Vr Surndra Sai Univriy of Tchnology, Burla Dparmn o f E l c r i c a l & E l c r o n i c E n g g S u b j c : S i g n a l a n d S y m - I S u b j c c o d : B E E - 6 0 5 B r a n c h m r : E E E 5 h m
SYLLABUS OF SIGNALS & SYSTEMS-I 3--0 MODULE-I 0 HOURS Inroducion of Signal, Claificaion of Signal, Gnral Signal Characriic, Signal nrgy & Powr, Coninuou-Tim Signal, Dicr-Tim Signal Baic Sym Propri, Sym wih and wihou mmory, Invribiliy, caualiy, Sabiliy, Tim invarianc, Linariy, Linar Tim Invarian LTI Sym, Dicr Tim LTI Sym, Convoluion Rprnaion of Linar Tim-Invarian Dicr-Tim Sym Convoluion of Dicr-Tim Signal Convoluion Rprnaion of Linar Tim-Invarian Coninuou-Tim Sym Convoluion of Coninuou-Tim Signal, Propri of LTI Sym, Caual ym MODULE-II 0 HOURS Fourir Rprnaion for Signal: Rprnaion of Dicr Tim Priodic ignal, Coninuou Tim Priodic Signal, Dicr Tim Non Priodic Signal, Coninuou Tim Non-Priodic Signal, Propri of Fourir Rprnaion, Frquncy Rpon of LTI Sym, Fourir Tranform rprnaion for Priodic and dicr im Signal, Sampling, rconrucion, Dicr Tim Procing of Coninuou Tim Signal, Fourir Sri rprnaion for fini duraion Nonpriodic ignal. MODULE-III 0 HOURS Modulaion Typ and Bnfi, Full Ampliud Modulaion, Pul Ampliud Modulaion, Muliplxing, Pha and Group dlay Rprnaion of Signal uing Coninuou im Complx Exponnial: Laplac Tranform, Unilaral Laplac Tranform, i invrion, Bilaral Laplac Tranform, Tranform Analyi of Sym MODULE-IV 0 HOURS Rprnaion of Signal uing Dicr im Complx Exponnial: Th Z-Tranform, Propri of Rgion of convrgnc, Invr Z-Tranform, Tranform Analyi of LTI Sym, Unilaral Z Tranform. BOOKS [] Simon Haykin and Barry Van Vn, Signal and Sym, John Wily & Son. [] Alan V. Oppnhim, Alan S. Willky, wih S. Hamid, S. Hamid Nawab, Signal and Sym, PHI. [3] Hwi Hu, Signal and Sym, Schaum' Oulin TMH [4] Edward w. Kamn and Bonni. Hck, Fundamnal of Signal & ym uing Wb and MATLAB, ***
Diclaimr Thi documn do no claim any originaliy and canno b ud a a ubiu for prcribd xbook. Th informaion prnd hr i mrly a collcion by h commi mmbr for hir rpciv aching aignmn. Variou ourc a mniond a h nd of h documn a wll a frly availabl marial from inrn wr conuld for prparing hi documn. Th ownrhip of h informaion li wih h rpciv auhor or iniuion. Furhr, hi documn i no inndd o b ud for commrcial purpo and h commi mmbr ar no accounabl for any iu, lgal, or ohrwi, ariing ou of hi documn. Th commi mmbr mak no rprnaion or warrani wih rpc o h accuracy or compln of h conn of hi documn and pcially diclaim any implid warrani of mrchanabiliy or fin for a paricular purpo. Th commi mmbr hall no b liabl for any lo or profi or any ohr commrcial damag, including bu no limid o pcial, incidnal, conqunial, or ohr damag.
Conn Lcur - Inroducion of Signal and ym Lcur - Claificaion of Signal Lcur 3- Claificaion of Signal coninud Lcur 4- Lcur 5- Lcur 6- Lcur 7- Lcur 8- Lcur 9- Gnral Signal Characriic Opraion on ignal Fundamnal of Sym Sym propri Sym propri coninud Linar Tim Invarian Sym Lcur 0- Convoluion of Linar Tim-Invarian Dicr-Tim Signal Lcur - Convoluion Rprnaion of Linar Tim-Invarian Coninuou-Tim Sym Lcur - Propri of LTI Sym, Caual ym Lcur 3- Fourir Rprnaion for Signal: Lcur 4- Fourir Rprnaion of Coninuou Tim Priodic Signal Lcur 5- Fourir Rprnaion of Dicr Tim Priodic ignal Lcur 6- Fourir Rprnaion of Coninuou Tim Non Priodic Signal Lcur 7- Fourir Rprnaion of Dicr Tim Non Priodic Signal Lcur 8- Propri of Fourir Rprnaion Lcur 9- Propri of Fourir Rprnaion coninud Lcur 0- Frquncy Rpon of LTI Sym Lcur - Fourir Tranform rprnaion for Priodic and dicr im Signal Lcur - Sampling Lcur 3- Rconrucion Lcur 4- Dicr Tim Procing of Coninuou Tim Signal Lcur 5- Fourir Sri rprnaion for fini duraion Nonpriodic ignal.
Lcur 6- Modulaion Typ and Bnfi Lcur 7- Full Ampliud Modulaion Lcur 8- Pul Ampliud Modulaion Lcur 9- Muliplxing Lcur 30- Pha and Group dlay Lcur 3- Rprnaion of Signal uing Coninuou im Complx Exponnial: Laplac Tranform Lcur 3- Unilaral Laplac Tranform, i invrion Lcur 33- Laplac Tranform Propri Lcur 34- Rprnaion of Signal uing Dicr im Complx Exponnial: Th Z-Tranform Lcur 35- Propri of Rgion of convrgnc Lcur 36- Invr Z-Tranform Lcur 37- Invr Z-Tranform coninud Lcur 38- Z-Tranform Propri Lcur 39- Z-Tranform Propri coninud Lcur 40- Tranform Analyi of LTI Sym
Inroducion o Signal: Wha i a Signal? A ignal i formally dfind a a funcion of on or mor variabl ha convy informaion on h naur of a phyical phnomnon. Whn h funcion dpnd on a ingl variabl, h ignal i aid o b on dimnional. E.g.; Spch ignal Ampliud vari wih rpc o im Whn h funcion dpnd on wo or mor variabl, h ignal i aid o b mulidimnional. E.g.; Imag D Horizonal & vrical coordina of h imag ar wo dimnional Wha i a Sym? A ym i formally dfind a an niy ha manipula on or mor ignal o accomplih a funcion, hrby yilding nw ignal. i/p ignal Sym o/p ignal.g.; In a communicaion ym h inpu ignal could b a pch ignal or compur daa. Th ym ilf i mad up of h combinaion of a ranmir, channl and a rcivr. Th oupu ignal i an ima of h informaion conain in h original mag. Mag ignal Tranmid ignal Rcivd ignal Eima of mag Tranmir Channl Rcivr ignal Th xampl of ohr ym ar conrol ym, biomdical ignal procing ym, audio ym, rmo ning ym, microlcro mchanical ym c. Gnral ignal characriic: a Mulichannl & mulidimnional ignal: A ignal i dcribd by a funcion of on or mor indpndn variabl. Th valu of h funcion dpndn variabl can b ral valud calar quaniy, a complx valud quaniy or prhap a vcor. Ral valud ignal x A = A in3π Complx valud ignal x A = A j3 π = A co3π + jain3π In om applicaion, ignal ar gnrad by mulipl ourc or mulipl nor. Such ignal can b rprnd in vcor form and w rfr uch a vcor of ignal a a mulichannl ignal. E.g.; In lcrocardiography, 3-lad & -lad lcrocardiogram ECG ar ofn ud, which rul in 3-channl & -channl ignal. On dimnional: If h ignal i a funcion of a ingl indpndn variabl, h ignal i calld -D ignal. Amp.g.; Spch ignal Tim
Mulidimnional ignal: Signal can b funcion of mor han on variabl,.g., imag ignal D, Colour imag 3D, c. Claificaion of ignal Broadly w claify ignal a:. Coninuou-im ignal: A ignal x, i aid o b coninuou-im ignal if i i dfind for all im, whr i a ral-valud variabl dnoing im. Ex: x = -3 u Dicr-im ignal: A ignal xn, i aid o b dicr-im ignal; if i i dfind only a dicr inan of im, whr n i an ingr-valud variabl dnoing h dicr ampl of im. W u quar brack [ ] o dno a dicr-im ignal. Ex: x[n] = -3n u[n]. Evn and odd ignal: X[n] -3-6 -9 0 3 A coninuou-im ignal x i vn, if x- = x and i i odd if x- = -x. A dicr-im ignal x[n] i vn if x[-n] = x[n] Exampl : x = - 40 i vn. and i odd if x[-n] = -x[n]. Exampl : x = 0. 3 i odd. Exampl 3: x = 0.4 i nihr vn nor odd. a b c Figur: Illuraion of odd and vn funcion. a Evn; b Odd; c Nihr.
Dcompoiion Thorm Evry coninuou-im ignal x can b xprd a: whr y i vn, and z i odd. and 3. Priodic & non-priodic ignal: y = z m = x = y + z x + x x x A coninuou im ignal x i priodic if hr i a conan T > 0, uch ha x = x + T, for all A dicr im ignal x[n] i priodic if hr i an ingr conan N > 0, uch ha x[n] = x[n + N ], for all n Signal do no aify h priodiciy condiion ar calld non-priodic ignal. No: Th mall valu of T N ha aifi h abov quaion i calld fundamnal priod Exampl: Drmin h fundamnal priod of h following ignal: a j3π/5 b j3πn/5 Soluion: a L x = j3π/5. If x i a priodic ignal, hn hr xi T > 0 uch ha x = x + T. Thrfor, x = x + T j3π/5 = j3π+t/5 = j3πt/5 jkπ T = 0 3 = j3πt/5 k = b L x[n] = j3πn/5. If x[n] i a priodic ignal, hn hr xi an ingr N > 0 uch ha x[n] = x[n + N ]. So, x[n] = x[n + N ] j3πn/5 = j3πn+n/5 = j3πn/5 jkπ = j3πn/5 T = 0 k = 3
4. Enrgy ignal and powr ignal: In lcrical ym, a ignal may rprn a volag or a currn. Conidr a volag vdvlopd acro a rior R, producing a currn i. Th inananou powr diipad in hi rior i dfind by Dfin h oal nrgy of h coninuou-im ignal x a E = lim T/ x T T/ d x n = lim and i im-avragd, or avrag, powr a d P = lim T T T/ x T/ d From abov quaion, w radily ha h im-avragd powr of a priodic ignal x of fundamnal priod T i givn by P = T T/ x T/ d Th quar roo of h avrag powr P i calld h roo man-quar rm valu of h priodic ignal x. In h ca of a dicr-im ignal x[n], h ingral in abov quaion ar rplacd by corrponding um. Thu, h oal nrgy of x[ n] i dfind by and i avrag powr i dfind by E = x [n] n= P = lim n N + N n= N x [n] A ignal i rfrrd o an nrgy ignal if and only if h oal nrgy i fini.i.., 0 < E < A ignal i rfrrd o an powr ignal if and only if h avrag powr i fini.i.., 0 < P < No: Enrgy ignal ha zro im avrag powr and powr ignal ha infini nrgy.
