r aay ip We j bi 49663 Saal ia VTL GlOWAL PRoceasings oe signal Angtting thal lve oy tudes Saya we alld ag Chyna ee dave Viale aan ri Jd d Physfal —Juantity wth depect to any the pendent 4 ot SsLempeuatire Saves $s voltage Ly Leabon Ls apace too Sina Processing r trator o. fre > ane pe aaa ‘ ‘ tled ipa the 41gn eee o sign spor iar In Sfana p | procerst 0 Ha : Sat Tarra 54 ‘ ee Br at 3 ao : Ir Analog VF PEE on thee sfpnctl 1 : purpor PF jpctol PT ; 4 ole al «1, 2%, cash 7 ,» laa Pod pnalog IP | AY pePy alge atual BO tar Ne Gia. Ae ar Co ) a > ust na we Hn fo over stqnot Ae Mot $ fre gots Into oo ee AnePre pressing re leer [ave He y of rl = of ae ‘Analog loigh EMR 6 T | sfemnal a 4 psp, | Digftat 5 pate ! ( ' ie a 2 hea fa oP ! at advaniages : o 3 Cfradt betomes tomplex due b& Ppt kek, as and post proces elements we > Mere — pou 2 Mintted — Bards to Sampling sail Advartages a > defiel Aigral ProceseIng 4 den sersfeda He Lomponent —éolevarnces ade 7 Atewracy fis greater ies g > exock (> Ay desired 2 ny es - ty innpeoving i: > Nok Wood length > ro of bit 2 wpre > Full Ai Eee shartro : > Jew | SS ke 4 Ga > sping > of Smal ¢ aumpl i ee > gral 1 amped therensed by Y Mol 2 enpb-ade Ba Jn ASPbe preprecessting a Sampting Aexto ‘ras > Cannmt, be mixed’ th AsPing ductor capueltoy Pirmttanecnsly Bh haBased on channel P Generated by pte dein steal 8 keads are used |Represented ooo | { o> | BAEC | Cons Pac | thunder st | nd? Signa! L tet | 4 ui chonnd * 4 tha Stned kp myo Mime . There pe . agntty + Hany Sou ale [nose | _ q havacerisves Of tay fo cact (cont Ame mode lled t . An i paobabt slic tem, Ls pron sre yersvables putts? Lfmerstor ot ae cP a all fe, ent p, 3, + Any preture Cx) ‘ / 5 fot F analec an Midem (xy, +) rot prs Hi chonnel $ig72! sprected b one C6) sfynol Yeads gre usecUNIT-4 - TNTRODUCTION tal stgnal Processing : * aaa ® Anti-atiasing Srey i signal Th js a filter befo’ Bu gtcts att) % sempled and values ar an PT ase taken irtegey Valhe,nstder two analog ee Mart) = Cogam int wast > Cos IT G0t' Obtain korrempending discret fame signal the Bampling frequency 4 40 Hy ue ER coy ait 540K | t= br xi (ny * osomiolrr) » ces aTion( a! fs gn yan [1 \ (az) C5 common’) By Sequence Ac presen a te _ wae depresenta:ere eee a ‘ Cnet true) Fe Za yo sa hoo Atgnaun are tnelsngcisabte So, the bo 4% Fe = to Hg e choose Now, ws i(n)= Ka, (4) lo nT 2 coum (nt) 5, term 95 (Sy al) Beta) = xan it) | ¢=ns soars 1p stg» ne _Ujseemple = lesan 5o(nt)= ‘ceszTt pon (L : ie wth) = srr fs no atfasing ice aa : & 3. An anal g uF Amplitude % of Sampling (e55ont . What vepyp=0 = tn», Sable ela uP ele, losin 300 TT = f- Sampling > Wa lin ionat mete” - of "eatin Fi+kE — Biee eS pues sane * Tne, eI yout pe proxd Gere eed 5 a o Ke2 1oH.2/ ado. saps go, atk al :. adel Pe) = Aso @ sare