Chapter 3. Hence, 3.2 (a) ( ) dt. (b) (d) using the. linearity property of the CTFT. Next, using the shifting property of the CTFT we get
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1 t t w, Cptr t t t <, t t t t t t t t < t t Y t t t t s t U t t t t t t t t t t t t V t t t t t t P t lt t t lt t usig t l l lirit prprt f t CTFT t, usig t siftig prprt f t CTFT w gt P l lt wi ltrtl prss i t fr l P ig us f t rsults f Prl T T l t t t t t t τ τ t t T futi t is ls t rt t Tus, t t t si si V t t si t fr sl
2 t, t <, t r vit w t tri t CTFT f, t t is t iffrtit it twi wit rspt t t, tri t CTFT f t t t us f t ti-iffrtiti prprt giv i Prl 6 ti-siftig prprt giv i Prl 6 w, t, t <, s t s up t, t t istiuitis wit psitiv up f vlu t t ±, gtiv up f vlu t t, zr vrwr ls, t s l ipulss f strgt t t ±, t t t t t t ipuls f strgt t t, i, Y ts t CTFT f t, t usig f t ti-iffrtiti prprt w v t t CTFT Y Usig t ti-siftig prprt, w rriv t t CTFT f t giv Trfr Y t s, i, Y Y 8 s si si t σ t σ t t Tus, t σ σ Mig g f vril τ σ τ t τ w gt τ σ τ σ τ σ τ σ σ σ σ Fr zr ipuls rsps, w t v t CTFT pir CTFT σ σ t σ t t t si t t t If t fr sl 6
3 6 t τ t t t t τ τ ti usig g f vril t t τ t t τ Trfr t t t t τ τ t t t t t t t t t t Trfr t t Itrgig t w gt t t Fr psitiv rl stt t CTFT f t is giv t τ t t τ τ I siilr r w sw tt fr gtiv stt t CTFT f t is giv CTFT Trfr t Diffrtitig t sis f t t gt t t CTFT t Trfr t t t 7 t t, wr rg{ } Tus, 8 t t t If t is rl futi f t it fllws fr t fiiti f t prssi fr tt r pl ugts Trfr Or i tr wrs, fr rl, t gitu sptru is v futi f t ps sptru is futi f t fr sl 7
4 9 wr ˆ τ τ τ t t T t T is t ipuls rsps f t ilrt trsfrr Tig t CTFT f t sis w gt ˆ T wr ˆ T t t CTFTs f ˆ t, t T rsptivl Rwritig ˆ p p T s t gitu ps f r v futi, is s t rl sigl Csir t pl sigl ˆ ˆ t ˆ t t t Its CTFT is t giv ˆ Y p T ttl rg ε t t t t t T ttl rg ls put usig usig t Prsvls tr ε Trfr, t 8% wit fu vlutig t t t t 8 t Trfr, 88 t 8 wr, v v w, v Y Si, s rsult, Tig t DTFT f t sis f t v quti, w gt Y Y r Y, v Y Y Y T ivrs DTFT f is giv t fr sl 8
5 Y wit < Rwritig w gt Y s si LP G G, * * Si is rl v, w v Tus * Trfr, s s is v, s rsult, t tr isi t v itgrl is v, s s Si is rl, w v Tus, si s rsult, t tr si isi t v itgrl is v, si t fr sl 9
6 6 si Trfr, t DTFT f is giv 7 Lt wit < Its DTFT ws put i Epl 6 is giv wit < Its DTFT is giv wit < t Trfr, usig t iffrtiti-i-frqu prprt i Tl w gt wit < Its DTFT is giv wit < Its DTFT is giv Fr t rsults f Prt w srv tt, wit > Its DTFT is giv t fr sl
7 f trwis,,, 6 M Its DTFT is giv 6 M M M M M M M M 8 Lt t t DTFT f Usig t tisiftig prprt f t DTFT giv i Tl, t DTFT f is tus giv Fr Tl, w v Trfr, Lt wit 8 < Its DTFT ws put i Epl 6 is giv w Usig t ti-siftig prprt f t DTFT giv i Tl, t DTFT f is tus giv wit > W rwrit it s T DTFT f ws put i Prl 7 is giv t DTFT f ws put i Epl 6 is giv Trfr 9 T trwis,,, Y si si t fr sl
