Chapter 5. hence all the terms which are not in the range 0,1, can be accumulated to ψ

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1 Cpt 5 5 t T Sic is pidic i wit pid Tf 5 c is s pidic i wit pid Tf { } b { } 5 Sic ψ ψ c t ts wic t i t K c b cctd t ψ w c i tis cs t Fi sis pstti ivvs cp pti sqcs t t w f Eq 5 t i sti is q t if twis it is q t Ts t w bsv t f s 9

2 5 { } cs T pid f is w f Eq 5 w bsv d twis f twis f c twis f b } { } { cs si T pid f si is d t pid f cs is c t pid f is t CM f d is c twis 55 t dt t cfficits f t Fi sis pstti f Sic is pidic wit pid P p p t f Eq 55b w v c f Eq 55 w t p P P p 5 < < w t f s

3 iwis t T Bt c 5 w Tf t t b 5 si Tf F Eq 5 t fist s is q t w d tis iwis f Eq 5 t scd s is q t w d twis Tf twis b cs cs b w t pit DFT f is f d twis F Ep 5 t pit DFT f cs is f d d twis Tf t f s

4 twis b c cs cs cs c F Ep 5 t pit DFT f cs is f d d twis iwis f Ep 5 t pit DFT f cs is f d d twis Tf twis c 59 α α α α α α b Ass fist is v i T dd v b b t ss is dd i T b 5 cs Tf t f s

5 si si si si 5 5 F w t d f w t w 5 Svi f d w t d Tf t f s

6 5 wit Svi f d w t d Tf 55 F v i b t T F v i 5 F v i F dd i d c f t f s

7 5 Ts M M M M M M 5 59 w c if d is v t b c if t c M M M c if M t f M 5 Tis ipis 5 si t cic cvti ppt f t DFT iv i Tb 5 w t d } { DFT } { DFT c } { DF w T b si t cic cvti ppt f t DFT iv i Tb 5 w t { } } DFT{ c si t cic cvti ppt f t DFT iv i Tb 5 w t } { DF T t f s 5

8 5 { } DFT c { } v f dd f DFT } DFT{ b DFT } DFT{ v V c c si t cic fqc-sifti ppt f t DFT iv i Tb 5 w t { } DFT } DFT{ 5 F t cic fqc-sifti ppt f t DFT iv i Tb 5 w t d c } { IDFT } { IDFT } } IDFT{ } IDFT{ w β α β α β α b si t cic ti-sifti ppt f t DFT iv i Tb 5 w t } IDFT{ c si t dti ppt f t DFT iv i Tb 5 w t } IDFT{ 5 b t f s

9 s 55 * b wic is c wic is 5 Rpci b * * i t sti w bti Ts * * * * } * DFT{ } * DFT{ b } * { } R{ Ti t DFT f bt sids d si t sts f Pt w t { } { } * } R{ DFT c } * { } I{ Ts { } { } * } I{ DFT d } * { cs si t iit ppt d sts f Pt b w t } R{ } * { } DFT{ cs } * { c si t iit ppt d sts f Pt b w t } I{ } * { } DFT{ c t f s

10 5 Sic f sqc * ti t DFT f bt sids w t * Tis ipis R{ } I{ } R{ } I{ Cpi d ii pts w t R{ } R{ } d I{ } I{ As } R{ } I{ } R{ } I{ } d { } t I{ } t R{ } I{ R{ } } { } 5 9 { 5 5 } Ts is cic v sqc d c it s -vd 9-pit DFT b 9 { 5 5} Ts is cic dd sqc d c it s ii-vd 9-pit DFT c 9 { 5 5} wic is it q t q t Ts s cp-vd 9-pit DFT d 9 { 5 5 5} Ts is cic v sqc d c it s -vd 9-pit DFT 59 5 c { } b { c 5 5 α β Tf 5 5 } 9 9 α β α β 5} b Tf t f s

11 c Tf < if if Ts if if < d Tf T scd s is z if s if d c if if > Tf 5 * * * * 5 R { DFT } t f s 9

12 * 5 t is t M pit DFT f t sqc btid f b ppdi it wit M zs Ts t t- M sqc is iv b t M M M Ti t M DFT f bt sids w M M M b 5 9 c 5 d T ivs DFT f is 9 Ts 9 F Psv s ti * *5 * * 9 *9 * * * * * 5 b c T d t Ts DFT{ } F Psv s ti t f s