A sdg)n= 9 rod |seemp Quantiser the fart : cavT) : Pica Time Amplitrae fendi vA bee ‘| Dien — Amplihid La]! : \ t (5 MiGuantizer, eth by truncate Ly Paund off The eaves feel i ed Cleon athe difference’ ly fwo euccenstie 9 ie ‘onss quantization . steed | S quanti gation ) ons Lew, y i , a binaey 3 ep age) <4 eam Too types fF — Quantization y = ef Une ee ices es ® Won ~ Untgerom UNIFORM Quantization Ro etetee | eon Gime. Sk Hep Az (QhA > Xmox ~ Xe tins | ; 2Hl a Kar Xf, —> Dynamic Vang Db Cent quantizer| coper Coder anstgns a unique Linooy mimlys js - “ngs ees Fate | cach quantization — level | fox dtgfeven level tp aa cifjenne — Woney levee 8€ 9 qutred : _ eras 9 leuk ae there , LS afyrtat - Be alo a 2oes fons (3 ~ Ue are wed )Lonsfats of ALD, Shite, of: adctheys newton. age I rs ea Coe eRer Te te Airfar ke miwopmesesey atk baye anchiectnal featges Aue to Whit Satan Funders a a a. Wavefort ae a5 Compared fe minepretexsor , 2. Tabular 4. Smyence DAC CPiitcs 4 Analog Converter) 5. eliioal ofr tal & Pralog Bnveakr Comers lig tal SP d a he © FONCTIONAL in Anoler Sigrel r3 20) - oie CstaPreae ~ waveform) ye ory \Daee 4 seq fb Arequencey L Obmupt soquency song)! B® Sequence @ sepol ™ Yn 47 Qn pUhde iSIGNAL CoIERETE | Time ) REPRE SenrmT ion} 4. Fundtonal — Representation 2. Wayeforny — Representatien 2. Tabuloy — Reprwsertation 4. Smuence — Represeritatunrs §. Signal Representation ee} Sum of welgttea Impulse Sequence ler, FUNCTIONAL REPRESENTA TIO 0 xire i, waveform > ; ae ,oene® otherwise waveform — Represertcien @ Sequente 2-3 @ sepod pepeBnLlEen lo) ' ' eq valueLDgsIFICATION OF DISCRETE Te ENG, Nay, metric (een) | trsyrneretit ant tou ~syrmet, og - , b ay t % 2e= NY e- xin), zen) Right hand | left bondeigral FOS 0, 84} an ; 1 elt and Xeni 1 ~ — pottodte | Nor tertodie ep ») xr fofd te be a ee seen fodic Ce ae w22 th a poledic PR eri i pereate with poral a 20 mae" x Orn) zinin) 2 2tr> LN) # try f 12,3, 4 Ritoie fh, 2 2 (0 a1 i2 34,1 fi +0 = ota 44 7, 4 i, °, aa a xtoy- edb ore ae a s, oa Hence xlo+ 4) D214)=1 H+ 3) = x65) # lo) x07) = we 2m wrted of pellecd on 3 Deteimine the fundamental Pf of fell a Sond, ‘, ee, a e aoe potod'e ) xn) = $7 ' #2 J f) xtnp 2 ¢@ Jan 3 ® 2 i Z A y ) . Spy fi ‘a 5 multiples oF @) it =2nm 27m 2 Feo N= 8m all ‘alistausjee ROU la wed xeae, ; een nD. alm) nen toate b Xfny ; étynal ee ss ; > 7 A Oe th teh peatedt Xt. es (Zn ae ww 00) 3BN, = BN (ee), a vi) ENERGY AND Power = SIGNAL Total Powe a any pra @ Bounded __ Signal B unbounded 54 lam] 2 2 ¥n on eee au © th the kon) 4M Peas. +, 4) absolutely Summble signal = kktwl¢# aes [x2 >. yk ie onegy Dp xt6) ele i552 ; Beto 2 be el nee ! 