Exampl: xn = - 0.5 n u[n] Soluion: E = n=0 n= x [n] = 0.5 n = 0.5 = 4 3 < P = lim n N+ N x n= N [n] = lim n N+ N n=0 0.5 n = + N n=0 0.5 n = 0 W go powr zro and fini nrgy. Hnc i i an nrgy ignal. 5. Drminiic ignal and random ignal: Th drminiic ignal i a ignal abou which hr i no uncrainy wih rpc o i valu a any im. Th drminiic ignal may b modld a complly pcifid funcion of im. Exampl: x = co π A random ignal i a ignal abou which hr i uncrainy bfor i occur. Exampl: Th lcrical noi gnrad in h amplifir of a radio or lviion rcivr. Baic Opraion of Signal Opraion prformd on indpndn variabl: Tim Shif For any 0 and n 0, im hif i an opraion dfind a x x - 0 x[n] x[n - n 0 ]. If 0 > 0, h im hif i known a dlay. If 0 < 0, h im hif i known a advanc. Exampl. In Fig. givn blow, h lf imag how a coninuou-im ignal x. A im- hifd vrion x - i hown in h righ imag. Figur: An xampl of im hif. Tim Rvral Tim rvral i dfind a x x- x[n] x[-n], which can b inrprd a h flip ovr h y-axi.
Exampl: Figur: An xampl of im rvral. Tim Scaling Tim caling i h opraion whr h im variabl i muliplid by a conan a: x xa, a > 0 If a >, h im cal of h rulan ignal i dcimad pd up. If 0 < a <, h im cal of h rulan ignal i xpandd lowd down. Figur : An xampl of im caling. Dcimaion and Expanion Dcimaion and xpanion ar andard dicr-im ignal procing opraion. Dcimaion i dfind a y D [n] = x[m n], for om ingr M Whr, M i h dcimaion facor. Expanion i dfind a y E [n] = x[ n ], n = ingr mulipl of L L 0, ohrwi. Whr, L i h xpanion facor.
Figur.8: Exampl of dcimaion and xpanion for M = and L =. Combinaion of Opraion Gnrally, linar opraion in im on a ignal x can b xprd a y = xa-b. Th rcommndd mhod i Shif, hn Scal. Exampl: Th ignal x hown in Figur of kch x3-5. Figur: An xampl of Shif, hn Scal Opraion prformd on dpndn variabl: Ampliud caling: L x dno a coninuou im ignal By ampliud caling, w g y = cx Whr, c i h caling facor. Exampl: An lcronic amplifir, a dvic ha prform ampliud caling. For dicr im ignal y[n] = cx[n] Ampliud addiion: L x and x i a pair of coninuou im ignal By adding h wo ignal, w g y = x + x Exampl: An audio mixur For dicr im ignal, y[n] = x[n] + x[n]
Ampliud muliplicaion: L x and x i a pair of coninuou im ignal By muliplying h wo ignal, w g y = x x Exampl: An AM radio ignal, in which x i an audio ignal x i an inuoidal carrir wav For dicr im ignal, y[n] = x [n] x [n] Diffrniaion: y d x d Exampl: Volag acro an inducor L v d i d Ingraion: y= x τ dτ Exampl: Volag acro a capacior C y= C Elmnary Signal i τ dτ Svral lmnary ignal faur prominnly in h udy of ignal and ym. Th ar xponnial and inuoidal ignal, h p funcion, h impul funcion, and h ramp funcion, all of which rv a building block for h conrucion of mor complx ignal Exponnial Signal A ral xponnial ignal, in i mo gnral form, i wrin a x = B a, whr boh B and a ar ral paramr. Th paramr B i h ampliud of h xponnial ignal maurd a im = 0. Dpnding on whhr h ohr paramr a i poiiv or ngaiv, w may idnify wo pcial ca: Fig: Growing xponnial, for a > 0 Dcaying xponnial, f o r a < 0 In dicr im, i i common pracic o wri a ral xponnial ignal a x[n] = Br n
Fig: Growing xponnial for r > Dcaying xponnial for 0 < r < Impul funcion Th dicr-im vrion of h uni impul i dfind by δ [n] =, 0, n=0 n 0 Fig: Dicr im form of uni impul Th coninuou-im vrion of h uni impul i dfind by h following pair of rlaion: δ = 0 for 0 and δ d = δ 0 Fig: Coninuou im form of uni impul Abov quaion ay ha h impul δ i zro vrywhr xcp a h origin. Equaion ay ha h oal ara undr h uni impul i uniy. Th impul δ i alo rfrrd o a h Dirac dla funcion. Sp funcion: Th dicr-im vrion of h uni- p funcion i dfind by:
Th coninuou-im vrion of h uni- p funcion i dfind by: u =, 0, >0 <0 Ramp funcion: Th ingral of h p funcion u i a ramp funcion of uni lop. Fig: Ramp funcion of uni lop Th dicr-im vrion of h uni- ramp funcion i dfind by: r[n] = n, 0, n 0 n<0 Inroducion o Sym Sym ar ud o proc ignal o allow modificaion or xracion of addiional informaion from h ignal. A ym may coni of phyical componn hardwar ralizaion or an algorihm opraor ha compu h oupu ignal from h inpu ignal. A phyical ym coni of inr-conncd componn which ar characrizd by hir inpu-oupu rlaionhip. Figur.: Coninuou-im and dicr-im ym: Hr H & T ar opraor. H = Hx T = Txn
Propri of ym: Claificaion of ym: Saic Mmoryl & Dynamic wih mmory: Saic: A ym i aic if h oupu a im or n dpnd only on h inpu a im or n. Exampl:. y = x - x i mmoryl, bcau y dpnd on x only. Thr i no x -, or x + rm, for xampl.. y[n] = x [n] i mmoryl. In fac, hi ym i paing h inpu o oupu dircly, wihou any procing. 3. Currn flowing hrough a rior i.., i = R v Dynamic: A ym i aid o po mmory if i oupu ignal dpnd on pa or fuur valu of inpu. Exampl:. Inducor and capacior, inc h currn flowing hrough h inducor a im dpnd on h all pa valu of h volag v i.., i = vτdτ and v = iτdτ L C. Th moving avrag ym givn by yn= x[n]+x[n-]+x[n-] 3 Sabl & unabl ym: A ym i aid o b boundd-inpu, boundd-oupu BIBO abl if and only if vry boundd inpu rul in a boundd oupu, ohrwi i i aid o b unabl. If for x M x < for all, oupu i y M y < for all ; whr M x & M y ar om fini poiiv numbr. Exampl:. y = x -3 i a abl ym.. y = x i an unabl ym. 3. y[n] = x[n] i a abl ym. Aum ha xn M x <, for all y[n] = x[n] = Mx = fini Sabl 4. y[n] = r n x[n], whr r > 3 Caual and non-caual ym: Aum ha xn M x <, for all, hn y[n] = r n x[n] = r n x[n] a n r n o y[n] hnc unabl. Caual: A ym i aid o b caual if h prn valu of oupu ignal dpnd only on h prn or pa valu of h inpu ignal. A caual ym i alo known a phyical or non-anicipaiv ym. Exampl:. Th moving avrag ym givn by yn= 3 x[n]+x[n-]+x[n-]. y = xco 6
No: i Any pracical ym ha opra in ral im mu ncarily b caual. ii All aic ym ar caual. Non-Caual: A ym i aid o b non-caual if h prn valu of oupu ignal dpnd on on or mor fuur valu of h inpu ignal. Exampl:. Th moving avrag ym givn by yn= 3 x[n]+x[n-]+x[n+] 4 Tim invarian and im varian ym: Tim invarian: A ym i im-invarian if a im-hif of h inpu ignal rul in h am im-hif of h oupu ignal. Tha i, if x y, hn h ym i im-invarian if. x - O y - O, for any O. Exampl. Th ym y = in[x] i im-invarian Figur.: Illuraion of a im-invarian ym. Proof. L u conidr a im-hifd ignal x = x - O. Corrpondingly, w l y b h oupu of x. Thrfor, y = in[x ] = in[x - O ]. Now, w hav o chck whhr y = y - O. To how hi, w no ha y - O = in[x - O ], which i h am a y. Thrfor, h ym i im-invarian. Tim varian: A ym i im-varian if i inpu oupu characriic chang wih im. Exampl : Th ym y[n] = nx[n] i im-varian. Proof: Oupu for a im hifd inpu i y[n] xn-k = nxn-k hn h am im hifd oupu i yn-k = n-kxn-k h abov wo quaion ar no am. Hnc i i im varian.
4 Linar and non-linar ym: Linar ym: A ym i aid o b linar if i aifi wo propri i..; uprpoiion & homogniy. Suprpoiion: I a ha h rpon of h ym o a wighd um of ignal b qual o h corrponding wighd um of rpon Oupu of h ym o ach of h individual inpu ignal. For an inpu x = x, h oupu y = y and inpu x = x, h oupu y = y hn, h ym i linar if & only if T [a x + a x ] = a T [x ] + a T [x ] Homogniy: If h inpu x i cald by a conan facor a, hn h oupu y i alo cald by xacly h am conan facor a. For an inpu x oupu y and inpu x = ax oupu y = ay Exampl : Th ym y = πx i linar. To hi, l conidr a ignal x = ax + bx, whr y = πx and y = πx. Thn ay + by = a πx + b πx = π [ax + bx ] = πx = y. Exampl. Th ym y[n] = x[n] i no linar. To hi, l conidr h ignal x[n] = ax [n] + bx [n], whr y [n] = x [n] and y [n] = x [n]. W wan o whhr y[n] = ay [n] + by [n]. I hold ha Howvr, ay [n] + by [n] = a x [n] + b x [n]. y[n] = x[n] = ax [n] + bx [n] = a x [n] + b x [n] + abx [n]x [n]. 5 Invribl and non-invribl ym: A ym i aid o b invribl if h inpu of h ym can b rcovrd from h oupu. L h of opraion ndd o rcovr h inpu rprn h cond ym which i conncd in cacad wih h givn ym uch ha h oupu ignal of h cond ym i qual o h inpu applid o h givn ym.