8 ,,, trwis si si T Y,, *, trwis O wr ssu t T w prss,, Trfr,, trwis Y Y Y Y w, fr t rsults f Prt, w v si si Y, Y si si t: T v rsult ls ls fr v,,, wr is t squ, trwis, sir i Prt is t squ sir i Prt, Y Y Y Y s,, si si si si, trwis T si si si si! Dt wit < W sll prv iuti tt t!! DTFT f is giv Fr Tl, it fllws tt it ls fr Lt T! Trfr,! t fr sl
9 , it ls ls fr w, ssu tt it ls fr Csir t!!!!!!,, Lt Csir t DTFT Its ivrs is giv Trfr, t ti-siftig prprt f t DTFT, t ivrs DTFT f is giv trwis,,, trwis,,,, s l l l l < trwis,,,,, wit < W rwrit s wr Fr Tl, t ivrs DTFT f is giv Fr Tl, usig t iffrtiti-i-frqu prprt t ivrs DTFT f is tus giv t fr sl
10 si Trfr, },,,,,,,,, { s Trfr, },,,,,,,, {, si Trfr, },,,,,,,,,,, { s Trfr, },,,,,,,,,, {, s s Trfr, },,,, {, } { s s s, },,,,, {, } { si s s, },,,, {, } { t fr sl
11 si s s 7, },,,,,, { } { Lt t t DTFTs f t squs, rsptivl G g Lirit Tr: F { } β β g g β g β G Ti-rvrsl Tr: Ti-siftig Tr: Frqu-siftig Tr: Lt F F { } { } Fr Epl 8 w v si <,,, Fr t rsult f Prl w gt <,,, si s t ipuls rsps f t s is giv O *, usig t vluti tr w ti t DTFT f t s: t fr sl
12 < <,,,,, _ 6 Y w,, Y Trfr ± ± ± trwis, 6, 8,,,, K 7 Trfr, Tus, w writ { } Y Tus,, fr, v, fr, fr, v, fr, 8 F{ } * * * * * 9 * i r i r, i F { } i t fr sl 6
13 6 Fr Tl w srv tt v rl-vlu squ s rl-vlu DTFT rl-vlu squ s igir-vlu DTFT Si is squ, it s igir-vlu DTFT Si is v squ, it s rl-vlu DTFT si si si Si, is v squ, it s rl-vlu DTFT Si Si is squ, it s igir-vlu DTFT is squ, it s igir-vlu DTFT Fr Tl w srv tt v rl-vlu squ s rl-vlu DTFT rl-vlu squ s igir-vlu DTFT Si Y is rl-vlu futi f, its ivrs is v squ Si Y is igir-vlu futi f, its ivrs is squ Si Y is igir-vlu futi f, its ivrs is squ Si is rl-vlu futi f, its ivrs is v squ Si is rl-vlu futi f, its ivrs is v squ Lt u *, lt U t t DTFTs f u, rsptivl Fr t vluti prprt f t DTFT giv i Tl, t DTFT f O* u is giv Y U Fr Tl, U * futi f Trfr, * Y wi is rl-vlu Fr t frqu-siftig prprt f t DTFT giv i Tl, F { } st f tis DTFT is sw lw t fr sl 7
14 _ 6 } { Fr, > s Fr Prsvl s rlti, s Trfr, w, s Trfr, w, Usig t iffrtiti-i-frqu prprt f t DTFT s giv i Tl, t ivrs DTFT f is, t ivrs DTFT f is Y Trfr, w, t fr sl 8
15 Usig Prsvl s rlti wit iffrtiti-i-frqu prprt Fr t iffrtiti-i-frqu prprt f t DTFT giv i Tl w v wr F { } Trfr, Fr t fiiti f t DTFT t fr sl 9
16 it fllws tt Trfr, Cg Fr Tl, F { }, < s rsult,, C g Lt G F { g } t g g g, F g } G { G G G { F { g } G G G t g g g w, F g } G, t g g g 7, F { g } G 7 G G Y, i, Sttig i t v w gt Sttig w gt, < Fr Tl, F { } T ttl rg f is E T tri t 8% wit f t sigl, w st E, 8 8 E 8 slv fr, i, st E,8 s si uril sluti f t v quti ils 8 t fr sl 6
17 Rll, wr s is usl, v v fr < fr >, tr is vrlp tw t zr prtis f pt t, w v Mrvr, si v v { v } r is rl, it fllws fr Tl tt F F { v rriv t } i Fr Tl w v Tig t DTFT f ν r w ν v ν F { } t Sustitutig t v i t quti prig it w gt ν ν ν r t ν r ν ν ν ν r r t ν ν ν r Cprig t lst quti wit i r r ν t i ν v, w rriv t ν Liwis, tig t DTFT f w gt ν i ν ν Sustitutig t prssi fr giv rlir i t v quti w gt i i ν ν t ν Cprig t lst quti wit ν ν ν ν ν i ν i t ν ν i r i, w rriv t t fr sl 6
