* v () = x( λ) v ( λ) dλ = x( λ) v ( λ) dλ+ x( λ) v ( λ) dλ = x( λ) v( λ) dλ ( by causaly), < = x( λ) v( λ) dλ, (3.34) The negral n (3.34) exss for all > f he funcons of x() and v() are negrable for all >; ha s, x( λ ) d λ < and v( λ ) d λ < for all > Graphcal llusraon for he convoluon of wo connuous-me sgnals 13- 信號與系統 4

25 13- 信號與系統 5

26 Properes of Convoluon for connuous-me sgnals Properes of he convoluon operaon Assocavy: x()*( v()* w()) = ( x()* v())* w() Commuavy: x()* v() = v()* x() Dsrbuve wh addon: x()*( v() + w()) = ( x()* v()) + ( x()* w()) Shf propery: w( c) = xc()* v() = x()* vc() where xc( ) and vc( ) are c - sec ond rgh shfs ( c - sec ond lef shfs) of x ( ) andv ( ) whenc> ( c< ). Convoluon wh un pulse: x()* δ () = x() Convoluon wh he shfed un pulse: x() * δ () = x( c) c Dervave propery: If he sgnal x() has an ordnary frs dervave, d [ x ( )* v ( )] = x ( )* v ( ) d d [ x ( )* v ( )] = x ( )* v ( ) d If he sgnals x() and v() have an ordnary frs dervaves, d [ x( )* v( )] = x ( )* v ( ) d Inegraon propery ( 1) ( 1) Le x () and v () denoe he negrals of he sgnals x() and v() ; ha s ( x 1) () = x( λ) dλ and ( v 1) () = v( λ) dλ The convoluon of ( 1) ( 1) x and v () () s represened by ( 1) ( 1) ( 1) ( x()* v()) = ( x ()* v()) = ( x()* v ()) Proof: 13- 信號與系統 6

27 ( x ( )* v ( )) = λ= + τ=- ( 1) [( x λ)*( v λ)] dλ + = [ x( τ) v( λ τ) dτ ] dλ λ= τ=- = { x( τ ) [ v( λ τ) dλ ] } dτ λ= (A-1) Because of ( 1) ( x ( )* v ( )) ( v 1) () = v( λ) dλ + ( 1) { ( ) ( )} τ =- = x τ v τ dτ = + τ=- + τ=- τ { x( τ) [ v( λ) dλ ] } dτ λ= = { x( τ ) [ v( λ τ) dλ ] } dτ λ= (A-) Comparng he (A-1) wh (A-) yelds ( 1) ( 1) ( x( )* v( )) = ( x( )* v ( )) The deal of provng ( 1) ( 1) ( x( )* v( )) ( x ( )* v( )) = s lef o he reader. Example: gven a causal lnear me-nvaran connue-me sysem wh un-mpulse response h() and s nal energy beng zero, when he npu wh x() = for < s appled o hs sysem, he oupu response y[n] wll be y() = x()* h() = x( λ) h( λ) dλ, (3.46) By he commuavy of convoluon, y() = x()* h() = h( λ) x( λ) dλ, (3.47) Noe ha he sysem s causal and he npu sgnal s zero before =. Le g() s he oupu response of he sysem when he npu x() s he un-sep funcon u() wh no nal energy n he sysem a me =. From he defnon of convoluon, g () = h ()* u () Dfferenang he boh sdes of he foregong equaon and usng he dervave propery of convoluon gves 13- 信號與系統 7

28 g () = h ()* u () = h ( )* δ ( ) = h ( ) Hence he mpulse response h() of a lnear sysem s equal o he dervae of he sep response g() of he sysem. Also, he sep response of he sysem can be sad o be he negral of he mpulse response of he sysem. Noncausal sysems: y() = x()* h() = x( λ) h( λ) dλ 3.5 Numercal Convoluon Consder a causal lnear me-nvaran connuous-me sysem gven by he convoluon relaonshp y() = x()* h() = x( λ) h( λ) dλ For me nsan a nt, ha s, =nt y( nt ) = x( λ) h( nt λ) dλ The negral of he above equaon can be obaned by = = = = ( + 1) T λ = T ynt ( ) = x( λ) hnt ( λ) dλ Smlarly, = ( + 1) T λ = T xt ( ) hnt ( T) dλ ( + 1) T λ = T xt ( ) hnt ( T) (1) dλ Tx( T) h( nt T ) y( nt ) T h( T ) x(( n ) T ) In general, he approxmaon s more accurae he smaller he duraon T s. 13- 信號與系統 8