6 La a ae a) Erevgy When the signal &% Surrnalle; en ——_ hall falled en etgy signalee ott a a8 ener a ha =p, ‘i io > ' a, - es i oy ees) al ee mmnelt at ? e | ian Si meee -— an x Pen i 4 unit IepulseOY ape (sy un \ xn)” Re cee S sign a, - 3 [atom larel + 2 Sia a wt le f ) 0 | Bol neo ! vi 4 + aes ; a neo ae (0) = x D) eypal > “at Ene: val Es Jer Fowbe hophiat Yte : Pe t SL esse 4@) a \ =1) (any © acne)on causal gral = a ; ahe yine é Set Ca) oe “— 2 — ” < " MEG 2 dg Lins i oa ree BattesTect OperntionS OM we th ty a win) 16 j \ W } = Causal / Arti comol | Dor causal ety Vrausal, xfqrals tho: ce) ae or @® Tuefm causal $fyrat Above tus SFgvol, Oe, nat Satcfrect then the pal 7, * rl fs nem — Causal OpeRDTIONS ON SIGNALS @7 Boteting se anita he value ement of Independent bf the Ascrete Ure Algol 4g varfablenea; pion 252) exo) = 4 em etoe. eke) - ? pe. Shin Seale oom "foo het, Yall exe 2 fe? | pays Q)= 4 pend f P(-3) = XG) kat n-02) (0) + 3) Tre Saléng s 7 ae nid Yod= [ ] —— fe an posh > yee coae") hon fe o> Yee [ie Daaeene” | | oe Ya (rn) = xC2n) : - io). 2 ‘ Yelm = 4 nei 5 fst = *(2) © F eee (> siGinel este cep =axe(- 2) 2, a7 o + Ele geen joa hp5) Signal mnelliplied eeREPRESENTATION OF AN PRBITRERY coma sampre 'segbenct 7 Stor? urbe. Fopslse 4, Janie Sample sy. £ sample sequeee , mrotwe xn) din) bo merrdinny 22 . i de, additoBhijted and wiied dmputse > HN ga thud pet tnok He) Slme ad nadine dy 21-9 dems) + 110) SC) + 260 1 Ane =n) : e x(k) Sk) = ae S eniatien'@ ‘amprhasbitaty using in=1) hal BO tein 7 i “et ; : (webped sum of Sbhfted, impy jemple Sequeree of of Troe _Systern pe pie piateele The 7 pas aa f 46 E eee fe le oe : ——. ae F a sts > % 4 40) oa Or YOR © Sia xOCONVOLUTION » sum % found austng 1. Yabulas method 1 he b for find 2 { + ort Yrophieat method |, es a irre %y 3 Matatx method _) * Aehteg : x a Analific method > fer lading — tan hn | 24 aun eae aoe 2 Neng a) -4 Detounine the tespernse 0 2 System deserted y 4 donpule Xespens ic % on tall “ > pinfte lu o 4 z aquente, —*t0)+ Kei nei ¥ Pr 5 ap a te system , y(o) = xin + bind yor 1 do malsfe methed, the 2m + hoo) sample gineton Merten ae) fend Melia ted golumnn wets him — Snpulit natn — nethocl ’ - Zz set the _— an > spat seal bly) ee .. pep = 2 plea Gin) 2 dU -OO ea| -4 22-3 EB x0 age at, 2 8 = | a 4 2, wi ‘4 3/3 é : sjrea Impul ( ay tat he 3 =a go fn a a yore dar, x 3 e, 3} : sanalb ts ghig % He Aemvelution sum ef tuo mie 4 2.00) = pohetter dhe tenvolution unr h We tan check | Ds by sing, Ne tN, -! fyi ~ 8t4-] = 6 Lorrect oYplat of the Ayal Te Gian by m4 Nye Mr OK orl =-) htny Zamples ayy yo egies whee him —> Sopa apa cvautiin 4, “Ta la ci) > pak Ayal ad(n-2y — 40073) 3 + ed) Gwr2 ay +1 : : J ae yacxorgin* et. 400-1 > agian) S102) » -dln-Mots’x — aethed x30ry| of eCH)F Ur) 5 hoe BA Ma, plore xin) x bes =F uk) bin-k) Bat fe Tavobwed|, TheD hey are Infinite Lays Bbquerde (@) insinte — ebaaleen gin = 4 uted) = 4)° ok | ) i, K 20 Lj a N-KeO ‘ : nro amet ak) 2) yon Sa yon > I Be = 3° uch) gore : kW120 tsintee engin)onlay in hy “ulna ®coroluction af ah 4 Cn, ali values et 0 Foy Wy arb te = bP utnyCay raCONVOLUTION 200) 4 (0) = hi ainy ProperTies OF cori fies 1. Commutoloug laws Ss . 