L H h coninuou im ym x inpu ignal o h ym y oupu ignal of h ym H inv h cond coninuou im ym x y x H H inv Th oupu ignal of h cond ym i givn by H inv {y} = H inv {Hx} = H inv H{x} For h oupu ignal o qual o h original inpu, w rquir ha Whr I dno h idniy opraor. Th ym who oupu i qual o h inpu i an idniy ym. Th opraor H inv mu aify h abov condiion for H o b an invribl ym. Cacading a ym, wih i invr ym, rul in an idniy ym. Exampl: An inducor i dcribd by h rlaion H H inv = I y = L xτdτ i an invribl ym bcau, by rarranging rm, w g x = L d d y, which i h invrion formula. No:i A ym i no invribl unl diinc inpu applid o h ym produc diinc oupu. ii Thr mu b a on o on mapping bwn inpu and oupu ignal for ym o b invribl. Non-invribl Sym: Whn vral dibffrn inpu rul in h am oupu, i i impoibl o obain h inpu from oupu. Such ym i calld a non-invribl ym. Exampl: A quar-law ym dcribd by h inpu oupu rlaion y = x, i non-invribl, bcau diinc inpu x & -x produc h am oupu y [no diinc oupu]. Linar im convoluion ym LTI Linar im invarian LTI ym ar good modl for many ral-lif ym, and hy hav propri ha lad o a vry powrful and ffciv hory for analyzing hir bhavior. Th LTI ym can b udid hrough i characriic funcion, calld h impul rpon. Furhr, any arbirary inpu ignal can b dcompod and rprnd a a wighd um of uni ampl qunc. A a conqunc of h linariy and im invarianc propri of h ym, h rpon of h
ym o any arbirary inpu ignal can b xprd in rm of h uni ampl rpon of h ym. Th gnral form of h xprion ha rla h uni ampl rpon of h ym and h arbirary inpu ignal o h oupu ignal, calld h convoluion um, i alo drivd. Roluion of a Dicr-im ignal ino impul: Any arbirary qunc xn can b rprnd in rm of dlayd and cald impul qunc δn. L xn i an infini qunc a hown in figur blow. Figur.3: Rprning of a ignal x[n] uing a rain of impul δ[n - k]. Th ampl x0 can b obaind by muliplying x0, h magniud, wih uni impul δn i.., x[n] δ[n] = x0, 0, n=0 n 0 Similarly, h ampl x-3 can b obaind a hown in h figur. i.., x[-3] δ[n+3] = x 3, 0, n= 3 n 3 In h am way w can g h qunc x[n] by umming all h hifd and cald impul funcion i.., x[n] =. x[-3] δ[n+3] + x[-] δ[n+] +. + x[0] δ[n] +.+ x[4 ] δ[n-4] = k= x k δn k Impul rpon and convoluion um: Impul rpon: A dicr-im ym prform an opraion on an inpu ignal bad on prdfind criria o produc a modifid oupu ignal. Th inpu ignal x[n] i
h ym xciaion, and y[n] i h ym rpon. Th ranform opraion i hown in h figur blow. x[n] T y[n]=t[x[n]] If h inpu o h ym i h uni impul i.., x[n] = δ[n], hn h oupu of h ym i known a impul rpon rprnd by h[n] whr h[n] = T [δ[n]] Rpon of LTI ym o arbirary inpu: Th convoluion um From h abov dicuion, w g h rpon of an LTI ym o an uni impul a h impul rpon h[n] i.., δ[n] δ[n-k] xkδ[n-k] h[n] h[n-k], by im invarian propry xkh[n-k], by homogniy principl k= xkδ[n k] k= xkh[n k], by upr poiion A w know h arbirary inpu ignal i a wighd um of impul, h LHS = x[n] having a rpon in RHS = y[n] known a convoluion ummaion. i.., x[n] y[n] In ohr word, givn a ignal x[n] and h impul rpon of an LTI ym h[n], h convoluion bwn x[n] and h[n] i dfind a y[n] = xkh[n k] k= W dno convoluion a y[n] = x[n] h[n]. Equivaln form: Ling m = n - k, w can how ha xkh[n k] = xn mh[m] = x[n k]h[k] k= m = k= Propri of convoluion: Th following andard propri can b provd aily:. Commuaiv: x[n] h[n] = h[n] x[n]. Aociaiv: x[n] h [n] h [n] = x[n] h [n] h [n] 3. Diribuiv: x[n] h [n] + h [n] = x h [n] + x[n] h [n]
How o Evalua Convoluion? To valua convoluion, hr ar four baic p:. Fold 3. Muliply. Shif 4. Summaion Exampl: Conidr h ignal x[n] and h impul rpon h[n] hown blow. L compu h oupu y[n] on by on. Fir, conidr y[0]: y[o] = x k h[0 k] = k= k= x k h[ k] = No ha h[-k] i h flippd vrion of h[k], and k= x k h[ k] = i h muliplyadd bwn x[k] and h[-k]. To calcula y[l], w flip h[k] o g h[-k], hif h[-k] go g h[l-k], and muliply-add o g k= x k h[ k]. Thrfor y[] = x k h[ k] = k= k= x k h[ k] = + = 3 Th calculaion i hown in h figur blow. Sym Propri Wih h noion of convoluion, w can now procd o dicu h ym propri in rm of impul rpon.
Mmoryl A ym i mmoryl if h oupu dpnd on h currn inpu only. An quivaln amn uing h impul rpon h[n] i ha: An LTI ym i mmoryl if and only if Invribl h[n] = aδ[n], for om a. An LTI ym i invribl if and only if hr xi g[n] uch ha h[n] g[n] = δ[n]. Caual An LTI ym i caual if and only if h[n] = 0, for all n < 0. Sabl An LTI ym i abl if and only if k= h[k] < Proof: Suppo ha k= h[k] <. For any boundd ignal x[n] B, h oupu i Thrfor, y[n] i boundd. y[n] x[k]h[n k] k= = x[k]. h[n k] k= B. h[n k] k=
Coninuou-im Convoluion Th coninuou-im ca i analogou o h dicr-im ca. In coninuouim ignal, h ignal dcompoiion i x = x τ δ τ dτ and conqunly, h coninuou im convoluion i dfind a = x τ h τ dτ Exampl: Th coninuou-im convoluion alo follow h hr p rul: flip, hif, muliply- add. L u conidr h ignal x = -a u for a > 0, and impul rpon h = u. Th oupu y i Ca A: > 0: y = x τ h τ dτ Ca B: 0: = ar uτ = ar dτ 0 = a [--a ] u τ y = 0. Thrfor, Propri of CT Convoluion y = a [ a ]u Th following propri can b provd aily:. Commuaiv: x h = h x. Aociaiv: x h h = x h h 3. Diribuiv: x [h + h ] = [x h ] + [x h ] Coninuou-im Sym Propri Th following rul ar analogou o h dicr-im ca. Mmoryl. An LTI ym i mmoryl if and only if
h = aδ, for om a Invribl. An LTI ym i invribl if and only if hr xi g uch ha Caual. A ym i caual if and only if Sabl. A ym i abl if and only if h g = δ. h = 0, for all < 0 h τ dτ < Inrconncion of LTI ym:. Paralll conncion of LTI Sym: Conidr wo LTI ym wih impul rpon h and h conncd in paralll a hown in h figur blow. Th oupu of hi conncion of ym, y, i h um of h oupu of h wo ym i.., y= y + y = x h + x h = x [h + h ] Idnical rul hold for h dicr im ca. xn h n+ xn h n = xn [h n+ h n] Fig: Paralll inrconncion of wo LTI ym & i quivaln ym. Cacad conncion of LTI Sym: Conidr h cacad conncion of wo LTI ym a hown in h figur. Th oupu of hi conncion of ym y= {x h h } Uing aociaiv propry of convoluion, w g y= x {h h }
Fig: Cacad Inrconncion of wo LTI ym & i quivaln ym Sp rpon: Sp inpu rpon ar ofn ud o characriz h rpon of an LTI ym o uddn chang in h inpu. I i dfind a h oupu du o a uni p inpu ignal. L h[n] b h impul rpon of a dicr-im LTI ym and [n] b h p rpon. Thn, [n] = h[n] u[n] = k= h[k]u[n k] Now, a u[n-k] = 0 for k > n and u[n-k] = for k n, [n] = h[k] k= i.., h p rpon i h running um of h impul rpon. Similarly, h p rpon of a coninuou-im ym i xprd a h running ingral of h impul rpon: = h τ dτ No: Th rlaionhip may b invrd o xpr h impul rpon in rm of h p rpon a h[n] = [n] [n-] and, h = d d
Fourir Rprnaion for Signal In hi chapr, h ignal i rprnd a a wighd uprpoiion of complx inuoid. If uch a ignal i applid o an LTI ym, hn h ym oupu i a wighd uprpoiion of h ym rpon o ach complx inuoid. Rprning ignal a uprpoiion of complx inuoid no only lad o a uful xprion for h ym oupu, bu alo provid an inighful characrizaion of h ignal and ym. Th udy of ignal and ym uing inuoidal rprnaion i known a Fourir analyi namd afr Joph Fourir. Baing on h priodiciy propri of h ignal and whhr h ignal i dicr or coninuou in im, hr ar four diffrn yp of Fourir rprnaion, ach applicabl o a diffrn cla of ignal. Complx inuoid and frquncy rpon of LTI ym: Th rpon of an LTI ym o a inuoidal inpu lad o a characrizaion of ym bhavior rmd a frquncy rpon of h ym. Thi characrizaion i obaind in rm of h impul rpon by uing convoluion and a complx inuoidal inpu ignal. L u conidr h oupu of a dicr-im LTI ym wih impul rpon h[n] and uni ampliud complx inuoidal inpu x[n] = jωn. Thi oupu i givn by: W facor jωn from h um o g y[n] = h[k]x[n k] k= = h[k] k= y[n] = jωn h[k] k= jωn k jωk Whr w hav dfind = H jω jωn H jω = h[k] k= jωk Hnc, h oupu of h ym i a complx inuoid of h am frquncy a h inpu, muliplid by h complx numbr H jω. Th rlaionhip i hown in figur blow: Th complx caling facor H jω i no a funcion of im n, bu only i a funcion of frquncy Ω and i rmd h frquncy rpon of h dicr-im ym. Th rul obaind for coninuou-im LTI ym i imilar o h abov.
L h impul rpon of uch a ym b h and h inpu b x = jω. Thn h convoluion ingral giv h oupu a Whr w dfin, Th abov quaion i rfrrd o a frquncy rpon of h coninuou im ym. Wriing h complx valud frquncy rpon Hjω in polar form Whr, Hjω = Hjω jυ Hjω => magniud rpon And, φ => pha rpon = arg{hjω} Exampl: Th impul rpon of h ym givn h figur blow i Find an xprion for h frquncy rpon and plo h magniud and pha rpon. Soluion: Subiuing h in quaion of Hjω, w g
Th magniud rpon i: Whil h pha rpon i arg{hjω} = - arcanωrc Fig: a Magniud-rpon b Pha-rpon Eignvalu and Eignfuncion of an LTI Sym Dfiniion: For an LTI ym, if h oupu i a cald vrion of i inpu, hn h inpu funcion i calld an ignfuncion of h ym. Th caling facor i calld h ignvalu of h ym. W ak h complx inuoid ψ = jω i an ignfuncion of h LTI ym H aociad wih h ignvalu λ = Hjω, bcau ψ aifi an ignvalu problm dcribd by H{ψ} = λψ
Th ffc of h ym on an ignfuncion inpu ignal i calar muliplicaion. Th oupu i givn by h produc of h inpu and a complx numbr. Thi ign rprnaion i hown in h figur blow. Fourir rprnaion of four cla of ignal: Thr ar four diinc Fourir rprnaion, ach applicabl o a diffrn cla of ignal. Th Fourir ri FS appli o coninuou im priodic ignal, and h dicr - im Fourir ri DTFS appli o dicr im priodic ignal. Th Fourir ranform FT appli o a ignal ha i coninuou in im and nonpriodic. Th dicr-im Fourir ranform DTFT appli o a ignal ha i dicr in im and nonpriodic. Rlaionhip bwn im propri of a ignal and h appropria Fourir rprnaion i givn blow: Coninuou-im priodic ignal: Th Fourir ri Coninuou-im priodic ignal ar rprnd by h Fourir ri FS. W may wri h FS of a ignal x wih fundamnal priod T and fundamnal frquncy ω 0 = π/t, a Whr, ar h FS cofficin of h ignal x. W ay ha x and X[k] ar an FS pair and dno hi rlaionhip a Th Fourir ri cofficin ar known a h frquncy-domain rprnaion of x,bcau ach FS cofficin i aociad wih a complx inuoid of diffrn frquncy. In h rprnaion of h priodic ignal x by h Fourir ri, h iu ari, i whhr or no h ri convrg o x for ach valu of, i.., whhr h ignal x and i FS rprnaion ar qual a ach valu of.