29 3.6 Lnear Tme-Varyng Sysems For dscree-me me-varyng sysem 13- 信號與系統 9

30 I s IMPOSSIBLE o express he oupu response of a lnear me-varyng sysem as he convoluon of he npu sgnal wh he mpulse response of hs lnear me-varyng sysem. Gven a causal me varyng sysem, le h[n,] denoe he oupu response when he un pulse s appled o he sysem. By causaly, hn [, ] =, n< For an npu sgnal x[n]= for n<, where n s neger, he oupu response y[n] resulng from x[n] wh no nal energy a me n= s gven by yn [ ] = x [ ] hn [, ] = n yn [ ] = x [ ] hn [, ], n = (3.69) Noe ha f he gven causal me-nvaran dscree-me sysem can be possbly and reasonably reduce o a causal me-nvaran dscree-me sysem, he foregong equaon can be rewren by n yn [ ] = x [ ] hn [ ], n = because of hn [, ] = hn [ ], for all n. For connuous-me me-varyng sysem Gven a causal me varyng sysem, le h(,λ) denoe he oupu response when he un mpulse s appled o he sysem. By causaly, h (, λ) =, < λ For an npu sgnal x()= for <, he oupu response y() resulng from x() wh no nal energy a me = s gven by y() = x( λ) h(, λ) dλ, (3.73) 13- 信號與系統 3

31 Noe ha f he gven causal me-nvaran connuous-me sysem can be possbly and reasonably reduce o a causal me-nvaran connuous-me sysem, he foregong equaon can be rewren by y() = x( λ) h( λ) dλ, because of h (, λ) = h ( λ), for all λ. 13- 信號與系統 31

32 Sgnals and Sysems Texbooks: Fundamenals of sgnals and sysems usng he web and Malab Auhors: Edward W. Kamen and Bonne S. Heck Prence-Hall Inernaonal, Inc. Web se: hp://users.ece.gaech.edu/~bonne/book/ Domesc dealer: Chwa books Corp. CHAPTER 4 THE FOURIER SERIES AND FOURIER TRANSFORM As wll be seen, he frequency specrum s used o fech ou he frequency counerpars of a real sgnal. In general, he frequency specrum s a complex-valued funcon of he frequency varable, and hus s usually specfed n erms of an amplude specrum and a phase specrum. Fourer seres s a powerful mahemacal ool o analyze he specrum of a perodc sgnal. Opposely, Fourer ransform s only suable o analyze he specrum of an aperodc ( a nonperodc) sgnal. 4.1 Represenaon of Sgnals n Terms of Frequency Componens READ SKILL: Wha s nroduced here does no nduce he orgnal dea abou he decomposon of a sgnal by Fourer seres. For a large class of sgnals, he orgnal me waveform can be synheszed (decomposed) by (no) he composon of snusodal sgnals. N x ( ) = Ak cos( ωk+ θk), - < < (4.1) k = 1 The characerscs or feaures of a sgnal gven n (4.1) can be suded n erms of he frequences, he ampludes, and he phases of he snusodal erms comprsng he sgnal. In parcular, he ampludes A k, k=1,..,n, are he major facors n deermnng he shape of he sgnal. NOTE THAT he drec componen of a sgnal s no ncluded n hs Secon. Examples Sum of Snusods x ( ) = Acos( ) + Acos(4 + π / 3) + Acos(8 + π / ), - < < 信號與系統 3