2, Adpsockative xaw —> xo [MO shat) aaa 3, Distahbutwe ta cE noone I i 1 ston al bz * (ho) the 13] = OD bled corel 7 station [xox mato) 1s GM ar) LATION * t, wed to Compe CONVOLUTION py tt fs used to finch scrcte de ctespmse of tere ayster ag moasuse lo ahh Aienthar , Yinr- {4 cord Ply ony 4 & Fipretsion for tonvolution + Gipeision ~ > yk) %2 Co-kD 4 © i. ag (ny) = Sy Glee G Ee 6; ee 3 +4 5 > ee, = eg 2 a es , Cae 4 d Ze 1 ey cy ee atm # %al-9) a8 no 5 ols .en a ae 2 ota - mS Me Wy & Heme fiom y ! t 27 J 3] ee . a 1 2 3 Vt r Wey oe eae 6 - ee W/) come prea “> Cae eM 1-#, 7, & 9, 8, 8h fh - y 2% (N)= %y Cn) ¥ Xe(-n> " 3 S20) at sae astny -f4 4Pag nay thm +m! Yh) cormall ¥ &, %!® BD sind wen and gd “pacts 1) = CStown +1) * zen)= Cfo. +1)? Pe S10 1 + Asta on 2 ep) = (Bout }. 6+ a6pwen) a a zon) FA = Fists] + 14 Qhinwe-n ay ig, pind = hoon | ~ abinocn> ee Hetn)= Xr KO-H = IF show, = . GEE Xn) = RC) 2 staan @ Fnd —eshethen tho giver pour — AT gratl (0) = axe ently a 0 = eDanies ee) (36 €J4tno een)ind A peodic or nm- produc eC m+ 14)fy 3d) 2ytr> toe anare! 70 noun Arnal. > od whetfor the Aipol xt) = Hn +b ot * fi periodic & pert Hynal hs a | LO) = HAND Any we hae & cnfdec tre 0-8 % (of An on x 1 2 tfiren geo? = d(n~i) 4 gon) = yor? Bro, 0, 1% 1) 400-5)gin-S) > +tw UB) ey v9-19 =Yert2y utn) uc) =) ene ia] if at DISCRET ile hies yor > 7 pisnete In ; device ov Naas er os Me atarat and re post 2 —2 x0? Find Lhe Ayauiat fake k ngquise tatoned of eee ea oer: urs ATION a. Stat may = cos Caocorrt) cesCippent), ... an 1 8 Hoes et, f ca lawont) + Caocont) | on | 2 3 4 a Ss: he | S aa [as ooo mb + Ws 3000 | 2brun = 99007) cetate oa eal Re i \ Smee = 2500 Hy of are aunesi FICATION quate & dypanic U8 2. to ee comsad system g rd cron bineort aytem dont ond tee Invardant Ayse™ depend > Grohe& fer state Systems () = 5 x70) ci ¥ @ + xen) 1) Blogs yp to) es 5x) Feta) Pon Be apes tr oly vee Pe ‘ie YO= sxHy oo dio = 'y a ceusne Bo NOM cqueay « SYSTEN 5 a(n) = caused: A byptem fs adud ke ag ; depends om erly PHORM Ep ar te olP We ppetecmire — bohathes non causal -- Bary apstem's olf depends 3 ) yo + zee on futwwe dip @ sabe 07% nu static spsiems atte fasts dp tousal 7 mt Giv tine a 2 Linen A NON Line 2 x0) 1 ln? ae a ar i yond = tio yiloye 43009 = Aiporpetion prindple States that. SYP ¢ ta 4] mn hi ( i oe the fo welghted — ain nf Anputs & gato) fo the wxclpated sum of Eclat of: 2 ummm 9 Yat) 7 | indittduel =p BinalPane ony epends % of acaport a, equel a to ee oe Ys (9) 4.007 ee 54,26 (0) ta Balm > Ji6r) = yoo 0) = Ain) Bol = Hor 1a retest pled pate by sherry lente APP By detour? XCM) - wln-8 > Kop giane cy Sgn causal PAP causal or eno a te dip we Pg, aid “i ir gin v SD i dn) 4 (nr) eae 7 HS yin 2% gym 40 | openn 2x00) ¥ —® gato = Xan) ~ %alo-2) ie 1% ; _ itn), (aa ato = EP | py 2m Pe Jo Ay ia @ fa, x (nr+4, ) $A. YP £22400) 7 4D aloy ~%sto-059 a Puen - Sl 2 thay 24, X60 ~ Qs Xp + @-@ The sprew 5 Eneor i) (oi |. 