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DSP Unit 1

The document appears to be a technical text discussing concepts related to digital signal processing, including sampling, discrete Fourier transforms, and properties of signals in frequency domains. It includes mathematical equations and transformations relevant to signal analysis. The content is fragmented and contains numerous typographical errors, making it difficult to extract coherent information.

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0% found this document useful (0 votes)
13 views86 pages

DSP Unit 1

The document appears to be a technical text discussing concepts related to digital signal processing, including sampling, discrete Fourier transforms, and properties of signals in frequency domains. It includes mathematical equations and transformations relevant to signal analysis. The content is fragmented and contains numerous typographical errors, making it difficult to extract coherent information.

Uploaded by

abi20052023
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
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Joo- 2400 O-4 HZ ~voin Fraquiniag ALIASIN GY fs o 24m pragh Ber Component in overlapping wh low $24 Corp wid = { 1,054 * yl = ere 3h t remy 49 nd = (3 on] IPIAAPIPOROKEEHEOEOKRHOKRAAKBEOSSSSSSAsesE ISVS PVPVVRVE DED BW WW We la le la la laa lo te te ee ee S01. 24 UNIT-L DIscreTe TRANSFORMS ACH) > dtsovete signal N+» -co tH 00 L Lnteg ee Victor: Variable HON > freq domarn pret = possible fey Component ain xtn) sia UM) 2 10 Sw, n +20e6W2"M T | 2Kfy anh Peg nimnogy domain kos -1wn Xfw) = BZ atnje nz-00 . . jum ee 25 ( eur 1 Aa) 4 12 (e -e ) J juin 00 Juin _jwn et(e? ae zee < nz-20 oo -j(w-wiyn - ze nee FO: an 6lw) CIM) am 60 HP tag SY rer) ¥(w) = 5( en 6(e-w1) +24 (wri) ) + 2 [20 So (w- we) - 2n8 (wo +we)) J != on Slw) a Ly on wo ° w tO Slwswy 2 we + ww, ° wey tw, Sl wi we) =) Coleen zon) =o w 4 twee yon low \ | { \ cw, oP) we | |. - 7 DIET wo / -Jwn Rw) = = xm €? Nni-o KW) Ab owRnnouns yonction of Ww (reh doe , nen ew iodic ow w ov 2 ' ev MN Synmaly of > mag mbade Ynpowse is. =}(vo tan D phose open is 5 Odds : Xen) & yrmnolty X( ween) = Zz Cnt ee y oa -jwn -jann =F wn) & e » = %(w) ni Ww vA » SO PAAOOOOOOROSSESKHSSHHSSSHHRHHRHAAAAEEE SS SVVVVVVVVDEEEVEVULUU eee eee eae Ve atny fy \ vj w x les) t n \ jw Jw ", ' xlw). @ 41 4e 14 2 ww 0 4 Shy \ XC Ww) = x(w) a ! wn Sy mm obvy mn ate {104 , jw x(w)2 4 2@ 4 Cw -j Ain 2 - yr) : Joo tosw) + Sin® w - + tsrwy 2 brew + ine \o 4 2 Ww in Ar { « Zt) = tan '{ ae w anf Cae x ( iaim ) vo Ixtw)| 2 x(w) : -h 0 ° KR ow & wn" ko? wr Nh, v2 " s" ot jor ° 2 ° ni nl, in - | n 0 0 |xéw)1= | xtw)] - enndines o vo CxX(-w)- Zxlw) odd func « a r(n) - n y inp def vin (ZY oem , (t nZ0 z) G o neo 00 . ~ n X(w)y 2S ppy~ j-jwn rel eB 2 n=0 @) 7 (aR G ) jw -)2w \+ Loo eet: + z 4 = 1 . Ie yer” FEJay wen e Laer ye Zz, . \ \-ae’ \ San ae 2( aam). 2 od Z-& t-az" La 1” wn F (fim) = = Soe nz +e = 1 (mre) for Mw from - 4 hon FoCij = 6(n) 9I9 RB RR BBB eae e2C4eneaseeeaseeeeeaeeesesee PI VSVVVVVV DV VDD WW Us lala la la la la ta la ta te ts ee he a Proparin > Time shapting (7 IMegor conat- yim = a(n-4) Weg vaMiable ylw) 2 ee mene ere" n=-00 n-L =m, * jw (n, +2) Yloy= SF wlnjye N--o H . co 5 eels xe n.-20 : ee Xlw) By Frepuenen sig ting jwn oo jwin -J Fle men) an xen) € nz 0 oo ~(w- wr)n = F mnye eer x (w-w) Convolukiow | AU) | + nto) eo) x(w) Hew) Yfw) 4 = wind * A(n) Y(w)= x fw) -H lw) wn | 1-07.24. 4. Vavsevas's idowbity = lacey) = " oh fl xc dvo aK Nz -co <2K> DISCRETE FOURIER TRANSFORM (pFT)> Foy fiwitt Auvatwn dequrnc 4fn) —> A pt Aea, N- ‘ ' -j ankn x(k) = SZ xny ean kso to N-1 4 ns0 7 - en aude (Analigns eqration) Yor wv V X(w) Nor Jenn AW) = TT = x(k) On mM=0 ke N-) Keo 'SyMtnuns 24 radon) ing { ~ Hime vavidede k- Freq Variable we DAK Kz 0 to N-1 (eK) = xfw) Kh) = *z) | jon zr ery O7ISSPPAPAKORRAPeKRenasenneRneeaeeeeese OSV VVUUVVUDDDEVVUUUU see Ube Twidelle factor . kn Jinn woes ee ON N a pe PPT 22" Cp one 4 Nee tn) —» n > Oly ’ Ke) = xy we” no o kc xc) = [ 4 (0) Wa + 40) We J Ez 0,) x(e) = oywy + XO) we ' ° KO) = Alo) we # OD WD 4 ! * Ww ° x10) ° We wr ee ’ aud t VL we We xO) Propukies of twiddle factor: KAN We > Wy Cporiedaesty ) hei ack . . O hee ~j2K (e4n) “J2KE -j2x pe oN > e ne -jrn k 2 er > Wy (han) t Wa 2 Ww, W SSRRROSEE SOE e EEK Osos EEE E EEE Kdddd: ¢ “ | | — 6/2 oy 8 | —— | 'y ion . a ‘ \ —, | | oe aes p ~ « 4 { eee. " — 2 | s 0 in = | 3 ~ | a | a 2 3 , 0 u 3 ’ ; = } s 7 2 | 7 3 y £ » ~ \ _ 8s ¥ ; 2 \ s ; 4 - t a = ¥ S )u 2 a 8 ° & £ + “a PISVVVVVVVDUVUBUVUUL elses ee wees x le) xa) x (2) wI3) - \ Wa We mM = aw) het > vp Wy vq” wy? wy” “0 | 1 Ee wy Wy -w,! vi,” —wq? wm =? atr) 1(3) Wee wy! KO) -242} x(2) 2-2 *(3)>-2-2) PAD = x Oy x(k aN) = x(k) ‘ > Priedie ' Nat *Cer ny = x (ry VY Ite = | } ir re) jnce)) ce i Ww, 2.