33 13- 信號與系統 33

34 Euler s formula j( ωk+ θk) e = jωk + θk + j jωk +θk cos( ) sn( ) j( k k) cos( j k k) Re[ e ω + ω + θ = θ ] Hence, N x ( ) = A cos( ω + θ ), k = 1 N k k k j( ωk θk) x () = Re[ A e + ], -< < k = 1 k In addon, s = a + jb, s = a jb yeld a = s+ s. Thus j( ωk+ θ ) A k k j( ωk+ θk) Ak j( ωk+ θk) Re[ Ak e ] = e + e Ak j k Defnng ck = e θ, yelds N N jωk jωk k k k= 1 k= 1 x () = ce + c e N j Then () k x = ce ω, < < k= N where c k s n he complex form as c k and c -k are equal. k θk nπ, for some neger n, and he amplude of DON T BE SILLY TO THINK A REAL-DOMAIN SIGNAL IN THE E 4. Fourer Seres Represenaon of Perodc Sgnals Le x() s a perodc sgnal wh perod T, ha s x ( ) = x ( + T), for all, < <. Then he sgnal can be expressed as a sum of complex exponenals jk x () = ce ω, < < (4.11) k = k where c s a real number and c k for k are n general complex numbers, and ω s he fundamenal frequency by ω =π/t. Obvously, he frequency conen of a perodc sgnal s lne specra. 13- 信號與系統 34

35 NOTE THAT he drec componen s ncluded because of k= beng consdered. In parcular, (4.11) s no equal o (4.1). The coeffcens of c k for k are compued usng he formula k T T () jkω jkω T (), c = x e d = x e d k=, -, -1,, 1,, (4.1) and he DC erm c of s gven by ck T = x() d, (4.13) The proof of (4.1) and (4.13) s lef o reader. I s dffcul o beleve ha a pulse ran sgnal wh corners can be represened as he sum of snusods ha are nfnely smooh funcons. I should be noed ha he fne sum x N () CAN BE CALCULATED BY TRUCATING THE EXPONENTIAL FOURIER SERIES DIRECTLY: N jk x () = ce ω, < < k= N k A perodc sgnal has a Fourer seres f sasfes he Drchle condons gven by 1. x() s absoluely negral over any perod; ha s a+ T a x() d <, for any a. x() has only a fne number of maxma and mnmum over any perod. 3. x() has only a fne number of dsconnues over any perod. IMPORTANT SENSE Fourer seres represenaon of a perodc sgnal can be n hree dfferen forms: (1) rgonomerc form, (3) compac rgonomerc form and (3) complex exponenal form. (1) x() = a + ( ancoskω+ bnsn kω) (LATHI, p. 595 rgonomerc form) () (3) k = 1 k ω k = 1 jk cke ω k = x ( ) = A+ A cos( k + θ ), - < < x() = k 13- 信號與系統 35

36 The rgonomerc represenaon of he pulse ran shown n Fg. 4.6 s 1 π = + k + < < (4.16) kπ ( k 1)/ ( ) ( cos( π [( 1) 1] ), x k = 1 k odd To approach he plo of x(), le x ( ) denoe he fne sum N 1 π x = + k + < < N N () ( k 1)/ ( cos( π [( 1) k = 1 kπ 1] ), k odd 13- 信號與系統 36

37 Gbbs phenomenon I can be seen he magnude of he overshoo s approxmaely equal o 9% a he corners of pulse ran. 13- 信號與系統 37

38 Parseval s heorm Le x() s a perodc sgnal wh perod T, he average power of he sgnals s T P = x () d = c k k = 4.3 Fourer Transform How does he aperodc sgnal be represened n erms of frequency conen? Is also rue o deal wh an aperodc sgnal usng Fourer seres? If no, wha do we do o ge he specrum conen of an aperodc sgnal? Fourer ransform, n conras o Fourer seres, s used o he above quesons. Agan le us dscuss he pulse ran of x() shown n Fg. 4.6, as he perod T s nfne, wll be one-second recangular pulse. I s known ha x() can be expressed n he form, jkω x () = ce, < < k = k where he coeffcens c k s equal o c k kω = k ± ± kω T ( )sn( ), = 1,,... and snce ω = π /T, he foregong equaon can be rewren by 13- 信號與系統 38

39 c k 1 kω = ± ± kπ ( )s n( ), k= 1,,... The plos of c k for T=, 5, 1 and nfne are dsplay below. I s obvous ha as T s nfne, he specrum of he one-sho pulse sgnal appears n connuous form. 13- 信號與系統 39