2 ). xlm= Sxtn-7) - rs, yu = na®™ direc oberValion yor 556 dapat uml a Boley 2 HO |rtoy =x6 YQ) = 5x,(n-a? - -— an 1.0) = B= B19 Jum - mln) 3xuln $y) = S%2(n-a)- __1 > = gtoy it - nae 4300) =s[o, 1 69 + +24s 1X (0-194 aaXalnet) gir = Ylo . 492 F gxst0-8) | -aa % jx As ¥5 in 4502 Xp (nA) = ——— ycnd= 4 3% (n=!) me 3 yy = s a > a [ay m0 Cp-ay + 2axs(0-2)) - 2fse +74) 1 Bx0-)) alOr+7@ ashen 1% en ~ Linco aay, . The = Syste % a 1 E “ yo n@ [ge te At eousal yshore np) = ylo? ac 8 | xem extn) yum = nag * —_,~o aMuc £ ceursel a. ye $207 = HO) (seo a mato) Xan $000 = no =) Gat = gir) €,(0-19 4 a2%gln-420) = 1 Vans ergen Yh wr) ausgi(n) + asys(n> = Lato) J +22 Cxatnn) Mie g Jad = 4 try J Vtisda ateot M%L(NY Yao) = Gy X(N) 4.02 %2(P') > @ &-@© pe io Umar if the SY y(n 6) Sian on ’ yo Fee go ay ek) «0 a ee ie gece Je dynamt - a \ ‘ Yim = yor ) e0y~ xl) Jo gator 3 4 On | xm = xa) n+! | at) = fou yu Ss pin)= Sm) pa - t obs yila= gf Y3(m) = Cod } 4 j Hin) = Oy) x (0) +r %aCn r y25) = yoo nat EY n+ " Ss (e,| Heed re ont, |e } f D2 [meth 4% te) [2 2 0 = 9) Da kee A te ae Anil tOsygo tnd = 1 ft) +a) Z “a ] fue) £00920 System Jo Erape Tive Invariant and Vartang ; System x In tine bovasdant systerm, —dfructeriies of System ) de vey With, thine & In Vasdant dystern chor the aysion a wts HN Mine, adneyone® ) dleloyed response xtn> 1 yio-bd Rear fy delayed doput Pcie bey TR gio = fio-k) => TI Fy) D> wW wt ayaa (0 [ono 70 [ocpimeeteD ie ime Srveotank — eystem a 2p) = xalb) pire «CP) 4K to-e ae J °, Aratie y Causal Yeerts ain d x, tn) y diuct obs , seit (o)e.(n Cee yi (r= JOO Jaen ate rnd a a2” GE i Pp) Yo = YCO? Jaem=roe9 Fle OE ere, x oN a a / eee j . oy , Lee PO =F Jax j4rt BE”? ue ae ou note) 5 aa + 2 [en-td & i Qi gi0) #029 200) > ai [oo [@ aia en ~Lint oy 8 Oro % petdtn-K) = yl is > Hk = rane Sed = (n-k-) @ ce to ate tept Reapons glmikd = Be lea —5 x(n-FD Py xaln-F) YOU -@-vD & @ @ The system Ps Fie Viniant- System tm 2 5, FIR (Finte Tnpule Raper) Jd. ~ JR ( Infinite Tenpule Rep yor) 4 , Igtm| = = iny= yr? 2 wig hin)= } Pn an a exper) |4¢n> | a 17 wie > yoy = ht) en) =d10> yi) bO » x07 REISER fe? finite > D(A Sal) ee THR wpttom ~ 1p iy fryinite. Bin=4 ~~. 