¥2b, 2 2.29 f fu va fe, Uso, sh 7-074 > to, “ay X, Tq f VY x yer \ ‘ KON-1) = ¥ CR) Comj tay ate SYwametin, tO xs 0-123 Tees 7 ~ ¥ x(q) > x* to) (= x) xfs) = TCV) x(z) = #8 (7) RAPP PPOOOORER PO RORERERORERREREEERE FUSIIVVUVUVIVEVUVUVVUVVYUVDEDDYDEDII.OdSS Px ON 2 [x Ce] = taen fumes of & et ae Lece) — odd fume. ok (kD A Jrce| Wc pe 7 Ve NB . J}; % | | Aly Li | Ltt ‘ |_—= T OS a } * > Th Spt DFT ki0 167 xk) = { xe) Kcr BC2d x03) x04) a xa) x") 800] 1 a 4pe DET = 4 At Hen) SEL Kloy= C0) 4d 4X2) + *03) yz) = 1) = L004 402) - HG) ¢ + 2 wd “of connpusaoverl Conny Lveiby oy, darect mn pent DFT fa 4 No: oh Couple nulbaplicedion 90 “4 a additon = 7 Ke) Your r um ror papa j a x2) J lay \-1 ae x(3) iv yh ye XC) Ve xo) 4 1x UD 4 EWE) tL xO3) Lg pe g § add wh ROD = 4 math £3 aad ¥(2 2 4 . \s X09 2 4 a ae yedlg RU ‘Tohatly , Me Compler wuretipliadtion d- 1% complex « ? « © « + + a 4 aadihow N=pt DFT te tomple le > N counp-lix vill ication 2 4. (- yu y, € NON-@) con AdAiHOn 7 napley Ds € Raion bow kf, wn Seog fs < Oo ; 5 € € Ligne ¢ ity ¢ os « \wt Ue wis : e o a Conk) ® = 4 PVVVVLVV UV UU UUW ls ls sls ls le la ls ls 8s 8 8 DET properties \ Veriodiciy ‘ xCNak) = XCk) Ne) -J2n = ON x (mak = > xen e ON ne N=0 nat yh be 7 j 27% Kin ~)2R fp 7 ™’, Watts, e° yw en’ an i v for intage, xCe) Agents for every N whin iS an Indegn 2e Symmetry + XON-&) = X "Ck) Nv Jj2K tw. K(N-k) = Damjye Solan nz0 ° & Ri watoey DeT( g(t) +H) = ver GO) +der (Au) > GH + Hee) HOD & Alm > Np Kequnrte eee DFT ( alm) #¥X2 0) > PFT (end) 4dET (Om) 4 am=fr 1 1a) detfxatny]= (Ce 2 ee] mn Hay Aord Done Pret vin ee (DC peak Ak wr Kee \ Sqn) Seeer=--.,. acne fry ~) per (acm] =( 0 22), o2tej) es PP PARABDADDD eo a 4 2 2 a > = 2 > a > a > a > 2 > BV VVVVUVVVUVUUUUUUI CEM), \) 4 ve ef % (ny 2 ( 41 00d WO % (nye (2!) 4208) (2) 4.43) CV OM OL Senn ata) @ fellows cavalo spend a Jna amkclotk: Ava chon iw OV > a wre pee apt A x(n) = {ate au) 42) asf tu) LA ( \ air) “| panto) we" Hs) 1A) neoy 26 I S) ae, cot peaks 4CCW), . xiwon)) ~ I % (em), pecan Lens > dot wise ACI) = (G0) = C4) LOGV) 4 = % (4-1) = aes X(HD)) 4 = Ay 2) = eS AC (EM) y= Ul-H = FO LOM) 103) K(2) Cx xu) Ay the dey is 9 Unde 3, monk clade wae Tr becom V4 yew Wank vires nwarye t ike owl cach t XCEM) 4 = (x60) 163) 4029 X09 J if we Lpve sek ly sha Clack wir f « dye oo oe oe oe ta e « e e e e e e e e e e e e e e e e e e « s a « s a ° y Db ebb SIN VVVVVVVUVVE UUW WW lo lala lo ls lola la + Der ( lem), | 2X (cw), er =) 2A kK DET (elem). = Z a(n-n) @ oN N-nN2 -j) =ME (neni) = Zt N ”, END jy 2rk +) Ey, e we bs J oper PUUARAR IS cle -) 2h wn, jee ee eo NW ow KM Gq pet{ «06 ] 2 S atny eI OMn, ny END x (ner) = (CD) y () () rove the proputy CY erlon, Sym multug take «OM * gph Mynee mm) = Cry 2, 3.4) xCey = fw, sae -2% -2-2)] % COM), > [ xc) %0g) xp exo J ‘ 4 3 a a ,- BS RPPPPOORP HOBO PORO RASS HOSBHEeBeesSse ow wd andes oad x Cond), 2-20 avabbw weap: {12 3-2 ald aoe din) - 107) ee —— ee etn 2 at) +9 = \e : L vv TU an per fretnj= tle) + ¥(-KD)y _ = a + = 4 = - per (Aer) = xlk) — x Cc-e)y DET Ob prot £ odd SAR \y porety, ian gead. way fr r,s hy er (axel) > PFT (40m) 4 ver (a e-R) _——_ ena L = = ey > > => — — = = > > > =) > 2 > ZZ 3 wa (lo, -2+2y- 2-2 2} ) (wo, -2-2; 2, 2a VA = {io -2-2 -2) Aud of Civ ag ww DET (to) = x(e) _ CCE) y DET (mo(n)) = o- -24j -2-2 4) - fw. an, -t- 2 a 8 ety ol pv choy jana pinoy Leey wlnty oda , 22-014 RCULAR CONVOLUTION: et conver etl — MN) bX.) prnear Convolution’ — 10M) x Ta (") = s ay (KB) > 2 (nk) K-00 2 culo convowunen yon) * LO) » , PPPEAPOROROORERAEAHROKROHHOBRHHRRHAHE Fae ee es ee PY IVVVVV VV UV bub carole Convoluhen * 1 : . rom) a, (Cn gtr ® ODE Zeon m (OM), N-\ . 