Th Dirichl condiion guaran ha h FS will b qual o x, xcp a h valu of for which x i diconinuou. A h valu of, FS convrg o h mid-poin of h diconinuiy. Th Dirichl condiion ar:. Th ignal x ha a fini numbr of diconinuii in any priod.. Th ignal conain a fini numbr of maxima and minima during any priod. 3. Th ignal x i aboluly ingrabl boundd i.., x d < T If x i priodic and aifi h Dirichl condiion, i can b rprnd in FS. Dirc calculaion of FS cofficin: Exampl: Drmin h FS cofficin for ignal x Soluion: Tim priod T =, Hnc, ω 0 = π/ = π. On h inrval 0, on priod of x i xprd a x = -. So, W valua h ingral o g
Fig: Magniud and pha rpon of X[k] Calculaion of FS cofficin by inpcion: Exampl: Drmin h FS rprnaion of h ignal x = 3 co π/ + π/4 Soluion: Tim priod T = 4, So, ω 0 = π/4 = π/. W may wri FS of a ignal x i, Uing Eulr formula o xpand h coin, giv Equaing ach rm in hi xpanion o h rm in quaion of x giv h FS cofficin: Th magniud and pha pcra ar hown blow.
Figur: Magniud and pha pcrum Exampl: Find h im domain ignal x corrponding o h FS cofficin Aum ha fundamnal priod T =. Soluion: Subiuing h valu givn for X[k] and ω 0 = π/ = π ino quaion x giv Th cond gomric ri i valuad by umming from l = 0 o l =, and ubracing h l = 0 rm. Th rul of umming boh infini gomric ri i Puing h fracion ovr a common dnominaor rul in Dicr-im priodic ignal: Th dicr-im Fourir ri Dicr-im priodic ignal ar rprnd by h dicr-im Fourir ri DTFS. W may wri h DTFS of a ignal x[n] wih fundamnal priod N and fundamnal frquncy Ω 0 = π/n, a Whr Ar h DTFS cofficin of h ignal x[n].w ay ha x[n] and X[k] ar a DTFS pair and dno hi rlaionhip a
Th DTFS cofficin ar known a h frquncy-domain rprnaion of x[n], bcau ach DTFS cofficin i aociad wih a complx inuoid of diffrn frquncy. Dirc calculaion of DTFS cofficin: Exampl: Drmin h DTFS cofficin for ignal x[n] hown Soluion: Th ignal ha a priod N = 5, o Ω0 = π/5. A h ignal alo ha odd ymmry, i can b um ovr n =- o n = in h quaion and w g, Uing h valu of x[n], w g From h abov quaion, w idnify on priod of DTFS cofficin X[k], k= o k=-, in rcangular and polar coordina a Fig: Magniud and Pha Rpon of X[k]
Th abov figur how h magniud and pha of X[k] a funcion of h frquncy indx k. Now uppo w calcula X[k] uing n = 0 o n =4 for h limi on h um in qn.of X[k] o obain Calculaion of DTFS cofficin by inpcion: Exampl: Drmin h DTFS cofficin of h ignal x[n] = co πn/3 + υ Soluion: Tim priod N=6.W xpand h coin by uing Eulr formula a Now comparing wih h DTFS quaion wih Ω 0 = π/6=π/3, wrin by umming from k= - o 3 Equaing h rm, w g Th magniud and pha pcrum i givn blow
Th Invr DTFS: Exampl: Drmin h im ignal x[n] from h DTFS cofficin givn in figur blow Soluion: Th DTFS cofficin hav priod = 9, hnc Ω 0 =π/9. I i convnin o valua x[n] ovr h inrval k = -4 o k = 4 o obain Coninuou-im nonpriodic ignal: Th Fourir ranform Th Fourir ranform i ud o rprn a coninuou-im nonpriodic ignal a a uprpoiion of complx inuoid. W know ha h coninuou nonpriodic naur of a im ignal impli ha h uprpoiion of complx inuoid ud in h Fourir rprnaion of h ignal involv a coninuum of frqunci ranging from - o. So h FT rprnaion of a coninuou-im ignal involv an inygral ovr h nir frquncy inrval; i.., Whr, Exampl. FT of a ral dcaying xponnial: Find h FT of x = -a u hown in h figur blow.
Soluion: Th FT do no convrg for a 0, inc x i no aboluly ingrabl, i..; 0 a d =, a 0 For a > 0, w hav Convring o polar form, h magniud and pha of Xjω ar rpcivly givn by and a hown in figur blow.
Th magniud of Xjω plod again ω i known a h magniud pcrum of h ignal x, and h pha of Xjω plod a a funcion of ω i known a h pha pcrum of x. Exampl : FT of a rcangular pul: Conidr h rcangular pul hown figur blow and dfind a Find h FT of x. Soluion: Th rcangular pul x i aboluly ingrabl, providd ha T 0 <. So w hav For ω = 0, h ingral implifi o T 0. L Hopial rul raighforwardly how ha lim ω 0 ω in ωt 0 = T 0 Thu, w uually wri Xjω = ω in ω T 0 Wih h undranding ha h valu a ω = 0 i obaind by valuaing a limi. In hi ca Xjω i ral and i hown in h figur blow.
Th magniud pcrum i and h pha pcrum i Uing inc funcion noaion, w may wri Xjω a Invr FT of a rcangular pcrum: Exampl: Find h invr FT of h rcangular pcrum figur blow givn by Soluion: Uing quaion for x for invr FT yild Whn = 0, h ingral implifi o W/π. A
W uually wri or Th valu a = 0 i obaind a a limi. Th x i hown in h following diagram. Dicr-im nonpriodic ignal: Th Dicr-im Fourir ranform Th DTFT i ud o rprn a dicr-im nonpriodic ignal a a upr poiion of complx inuoid. A raond prviouly, h DTFT would involv a coninuum of frqunci on h inrval π < Ω < π, whr Ω ha uni of radian. Thu, h DTFT rprnaion of a im-domain ignal involv an ingral ovr frquncy, namly, Whr i h DTFT of h ignal x[n]. A X j Ω and x[n] ar a DTFT pair, w can wri. Th ranform X j Ω dcrib h ignal x[n] a a funcion of a inuoidal frquncy Ω and i calld h frquncy-domain rprnaion of x[n]. Th quaion for x[n] i uually calld h invr DTFT, a i map h frquncy-domain rprnaion back ino h im-domain. If x[n] < n= i..; if x[n] i aboluly ummabl, hn h um in qn. X j Ω convrg uniformly o a coninuou funcion of ω. Exampl: Find h DTFT of h qunc x[n] = α n u[n].
Soluion: Uing h abov quaion, w hav Thi um divrg for α. For α, w hav h convrgn gomric ri If α i ral valud, h dnominaor of h abov quaion may b xpandd. Uing Eulr formula, w g From hi form, w ha h magniud and pha pcra ar givn by and, rpcivly. Th magniud and pha pcra for α = 0.5 and α = 0.9 ar hown in h figur blow. Th magniud i givn and h pha i odd and boh ar π priodic. Invr DTFT Exampl: Find h Invr DTFT of h following rcangular pcrum
Soluion: Subiuing X j Ω in DTFT rprnaion, w g For n = 0, h ingrand i uniy and w hav x[0]=w/π. Uing L Hopial rul, w can aily how ha And hu w uually wri Fig: Invr DTFT in Tim domain Propri of Fourir rprnaion:. Linariy: All four Fourir rprnaion aify h linariy propry.
In h rlaionhip, w aum ha h uppr ca ymbol dno h Fourir rprnaion of h corrponding lowr ca ymbol.. Symmry: 3. Convoluion: a Convoluion of nonpriodic ignal: Convoluion of wo ignal h & x in h im domain corrpond o muliplicaion of hir FT, Hjω & Xjω in frquncy domain. A imilar propry hold for convoluion of dicr im non-priodic ignal.if DTFT x[n] X jω and h[n] H jω,hn DTFT DTFT y[n] = x[n] h[n] Y jω = X jω H jω b Convoluion of priodic ignal: Th priodic convoluion of wo priodic CT ignal x and z, ach having priod T, a Subiuing h FS rprnaion of z ino h convoluion ingral lad o h propry Similarly in DTFS 4 Diffrniaion and Ingraion:
a Diffrniaion in im: Diffrniaing a ignal in im domain corrpond o muliplying i FT by jω in h frquncy domain i.; b Diffrniaion in frquncy: Diffrniaion of FT in frquncy domain corrpond o muliplicaion of h ignal by j in h im domain i.; Commonly ud Diffrniaion and Ingraion propri: 5 Tim Shif: L z = x- 0 b a im hifd vrion of x. Th goal i o rla h FT of z o h FT of x Puing τ = 0, w obain Tim-hif propri of Fourir rprnaion:
Frquncy-hif propri of Fourir rprnaion: Muliplicaion propry: For nonpriodic ignal: For priodic ignal: Scaling propri: L u conidr h ffc of caling h im variabl on h frquncy-domain rprnaion of a ignal. Bginning wih h FT, l z = xa, whr a i a conan. By dfiniion, w hav W ffc h chang of variabl τ = a o g Th can b combind ino a ingl ingral W can conclud ha Scaling h ignal in im inroduc h invr caling in h frquncy-domain rprnaion and an ampliud chang, a hown in h givn figur:
Parval Rlaionhip: I a ha h nrgy or powr in h im-domain rprnaion of a ignal i qual o h nrgy or powr in h frquncy-domain rprnaion. So h nrgy and powr ar conrvd in h Fourir rprnaion. Th nrgy in a coninuou-im non-priodic ignal i Wx = x d Whr i i aumd ha x may b complx valud gnral. A x = x x, aking h conjuga of boh id of Eq, w may xpr x in rm of i FT Xjω a Subiuing hi formula ino h xprion for W x, w obain Now w inrchang h ordr of ingraion: Obrving ha h ingral inid h brac i h FT of x, w obain And o conclud ha Hnc, h nrgy in h im-domain rprnaion of h ignal i qual o h nrgy in h frquncy-domain rprnaion, normalizd by π. Th quaniy Xjω plod again ω i known a nrgy pcrum of h ignal. Analogou rul hold for h priodic ignal i known a powr dniy pcrum of h ignal.