18 t ν ν ν i r If is t iput t t LTI isrt-ti sst, t its utput is giv z u z z z z z u wr is igfuti f t sst z z If is t iput t t LTI isrt-ti sst, t its utput is giv z v z z z z v Si i tis s t suti ps up is t igfuti f t sst 6 F F } {, } { F F } {, } { F F } {, } { F F } { } { T vrll frqu rsps f t strutur f Figur is giv 6 Dt F i }, { i T vrll frqu f Figur P is t giv i T strutur f Figur P rrw s sw lw wr t l wit ipuls rsps rprsts t prt f Figur P wit f lp s sw lw t fr sl 6
19 u v w Lt F F F T w v U }, { u V }, { v W } { w V U W Eliitig fr ts tw qutis w gt W V W U V wi ls t t frqu rsps f t f strutur giv U V T vrll frqu f Figur P is tus giv 8 F F } {, } { F F } {, } { F F } { } 7 { 7 T vrll frqu f Figur P is giv w, is t ivrs DTFT f Rwritig w gt v r r Its ivrs DTFT is v Si is rl usl, its DTFT ists, it is ls slutl sul, w rstrut fr s v v v t fr sl 6
20 si s si si, t gulr frquis prst i t utput r ± s s s s s s s s s s s s s, t gulr frquis prst i t utput r s, t gulr frqu prst i t utput is R F { R} Lt T t iu vlu f is t iiu vlu is Tr r R ps f lt t R, R, R ips lt t R, R i t frqu rg < Mgitu Ps, i ris - M M - M G t G fr I rr t v G, t ipuls rsps sul ultipli ftr K, wr K M s s si si T frqu rsps will v zr-ps fr t fr sl 6
21 If t w rwrit t v quti s, wi is s t v lir ps s s, will rl futi f if i wi s w v, 6 F, } { F, } { F } { T vrll frqu rsps is t Trfr, s s s s s s s s,, if,, s Sustitutig i t quti t lft w gt 7 rg Y Trfr, Y Si is rl futi f, it s zr-ps 8 R Tig t DTFT f t sis w gt, R Y Y R Y t fr sl 6
22 T iu vlu f is t iiu vlu is Tr r ps ips i t rg < T ltis f t ps t ips r giv ± R r ± R T ltis f t ps r giv R t ltis f t ips r giv, R R Plts f t gitu t ps rspss f fr 8 R r sw lw: Mgitu - - Ps, i ris I tis s t iu vlu is 8 t iiu vlu is Tus, w st * s s Sluti #: ± sg I wi s, ± trivil sluti Sluti #: ± sg I wi s ± 6 B B Tus, w st B B B B * s s B B B B t fr sl 66
23 Sluti #:, sg, ± B B B I wi s ± ± ± ± iplig trivil sluti ± Sluti #:, sg, ± B B B I wi s ± ± ± ± iplig ±, -trivi sluti is * 6 si s si s t t s Trfr, t iput-utput rlti is giv s s Trfr, t iput-utput rlti is giv si s t Trfr, t iput-utput rlti is giv s si t Trfr, t iput-utput rlti is giv 6 Fr Eq, t iput-utput rlti f ftr-f- L up-splr is giv T DTFT f is tus giv ± ± ± trwis,,,,,, K L L L L t fr sl 67
24 Y F { } L L L L, wr 6 G, < Tus, w writ L G, wr L Fr Tl, t ivrs DTFT f is, fr t rsults f Prl 6, it fllws tt L,, ± L, ± L, ± L, K g, trwis 6 Fr Tl, Tus, s si ± rg{ } t 666 Trfr s ± 9 ± 8 ris w, fr iput si, t st-stt utput is giv si Fr, t st-stt utput is trfr giv si 9si si 8 6 s W rquir s 6 s6 Slvig ts tw qutis w gt s W rquir 6 s s6 Slvig ts tw qutis w gt t fr sl 68
25 s s W rquir s s, s s, 8 s6 s8 Slvig ts tr qutis w gt 866, 8, 6889, i, { } { 866, 8, 6889}, 69 Trfr, 7, 7, Si t ipuls rsps is rl, t vlu f t is t ugt f its vlu t, i, * 7 Writig t fur qutis i tri fr w gt 7 7, 7 7 { {,,, },, Trfr, 8 Slvig ts tw qutis w gt, { } {,,, }, p 7 T tw itis t stisfi t filtr r: 8 Slvig ts tw qutis w gt t fr sl 69