2 eM) CONN m:o (ane goivid bY mMatrrx wetrod concentric carele metnod DET DFT pokmenkoned choose mabix metrod as ies Lag %2 (Crm) 2% (C14 n=) 14 eas Cam wis 1) GD 2M , vagnrt - wyid= Zt) x2 (Cr), . mso Nog : nee bB wt (n) > ( ie 2) alm. C12 303 Los exppurd core ak fla acu yen U 40 Rise ate eg somac engl apo) { %gt07 GI9 (3) 22) 2200) [rete dp WOOD) 2. | aa) Wile) WG) Ble) laa Aa (2) WD yD ale) Xe (3) | OE up 7 b ay, OY x2 (8) Ml 2D AL aa (07) (ALO - . \ f &, fo / AZlo>D ( % top 4, (9) ply ple 4) | . at %201) - | an he Ce 2 (\fn (2) me ~ >) a, (2) xy (2) ae (29 {eae OD ee Hote n | (sy Wan 2 Oy (0) HES X32 (3 Ar a a5 _ fr 2 3 a 'o Bo oe 8 yr 1 10 2 3 tt We crt eras" Parton 5 4 5) ml”? “(L237 4p SAAAHAPE EOE n aS aaa - ) +0 OS le (' | ee \ 410 o 2 | oO yY 3 = 24250 tM }o 5 y 0 4 \\F B44 434TH Lo ; a | 0 4 ee iY X 4 2 I 5 OFETATL +4 coo ON \ AYYYVVVVVWVV UV VU EEVE EUW Wl e eee DBS Content ard mutnod. NA iM s Zi aim) a, ( (n-m)) yy M20 N= 0 toN-l aA IN)= [ ey 2) a, t= (1, 2,73) ain: (1, 22) aztne C1,% ~322) append cere nz ohod ayCm) > SeRuince Cepvesentec in amk clockwise divtthers, Ay (o- ™)y My (mr). A Cm) = ary ho: ° ° pe a . ma a) 1,(0) END) Aa (d=) a0) Ali) 22 lee and chee waa 2°" Clenime in che oh dae dn? me 2 tif In oul aé pe “ oh Clo de wise dive chon - rae @ y & o { +2 = Io e 1 4 “ } Aa(22d 6 e & ee moe . J." : A, (2) 23 e ato = Jaleo é a az neta . e é é € = 2 agene | gk 105 88 Furperm the omrolar Conv bdo aegumer ato) [Lu muttod a (8) = {ues 4 §, wing Conc. Orv ele : « 4, (om) yu e wt, “4 rte (992 4) ++ é& 5 «& ele) b \ @ wa Saw wl L a e Tv a 2 CN) a at? . a i 7 = wy I oe . > a PVIVVVVUVVUVUUEDUV VEU WV ee eee ae v z ° 3 3 : \ \ ; -O 2) * a . aa 1 Ing DFT & IDFT artnyetitny () % Cn) eel EA Cm) a Cam) m=o me Ae (wan-m Yr) = DET (TM) @ 2 ow) na Wel ~jamkn . s ( 7 Hy Com) ACN nem) @ NI nz2 lm-o ~T samp. yg = er CK) © ¥2 Ce) 2 ON (wen +m) NA soo : = me wm Rafe) = = 42 (Nt nm) on ae CN aR D "0-1 , = X Om) ew ” M20 ODE FLO» x6 — [YF OD = 1c8) | gor 4094 bey \ (om -s0¢r( 60} ee = XC) Coo) \ Vl2d2 x leo HC) ) DFT ef the cromlor Convolution EF product af theiy DFTs 24:07:24 bpd mile) D> dET ( x,t) ) yk) > pet [ a> (n)) go. Find x3 CK) ® = XC). ¥2 Ue) toy K=O WN-) poink wist nau pl cchon ¥3l0) = wy 0) «x, (0) By (MDS xy (a) - X2 (W-1) 3 Find 03(") = Der Xs cey) 3k) © Neo kone PVIVIVVVV VV VUUUVV EUV De puso Ant crovlarg, convolukson between tre AX3puL BA wee few) %2l)* (ipacence 4) gs” meee fe] fr erate) HO} t li ja j z x) (2) ta oyc-t ' > 1132. ry = 4 L - {+ | (-2)4 +2) 2 6 + I-2ti-22- l+ry -1-2) 20 * (42 bone ry) x(k) - [ie -2+a, “4,-2-%) kyes ¥5 Ck) = xy Ce) ee (ek) Ctlement wie mut) Ww, ‘ ie 2 syd \ Ig (e) | [ '3!e) | Yo wy wy’ My Azar] = ° -1 - - | fre weg? | %3 (2) . .2 at [ 3) WE wy wygit wy tz f? Lowe? 3 t -4 we ow we ~ (ean) Wy : wg -(h4nh) “k “% a 2 Ww, 4 jon, oo N=4 we ret -2 - (042) a 5 Wa = Wy : Wy -W4 = -\ ey -Cite) . Wy” = Wey . -w4) 24 = (044) oo. 4) a a we Ms wg = Mh a Ghee _b -(244) ut we wy > % 2M 2 a ~ (S44) = ~ Cth) -| . 4. We = Me > WE = My at eae 12 + wh th bo 0 Y yee > -! a 432) J f abl de “ Bevomus twidde, t four motriy > 1] 6 | al Ter T\ vo -4 4 64 bo +4 5b bo ~4 » : (* yy, Wee 4) : Campo agate) 2 Snir how 5 a4 g gtr vow of twiddle tact mabty you wil gt bwiddte fader matrip of the IDET. yy as A co) 2 = = > > a POVYYVVVVVIIVIIVUVIE DW Wie DFT eo x6 CW)? PRY Sy Ny ~JAnkn pet(atomjes 2rme N nzo NA 2 am tn me =/ S xm) & N n=0 \ a PRC N-E)N nN x a [2m ganna nN ) n ° neo =X Cw-e) = { x (ew), | conjnge soy He Airey saa Cov tls) Tf Sequinee is chock wis wrik fh anticlockes Fe we get marie image » we DET of Civonler shifted Seqrmcee | L( (-)y = : > DET Setnenur ( x 4 (Nan-L) lee big ane - =) 4 (Ntn—g) oe ; Ic = 6 de NA) -J2KK (N an-L4t) eo ON PEE Cuan ot - aut EZ xinen-t) e wan ain) yin-Oy, F gNg abe (ocr domain vant s ee Z ( Zrorytern- J 7 ER (Ntn-Q- wn) prt (Tay Y= 120 mJ JVPIVPIDIDPBARPAHAAAEAAASSHPHOHRAOSBSSB8E ry VY VP VYVYVVVVVVVUV VV UV veers -j2mkn NA Jom (rien-L) : Fam 6 N > ylnan- oe w20 deo S xt) oe [ Zacwen aye n=0 ul * KC) + ¥ Ce) Car wlan frequency Aba DFT (e a xn) ) NA j2mbn -jonk a7 2-5 ew amen n=o0 NA jam Cwte-kn) > DB une ae n=O >® ( NaE-K)S =x (eM) y ree = [ \ 2 3 4) X(0) xt lz) #08) ¥(EDy > ( ey K(o) x0) ray) wre) _— 7 OE . « or) wf ky > fort: indeger in FT domaan per index Wwe our doing He Shift 19 fra danrain aii lod itt shifting sa Owls clock wutte dave ube fa et ”n y¥lay (Sm y(b) r Ves) eh Pyoduyr o| dequeue PFT Cam. a2 thy) ud ant > SS men) mt, tn) e -N “20 v4 nk A Janka - Z too b 2 42 0) ew et) eo W20 K we jan nar -b0" ts 'xe (KD = poe) e a N & N+ 2 Ra (ey) xy Cte -€1) N k20 = + Ss X2 Ck) x) CEN) aa 5 Y ce)] = - or Convo ow betwen x, (Ke) 2X2 iN ( wrot protedue Lenvoluhin 1S Shme In be th Ne . sf time & fra domary ¥, W) yO HD Hl) yx (07 x toy oy 13) 2) x2 x109) x2 (2) xi(o) x3(>) x, a) x (D x, 0) x) (0) ere P41 MITIVIYVVVVVIVVDHOVIVUVUVUD EDD UE TEESE ns ars MA Pans nardy prdodvon dot deer ohern pean s uel 2 Z [amI" -JlZ | xe] n-o N eo ( Wwe 2 2 an a*n) no a wl ans gine ? : in a = xt cr > (a fae nev ne) nA j2nkn 2 xe) Z ame "N KE ae NA 4 Lz ce) rey N Keo Nol Lez | te) NO OKO Pavceval's solabon %(ny> (Cv 2,374) Prove the Pay ae Lis? x(k) = Cio-242, -2,-2-9) S L Z (MWe vagy att neo = 40 RUG zs (wNb, 2,0) 2 4 (worery +8) r + = Po. Ww 4 aS OT Dy Computational wovaplerivg 2 DPT ddveut Conaputaion, fh Npt pet Woof, Comply raalt pli action 2 N™ No - Of woe aadinen = N(w-1) Faby Fouvity Tramspoewn Cert) (Atger' thaw) atts an odgerittne wed tp seduce connpudabienra! . nt pli cattén Nn N Wo: Ok Complex rath pli > NL eg no. ob 4 addition > ON dey Bost of 2adry ~ 2 abgoritann \. pecimahen Ww time g. permahin in OH, Ape DET wring DIT [ decimation in time) mC { Alo), We, 2, x] » ~PPPPAAPPPPPPRPPRPPRPPeeePeFrrEeFEEhRaAbt PIPSVYVVVVVVUVUVUVUVUVI SEEDER az oly \ WU xt2n> j > opt per *ICED at) — Koy, U2) odd Ke Leone) j Rie) HOD, 403) pe DFT (4) a 4p eT x(k) rele) 3 K(ey = EB ttny wy," naa ) kon, \ = Zxlen) wy +t Erne) Wy Nico no K(antt) ' ' aa 2 BZ xeadwye ty Zazcme wien nzo nzo __ —_——————— ¥2 Ce) r le “tM jo, Ryle) Wy, akan k 2 ok xdtie) = AIC) 4 We kp Ce) ky = Wy ks 0,1 KUe) = x Cte) 4 wat kp (kK) Kc 4) : kee ~%y (le 42) 4 Na xp Char) wy (42) = *y Cte) M2 Chet) = x Ce) WME) g%2 (IC) Ave opt DET, 2 periodic wile peried 2 Kir pam(ety xj Be cin J aE 7 We => e = : ee ko, vy Xl) = xi Ck) + wit Xr ley — 5 x (0d, x0) Klee) x(k) _ Wi x Ce) —> x6) , x3) I File) _ > x(2n) wn” nro RVC) st C0) wy? 4 tir) 0 FO) = 1 (0) wy” + 462) wr) = KOO We” 4 1 (2) V2" xt ¥1 OD = neo) wm? 4 Ada) wi = Ale) wy? - i 1 sJan wr = 252 - aul) \ Xo CE) = SS games) wy noe ROK) = Uy we 4402) we ° oO ¥, (0) = Ker) wor 4+ 403) ues HO) = aay we 71) Ww! 5 6 = KA) wy? - FLD 9) SSVVVVVVVVPPVVVDDVUDEEDDEDETOVESS Paes Prabluty MUL fey 4 pt DET Wel ut overs a Xo) y «082 ama me 4 18) > Z ke ON FCW) = xy CED + wat ¥(k42) = x, Ole _ wrt welt) ce Ap = a 4 wk x xlo) = KyloD + we! x, (0) XCD = xt) ty! x2 U KIM 2 XE lo) — Wy °x 2 lo) X03) = xy CD wy! Kt) Nernst y or (0) wet" Ko) - x0) 20% Yel) oot” mg! ¥(2) ' . (0) we ete Ly! x(3) Nn \ Xz Ck) — 5 xle x0) 2Cke) —> x(2)_ xO) Nov og, corp ruth Sly t =4 Wo of, add Lin bit reversed Trin bit Awevsak eda SIP in normal ordy ALO) 4 (00) LU yw) XQ) Lv) 403) a) (0) o© ) oo fo) ay er [Lo ¢ey @) .0 Jor iy ao) wedi fy too [Pim ermal order Xo) n loo) xa) * (2) (3) ra Compute 4 pk DET of the Sequimce wen = (L354) = ~ wring DIT Algerith, 2 HI LY (2) XE) e MEM = (1% 3,4) 2 e \ _— x1 Code" xfoy- 0 we - 2 KOS 2427 & f yploye& —KQ):-2 « We ee -2 SCSIVVVVIYVEVEVUVEIEES ss BEESEEDL 24: 0124 Spe ori wring PIT ya n 4 ww awe ob UC22N) = HGH) ne 01 ae Obed { Alo) Le) Xue) —u2n) — . toy 1 Ute) AU) Lea[ 2.@atiy = cltnre) x Cn). xC2) Alb) * 20k) 4.(2W4)') al?) wecry HCI) 45) PAC2-2mat) =e fan at) |__—§ guy xcs) Farle) (2. Cent 41) 2x U4ne sy 409), HCP) Xan Ce) ke 01 kr0,\ 1 my kd = FS xlany wen n=o xy (0) = 1Co) + xy) xy OO= Ato) - ACK) x (e)= 2H alynar) win neo yr!) = tay atey x2) = 402) - Xle) | ror HO) = 2 Uqna wi" noo Ko (0) rai) = = 10) +25) Te) - x5) I X22 (k)= Zan 8) w,” For (0) = X(3) 42X02) Xo. 01) = L03)- (7) wy Cee xu) + Web x1o CE) wy llear) = race) wi ny, Ct) XL loy= xy C0) ¢ wy,’ Ka (6) 12 Xuld + wy) «1,00 ¥ ye Ku lr) Wg Xt (0) 10322 eu OD ~ ng xy, C1 x2) > x2) (tb) 4 wy mee Ck) Ko (Rte) = Kay Ce ~ wa ¥ 25 Ce) X20) = ¥2) (0) + Wy ° kar (0) ¥2Mexy Ct) 4 wa! x2) (1) ale) = %y (o)- we ¥a2 (0) | Xl) =r) ml xe ©) Kz 0b Me) = ACK) , wrk, CKD 2 X(K+q) > x, (8) wg a Ue) ¥ (0) > 100) + WE? x, 10) KU) = We xe) 49! x01) XQ) =. ei) + Wg? 200 *(3 =: x13) re we 5 (2) PLY = - Nem (0)@ we? x, (0) (5) = C1) we! xo CN) ¥lo)= ¥\ 2) — wy? %2(2) XO) = x1 (a ~wy'x, (2 = > OORRRRRRRARARARD FIIVIVVV TIT TT GGG GY yw ie la le lets lee te te io es, Spoiny DIT FFT BUTTERFLY STRUCTURE 40) xX lo) 7 (oy 7 ROD xc yu ra) we) 4) we 42) x ¥ lo X nz) * (3) “SB 1 x, ¥, U8) ¥ (4) ee ytd ‘ 4U2 ¥, (0) x(s) / as) m0 (6) “s ms) 2) ~~ x7) pls) ms) SA — Datawune Erk DET Fer the degree § 12 34, + 42,13 hen eo) fi xno). 4G 25 wend) =l-4 2-3 Rn (0) FP2aydes xpt)=3-2=5 Foy (= p4a-5 Xai) = 2-S=-) For lOV= le Gee Kars 4-123 ¥1 C0) = xn (04 my °X)7 (0) = § +52 eV = eC ty xin GD = 3B ete -22) x(2) = (0) - wy °¥i2(0) = $-S=0 x OS) = ww C0) Hwy! X12. 