320 Be np Y= bl »FP cme ON ae o_o” “amd SR , nase 2 TE bounded 1 protucey bounced ofp enen we seem ta Biles BIE Bible aster, ; grandee Pie xen) | 2s Mee poet O17 1903) cae D> hl absolutely Aunmable pi Sixble System 2 ) md = [hm [zs wae Wea ‘el 5 _ myc RO we lh) Tofinite Terpulse Reson, go) a) + ae yoo) = | tee ol aaing | aD, (terry H- +o yo F Nee pled dtn-¥) ae Un) > h(n) * ee) — a Ze + > oe ; ' ee hee xtn-¥) * is a 0 Bo Ae 1px Yo peo 2l6rr bu joerg) groundFor causal, frre Bowld not ep fo, “hein Be-39 ancl do on ore) pene hig Yo) = [htop xen bli) X>-D4 - - J - > : om bee (a Very dauatty A stabuty f, y GBs othe. Auster J deseited by Impule Pesport ys Stal ) bi) - 2” un) ee TER gyste : + 3 ‘1 114 hind = ot Me) 3g hese «l-)7 al EEL a = 27! (07 3° ia. as Am=4 For all negabve Instances, Ul) fy negative a, a oe satrusiea = hin) =0; no CNo RU cause So, # FA causal 5 2 6 “a = Ik | = é ; , ke-0 Dee kK) > 2 Ulky tute) 6 6h Bs ke~0 att op) 1 © al aS k “ i pean UPL) , BS ke 4 Gece - 2 2é Lal Bao. | ee ak keo pe O22 peeree System.Ber, Bere of n wh b0 = (S)” weny stom , Lee at 1p causal 4:0 teow, hinder, neo » oS ‘ her) 3 Si Wray 0 = (ADF uty oe es te =e (h es (A) a) “ER CAL: cea 1 a 2 \ ese I-% 2B the. (Au Bystem Stable System , : ere Aince Ch) fx Tnveluved { OH) + RTD 5 a sin hesorme eve dev al rales ] ¥ 4) at 14, towekoed a pandd CNo duture > ‘as 5 SE) 4 singe) tute #5 2 ee all age) sue l= : thom ceantnded & - gable 5st@. pdrmine wheter He ptm ae H 1 yor = [xt] * Nesp xo te Harte =), glo? 8 (Hibte) = fate ZI OY 2 el 2 fw = k * Dyin & (27% xen a k=0 2 a) <1, fm we will apply Ss a*: a Fro. oo do, the values will be finite &, qo ts Arable ne Mable iegin= > 2" xe (9) pions Keo eee 7 abs. / gy cas oe ne ee add 2% fox tntinty Lame aims 5a yor % ursiable *t sl condition fer Beal vy) Yin? 4 ces &on) - Xk nee 5 Ss he n> a VW) Yr = NGS (un). KOH ») unstable ut ohm 4 - shen» Ghoble “ai 0) 4 = Dxcny =) unstable | us) ae aber? 4 7 Re IK J then, qin) = ©: famine the Stability ef ane fopulse WOR SA for Stability EB huy <0. a* uch) Gai eee 6 Sistem alesculea, uc-ny UGK 215 -k 20FREQUENCY ANALYSIS. OF iene an AES @ pps CPomev Density @ Four Parfom » Fours andl aN may ope DISCRETE Tue SIG WpLey ‘propertiis specu) pupoctis Honrne fealy xy Fourte” lopmibesis outs em averted PO phase 4pusntalion apne safer setios Fos OMS rE Time SiG nieve ‘ Haxmon ically ) orFs pie Cy ana if 3 Slowesds (0) aafng — Aermmonically related Representation jamn x Sara Loe} filer s a moter SOR F > qprasstotin °F / ors ore apraenbd by Ue Lorniporets telateelSPECTIRUN Frepoenty F) ria, -apedtettn i) Pes Neel a % rt ee, / ! = | a Si) Phase Spectrum .. 4 } le » [72 e | ag ~~ ie te : a -1 S Xb) x*(p - = aLTe 4 Roy» Sy - neoy tty a ‘ig wr) > $1,100} a ie 7401 Ghee iheaa Se j p vil aepeat tontfnodsly , a ute Tefintte er , tc ~ fink Ce ly an Pye. 3§ p20 F 3 am Cy Saxton 5 keo,!,2,2 +” nae a Me, a Pe F(p) pain x * ge). » [14 (8 tord] fie kee ee 4 . 