26 Mgitu Ps, ris pi pi 7 s s T tw itis t stisfi t filtr r: s s 8, s7 s Slvig ts tw qutis w gt Mgitu pi Ps, ris pi 7 si s si T tw itis t stisfi t filtr r: si7 si s, 8 si s 7 Slvig ts tw qutis w gt t fr sl 7
27 Mgitu Ps, ris si s si T tw itis t stisfi t filtr r: si8 si s 8, 8 si6 s8 Slvig ts tw qutis w gt Mgitu Ps, ris , B t fr sl 7
28 B Mgitu Mgitu It s fr t v plts tt is igpss filtr, wrs is lwpss filtr B C B 76 Tig t DTFT f t sis w gt Y Y, t frqu rsps is Y giv trp 77, sips Trpzil Sips Mgitu 6 8 t: T pr t prfrs f t Trpzil uril itgrti frul t wit tt f t Sips s frul, w first srv tt if t iput is t, t t t rsult f itgrti is t Tus, t sir il frqu rsps is t fr sl 7
29 78, w t t rti f t frqu rspss f t ppriti t t il, plt t tw urvs s iit t prvius pg Fr tis plt, it is vit tt t Sips s frul plifis ig frquis, wrs, t trpzil frul ttuts t I t vr lw frqu rg, t frul il rsults ls t t il wvr, Sips s frul is rsl urt fr frquis ls t t i rg G g g g g g g g g g s g si g s g si g s g si g s g si g gs g g s g gsi g g s Tus, if g g g, t g G g g s g g s,, stt grup l ltrtl, if g g g, t g G g gsi g g si,, stt grup l 79 s si Tus,, τ si s wi s lir ps wi s lir ps si s t si s s s s si si s si s s s si s s s si s t fr sl 7
30 Lt Fr t rsults f Prt w v si s G s s τ G Si, G w v, G s s τ τ G, wr is t frqu rsps f Prt is t frqu rsps f Prt Tus,, trfr, τ τ τ s s s s, wr is t frqu rsps f Prt is siilr i fr t Tus,, trfr, τ τ τ s s s s 8 T grup l f usl LTI isrt-ti sst wit frqu rsps is giv τ g w,,, r, quivltl, T first tr t rigt si f t v quti is purl igir, t fr sl 7
31 R τ g 8 Si is t Furir trsfr f G, g G Rwritig Eq 7 w gt τ * g * * * * * i r i r G G * i r i r G G i i r r G G i i r r G G 8 tus, si s { } s R { } si I si s G Tus, { } s R G { } si I G Trfr, usig Eq 8 w gt τ si s si s s 8 s 6 s 6 8 s 6 si s 6 s 6 si s 6 t fr sl 7
32 Lt G 6 T τ G τ T usig t 6 6 s s prur s i Prt w gt τg Trfr, 6 s 6 6 s τ 6 s Lt G, G wr G G Trfr, τ τ τ T usig t s prur s i Prt w gt G G s τg usig t s prur s i Prt w gt s 9 s s 9 s τg, τ 9 6 s s 9 6 s Lt G, G wr G G Trfr, τ τ τ G G T usig t s prur s i Prt w gt 9 s s τg τ G 9 6 s, s τg 9 s s 9 6 s s 8 Fr Tl, s si s si s Tus, si si rg{ } t t w s s 68 7 Trfr, s si t fr sl 76
33 rg{ } t 6 ris 68 Si fr frqu rsps wit rl ffiit ipuls rsps, is v futi f is futi f, w v w, fr iput si, t st-stt utput is giv si Tus, fr, t st-stt utput is giv si 6969si 6 8 s W rquir s s7 Slvig ts tw qutis w gt 68 Mgitu 6 8 t fr sl 77
34 M r 9, 7 8 Rl prt Igir prt 6 plitu plitu Mgitu - Mgitu Sptru 8 6 r 7, Ps, ris -8 Ps Sptru Rl prt Igir prt plitu Mgitu Sptru plitu Ps Sptru Mgitu Ps, ris M It sul t tt Prgr _ uss t futi frqz t tri t spls f DTFT tt is rtil futi i, i, rti f plils i Tir ivrs DTFTs r tw-si squs wvr, ll squs f t fr sl 78