01) = -3 4) “elo =m ( Wy ohe (0) = Ss Vo) pi) = -1-3) t nL aoe ¥ 2 C1) Wa Fx ep (2) = Fay (07 — MG" Ree (07 = © z = -\ +3) eat?) = x21 (0 X22C0) = 2 Spr DET k= od» 3 xe = mr Ck) + wee Ck) RIOD = Xil0) twee X2l07 > 10TID =e wer ex C9 403d 1S ha) rai) 2 —S-B2F-J2-414 czy = x1 (2) +E (2) =O x3) ¥) 7) + we? x9) = 34s) + (-0-101-j0-107) C=143)) wt = OITA) orig ae FPHLPAPDADIBIDRDBDHPHMHOHOHOHHSHOOBRRARAAT wie a le a le ee ee ee ee x Ck +4) ey Ce) ~ Wei ee CE) XG) = &1 (0D - we’ x, (0) > vis) 2 1G) = wel x2 OD = (-3-J) ~ (0-707 -j 0.701) (-1-3)) = —Ot2 + fj O-4tgr x ley 2 xa) ~ Wg HAC > Ow KID = x, Od ~ wg? Xe (3) = (3) - (- 9797 Jo 9 NC-4 43) SHB) Z 4 xb = (2%, -S828-J2-41y4, & HONS ake, O, - ONILAPO MND , 0, - sport }o-uin) ( Lue ald . (ny > Find the B ph PFT ofthe Agueme n) aS n Ky (O)= Coy y AEAD > 144 ent) = 46) 24) =e 6 Kp (0) = C22 AAI) 2p 4d xpM= 0 =e ap =e ry (= arate) = 2p2 7% 24 x5, 00 sO Kop lO) = UIA ACMI =24e acy el Jenks mee j2n _ eit x) (oye ¥ lO) + Mee X12. (0? > 242 2q xa = eu (YO twa! a ae 0 + jle) =O i = 2-1) =9° ea (2D Sx (0) amy X12 (O x OSD = kw C0 = wa) ee ee —— , / fp (0) > X24 C0) + Wy kre lO> 4F 428 xp A = Xa + wy ar D = OF@=O yn (2) = ¥21 (0) ~ WG? Kar 0) = 4 - GeO Ka (4) = Fay (PD ~ GP Kae (1) = 9-020 K(o) > %) 10) +We" ky (0) = ate = (2 ! . XCD = YOO ag CI = o4(b-id)(o>=0 A) = ¥, (27 VWe™ x2 (2) - 040 +0 (B= (3) 17H) 2 GO 40 = O° at APP PPHPAPYVBUBDBDHDVDVABDWMNMHMDVMHDOVYVNNYVOE SFSIVIVVVIVVIUUVUVVVVVVEVOVIVUIIIIYDY XH) = elo) ~ we 2 lO ~ g- bo -G FS) = xy - Wg HCD = © (U7 = x 2) > Wg CD = & x7) = ¥ (3) 7 Wyre) = © SPECTRAL ANALYSIS OF SIGNALS USING DET oe”) > 26b Sampls ——« 4D sey fs = Kur i” & Fove saraplls [4 Spe ignod fm = 4th ISL semaply a(M) Lod ko fm L, 25upt DFT xe) Lor k fom oto 65 [xo] = [x cae) Ppt PFT >) 1xa7l prot [ preerl, Peel, Deed, tera, Pl jriay on} | Cee rye X Cr eMeteRs pad sy XY (N-K) peng ce) \x (n-¥) | {ikea phoote suaperst - edd agen [X CN) = - Lele) - en 2 xen) = (U2 V2,Vt,y J <4 vey = (12,0,0,68°,%9) (12,0, 08, 4, er 20) IxCe) = Ix] —, eal fon fs = f.= 4 yp, ~ fs = fm 4 z 26k Aowaplts 4 Apes pagal, duit Aampling pate fs = 8000 oko Ake (2bo Soampls Dor ok gic). Sonnples rmdagriboat lees Syrantdnny mA AAFFHFSSSHSSPHPPHPHOHHSHHHHHEHSHGHHGHHLRHMHOMROE tii ee eee ee ee 2cb pk PFI Wu Sanrapln lav! frorn | mal 0) 0 fo Cru olrel xCK)| for Keo to 127 . | fs wade | 7 Find he man Vi y stad in | 1” 128 Values abothe, 123 Vals WE jut Mee Con jag capeanallay ony take only 4 Zebact indians Ck) at did |)x 9) 2 OS may, | — eae —— 1) ay th Hu Ir - asnune 5,8. Ibe au dhe indies @bvedvth aror ¥ k- [ 5,8 Iv, 126) f, =f 2 ¥ £000 pS x00 , It xxs000 k “| Dep 256 Zsb (2b y soce ) 2b ~ “oH fe: [1st 25 Hz, aeowr, Soome , 3437-5 x) (tet ible procedwee togred the fry A the 10 , 0)-08 24 FAST FOURIER TRANSFORM (FFT) wing dedmation 40 Tarynn CDF a pt DET 3 PA XCKD= DB xn) Ny Kz o4D3 n=0 3 \ En en 2 Zax wP 4 Zoom, neo nee > nen-2 Ken+2) ' ) xe Bun) wa” 4 Ee x0n42) wy ° nz0 ne en = = (x swe Pa cnerd Joe wee ee -Jare2e = e Le ee = Ck \ my Caen) + Cotas, 2 neo x(2e) x(o) x(2) x) — x (2k 41) xD ¥(%) VVBDBBPBBVBHV9H9VHHHOSOHO HOH HHOKHSBHOKRESBSOSHOAHEUHE = = 5 5 Ss S s Ss = ” ia) ” ” ” » » » ” 2 Aa 2 a 2 a a 2 A A A A A a ~~ 1 K(2E)2 SCX) 4+ coy (m429) We nro C1 tor inbegee 21 2en kn We = Wy -j 28 5 2 eo tN Aten en ' X(2e) = Fin 2tn42)) wn nro —————_______ 2 nen) \ X(2et = = (xn ay xene2y) wl2kton n=0 4 2F+) Oy yey =e 7 2en ne en n Wy Wy > WW \ x y(read = BZ ( etn) -xCn+t2) wy] w,*" neo po k= Oo) U kn x02) = = Aun) Wr n=0 Hoy ¥le) \ Ki Kk4N) > 29 ON Wy ” neo xy x0) mz 04>3 n> ole) kK: ote) 9) (wy —— rer) 4tn) 92. (n) ——> X(2E +1) a spr C x00, ear, x(2), x03) mond n=o-1 QiM)= HON) 4+ Unter) ilo? = Kl0d4 %XE2) FD = ay 44X03) n gut 2 (EOD 901129) og ° ae (0) = (x00 - 1020) 42 =r ~ 039) k= 0,) \ xanrZ quem wae? nee — . to) * sow = Gl *9 xX Co) = qo +i) ¥ (2) = grt 9 4 \ x 2 SGr(nywy x (ze AND 24 29 +5 we *OD= gute) + qe) SIGN yo ~4r 0) ert et 9O ORC ONMARRDHOOSLEBeeeE PPP EF EF FFE PRS FIDIIIID DB». x0) yay) K2d xs) io es “« 9) = i) ¥ CID A way? : xX wyl qr a) =I we? '[p- Normneb orb OlP - bit Awevsep Oda Salient featwwes of DIT: Le Noh tomaplix, Weatitipli catcon > bog JN Ato oh Covepe adden = N kay N o. Tip in bit pavewsal. ay dar Dee olp an neumnel eA 3B. gm place abgorithun 5 comp dene witlon Hot ATAU + tipliateon tae place byore . addi Ken and & poverpt alti r salir pentane op DIF: and a £lp un nounced od el in bit Aeversal ordi» afha Puibtro hen. eee mR DRDRAPOHOHDODOODO PR OOROOKRRGS torrpoks apt PFT of He secmenu Cn) “Ce, Lav) ¥ (0) 2 Aw) ot ye '° 4 ge _ SN — x (uD a7 xO 4 vn ge 0) 2-2) C1) \7 al) qr) vary ¥ 19) wae} ~Y veers lw, ey ety 06-08-24 iva, DIF) Bpt ver twang neo bl Ann) xCqey Wr0 raae) gir) xCeerr) oT an), gam Che +t) L4 Cn) 4 ge) xCak +3) ge nO) 44 M44) 2 m= ( tn) - x(44)) we — SVISIOOSSKOSHOVVTVIVUDEIIIIIISD Da» Oe OS a. gu) = Qe + grcnt2) qr = (gr (Md ~ 4,427) we ors qa CMe Br) 4 Ge lnar) qu Os (@- qzinte)) Wa . \ rH e20)) ROME AVS Z gir (now neo \ XCM) = 2 Sylmw,*? nro _ ¥(2) = gute? +9 a ap) nib) = qn!) agu lo xb) =-gie 'O) — 9 - = ¢ xC4)= gn (09 a" ) \ vo w KOH B= Z Que ve ate 870" teal \ en “2. 