2 t =| rt (eee . PRA” ie Va) > Coe pekar; Geof 7 3 3 (hs o> eae f = jen J wn” -¢, Mag — Speetsum Pe oa. Iq) > OF to = aes prose — Spechrum : ‘i Pot Comparing. renal Ee, pete ? Aendiyy pues nin 6 § ) 7 4 4 fam) > ¢ He xen} ors «tm (3 Ty : J One 7 a woe Y% 21M a Near gx m Ts 2 Jo ae unter Serces Repnesentatun i“ _ famko i % 2° N pion *. i Mas } es a man og ere rie Te ) x v j 20 n 4s ao ‘ + ¢28 + ce ¢ | poth | Gampeang 7 } } 1 eee G0 4:6 i Pos f Y% y ; 2-0-9 @ 1 @-§ ’ °@ 210) = Scosty ye” Te ah perbecl 5 we ™% Ub = gorse et oN N, = 21m FS — , 4, zip (| natn i %, 7 Tira EB $ eo a N, 2am ‘ = Partod » N28 0 c Representation 1 fom . Fourfer SedesouRIER pesiorle i _ PERIOD. . S & Sy zip &ec7y) : on a o1P Cty ri Oo 2p at ~ a Thm a Ti Time laps a D e D ¢ xsPp » c c Ee C c c x D e c e ee > c 2 ¢ . EgaTion Syrttests 2 Frequerty _, ‘itme Rr: Gated domaine Jorraie j = aa leven 4 xy ¢ 3 mrCuni) v7 Feta) denoted 24 x(e) 2 x(w) sce Iori Pes Tee a aes ne apd, Agire Fou SYNTHESIS EQUATION ;- 50" i ae n- xm) 2 I prev (or x Fr] x(wo)) . 4 =” =) )} + 5b JKeame eg 00) a mo . oe? oo D Dutexmine —“Fousfon “harkform — xin) > $2, 1, Re Non - perfedic | fintte feng th donol = a -Juon Fr [um]: ZF xwre ne z fon * nim € i nee Ea = as 0) jw aa neat OS xae oa 2 9% = xe x Je -2j0 | q ne) = 2+ e te A ae - re 0 mo) = dt+t2) : PO sy Be um={),0, © Tif Ren ~perbodic foe > Byna} Pay ers Sage hon mmc © 2-0 2 en 7 a an) @ Oted joc) foe) “pe SARE Ne MADE 4 xe + jat jase) ate) Brahe gow ee mo) + © 10 4 re)joclic , finite leagth Ps Aynal ate) ee nee, ce)D fowler” Wanton" 0 FO} = 20th ae FT [remy] s . Xen) ean a nae so be 2 5. BOGE 340M +1) ~ depen. O n2-0 3)e% a ? sunt yah? >= 2 when n=43 : oa HUD) = 4) cat when nee Lil a. ps (a MD #4 x) Bd i E es [xtnp. s “je? = ec LHS = €8f PT[xOR] > F * x0ny + S- Mme 18 = esliaj 02-3 ano na oo © == aaal I Fifa) - SF autrrie - F. Grtae jer mo ed — oe fst = Ogee ns0 2 only when ES) peo - 3-1 @ 7e" a “— «oe \ 3 er .-”)” ANS = RHS yp as aN n=-)3 (4) 2) +43 x2l0) %FSCaitmi) 44s s Cason} we! os } = na Kn es [xm] =G. = & xe p jz ‘ De HS = esJ y IHS FST a, xl +0, %2(m]2 2% Camo) 405 oz0 en Aaa pa - Jak -fizmkn N N-)= ¥ SMW2 FSfxtn) Je le Fe Pur) ) Cn wt ~f2mkgn) HS = lye @ Ree 2) 5. xen) @ We. - efx ee. S ae het c Tregiueneg Er 7 ~ jim ee ai See Fs n=O; so a pr! emeRe j Fs Paw oe xd) ew Hees = 12 ) I + iad Fs [xem € Se eee Lge = RMS “J Fs (xn @ >faq Q Ine fog i rs Teo Zoey ST 20nm) “ark : or- | > Lote eee ~farntnpeny ele ie me) oe Fe oi Riad EZ xtn-m amen Genkeesy a0 NN Re ee Ky bem -m >: Aas 4 none eee Samim eaeConfugate — Prope ty FS Lxto)] = Ce ETIBUE (np Jere -« ne a v “a Gm Erte” S[a tome, £ ,. No 7 be nm] fa = ite J2mkn ja * P htt Te Gre: ieee e 14 io. i poe . oe ft We ne 4 : lis: FS(x2)) =o ao EO FT [utny]- Kw) = x(e Tl) = rear’ @ Time Reversed FT [ay YO) 4 92 CH a, FT (wry ] +o PTL) me FSUxtt yo) bus. x(e’") = Frlarniny 