35 Prl 9 pt tt i Prt r tw-si fiit-lgt squs f lgt, tir DTFTs v t psitiv gtiv pwrs f s rsult, t frqu spl put usig frqz sul ultipli t vtr vlut t t frqu pits us i frqz I Prts,, t ps sptr r t plts f t uwrpp ps ti usig t futi uwrp Mrvr, t DTFTs f t squs i Prts, r rl futis f tus v zr ps Mr urt plts f t DTFTs r ti usig t futi zrps,, Y, trwis, Prgr _ r sw lw: si si T plts ti usig Rl prt - Igir prt plitu - Mgitu Sptru plitu - - Ps Sptru Mgitu Ps, ris T plt ti usig t futi zrps is sw lw: plitu t fr sl 79
36 ,, T Y, trwis ti usig Prgr _ r sw lw: Rl prt 8 si T plts si Igir prt plitu - Mgitu Sptru plitu 6 - Ps Sptru Mgitu Ps, ris - -,, si Y, trwis si usig Prgr _ r sw lw: Rl prt Igir prt T plts ti plitu plitu - Mgitu Sptru Ps Sptru Mgitu 8 6 Ps, ris t fr sl 8
37 T plt ti usig t futi zrps is sw lw: plitu, si,, si Y trwis si si Rl prt - Igir prt plitu - Mgitu Sptru plitu - - Ps Sptru Mgitu Ps, ris - - T plt ti usig t futi zrps is sw lw: plitu t fr sl 8
38 Y s,,, trwis si si T plts ti usig si si Prgr _ r sw lw: Rl prt - Igir prt plitu - Mgitu Sptru plitu - Ps Sptru Mgitu Ps, ris - T plt ti usig t futi zrps is sw lw: plitu M T plts ti usig Prgr _ r sw lw: t fr sl 8
39 Rl prt Igir prt plitu - Mgitu Sptru plitu - - Ps Sptru Mgitu 8 6 Ps, ris T plts ti usig Prgr _ r sw lw: 8 Rl prt 6 Igir prt plitu 6 plitu - Mgitu Sptru 8 - Ps Sptru Mgitu 6 Ps, ris - M % Prprt t fr sl 8
40 8; % ur f spls i squ g ; :-; p-*g*; p-*g*fliplr; % r - t r-- % s if pw is DTFT f -, t % pw Rpw pw-ypw Y,w frqz,,; p*w*-*y; :; w -pi*; frqz,,w; % Vrif % Prprt :-; p*g*fliplr; Y,w frqz,,; p*w*-*y;,w frqz,,; % Vrif % Prprt rl; Y,w frqz,,; :; w -pi*; frqz,,w;,w frqz,,; % Vrif Y * % Prprt *ig; Y,w frqz,,; % Vrif Y *- % Prprt :-; p-*g*fliplr; s *zrs,- * zrs,-; s *zrs,- - * zrs,-; Y,w frqzs,,; Y6,w frqzs,,; Y Y*p*w*-; Y6 Y6*p*w*-; % Vrif Y rl Y6 *ig M 8; % ur f spls i squ g ; :-; pg*; pg*fliplr; v *zrs,- zrs,-; *zrs,- - zrs,-;,w frqz,,; v,w frqzv,,;,w frqz,,; v p*w*-*v; t fr sl 8
41 p*w*-*; % Vrif rl v, *ig M iput't lgt f t squ '; :-; g -; g pg*; % g is ptil squ si*pi*; % is siusil squ wit pri G,w frqzg,,;,w frqz,,; % Prprt lp ; t ; lp*gt*; Y,w frqz,,; % Plt Y lp*gt* t vrif tt t r qul % Prprt ; % Squ sift spls zrs, g; Y,w frqz,,; G p-*w**g; % Plt G Y t vrif t r qul % Prprt w pi; % t vlu f g pi r 6; % t vlu f g i trs f ur f spls :-; g*p*w*; Y,w frqz,,; :; w -wpi*; % rtig Gpw-w G frqzg,,w; % Cpr G Y % Prprt :-; *g; Y,w frqz,,; % T put rivtiv w spl t pi -^*g; G G:' su'; lg G-G*pi; % Cpr Y, lg % Prprt vg,; Y,w frqz,,; % Cpr Y G* % Prprt 6 t fr sl 8
42 6 g*; Y6,w frqz6,,,'wl'; G,w frqzg,,,'wl';,w frqz,,,'wl'; % Evlut t spl vlu t w pi % vrif wit Y6 t pi fliplr:9' fliplr:''; vl *sug*; % Cpr vl wit Y69 i, spl t pi % C t tis t tr pits siilrl % Prsvls tr vl sug*; vl sug*; % Cpr vl wit vl st f isrt frqu pits,, M7 T DTFT f is, t grup l τ g put t s fllws: R DFT{ } τ g, DFT{ } wr ll DFTs r pits i lgt wit grtr t r qul t t lgt f { } M ;,w frqz,,; pltwpi,s; gri ll'\g\pi'; ll'mgitu'; Mgitu 6 8 M ;,w frqz,,; pltwpi,s; gri ll'\g\pi'; llmgitu; t fr sl 86
43 Mgitu 6 8 t fr sl 87
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