2) 6) W7' K (acy = 2 Qa n e ¥()=- grr ld s9er er = gn fe) +92) KEE grrl)~ Jor Od ) xCS)= jr lor -G ic aw AR RARHRARRBAHRHHOOHOOHROOOMROASBeReeE MSIPSVVVOVOVUDVODIVUVEDIDEIDEDIdOdDDOS: Find Bpt DET of Ane Sequente y tn = tot (200) 8 The Sey beer Arprate ey ° AY gg =n 27 /y — a(n) = les fonn (37) Aley= 1 LAY = 0-709 U0) =0 113) == 0-707) 2xtK) = =I C5)2 — 0-707 Alo) ~ 0 AC7) ~ 0-707 & no os gi(m= xn) 4 Uta) ploy= 2 q: lo) = Alert Al(eV= © pw of 2(0 = 3 ee TOE Dee gz (22° PCE: ~ zlby = 0 . Gi >= 127+ ~¢ gl2)= 14) 5 (42 403) 4 AOD = a Y- oto) gu (nye 91m) +g) Ente) fa = fru) +92 Intr) qn (00 = ailora gt (20 ger ( = gala grcr= 2 pr (1) 2 qt) +91 (3) =0 ries a gt ee (gutrnd - gered Joy Qrr lo) =2 gre (po) = Qilo) - Glows = 0 yz (l-}- ) (3) ay Uy = 719-7) we 22 qeu ele atm) = (0m) 244.) wg? G21 C1 = qrtry age lo 2 Ger) = (a (ny S5rh rug" ~ yD DARAPARBBIBBABPRASCASPHOMMOMOMOBRABBeBBeeeeee VUVPVIVEOKOOVEVUTIISVUVEVTIII IID DS ' Alo) id 0-707 XO) i 0 xXtey ° ~ 0-707 23) ae =! aly 2 — 0107 ty 1-5 okey ° © 707 27) it) xCK)= (0, 4,0,0 609, 4) : papa find gpk DFT of We Aegnin xen Cy’ é? oy ea & © Hess gomye TAI ATMAGY | gy (nde (HO — Ent Hg" - qe) > |-l =0 ples Vee ee 7 wt ays Vl =e LOS flav) fi ph are: \-\ 20. d f be Me , ~ . 2 ye 2), : glare tv 0 ve vO Pate ght $e. (2) (5-3) Qe = rey (2 Jk) 20401 FA bhaly aj eaty, gn (a) = 910 4g, Cntr) Qn (P= 920) tor(arr) qn (ey. O40=0 ger (o)= 2+ (-2))= 2-2) gr =e jn Cr) = (Ve tyfanj uy) + gatm= (gilm-§G ("42))ivy” : = Ay dhe = 0 woh =-) Paces. gir Lo) 20 9226 = 6. gaa (nny) wg? jerlol= 2-2 Biz) =e Q22Cieee wed -) opr vse Dee 06-08-24 DICOCRETE TIME LINEAR TIME TNVARIANT SYSTEM Dr UTI system —- — , oT iW) ae uT| dln) yon) xeon x Den) 2 tn « 10) “yen SF Wa) xln-b) NIFTPPPP PHP RRR ee eeRens: YVPSFOOOODDOVOOUHEVIVEBIEBEEDDdD DDS Shakic + Cowal; ACA)= 0 M0 Win) = nen wro nNZd mento nro a neo Aen) = Ftn+l) = renee. n to Frabeiliiy & [new] = yr nNr- h(ny= 2% wtn) ~ wnstabee sgt th (n)= bb)" er = crake rte umn Con it heath, ) ag ert sagtern 2 comple mepenerit Papennt jwon ‘ton Cansiveke omputod yin) eo” Bead ar qrer) 0 a te gives 2) My= Zz HL) xln-b) . fe y 4 gle” : oo Jwok ae “> nae . Sa oye " : ov" Wwe) = %tn) Hue) (input x fore, Feopred Sytem amd ov: ie ) \ L bigen voli ‘ gen Auvnchion fro xin) HC) inn “Odwwine He elp umponse — ten x(n ze 0nd - © &en- F (5 )> ) Hn} = CON 4 ECan) r(ide 2a8ta) sony 21 Atwy= &(n) 4 Fn) sr “jw pe 2nd) HlwW)y > 1 +e — > Pe cCoyw -jAinvwo f Wo +73 qin = tmp lwo) _——_ fy plwoy> 14 008(%) ~} sin 5) 21S jo bo 2 4 OF jr bbe J Hewey) = 79 . = i =-24-949 = OlWe) utes) = tom! an) —_ 14 My ~ a) 14 (wo) (Wo) | se TBE 538 5 ("5-90") (n) = mn xV7g Ke ae oe _ 4 i 2 (oortpt 8 ie a vows tan ? in am): cos, 1 amd the 543 (mn) fly" vl Vang Alay ore ohn a 6) wt) yn/4" ACA) = cos MY, 7 oe dag eg A) jem’ dadowint He ofr 4%) Jaen eanabhdeen meccanannnaanannppaneesceee SO PIFPOOOdIIHOOOEVVVVVBIBIIIEIDIDYD’: 7 a hewn ayy" ate a” alm) | (y= = ' (ae mend = % OND 4420) | ocacl ; we A \ afutm = 1 vinye ol Cry) Ab Le ae! 5 myn wee NA P (we 4 re) (-e Lt ' : | __ 4(™%) = ——y 2 -) Alu JSMA, ( Inape Vy, ml, +My 4 =) rs | = 454 Ie o3sas 403535 © by bg 4) Be 2 0. Be a5- = \ = 8s? E£7ee vogh L28°67 jm —j 28-67 3 (lyn ” pin oH" nea Dee Jl lg -28- 67") Hi Cn z AN gem: € t H (-%,) wey) 2 te Ed 1p ol I= ¥ves(nig) ~Uy (Me) 4b jos > 1 2 1.357 (28-67° 02-7367 (528 67 oO ~j lyn 328-67" “J (myn ~2h67°) ye tn) = x haste Byesitem i a 2 pon Briony a goon) = 1-367 elimeng - 20-67) z 5 (Me 8-28-47) + +357 @ zu ae E357 Tn - 26-47 po OP ter 57 ottlayed by 2B67* 3. Rakai the oly foe 2M) = It con) + 8 (%”) = ohn nO) *@y am) jm 04") JMO ron ) geome PERL) + i (TM) 4% -%) Una ~j UR)” +1 (e" hem) ~@ H(-m) ) 4 ros | #0 mAdg - een Spm Uw) >) > _| ——j > ——— -1e = yp wsee sj fine - - -) BOM I-45 € \-y Jh WMG) - Latte) amg - odd Syren - _ Ho) = . - . ~) 28-67 Wa): 1357 (- 286) = 357 & i a) . j2e-er" H(-Mg) = 387 (2tb1 = n3s7& = J2bse° # (th) : 1 »* °. SMe), 0 84-4 69? 1 yb +4 SH j26-56" W (Th) 0-844 e , ALERPDDPDPPRPPPPHAAHEMHOMNOOO HOKE YY PADS, ODDVPVDOTVIIVVVSITIIITIIIVIGOS” reer oN Ra : ; [v= 2 + tos (my - 28.67") 4 Sin(mn ~26-56)) - 7 | OUvaing We cl ytA) , forthe inpak olin) = 24 wm + hom (LY wen) 4) Sin ( nyt "Iy) spomee Ane aysiow pr xm) un giny C Aln) Prviodst cequunte trea fou wor N sowplis juwon ams 5 ae Kk: ar Wot 27 2 qT 7% JkwWon 20 te N-I Mnpe eae? ae ter © 7° kod ” Jk 2m rl xen) @ nN Nowe cn? Byaee Me x Ante = An DFT: ’ + -janen yCle s" ume ~ N- neo ; £ = Nag yetation bl PFT ag > a ‘ pres (ny ue 4 » Jkwo ln 2 ! = Ha se yen = S wr) acn-L) = = htt) ene Q=~00 Qz-0 . JE Wok an eee Ee nwe = Kk - Aa-00 ae erat ith we EW prey ot won) wite we kwe > davis A Wo jkanrn