40 The defnon of Fourer ransform s gven by j ω e ω X( ) = x( ) d, - < ω< (4.36) and he nverse Fourer ransform s expressed n he form 1 jω x() = X( ω) e d, π (4.36A) A sgnals x() s sad o have a Fourer ransform n he ordnal sense f he negral n (4.36) converges (exss). In oher words, he sgnal x() s well-behaved and he sgnal x() s absoluely negrable, x () d < (4.37) The erm well-behaved means has a fne number of dsconnues, maxma, and mnma whn any fne negral of me. Excep ha mpulse sgnal, mos sgnals of neresng are well-behaved. Example 4.5 Consan Sgnal Consder he consan sgnal x()=1, -< <. Please fnd he Fourer ransform of hs consan sgnal. Noe ha he condon of (4.37) s no obeyed for consan sgnal. Example 4.6 Exponenal sgnal Now consder he exponenal sgnal b x() = e u(). Please fnd he Fourer ransform of hs consan sgnal. ( 1 X ( ω) = b jω, 1 X ( ω) =, b + ω + ω ω b 1 X ( ) = an ( ) ) Usng Euler s formula X( ω) = Re[ X( ω)] + jim[ X( ω)] 13- 信號與系統 4

41 Defnng R( ω) = Re[ X( ω)] I( ω) = Im[ X( ω)] Thus he recangular form of Fourer ransform for he sgnal x() s X( ω) = R( ω) + ji( ω) (4.39) R( ω) = x ( )cosωd I( ω) = x ( )snωd T he polar form of a complex funcon X ( ω ) s X( ω) = X( ω) exp( j X( ω)) (4.4) The relaonshp beween (4.39) and (4.4) can be conneced usng he followng formula. X R I ( ω) = ( ω) + ( ω) 1 I( ω) X ( ω) = an ( ) R( ω) For even sgnals x()= x(-) R( ω) = x ()cos ωd R( ω) = x ()cosωd I( ω) = x ( )snωd= For odd sgnals x()= -x(-) R( ω) = x ( )cosωd= I( ω) = x ()sn ωd I( ω) = x ()snωd Example 4.7 Recangular pulse 1, - τ < τ pτ () =, all oher ωτ Please compue s Fourer ransform. ( X ( ω) = sn( ) ) ω 13- 信號與系統 41

42 13- 信號與系統 4

43 NOTE THAT here are wo ypes of defnon abou snc funcon (hp://en.wkpeda.org/wk/snc_funcon). 1. In dgal sgnal processng and nformaon heory, he normalzed snc funcon s commonly defned by sn( π x) snc( x) = π x. In mahemacs, he hsorcal unnormalzed snc funcon (for snus cardnals), s defned by sn( x) snc( x) = x 4.4 Properes of he Fourer Transform I s well known x () Xω ( ) and v () Vω ( ). Lneary: ax() + bv() ax ( ω) + bv ( ω), for any real or complex scalars a and b. j c Lef or rgh shf n Tme: x( c) X( ω) e ω for any real scalar c. Tme scalng: 1 ω xa ( ) X( ) for any posve real scalar a. a a 1 ω xa ( ) X( ) for any nonzero real scalar a, eher posve or negave. a a Tme reversal: x( ) X( ω) f x() s a real-valued sgnal, he Fourer ransform can be rewren as x( ) X( ω) where s X ( ω ) he complex conjugae of X ( ω ). Mulplcaon by a power of n n n d x () ( j) X( ω) n dω 13- 信號與系統 43

44 Mulplcaon by a complex exponenal x e X jω () ( ω ω) Mulplcaon by a complex exponenal 1 x () cos( ω) [ X( ω+ ω) + X( ω ω)] j x ()sn( ω) [ X( ω+ ω) X( ω ω)] The above equaons can be proved by Euler s formula. Dfferenal n he me doman d n x n () ( j ω) X ( ω) n d Convoluon of wo sgnals n he me doman x ()* v () X( ω) V( ω) Inegral n he me doman X ( ω) x( λ) dλ + π X() δ( ω) jω In general, he negral of x() doesn have a Fourer ransform n he ordnary sense, bu does have he generalzed ransform. Mulplcaon of wo sgnals n he me doman x 1 1 () v () [ ( )* ( )] ( ) ( ) X V ω ω = X λ V ω λ d λ π π 13- 信號與系統 44