Th ohr hr rprnaion ar ummarizd in h abl blow: Dualiy: In hi chapr, w obrvd a conin ymmry bwn h im and frquncy domain rprnaion of ignal. For xampl, a rcangular pul in ihr im or frquncy corrpond o a inc funcion in ihr frquncy or im, a hown in h figur blow. W hav alo obrvd ymmri in Fourir rprnaion propri i..; convoluion in on domain corrpond o modulaion in ohr domain; diffrniaion in on domain corrpond o muliplicaion by h indpndn variabl in h ohr domain, and o on. So by h conqunc of hi ymmry, w may inrchang im and frquncy. Thi inrchangabiliy propry i rmd dualiy. Dualiy propri of Fourir rprnaion i ummarizd in h abl blow:
Applicaion of Fourir rprnaion o mixd ignal cla W now dicu h applicaion of Fourir rprnaion o mixd ignal lik - Priodic & nonpriodic ignal - Coninuou & dicr im ignal Such mixing of ignal occur mo commonly whn on u Fourir mhod o - Analyz h inracion bwn ignal & ym - Numrically valua propri of ignal or h bhavior of a ym For xampl: If w apply a priodic ignal o a abl LTI ym, h convoluion rprnaion of h ym oupu involv a mixing of nonpriodic impul rpon and priodic inpu ignal. Anohr xampl: A ym ha ampl coninuou im-ignal involv boh coninuou & dicr-im ignal. In ordr o u Fourir mhod o analyz uch inracion, w mu build bridg bwn h Fourir rprnaion of diffrn cla of ignal. DTFS i h only Fourir rprnaion ha can b valuad numrically on a compur. Fourir ranform rprnaion of priodic ignal: Nihr FT nor DTFT convrg for priodic ignal. Howvr, by incorporaing impul ino h FT & DTFT in h appropria mannr, w may dvlop FT & DTFT rprnaion of uch ignal. Rlaing FT o FS: Th FS rprnaion of a priodic ignal x i x = X[k] jk ω 0 k= Whr, ω 0 i h fundamnal frquncy of h ignal. A,, uing frquncy hif propry, h invr FT of a frquncy hifd impul δ ω-kω 0 i a complx inuoid wih frquncy kω 0. i.,; If w find FT of x, hn by uing linariy propry of FT w obain Thu h FT of a priodic ignal i a ri of impul pacd by h fundamnal frquncy ω 0. Th k h impul ha rngh πxk, whr Xk i h k h FS cofficin. Th hap of Xjω i idnical o ha of Xk. Th FT obaind from h FS by placing impul a ingr mulipl of ω 0 and wighing hm by π im h corrponding FS cofficin. Givn an FT coniing of impul ha ar uniformly pacd in ω, w obain h corrponding FS cofficin by dividing h impul rngh by π.
Fig: FS and FT rprnaion of priodic coninuou-im ignal Rlaing DTFT o DTFS: Th DTFS xprion for a N priodic ignal x[n] i Th invr DTFT of a frquncy hifd impul i a dicr-im complx inuoid. Th DTFT i a π priodic funcion of frquncy. So w may xpr a frquncy hifd impul ihr by xpring on priod uch a Or, by uing an infini ri of hifd impul, parad by an inrval of π, o obain h π priodic funcion Th invr DTFT quaion i valuad by man of h hifing propry of h impul funcion. W hav Hnc w idnify h complx inuoid and h frquncy hifd impul a a DTFT pair uing linariy propry and ubiuing h abov quaion in quaion of x[n], w g DTFT of priodic ignal x[n] Sinc DTFT i π priodic, i follow ha, DTFT of x[n] coni of a of N impul of rngh πxk,k=0,,.,n-. Givn h DTFS cofficin and h fundamnal frquncy Ω 0, w obain h DTFT rprnaion by placing impul a ingr mulipl of Ω 0 and wighing hm by π im h corrponding DTFS cofficin.
Fig: DTFS & DTFT rprnaion of a priodic dicr-im ignal Fourir ranform rprnaion of Dicr-im ignal: In hi cion w driv, an FT rprnaion of dicr-im ignal by incorporaing impul ino h dcripion of h ignal h appropria mannr, ablihing a corrpondnc bwn h coninuou frquncy ω and h dicr-im frquncy Ω. L u dfin h complx inuoid x = jω and g[n] = jωn Suppo a forc g[n] qual o h ampl of x akn a inrval of T i..; g[n] = xnt => jωn = jωt n From which w conclud ha Ω = ωt. Rlaing FT o DTFT: DTFT of an arbirary dicr-im ignal x[n] i W wan o find ou an FT pair ha corrpond o h DTFT pair. Subiuing Ω = ωt, w obain h following funcion of coninuou im frquncy ω. Taking h invr FT of X δ jω, uing linariy and h FT pair yild h coninuou im ignal dcripion Hnc,
Th dicr-ignal ha valu x[n], whil h corrponding coninuou im ignal coni of a ri of impul parad by T, wih n h impul having rngh x[n]. Th DTFT X jω i π priodic in Ω, whil h FT X δ jω i π / T priodic in ω. Fig: Rlaionhip bwn FT and DTFT rprnaion of a dicr-im ignal Rlaing FT o DTFS: Th DTFT rprnaion of an N priodic ignal x[n] i givn a Whr X[k] i DTFS cofficin. Subiuing Ω = ωt ino hi qn. yild h FT rprnaion Uing caling propry of impul W can wri DTFS cofficin X[k] ar N-priodic funcion, which impli ha X δ jω i priodic wih priod NΩ 0 /T = π / T. Coninuou im rprnaion of dicr-im ignal a drivd in prviou chapr A x[n] i N priodic, o x δ i alo priodic wih fundamnal priod NT
Fig: Rlaionhip bwn FT and DTFS rprnaion of a dicr-im ignal Sampling: - Th ampling opraion gnra a dicr-im ignal from h coninuou-im ignal in ordr o manipula h ignal on a compur or microprocor. - Such manipulaion ar common in communicaion, conrol and ignal procing ym. - Sampling i alo frqunly prformd on dicr-im ignal o chang h ffciv daa ra, an opraion rmd ubampling. Sampling coninuou-im ignal: - L x b a coninuou-im ignal. To dfin a dicr-im ignal x[n] which i qual o h ampl of x a ingr mulipl of a ampling inrval T, i..; x[n] = x[nt ] - Th impac of ampling i lvad by rlaing h DTFT of x[n] o h FT of x. - Th coninuou-im rprnaion of dicr-im ignal x[n] i givn by - Subiuing x[nt ] for x[n] in abov qn. w g Sinc So w may wri Whr - Th abov qn. impli ha w may mahmaically rprn h ampl ignal a h produc of original coninuou-im ignal and impul rain.
- Thi rprnaion i commonly rmd a impul ampling and i a mahmaical ool ud only o analyz ampling. - Th ffc of ampling i drmind by rlaing FT of x δ.o FT of x. - A muliplicaion in h im domain corrpond o h convoluion in h frquncy domain, o by muliplicaion propry: - A impul rain i coninuou priodic funcion, o i FS i givn a P[k] = T By uing FT rprnaion of FS W g FT of impul rain a T / δ jk ω 0 T / d = T P jω = π P[k]δω kω k= P jω = π δω kω T k= Whr ω = π/t i h ampling frquncy. Now w convolv Xjω wih ach of h frquncy hifd impul o g - Th FT of h ampld ignal i givn by an infini um of hifd vrion of h original ignal FT. - Th hifd vrion ar off by ingr mulipl of ω. - Th hifd vrion of Xjω may ovrlap wih ach ohr if ω i no larg nough compard wih h frquncy xn or bandwidh of Xjω. - L h frquncy componn of h ignal x i aumd o li wihin h frquncy band W < ω < W for h purpo of illuraion.
Fig: Th FT of a ampld ignal for diffrn ampling frqunci a Spcrum of a coninuouim ignal, b Spcrum of ampld ignal whn ω = 3W, c Spcrum of ampld ignal whn ω = W, d Spcrum of ampld ignal whn ω =.5W - No ha, a T incra and ω dcra, h hifd rplica of Xjω mov clo oghr, finally ovrlapping on anohr whn ω < W. - Ovrlap in h hifd rplica of h original pcrum i rmd aliaing. - Aliaing dior h pcrum of h ampld ignal. - Th pcrum of h ampld ignal no longr ha a on o on corrpondnc wih ha of h original coninuou-im ignal. - Thi man ha w canno u h pcrum of h ampld ignal o analyz h coninuou-im ignal and w canno uniquly rconruc h original coninuouim ignal from i ampl. - Th DTFT of h ampld ignal i obaind from X δ jω by uing h rlaionhip Ω = ωt - Th caling of h indpndn variabl impli ha ω= ω corrpond o Ω = π. - Th FT hav priod ω, whil DTFT hav priod π.
Fig: Th DTFT corrponding o h FT dpicd in Fig:b-d. a ω = 3W, b ω = W, c ω =.5W Rconrucion of coninuou-im ignal from ampl: - Th problm of rconrucing a coninuou-im ignal from ampl involving a mixur of coninuou & dicr-im ignal - A dvic ha prform rconrucion ha a dicr-im inpu ignal and a coninuou-im oupu ignal. - Th FT i wll uid for analyzing hi problm, inc i may b ud o rprn boh coninuou & dicr-im ignal. - W fir conidr h condiion ha mu b m in ordr o uniquly rconruc a coninuou-im ignal from i ampl. Sampling Thorm: - Th ampl of a ignal do no alway uniquly drmin h corrponding coninuou-im ignal. - For xampl, h figur blow how, wo diffrn coninuou-im ignal having h am of ampl x[k] = x nt = x nt Fig: Two coninuou-im ignal x dahd lin and x olid lin ha hav h am of ampl
- No ha h ampl do no ll u anyhing abou h bhavior of h ignal in bwn h im i i ampld. - In ordr o drmin how h ignal bhav in bwn ho im, w mu pcify addiional conrain on h coninuou-im ignal. - On uch of conrain, ha i vry uful in pracic, involv rquiring h ignal o mak mooh raniion from on ampl o anohr. - Th moohn, or ra a which h im domain ignal chang, i dircly rlad o h maximum frquncy ha i prn in h ignal. - So, conraining moohn in h im domain corrpond o limiing h bandwidh of h ignal. - Bcau hr i on o on corrpondnc bwn h im domain and frquncy domain rprnaion of a ignal, w may alo conidr h problm of rconrucing h coninuou-im ignal in h frquncy domain. - To rconruc a coninuou-im ignal uniquly from i ampl, hr mu b a uniqu corrpondnc bwn h FT of h coninuou-im ignal and h ampld ignal. - Th FT ar uniquly rlad if h ampling proc do no inroduc aliaing. - Aliaing dior h pcrum of h original ignal and droy on-o-on rlaionhip bwn h FT of h coninuou-im ignal and h ampld ignal. - Prvnion of aliaing rquir aifying h ampling horm. - L rprn a band-limid ignal o ha Xjω = 0 for ω > ω m If ω > ω m, whr ω = π / T i h ampling frquncy, hn x i uniquly drmind by i ampl xnt, n=0,±,±, - Th minimum ampling frquncy, ω m, i rmd h Nyqui ampling ra or Nyqui ra. Th acual ampling frquncy, ω, i commonly rfrrd o a h Nyqui frquncy. - If f m = ω m / π and f > f m =>/T > f m => T </ f m
Idal Rconrucion: - Th ampling horm indica how fa w mu ampl a ignal o ha h ampl uniquly rprn h coninuou-im ignal. - If, hn h FT rprnaion of h ampld ignal i givn by: - Th goal of rconrucion i o apply om opraion o X δ jω ha convr i back o Xjω. - Any uch opraion mu limina h rplica, imag of Xjω ha ar cnrd a kω. - Thi i accomplihd by muliplying X δ jω by, a hown in h figur blow Fig: a Spcrum of original ignal, b Spcrum of ampld ignal c frquncy rpon of rconrucion filr. - W hn hav Xjω = X δ jω H r jω - No ha, muliplicaion by H r jω will no rcovr Xjω from X δ jω if h condiion of h ampling horm ar no m and aliaing occur. - Muliplicaion in h frquncy domain ranform o convoluion in h im domain. - Hnc, x = x δ * h r Whr,. Subiuing x δ in h abov quaion, w g Nx w u = T ω π inc ω π = inc ω π
So, - In h im domain, w conruc x a a wighd um of inc funcion hifd by h ampling inrval. Th wigh corrpond o h valu of h dicr-in qunc. - Th valu of h x a = nt i givn by x[n], bcau all of h hifd inc funcion ar zro a nt, xcp h n h on and i valu i uniy. - Th opraion dcribd by h abov quaion i rfrrd o a idal band limid inrpolaion, inc i indica how o inrpola in bwn h ampl of a bandlimid ignal. Fig: Idal rconrucion in h im domain Fourir ri rprnaion of fini duraion on priodic ignal: - A DTFS i h only Fourir rprnaion ha can b valuad numrically, o w apply DTFS and FS o ignal ha ar no priodic o facilia numrical compuaion of Fourir rprnaion. - Anohr advanag of hi rprnaion i undranding of rlaionhip bwn h FT and corrponding FS rprnaion.