jkKWon 4 ft2n) Oa > kk = ZA aA yime 2 te é H (leo) = 2 ( zx.) ke be Syniusis Gin. DTFS. coegigutent aa ae 4 (+25)\ py es 0 dont Nv pres comp oh in elp is OTE Coeff Tp wettiplidting the frugroney ithpewsd of He Apiem ot ves BE whee kvowin from ee ; rR PT PPP PPP PPP PPR ERR A AB EBBBA888E8kAH = Yi) = bewomes pevladic iQ AW) iS pertodar & Ila) ot wae ye odie \ fre ¢ Goo itwhvon (1) A by darn Lovee 7 pl dene PV POAIPOOVITHVOOVIVE EBSD dYdDD YS Rusporse of DT LT Syviem ty nen periedie inpuds x) ae *) antn) a penn we x Ain) F ty) = Ytw)= x(w) MCW) e 2 a Jyw)| = Jel - [How | Dear danatoy spcchrwen oy gi) Lytw>) 72 Sy (0) es of YOM ° 9 4 ped} = Sux (wy > D8 oh xn) Parseval's wlotion | 2 Ziad = LY petwy!? avo nz -00 220? ~ Spy (0) Gwin a parieaic prchen o 7 ; vith pevied ox th) = 5 2)" Ve 5 IY(w) | daw <2ny Syy (w) oe vw Find the EDS of Stinenee (0) for the inpat xCA)-(L) 20h) a Oo innpulre 9eatponne Ila) ay an) vain) is pri odic for veg ste a on we le wlan) > poniodic (yon penuent) (b° > nen -pericdic 2 tnd = (4)? u(r) 2 MY“ 74 ined h . [wt yo)2 1 on &) y 1-,e u (wy = Toe. ; > \ ) X(wW)e : (w) ym tite co ae ('-Fote) ayes 2 i nt Vary - vow +44," J xu) #1 125 - 50 : a, V4 Uy a y- e \ oe 4iw)) >= a hb Lilope | {! : 1), * ('s sinvs) = \ . Co \+ yaa -2 ww + |e, 19), ~ %/, eso pro! a STAN ALE LRRARADADNHMHOP HEH PEMRSSORVSWSE Sb] Sygl= Jy! = Cras tos) ( 0/4 — 7/2 vosvo) writ ow an 7 Athen dale invuge & terordoren wars po ad Caen ee £en) 0 - yt) . ot > Hw) ype [wy = xlw) Les ——— es ylw) ee xl) pale yi) + My yoy =xtn) Bala) = (- y wu) 2 sete boys De-OS-24 — DTLTI austen as Thequaney eee eee hae x) oe i A(n) yim? atin) « Alo) VORVIV DEVE SD DD DE y(z) = X(Z)- H(Z) H(z) > brome}. dncion ray “= rT owe ; ; fe . DTLTI sys? 2 (v= Y(z) ag oe (rasig Soll x(Z) we Se ~ BPE BSF pie Zz UF h ’ wo <8 ! (w)! (tw) | _ ee) ; >) t “e an ; , ' =: 1 = ~ 1 Lio EE 7 ° we x we OW wey We, “ WOW MO sate oh hres 02") ( oundarg Foye uum tober accourals vv a : 1 ip ae punonvinaly - wpe = ae 14 Fay2 ret yim eg tnt) eat) yey cap vie) = r!2) J allzere system yes ) 2. All pele system IZ) ead 3+ pelt coro syste & Aliawo Ayptow dow pans filler a LPE wiles) 2) NO) |. we? 2 - oP se Sh [wy = max AMRP yg yeh POS us qn ab wera wr ety ? i 1 Ae . 4 Y ZO gy lle =O ele Ie N Tm i w ye’: Ine 0 (w) Je d Ml). pane wv ‘ : ae say, oe Dh me sa LABDPHPHDHPHPHDDDPDPRAFERBRIBASSSSSSRSEEBSSE SSVSEAMIVAIVVIVDVDVVDIVVEEESESEDEEEYDS: |H ew» s (Alp 82 4 (h sine) aul 5g see AL w=0 jrewy|? 2 Ht! = Y Max 2 : 4 wen ] Ho] *S os "a hwo] a _ L 'h __—__— weny, LHe = 3% vol ° Mh : a wo pole cere plot Ima. domann af gow erent the A tm(2) eb filtew te Alte eeadar id O-freg- eonprontled , re freq - de omphannised tig pes pun: - N(z) = 1-1 oe vo] ; z 38 9, |Awl*= grin ad weo max al wen | I-petco ley al rT J. Jol 15 a ", Y Ee + € an wn Lo WW) = 1H LOY + 1 sw +ypsinw 2D [HC] = (tain) sire) i) ie & - ww 2 oe >0 |Hlw)) = : | Ak w=0 |Hlw) a = [Wealay y © : | wen leo) = Say 24 | mh | A + 4 Ml L Nn wit \Htw) > Ie nSml2) ° "h eG ily Z Dole covo pot ( inz domen, 7 7 bly * fxr is . A @uze de eonplesieed bod poss fill; fue yet wey (1-h2") Zed No 2-4 Za 7 . Jn val He ve = ye Woo, 7 [Hew] 2 winvaliae Mh putea|* - manvabene ~jrw 2 ryt u Blu) = I Wt - (1 ; rs e [Wl] ‘a y,te%) + (2 Sinaia) - Wy Us2wW Pty fk LU LeEe hE tate SIELTP ORO e Ren neeens KVAIDFIVOVODVVVVVIVVVVVESE TERRES DY wWw=0 wre a) ah “Ye hy, pert win | Wwe)! = ie ° Mh toe He a “ - J¥tw)|o> IL at = 25 lb 3 1 Rea Ie A euaphasical Bond slop filtew nied = Cai’) > Crayz")(-y 2) \ / Gr-) 7 devoes of the ry yn fille a Geet \Plw)| > max at wee,m the \ min a oe t Ah All pass dite . le aTt™ By / \ H(z) =Z Z=0 +—_j__}__ ilu n \ ° ow iy Hlw) = & r\ KF < | z | HOo)| =) all w \ ke(z) from o toan 2 Hansjoom done ia A wl cave 1 pitt Ware PoLe SYSTEM; H(z) = Ni) iz) N(z) =) nz + me) kot leu jee An dow po lis. (LPe)! H(z) 2 4 " . eo = 1.0% ws 0 7 dreamy snaphonnel At wo | Hw) “2 mo | imle) | ale : ab smh Y y ) HW(2)=2 Vieye zl el2) wenden Pala aie teapnisey the pred pevecs ~ deems to's ioe Feet cereg jw ye = ¥(gestos fine) Ceal : ¥ (edtu imaging « ysinw ° kelZ) SL TTT TLL LT TTP PRS O OHH OHRRORRREEARSESSE ~YAADVDVPNDDVVVVVVVDIVVVTIRTTTBTIVIIIVID?Y Mower ful me —_— bh Band poss fie: Hiz)=_ I lav laz*s 9 foe 2A eae . Pi yor. ga at yoo thy, |¥euor| — mar Bomd week filles W(t) =_ 4 l-z% 2-t) Zetle jel. ve” Jn j Zitz le = vel” ak Woh — \H (od) ~rmax / In(z) Ele “mt |e eee SL Vole gy . Y' Pore -Z ERO SYSTEM, giz)z NIZ) (2) dove pars Kir Wiz) 2 tae 2ev0 at z =-l -) wo Jn \-z ye +1e€ \ deemnphonited At wen |Hie)) 2 09 pole at 2=\ jo red Mse b ernphorated ak weo [Heyl mar ee ty [w) 2 14 bees panne [-losw +) 400 ee ee v e ae’ Jyres)) = 2 +2 Cow yo? wo ke g- GAY L RN Chic wen 0 tevo ok Z=| . jo yes \.