4a; e200) ere jah a : a, i005) Baath) J & 1 T [xn 4 neo : 20 a ~ jw wrxat) € S 41% (piesa =n thX(wen) = x(n) pr 02) nn it ete fens apttiadie Pyrat XO ae In eRe) — fatsnm x0 To prev M predic Reverscl md > Fs{xler cay MER). ee 5 a vets 2 an XU) oot Fyegueny s Fs{x > ei xt) ae cl 2 Sent 7 fee 1k @ Lex ition 5 jane Fs[xco-m)] > Bye oN ae « _jusm (-m)] = x (e348 _ Je xe) @ : Ft Teme ~juon FE: “ A(x(n-my) eunse 5 xin-me fee © i rant eS xe)? em o > nelim eS F3[x* () Jz ecesdy 4 A jul jum nee, 20 e xt) e et[at on Je? ° = Jum : Ce” ee : jen th “Lys = 24s de ae fy jem FiCxn mm]: xe JePe). Fre guasry rst eI. 6 i Exner > eal imo = ime 2 Sung, fem $e Xn) e Sion ne-o : LHS ae -jlw-m)n, km es ke a (ev 2) fusm AS = X% (wrm) = RHS +t Ty Soni ) ~jon FT Duce 2 > KX o-my ye (8) teRugection ju (Lt) 4 ui FS oe = 7 é fxt ny Je cg (@ (xf mje x° OH) ot wn 5 fin F prfaton]= bis Se ee 2 J J s ee £ i " ~jon)* 2 a 5 4 ~ : Py ee” ; 25 [rm J ( I Sale y Nog ‘ | ‘Sf-u0In e nee 4 (wn ‘ Ses ae any i (7 oe Sie 5 Cent fence prwedD Convelutior — (ume FTL xseny # Helm 1 Coe) Ma Ceo) a i % Prt # 00) J = ns S- [x7 #92200) 16 ee f= nos ) Zz oe fiz “ty 70" 4 “= [Z, POwwJe 2, x Sab pete day AS -O-K 2-0 nrttk [n-wy Le e- Ks 1 s : = Xi) 4 q) Differentfat FT[ns hes = SF xe eH ~ Xe). X2(w) Lits = RHs ? Taequertiy covolution ) = x la) FTL Gm > Lip] 2 Xsan s VS T FE ; LHS e PTL adn) seam) Ss [its wy Jeg n=o ss | 6 ue? ile Riis 2) J éxe a_i [a PE ais 9 Io) eG Ju (ele) gin 4, a one $0) site am < 6-8 )n a is A pee M0 ae et J x2 te) ( = xy) ee) 4, om ) © Ki() % Xo(w) = RMS Q) Diffexentfatien FT [n: xto>] frequency eas jak ¢to) Symmetay Peper SCD eee) Xr Cede) th Ny Gey) = Sm Ke) = fp Meo By ( then c 49 a ek ore ae propesty : =F x (Coston Frinton y Pix osm 2 xn) Gian ye XO) A) x “~ . = yo ° iS xe) 2 SF xem , Sewn = a x00) [resc-2>n ~ Jono) 1) Parseval ‘6 “Theorem l L ( Jxeie)]" es RHS ! rife : J x6 i) xe) a ¢ dw =e (Cae = ) | x¢c 3 mh ri) an ale lee xe”) x" (ce an aT ot er)me, Pe aye Xr (ede y a mF Bie zit © Tet [xk ce-Suy] ede j K° (GaSe ces + Cxcey ey di am J CHW ae Dn : 6 J » ae 1 tee Wee ae e }2%. an ae | AG) * @ p We is 2 a See ll 7 | ‘ ree ees) = Sgr nae es ton 5ide eyes dv egjugee edie’) 3 \ Pe (S are 2 jo) gu ee 2 aura pel) x E@ ple ae 7 jw akin Bert NaePropetiic °F foun Sortes al & Even Y Converurion om 3) 4 x a an Fr [aC ¥ MTs Xul00) . Xs (wy er po Fs[ aims 0 J= N Oy Cay ae ga un + maginodt Lis = FS [xuln) » mlm] Fs [roto Je oe” ae ye) 5 3 4 Sym ae F ~) omen NG = z [xe mim) goa ey cogugete d i yn fe pe dee $2 ores [cx J » (C.* ee ee % B Aen + ’ 7 (Fs trny)) Na = C.* =e mo - SS . ng :1 g = i n. ce K = ¥, (0) * xa(n) = LS De 200) ever Wo ger Pacey] > 27 = ‘ a ° jon 4. 18 i Sucei#y @ 1" dw e 26 i si 2 ° Qn a $0 su anf Held) 94 uy free xy SAL eis) 38) 7 ie - an ae a SO) fa aes J