Rlaing h DTFS o DTFT: - L x[n] b a fini duraion apriodic ignal of lngh M i..; x[n] = 0 for n < 0 and n M - DTFT of hi ignal i - L x[n] b a priodic dicr-im ignal wih priod N M uch ha on priod of x[n] i givn by x[n]. - Th DTFS cofficin of x [n] ar givn by Whr Ω 0 = π / N a x [n] = x[n] wihin on priod A comparion of X[K] and X jω rval ha a x[n] = 0 for n M - Th DTFS cofficin of x[n] ar ampl of h DTFT of x[n], dividd by N and valuad by a inrval of π / N. - Alhough x[n] i no priodic, w dfin DTFS cofficin uing x[n], n = 0,,.N- according o So X[K] = X[K] = /N X jkω 0 from abov wo quaion - DTFS cofficin of x[n] corrpond o h DTFS cofficin of priodically xndd ignal x[n]. - Th ffc of ampling h DTFT of a fini-duraion nonpriodic i o priodically xnd h ignal in h im domain. i..;
Fig: Th DTFS of a fini duraion nonpriodic ignal - Th abov rlaionhip i dual o ampling frquncy. - Sampling a ignal in im gnra hifd rplica of h original ignal in h frquncy domain. Dual: - Sampling a ignal in frquncy gnra hifd rplica of h original ignal in h im domain. - In ordr o prvn ovrlap or aliaing, of ho hifd rplica in im, w rquir h frquncy ampling inrval Ω 0 o b l han or qual o π/ M Ω 0 π/ M => N M Rlaing h FS o h FT: L x, an apriodic ignal hav fini duraion T 0, i..; x = 0, < 0 & T 0 Conruc a priodic ignal wih priod Wih T T 0 by priodically xnding x, h FS cofficin of x ar
Whr w hav ud h rlaionhip: Th FT of x i dfind by x = x for 0 T 0 and x = 0 for T 0 < < T Hnc, comparing X k wih Xjω a x i fini duraion Th FS cofficin ar h ampl h FT, normalizd by T.
Modulaion: Modulaion i baic o h opraion of a communicaion ym. Modulaion provid h man for. Shifing h rang of frqunci conaind in h mag ignal ino anohr frquncy rang uiabl for ranmiion ovr h channl.. Prforming a corrponding hif back o h original frquncy rang afr rcpion of h ignal. Formally modulaion i dfind a h proc by which om characriic of a carrir wav i vrifid in accordanc wih h mag ignal. Th mag ignal i rfrrd o a h modulaing wav, and h rul of h modulaion proc i rfrrd o a h modulad wav. In h rcivr, dmodulaion i ud o rcovr h mag ignal from h modulad wav. Dmodulaion i h invr of modulaion proc. Typ of modulaion: Th pcific yp of modulaion ud in a communicaion ym i drmind by h form of carrir wav ud o prform h modulaion. Th wo mo commonly ud form of carrir ar a inuoidal wav and a priodic pul rain. Corrpondingly, hr ar wo cla of modulaion: Coninuou-wav CW modulaion and pul modulaion. Coninuou-wav modulaion: Conidr h inuoidal carrir wav c = A c coφ which i uniquly dfind by h carrir ampliud A c and angl φ. Dpnding on h yp of paramr chon for modulaion, wo ubcla of CW modulaion i idnifid. i..,
Ampliud modulaion, in which h carrir ampliud i varid wih h mag ignal. Angl modulaion, in which h angl of carrir i varid wih h mag ignal. Fig: Ampliud and angl modulad ignal Pul modulaion: Conidr a carrir wav c = p nt n= ha coni of a priodic rain of narrow pul, whr T i h priod and p dno a pul of rlaivly hor duraion. Whn om characriic paramr of p i varid in accordanc wih h mag ignal, w g pul modulaion. Dpnding on how pul modulaion i acually accomplihd, h wo ubcla ar Analog pul modulaion, in which a characriic paramr uch a ampliud, duraion or poiion of a pul i varid coninuouly wih h mag ignal. Digial pul modulaion, in which h modulad ignal i rprnd in codd form known a pul cod modulaion.
Fig: Pul ampliud modulaion wavform Bnfi of modulaion In communicaion ym, four bnfi which rul from h u of modulaion ar:. Modulaion i ud o hif h pcral conn of a mag ignal o ha i li inid h opraing frquncy band of a communicaion channl. I i uful for long dianc and high pd ranmiion. E.g., Th lphonic communicaion ovr a cllular radio channl, whr 300-300 Hz frquncy ar hifd o 800-900 MHz frquncy, which i aignd o cllular radio ym in Norh Amrica.. Modulaion provid a mchanim for puing h informaion conn of a mag ignal ino a form ha may b l vulnrabl o noi or inrfrnc. 3. I prmi h u of muliplxing. Muliplxing prmi h imulanou ranmiion of informaion baring ignal from a numbr of indpndn ourc ovr h channl and on o hir rpciv dinaion which mak communicaion channl co-ffciv. 4. Modulaion mak i poibl for h phyical iz of h ranmiing or rciving annna o aum a pracical valu. Elcromagnic hory ay ha h phyical aprur of an annna i dircly comparabl o h wavlngh of h radiad or incidn lcromagnic ignal. Alrnaivly, inc wavlngh and frquncy ar invrly rlad w may ay ha h aprur of h annna i invrly proporional o h opraing frquncy. Modulaion lva h pcral conn of h modulaing ignal by an amoun qual o h carrir frquncy. Hnc, h largr h carrir frquncy, h mallr will b h phyical aprur of h ranmiing a wll a h rciving annna.
Full ampliud modulaion: L u conidr a inuoidal carrir wav c = A c coω c. For convninc of prnaion, w hav aumd ha h pha of h carrir wav i zro in abov quaion a h primary mphai hr i on variaion impod on h carrir ampliud. L m rprn a mag ignal of inr. Ampliud modulaion AM i dfind a a proc in which h ampliud of h carrir i varid in proporion o a mag ignal m. = A c [+k a m] coω c... whr k a i a conan calld h ampliud niiviy facor of h modulaor. Th modulad wav o dfind i aid o b a full AM wav. Hr, h radian frquncy ω c of h carrir i mainaind conan. Prcnag of modulaion: I calld h nvlop of h AM wav. Uing a o dno hi nvlop, h quaion may b wrin a a = A c +k a m. Two condiion ari, dpnding on h magniud of k a m, compard wih uniy:. Undrmodulaion, govrnd by h h condiion k a m, for all. Undr hi condiion, h rm +k a m i alway nonngaiv. Thrfor, h xprion for h nvlop of h AM wav may b implifid a a = A c [+k a m], for all.. Ovrmodulaion, govrnd by h wakr condiion k a m >, for om. Undr hi condiion, h quaion i ud in valuaing h nvlop of h AM wav. % modulaion = Th maximum abolu valu of k a m 00. Accordingly, h fir condiion corrpond o a prcnag modulaion 00%, whra h cond condiion corrpond o a prcnag modulaion > 00%.
Gnraion of AM wav: Variou chm hav bn dvid for h gnraion of an AM wav. L u conidr a impl circui ha follow from h dfining quaion. Thi quaion can b rwrin a : = k a [m+b] A c coω c. Th conan B, qual o / k a, rprn a bia ha i addd o h mag ignal m bfor modulaion. Th abov quaion ugg h chm dcribd in h block diagram givn blow for gnraing an AM wav. Baically i coni of wo funcional block: An addr ha add h bia B o h incoming mag ignal m A muliplir ha mulipli h addr oupu [m + B] by h carrir wav A c coω c, producing h AM wav. Th prcnag modulaion i conrolld by adjuing h bia B. Mag ignal m Addr Muliplir AM Wav Bia B Carrir A c coω c Fig: Sym for gnraing an AM wav Fig: Wavform of Ampliud modulaion for a varying prcnag of modulaion
Frquncy domain dcripion of ampliud modulaion: To dvlop h frquncy dcripion of AM wav, w ak h Fourir ranform of boh id of quaion. L Sjω dno Fourir ranform of and Mjω dno Fourir ranform of m. Th Fourir ranform of A c coω c i π A c [δ ω-ω c + δ ω+ω c ] Th Fourir ranform of m coω c i [M jω-jω c + M jω+jω c ] Uing h rul and invoking h linariy and caling propri of h Fourir ranform, h Fourir ranform of h AM wav i xprd a Sjω = π A c [δ ω-ω c + δ ω+ω c ] + k a A c [M jω-jω c + M jω+jω c ] 3 L h mag ignal m b band limid o h inrval - ω m ω ω m a hown in h figur blow. a b Fig: W rfr o h high frquncy componn ω m of m a h mag bandwidh, maurd in rad/. W find from h qn 3 ha h pcrum Sjω of h AM wav hown in h figur b abov for h ca whr ω c > ω m. Thi pcrum coni of wo impul funcion wighd by h facor π A c and occurring a ± ω c, and wo vrion of h mag pcrum hifd in frquncy by ± ω c and cald in ampliud k a A c. Th pcrum in fig b dcribd a follow: a For poiiv frqunci, h porion of h pcrum of h modulad wav lying abov h carrir frquncy ω c i calld uppr idband, whr a h ymmric porion blow ω c i calld lowr idband. For ngaiv frqunci, hi condiion i rvrd. Th condiion ω c > ω m i a ncary condiion for h id band no o ovrlap.