45 xve ()() jω d 1 jλ jω = { x( λ) e dλ} v( ) e d = π λ = 1 jλ jω = { x( λ) e v( ) e d} dλ π λ = = 1 jλ jω = x( λ){ ve ( ) e d} dλ π λ = = 1 j( ω λ) = x( λ){ ve ( ) d} dλ π λ = = 1 = x( λ) V( ω λ) d π λ λ = Parseval s heorem Dualy Propery x 1 1 ( ) v ( ) d ( ) ( ) ( ) ( ) X ω V ω d ω = V ω X ω d ω π π If v()= x(), x 1 1 ( ) d ( ) ( ) ( ) X X d ω ω ω X ω d ω π = π If x () Xω ( ) s sure, X() π x( ω) The dualy propery s easly proved by he nsuon of varables and he defnons of Fourer ransform and nverse Fourer ransform. 4.5 Generalzed Fourer Transform In he ordnary sense, here are no he Fourer ransform for consan sgnals, un-sep funcon and snusodal funcons because he absolue negral propery doesn hold for hem. x () d < Bu for he engneerng needs, he generalzed Fourer ransform s consdered here. 13- 信號與系統 45

46 The Fourer ransform of δ() s gven by (1) δ () 1 (4.75) Proof: j + + ω jω δ e d= δ e d δ d = = () () ()1 1 () 1 πδ( ω) (4.76) Proof: Applyng he dualy o (4.75) yelds he resul of (4.76). 1 (3) u () + πδ( ω) (4.81) jω Proof: Applyng he negraon propery o (4.75) yelds he resul of (4.81). (3) cos( ω) πδω [ ( + ω) + δω ( ω)] (4.77) (4) sn( ω) jπδω [ ( + ω) δω ( ω)] (4.78) j (5) e ω πδω ( ω) (4.79) Proof: Applyng he Euler s formula, (4.77) and (4.78) yelds he resul of (4.79). Ineresngly, f a sgnal x() can be decomposed and denoed by jk x () = ce ω, < < k = k Thus, he Fourer ransform of x() can be expressed by X( ω) = c πδ ( ω kω ) k = k 13- 信號與系統 46

47 Common Fourer ransform pars (1) δ () 1 () 1 πδ( ω) 1 (3) u () + πδ( ω) jω j c (4) δ ( c) 1 e ω, c s any real number b 1 (5) e u(), b > jω + b j (6) 1 e ω πδ( ω ω) 1 (A1).5 + u ( ) jω 1 1 Proof:.5 + u ( ) ( ) πδ( ω) + + πδω ( ), jω (7) cos( ω) πδω [ ( + ω) + δω ( ω)] (8) sn( ω) jπδω [ ( + ω) δω ( ω)] j (A) e ω πδω ( ω) (9) cos( ) [ j j e θ θ ω + θ π δ( ω+ ω) + e δ( ω ω)] (complex funcon) (1) sn( ) [ j j e θ θ ω + θ π δ( ω+ ω) e δ( ω ω)] (complex funcon) (A3) j( ω+ θ) j( ω+ θ) cos( ω + θ) = [ e + e ]/ = + j( ω+ θ) j( ω+ θ) [ e / e / ] = + jθ jω jθ jω ( e e ) / ( e e ) / j( ω+ θ) j( ω+ θ) sn( ω+ θ) = [ e e ]/ = j( ω+ θ) j( ω+ θ) [ e / e / ] = jθ jω jθ jω ( e e ) / ( e e ) / jθ j ( ω ˆ + θ) j ω( + θ / ω) j ω + j ω e = e = e = e ˆ =+ θ / ω [cos( ω ) sn( )] 1 jωθ / ω e = + j πδ( ω ω) [cos( ωθ/ ω) sn( ωθ/ ω)] πδω ( ω ) 13- 信號與系統 47

48 (Wha s he msake for hs deducon? Please fnd ou he cause.) 13- 信號與系統 48

49