€ Wo 9 de eunphmnrled woo [AU)) - won Ce EPTTTTTT ITT OPP HMM MOBOOHHRORARASRRES TT FIPIIPIPIIDVVVVDIIVVTTITRODIVYD dD? a > bob at z=-| ( ver. eopwited, nv [7 o Rel) ie facodl —max My _ Bond pus fil. H(z) 2wos of 2:4) 5 Jo reo” =[-2 or | ae lo jn oo ate emprtted ween jan) omer pols at z= 4) 5 Me Jo _ Ale ele 5 ye ye” ov BF ie KS ebhh In(2) Ab we JA) | Hew] - max a xn o kelzy Ye Bama neice fil: aCe es 2 H(z) = V4 z St? \-27> “os rua, Zeow at z=) C7 depen dw. Jrle IM ye =)]-e@ ey NI eh Me A100) ; nm 5 at vw: fA, |Muo) | — vbax ame ginle poltsak zea) | ip xX [ oo y= eee oy AL weo,m > | A(t)» may v L anne ed 2-08 Ly. one Hwo pole BPE that baste clnbe oi pars oon nwo ero nits Chavaduviskes 20, oe i abu: 477 { wats Le = ond ts may seapen x a ok al Ken po Yas age ' ) v py al zy a [nol yy \ mw - 0 ME mMOK Ws q Wa We, Wee or 2 Zz Seeomas . uae co wae (HN H(2) 2 (27) cee) Oy fives (rajazt) Cty? ) JL \ Ae Wh , ye > DPHPLPLELRLPDDPDDHDABHHBBBBHAOAKaaeeeasses FVAVEVIAOD DODD VVODIVUUUSE EDD OD dL D > s = & ° lever |n(a)} 2 \" (en) ct And @& \a(re)| =) Ve nd G&y Blu) = @ Ce ef) wee 2 e \Hlw)] = & Ca tes20) 14 V4 4 297 CHO JCM |: ety . a 7 Vackiy (-r)* 2 ate tery” Q: + : (41+ + ie)" 2 4” (2-2 0-44) 24° ° leyte 210-5 u z \ [eqwr Vl corks yoy) _ a \ a pay te BN 7. 2 Cy?” Ceay) = varie tea = Chavte2rt) (han) ASIN = latter” peri timer = S561 1-1 Har gEy™ 2 = 0 4ayt-2¥° 40.44 =0 2 —————_—_—_ a - V2 243 V4 - 4x0-94 x0-44 tb A044 L Ys dt 0.6923 en eer Ts a By [er Y*: 4a 01 vo sth] ,to. 837 DEAL, rete a Bul ct my ot of, Cor le Poles z= aja bypy rt AE 7 £5 . yew. ¥ e § radius Maaco Ge ey, 22 = ols TT z . w(z) = ols (1-2) L4oqet = 2 dt ay(nye (12-129 ari 202-021) bad 1 CM) = eae . gt ib YO" HL) AU) pled = act) # ot (H+) IP PPPLRPPHPPLRHPNMNYVDVVVDAAVAAVAOBGaeece 4 & + te > Conpdinthe signed «(1 Ainusoidal burst signal don seen) buvet AraH at bo, wnds ak Ob ms . Gena Ht Ae OH) if He Aignal xO) Jo bewaplad at poe Ym | Find He Der af tre gerrmled Sy rimee . yb) = $m (200 7E) > sm (geno) > Sin (En) sore Te 2 at 2 66m4 9 asgon = nro od xen) = Si (a ate fee “y ber a [o A, Pe) PDBPDPPPBPPRPRPPBBBBBBBEeeEESesGF (0) / ran} | tos? ¥(2) “ft-) t-) . = ¥(39 ryt = ° e . Vel , JJp oy = a = 3 > > > > > > > 2 : 2 > : > > ’ > : 20-08-24, Dtgtal xesonatyy Wo — Atbonank 190 224 ab We Wo yuna ante \H Cw) | 3 mar a weeded abd hoe Mane bode“ \ (ac é ple 0 awe So hoe 9 civeahy oh) ame (0 WAR) (qhy veg ¥w) 2 7) daormeacedw ah eye mE fy em2") H(z) as walualed on unit cole Hw) = j-2Me 2 yt) \ = 1 \- tos (w-wo) 4palner 2) 1+ Ws (w-Weo) - 2 lostw- wo) Sin’ (we - ' = —— “Re digslw-We) poe) ___—. \ — Fo : ——e (reese) A (Bia (o-) ) L -we) A - Aw» (w-Wo) bs ar WweW. Cw)" =e chal ALHONDY 45 0 Atgtal iter wbwch prodmes marin Aespor oe ab os wenpovRulon, Progeny . Digntal notch fillor: Aiizevo Biter (eve _deemphosries 7) Magwilrde wo wo at a par holon prun'g wo- jwo ll attpr l-e & jwe -j)w —j(w- We) wiwy = 17 & e eaaeae L : spews \ \Auo) > (\- wos (i-wo)) + (sinw-we) ) 2 = a&-2v0t[we-we) [veo)) ‘ - ~ At w= Wo , \ We Qe we 2 ) Hal ° CLT PPPPP PPP HHS HHPABHBABSHHSKRAgggasg ESL FNAVEVVPRVEVVHPHOVIDIVUVUVUDEDVEUUUUEDS Nig {lal comb filer : pilows contain Rreguemes er uhale Wocking eles « ” wiz) eS hte 7 kee MM order flr (=) -)2" Av) = (h tn ye 4)” SS align te Oe! UM lg hh w (2) = tah Za huny = Senay g(n-7) wiry =f, | 1 uk Weld * 2 Ate 2 OL Wlw) = Zz nck & Yn (w)) = wen vo ofwe2n Wy (0) = H (ew) W(Lw) = nm wero oée oe a 2m aan c SLES Hom paveh cond pllaris Produas wll periods 2 ak -_ jes Arkegen, mmubbples of 2A/_ Mal eal 369) | 2 Irodutesy new periods callay Ak vakeg rll pli» 2 aE. Oigital sinuwridal pec tatoy / G prodace o |p witlrond” tia H(z) Vo = -t baat + aot oud aun Silo Q,= - WeWo a>) H(t) = Ylv \ XL) \ inikal vals =0 We) 2 x(t) t+a,r7 44,0 Yee) Chawre 4 aoe) = rte) snwerse Z transfor mend 4ay ginrd +42 n-t) -2) -@ yim = ren) = Ay gine) = A244" Pealvace Cin put given at onby neo) =u) [ny = Sin we Bin) A108" 2y wr’ O,2 LWbWo a= ) ov 209) = wSinwy, din) Input applied at onhy nz0 yore -yGrzo. DPPPPNN PNP NL KHOK PHP EPEEAE: © ) ge i PPPBEPEPE -y 5 4 (o> = Ahim We 4 toe Wo o . D = f0) = Sinwe 19 Coro CY - 4% > 2 or a Lv ~ 2 Sinwo > yo Oo + 2wWRWo oe go") > : , > a Los SinWo Sin®Wo ch neo, 3 house Ue nen “ty > yen) = Sin (MIN opp ay HUE for ab esse 2 ree sense 6° Oo) yond? gin nt Ye 3 Bu &)——— | ) > (2) Ye 3 [BP] we hey 2 ee anaes > z < > eles 2 Av) yew) 2

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