b For poiiv frqunci, h high frquncy componn of h AM wav i ω c + ω m and h low frquncy componn of h AM wav i ω c - ω m. Th diffrnc bwn h wo frqunci dfin h ranmiion bandwidh ω T for an AM wav which i xacly wic h mag bandwidh ω m, i.., ω T = ω m. Th pcrum of AM wav a dpicd in fig b i full, in ha h carrir, h uppr idband and h lowr idband ar all complly rprnd. Hnc, hi form of modulaion i calld a full ampliud modulaion. Dmodulaion of AM Wav: Envlop dcor i ud for dmodulaion of AM wav, hown in h figur blow, which coni of a diod and a rior-capacior filr. Th opraion of hi nvlop dcor i a follow: On h poiiv half-cycl of h inpu ignal, h diod i forward biad and h capacior C charg up rapidly o h pak valu of h inpu ignal. Whn h inpu ignal fall blow hi valu h diod bcom rvr biad and h capacior C dicharg lowly hrough h load rior R l. Th dicharging proc coninu unil h nx poiiv half cycl. Whn h inpu ignal bcom grar han h volag acro h capacior, h diod conduc again and h proc i rpad. Hr i i aumd ha h diod i an idal diod, h load rianc R l i larg compard wih h ourc rianc R. During h charging proc, h im conan i ffcivly qual o RC. Thi im conan mu b hor compard wih h carrir priod π/ω c, i.., RC << π/ω c Fig: Circui diagram of Envlop dcor howing i inpu & oupu. Accordingly h capacior C charg rapidly and hrby follow h applid volag upo h poiiv pak whn h diod i conducing. In conra, whn h diod i rvr biad, h dicharging im conan i qual o R l C. Thi cond im conan mu b
long nough o nur ha h capacior dicharg lowly hrough h load rior R l bwn poiiv pak of h carrir wav, bu no o long ha h capacior volag will no dicharg a h maximum ra of chang of modulaing wav, i.., π/ω c << R l C << π/ω m. Pul Ampliud Modulaion: Pul ampliud modulaion PAM i a widly ud form of pul modulaion. Th baic opraion in PAM ym i h ampling ha includ h drivaion of ampling horm and rlad iu of aliaing and rconrucing h mag ignal from i ampld vrion. Mag ignal m Low-pa anialiaing filr Sampland-hold circui Sampld ignal Timing pul gnraor Fig: Sym for gnraing a fla-oppd PAM ignal Th ampling horm in h conx of PAM i in wo quivaln par a follow: A band-limid ignal of fini nrgy ha ha no radian frquncy componn highr han ω m i uniquly drmind by h valu of h ignal a inan of im parad by π / ω m cond. A band-limid ignal of fini nrgy ha ha no radian frquncy componn highr han ω m may b complly rcovrd from knowldg of i ampl akn a h ra of ω m / π pr cond. Par of ampling horm i xploid in h ranmir of a PAM ym and par, in h rcivr of h ym. Th pcial valu of h ampling ra ω m / π i rfrrd o a h Nyqui ra. To comba h ffc of aliaing in pracic, w u wo corrciv maur: Prior o ampling, a low pa ani-aliaing filr i ud o anua high frquncy componn of h ignal which li ouid h band of inr. Th filrd ignal i ampld a a ra lighly highr han h Nyqui ra.
Mahmaical dcripion of PAM: PAM i a form of pul modulaion, in which h ampliud of h puld carrir i varid in accordanc wih inananou ampl valu of h mag ignal. Fig: Wav form of fla oppd PAM ignal For a mahmaical rprnaion of PAM ignal for a mag ignal m, w may wri = m[n]h n T n= Whr, T ampling priod m[n] h valu of mag ignal m a im =nt h a rcangular pul of uni ampliud and duraion T 0 Th impul ampld vrion of h mag ignal m i givn by Th PAM ignal i xprd a m δ = m[n]δ n T n= = m[n]h n T n= = m δ h Th abov quaion a ha i mahmaically quivaln o h convoluion of m δ - h impul ampld vrion of m and h pul h.
Muliplxing: Modulaion provid a mhod for muliplxing whrby mag ignal drivd from indpndn ourc ar combind ino a compoi ignal uiabl for ranmiion ovr a common channl. In lphon ym, h ignal from diffrn pakr ar combind in uch a way ha hy do no inrfr wih ach ohr during ranmiion and o ha hy can b parad a h rciving nd. Muliplxing can b accomplihd by paraing diffrn mag ignal ihr in frquncy, or im, or hrough h u of coding chniqu. Thu, hr ar hr baic yp of muliplxing, viz: a Frquncy-diviion muliplxing: In hi yp of muliplxing, h ignal ar parad by allocaing hm o diffrn frquncy band. FDM favour u of CW modulaion, whr ach mag ignal i abl o u h channl on a coninuou-im bai. Fig: a Frquncy-diviion muliplxing b Tim-diviion muliplxing b Tim-diviion muliplxing: Hr, h ignal ar parad by allocaing hm o diffrn im lo wihin a ampling inrval. TDM favour h u of pul modulaion, whr ach mag ignal ha acc o h compl frquncy rpon of h channl. c Cod-diviion muliplxing: I rli on h aignmn of diffrn cod o h individual ur of h channl.
a Frquncy-diviion muliplxing: Th block diagram of FDM ym i hown blow. Th low pa filr ar ud for band limiing h inpu ignal. Th filrd ignal ar applid o modulaor ha hif h frquncy rang of h ignal o a o occupy muually xcluiv frquncy inrval. Th band pa filr following h modulaor ar ud o rric h band of ach modulad wav o i prcribd rang. Nx, h ruling band pa filr ar ummd o form h inpu o h common channl. A h rciving rminal, a bank of band pa filr, wih hir inpu, conncd in paralll, i ud o para h mag ignal on a frquncy occupancy bai. Finally, h original mag ignal ar rcovrd by individual dmodulaor. Fig: Block diagram of FDM ym b Tim-diviion muliplxing: Th baic opraion of a TDM ym i h ampling horm, which a ha w can ranmi all h informaion conaind in a band limid mag ignal by uing ampl of h ignal akn uniformly a a ra ha i uually highr han h Nyqui ra. Th imporan faur of h ampling proc i conrvaion of im i.., h ranmiion of h mag ampl ngag h ranmiion channl for only a fracion of ampling inrval on a priodic bai, qual o h widh T 0 of a PAM modulaing
pul. In hi way, om of h im inrval bwn adjacn ampl i clard for u by ohr indpndn mag ourc on a im hard bai. Fig: Block diagram of TDM ym Th concp of TDM i illurad by h abov block diagram. Each inpu mag ignal i fir rricd in band widh by a low pa filr o rmov h frquncy ha i non nial o an adqua rprnaion of h ignal. LPF oupu applid o a commuaor ha i uually implmnd by man of lcronic wiching circuiry. Th funcion of h commuaor i wo fold. o ak a narrow ampl of ach of h M inpu mag ignal a a ra /T i.., lighly highr han ω c / π, whr ω c i h cu off frquncy of LPF. o qunially inrlav h M ampl inid a ampling inrval T. Th muliplxd ignal i applid o a pul modulaor ha ranform i ino a form uiabl for ranmiion ovr a common channl. A rcivr, h ignal i applid o a pul dmodulaor which prform invr opraion of pul modulaor. Th narrow ampl producd ar diribud o h appropria low pa rconrucion filr by dcommuaor. Synchronizaion bwn iming opraion of h ranmir and rcivr in a TDM ym i nial for aifacory prformanc of h ym. Synchronizaion may b achivd by inring an xra pul ino ach ampling inrval on rgular bai. Th combinaion of M PAM ignal and a ynchronizaion pul combind in a ingl ampling priod i rfrrd o a a fram ynchronizaion.
Pha and Group dlay: Whnvr a ignal i ranmid a hrough a frquncy-lciv ym, uch a communicaion channl, om dlay i inroducd ino h oupu ignal in rlaion o h inpu ignal. Th dlay i drmind by h pha rpon of h ym. L h pha rpon of a dipriv communicaion channl i rprnd by: υ ω = arg{hjω}, whr Hjω frquncy rpon of h channl Suppo ha a inuoidal ignal i ranmid hrough h channl a a frquncy ω c. Th ignal rcivd a h channl oupu lag h ranmid ignal by υω c radian. Th im dlay corrponding o hi pha lag, which i known a pha dlay τ p : τ p = φω c, whr minu ign - dno h lag. ω c No: Th pha dlay i no ncarily h ru ignal dlay. L u conidr a ranmid ignal = A co ω c co ω 0, coniing of a DSB-Sc modulad wav wih carrir frquncy ω c and inuoidal modulaion frquncy ω 0. Expring h modulad ignal in rm of i uppr and lowr id frqunci, i may b wrin a: = A co ω + A co ω whr, ω = ω c + ω 0 and ω = ω c - ω 0 If ω 0 << ω c => id frqunci ω & ω ar clo oghr, wih ω c bwn hm. Such a modulad ignal i calld narrowband ignal. Th pha rpon υ ω may b approximad in h viciniy of ω= ω c by h wo-rm Taylor ri xpanion Th im dlay incurrd by h mag ignal i.., h nvlop of h modulad ignal i givn by: Th im dlay τ g i calld h group dlay or nvlop dlay. Th group dlay i dfind a h ngaiv of h drivaiv of h pha rpon υ ω of h channl wih rpc o ω, valuad a h carrir frquncy ω c. No: Th im dlay i a ru ignal dlay.
For wid-band modulad ignal, h frquncy componn of h mag ignal ar dlayd by diffrn amoun a h channl oupu. Conqunly, h mag ignal undrgo a form of linar diorion known a dlay diorion. To rconruc a faihful vrion of h original mag ignal in h rcivr, w hav o u a dlay qualizr. Thi qualizr ha o b dignd in uch a way ha whn i i conncd in cacad wih h channl, h ovrall group dlay i conan. Inroducion o h Laplac Tranform Fourir ranform ar xrmly uful in h udy of many problm of pracical imporanc involving ignal and LTI ym. Thy ar purly imaginary complx xponnial, =jω A larg cla of ignal can b rprnd a a linar combinaion of complx xponnial and complx xponnial ar ignfuncion of LTI ym. Howvr, h ignfuncion propry appli o any complx numbr, no ju purly imaginary ignal. Thi lad o h dvlopmn of h Laplac ranform whr i an arbirary complx numbr. Laplac and z-ranform can b applid o h analyi of un-abl ym ignal wih infini nrgy and play a rol in h analyi of ym abiliy Th rpon of an LTI ym wih impul rpon h o a complx xponnial inpu, x=, i y H whr i a complx numbr and H whn i purly imaginary, hi i h Fourir ranform, Hjω whn i complx, hi i h Laplac ranform of h, H Th Laplac ranform of a gnral ignal x i: X and i uually xprd a: L x X Laplac and Fourir Tranform Th Fourir ranform i h Laplac ranform whn i purly imaginary: An alrnaiv way of xpring hi i whn = σ+jω h x X F x j X j x d x x' d j j d d j d F{ x' } Th Laplac ranform i h Fourir ranform of h ranformd ignal x = x -σ. Dpnding on whhr σ i poiiv/ngaiv hi rprn a growing/ngaiv ignal
Exampl : Laplac Tranform Conidr h ignal Th Fourir ranform Xjω convrg for a>0: Th Laplac ranform i: which i h Fourir Tranform of -σ+a u Or If a i ngaiv or zro, h Laplac Tranform ill xi Exampl : Conidr h ignal Th Laplac ranform i: Convrgnc rquir ha R{+a}<0 or R{}<-a. Th Laplac ranform xprion i idnical o Exampl imilar bu diffrn ignal, howvr h rgion of convrgnc of ar muually xcluiv non-inrcing. For a Laplac ranform, w nd boh h xprion and h Rgion Of Convrgnc ROC. Exampl 3: Th Laplac ranform of h ignal x = inωu i: u x a 0, 0 a a j d d u j X j a j a 0 0 d d d u X j a a a a a X u L a } R{, 0, a j a j X u x a a d d u X a a 0 0 0 0 0 j j j j j d d d u X j j j j j j j j j j j j
Fourir Tranform do no Convrg I i worhwhil rflcing ha h Fourir ranform do no xi for a fairly wid cla of ignal, uch a h rpon of an unabl, fir ordr ym, h Fourir ranform do no xi/convrg E.g. x = a u, a>0 X jω = 0 a jω d do no xi i infini bcau h ignal nrgy i infini Thi i bcau w muliply x by a complx inuoidal ignal which ha uni magniud for all and ingra for all im. Thrfor, a h Dirichl convrgnc condiion ay, h Fourir ranform xi for mo ignal wih fini nrgy. Rgion of Convrgnc: Th Rgion Of Convrgnc ROC of h Laplac ranform i h of valu for =+jω for which h Fourir ranform of x -σ convrg xi. Th ROC i gnrally diplayd by drawing paraing lin/curv in h complx plan, a illurad blow for Exampl and, rpcivly. R{ } a R{ } a Th hadd rgion dno h ROC for h Laplac ranform Exampl 4: Conidr a ignal ha i h um of wo ral xponnial: Th Laplac ranform i hn: x 3 X u u Uing Exampl, ach xprion can b valuad a: 3 3 u u 3 X Th ROC aociad wih h rm ar R{}>- and R{}>-. Thrfor, boh will convrg for R{}>-, and h Laplac ranform: X 3 d u d u d
Raio of Polynomial: In ach of h xampl, h Laplac ranform i raional, i.. i i a raio of polynomial in h complx variabl. N X D whr N and D ar h numraor and dnominaor polynomial rpcivly. In fac, X will b raional whnvr x i a linar combinaion of ral or complx xponnial. Raional ranform alo ari whn w conidr LTI ym pcifid in rm of linar, conan cofficin diffrnial quaion. W can mark h roo of N and D in h -plan along wih h ROC Exampl 3: Pol and Zro: Th roo of N ar known a h zro. For h valu of, X i zro. Th roo of D ar known a h pol. For h valu of, X i infini, h Rgion of Convrgnc for h Laplac ranform canno conain any pol, bcau h corrponding ingral i infini. Th of pol and zro complly characri X o wihin a cal facor + ROC for Laplac ranform i zi X p Th graphical rprnaion of X hrough i pol and zro in h -plan i rfrrd o a h pol-zro plo of X Exampl: Conidr h ignal: 4 x 3 By linariy w can valua h cond and hird rm j j u 3 u
Th Laplac ranform of h impul funcion i: Th Laplac ranform of h impul funcion i: which i valid for any. Thrfor, ROC Propri for Laplac Tranform: Propry : Th ROC of X coni of rip paralll o h jω-axi in h -plan Bcau h Laplac ranform coni of for which x -σ convrg, which only dpnd on R{} = σ Propry : For raional Laplac ranform, h ROC do no conain any pol Bcau X i infini a a pol, h ingral mu no convrg. Propry 3: if x i fini duraion and i aboluly ingrabl hn h ROC i h nir - plan. Bcau x i magniud boundd, muliplicaion by any xponnial ovr a fini inrval i alo boundd. Thrfor h Laplac ingral convrg for any. Invr Laplac Tranform: Th Laplac ranform of a ignal x i: W can invr hi rlaionhip uing h invr Fourir ranform Muliplying boh id by σ : } { d L } R{, 3 3 4 X d x x F j X j } { d j X j X F x j } { d j X x j
Thrfor, w can rcovr x from X, whr h ral componn i fixd and w ingra ovr h imaginary par, noing ha d = jdω x j j Invr Laplac Tranform Inrpraion: j X d Ju abou all ral-valud ignal, x, can b rprnd a a wighd, X, ingral of complx xponnial,. j x X d j j Th conour of ingraion i a raigh lin in h complx plan from σ-j o σ+j w won b xplicily valuaing hi, ju poing known ranformaion. W can choo any for hi ingraion lin, a long a h ingral convrg For h cla of raional Laplac ranform, w can xpr X a parial fracion o drmin h invr Fourir ranform. X M i Ai a i L { A / i a i } A i A ai ai i u u R{ } a i R{ } a i Exampl : Invring h Laplac Tranform Conidr whn X Lik h invr Fourir ranform, xpand a parial fracion A B X Pol-zro plo and ROC for combind & individual rm L u, R{ } L u, R{ } L x u, R{ }
Exampl : Conidr whn X R{ } Lik h invr Fourir ranform, xpand a parial fracion A B X Pol-zro plo and ROC for combind & individual rm L u, R{ } L u, R{ } L x u, R{ } Laplac Tranform Propri: a Linariy propry: L If x X ROC = R L and x X ROC = R L hn ax bx ax bx ROC= R R Thi follow dircly from h dfiniion of h Laplac ranform a h ingral opraor i linar. I i aily xndd o a linar combinaion of an arbirary numbr of ignal. b Tim Shifing propry: If L x X L hn 0 x 0 X Proof: j j X j x d ROC=R ROC=R Now rplacing by - 0 x j 0 j j j j j 0 X X d 0 Rcogniing hi a L{ x 0} X A ignal which i hifd in im may hav boh h magniud and h pha of h Laplac ranform alrd. 0 d
Exampl: Linar and Tim Shif Conidr h ignal linar um of wo im hifd inuoid whr x = inω 0 u. Uing h in Laplac ranform xampl Thn uing h linariy and im hif Laplac ranform propri c Convoluion propry: Th Laplac ranform alo ha h muliplicaion propry, i.. ROC = R ROC = R ROC R R roof i idnical o h Fourir ranform convoluion No ha pol-zro cancllaion may occur bwn H and X which xnd h ROC Exampl : Fir ordr inpu & Fir ordr ym impul rpon Conidr h Laplac ranform of h oupu of a fir ordr ym whn h inpu i an xponnial dcay? Solvd wih Fourir ranform whn a,b>0 Taking Laplac ranform Laplac ranform of h oupu i 4 0.5.5 x x x 0 } R{ 0 0 X 0 } R{ 0.5 0 0 4.5 X X x L H h L * H X h x L } { } { } { H X H X u h u x b a a a X } R{ b b H } R{, }, max{ } R{ b a b a Y
Spliing ino parial fracion and uing h invr Laplac ranform No ha hi i h am a wa obaind arlir, xpc i i valid for all a & b, i.. w can u h Laplac ranform o olv ODE of LTI ym, uing h ym impul rpon Exampl : Sinuoidal Inpu Conidr h ordr poibl unabl ym rpon wih inpu x Taking Laplac ranform Th Laplac ranform of h oupu of h ym i hrfor and h invr Laplac ranform i d Diffrniaion in h Tim Domain: Conidr h Laplac ranform drivaiv in h im domain ROC = R ROC R X ha an xra zro a 0, and may cancl ou a corrponding pol of X, o ROC may b largr Widly ud o olv whn h ym i dcribd by LTI diffrnial quaion X x L j j d X j x j j d X j d dx X d dx L }, max{ } R{ b a b a a b Y u u y b a a b H h L co 0 u x u h a a a H } R{ 0 } R{ 0 X a a a a a a a Y } max{0, } R{ 0 0 0 0 0 a a a a u y co in 0 0 0 0
Exampl: Sym Impul Rpon Conidr rying o find h ym rpon ponially unabl for a cond ordr ym wih an impul inpu x=δ, y=h Taking Laplac ranform of boh id and uing h linariy propry whr r and r ar diinc roo, and calculaing h invr ranform Th gnral oluion o a cond ordr ym can b xprd a h um of wo complx poibly ral xponnial. x cy d dy b d y d a } { } { } { r k r k r r a c b a H y L c b a y L L y cl d dy bl d y d al u k u k y r r
Th z-tranform I play h am rol in h analyi of dicr im ignal & LTI ym a h Laplac ranform do in h analyi of coninuou im ignal and LTI ym. Th mo imporan on i, in h z-domain, h convoluion of wo im domain ignal i quivaln o muliplicaion of hir corrponding z ranform. Dfiniion: Th z-ranform of a dicr-im ignal x[n] i: X z = x n z n n= W dno h z-ranform opraion a x[n] X z. In gnral, h numbr z in Xz i a complx numbr. Thrfor, w may wri z a z = r jw, whr r i h radiu of h circl. Whn r =, 7. bcom X jω = x n jωn n= which i h dicr-im Fourir ranform of x[n]. Thrfor, DTFT i a pcial ca of h z-ranform. Hnc, w can viw DTFT a h z-ranform i valuad on h uni circl. figur blow Figur 7.: Complx z-plan. Th z-ranform rduc o DTFT for valu of z on h uni circl.
Whn r, h z-ranform i quivaln o X r jω = x n r n jωn n= = [x n n= r n ] jωn = F [r n xn], which i h DTFT of h ignal r -n x[n]. Howvr, from h dvlopmn of DTFT w know ha DTFT do no alway xi. I xi only whn h ignal i quar ummabl, or aifi h Dirichl condiion. Thrfor, X z do no alway convrg. I convrg only for om valu of r. Thi rang of r i calld h rgion of convrgnc. ROC: Th Rgion of Convrgnc ROC of h z-ranform i h valu of z uch ha X z convrg, i.., n= x n r n < Exampl: Conidr h ignal x[n] = a n u[n], wih 0 < a <. Th z-ranform of x[n] i X z = a n = az n n=0 u[n ]z n Thrfor, Xz convrg if n=0 a z n <. From gomric ri, w know ha n=0r z n = az whn az -l <, or quivalnly z > a. So, X z = - ax -l, wih ROC bing h of z uch ha z > a.
Figur 7.: Pol-zro plo and ROC of Exampl. No: ROC of caual and infini qunc i h xrior of a circl having radiu a.