13 5 w c C C 5 5 C 5 5 C 5 5 C 5 5 C 5 5 C C 5 5 iwis c Cpi C C C C C C 5 5 C C wit w bsv tt 9 5 Sic is t-9 sqc its DFT stisfis * Tf * 9 * 9 t f s

14 * 9 * 9 5 *5 9 * 5 5 * 9 * 5 * 9 * 5 * 9 *5 * 9 * * 9 * 59 Sic t DFT is -vd is cic v sqc i Tf Sic t DFT is ii-vd is cic dd sqc i Tf * * 9 * 9 * * 5 * 5 5 * * 9 9 * * 9 5 Cpi ts DFT sps wit t DFT sps iv bv w ccd 5 b dc v f { } c R{ 9 } R{ 5 } R{ } R{ } t f s

15 d 9 5 * * α * * 5 * 5 *5 5 * * * * * γ γ * b Sic is -vd sqc f t d st b Ts α d β As d v t s ii pt ε 9 d iwis s d 9 v t s pt δ d 9 Sii s d 5 v t s ii pt γ d 5 5 Fi s c dc v f is d R{ 5 R{ 9 } R{ } 5 } R{ } R{ } 5 9 Tf 9 Ts * 9 *9 c 5 * * * 95 * 5 t f s

16 9 9 w Tf 9 Ts * 9 * 9 * * b C C C C c 5 5 C 9 C Tf 9 C C t f s

17 9 9 C C C C d 5 5 C 5 5 C 5 5 C 5 5 C 5 5 C C 5 5 d t sw t d O O O O 5 Ts O b Rpci b t f s 5

18 i t fist s t it-d sid d b i t scd s t it-d sid w bti 5 Tf w c Sic sw Ts 59 v c } * { V V is t pit DFT f d } * { V V is t pit DFT f V * V Tf T IDFT f btid si MATAB is iv b V Cs t 5-5i - i -5 5i -9 - i Cs 5 t 5 5i 55 - i 5 5i -9 i t f s

19 T s st is btid b cpti t IDFT f d t IDFT f si MATAB d t fi 55 v Tf i 5 V V V V 5 V Ts 5 * V Tf V V * d 5 5 * V V 55 t Ts } IDFT{ P p p p p p d d d d Tf t Ts } IDFT{ P p p p p p d d d d Tf d w ˆ M t f s

20 ˆ M M ˆ M M M M M M M M Ts ˆ is btid b sifti i tips f M d ddi t siftd cpis Sic t w sqc is btid b sifti i tips f c t cv t ii sqc t csctiv sps i t f v f Tis wd b t if t siftd cpis f did t vp wit c t tt is if d if M 55 Tf c 9 M > si t sts f Pb 55 Sic M d d c is cvb f I fct { 5 } d is iv b t fist 9 sps f b c si t sts f Pb 55 Sic M d 9 M < d c is t cvb f I fct t T F { } Tf F{ } F{ F { }} F{ F{ }} c F{ F{ F{ F { }}}} t f s

21 55 O* O w f 5 Ts f 5 55 Ovp-dd td: Sic t ips sps is f t d t DFT siz t b sd is c t b f dt sps qid f c cvti wi b 9 9 Ts t tt b f DFTs qid f t t- dt sqc is 9 As t DFT f t ips sps ds t b cptd 9 c c t tt b f DFTs sd 9 T tt b f IDFTs sd 9 b Ovp-sv td: I tis cs sic t fist 9 pits st w d t pd t dt sqc wit 9 zs f tt t f 9 Ai c cvti wi st i 9 9 cct vs Ts t tt b f DFTs qid f t dt 9 5 Ai DFT is qid f t ips 9 sps T tt b f DFTs sd 5 T tt b f IDFTs sd 5 K 55 sw w F t T b Sic f stc f is ts s sw bw d Sic d stic sqcs it fws tt d stic sqcs d d tistic sqcs w csid t t sqc Its ct sqc is iv b * t w bsv tt t f s 9

22 Its ct sqc is iv b * B ddi t st sqcs w t * * F Tb 5 if t } DFT{ } * DFT{ * d } * DFT{ * } DFT{ Ts * * } DFT{ Sii * * c * * } DFT{ iwis * * Ts * * } DFT{ Fi * * c * * } DFT{ 5 b DFT b b b b b b b s f Eq 5 twis if b 5 Ts β α β α DCT cs cs cs cs DCT DCT β α β α t f s

23 b * DCT cs * cs DCT Tf * * DCT DCT c t tt twis d if if cs cs w cs cs * * DCT DCT α α Ts cs cs * DCT DCT α α si t tit ppt tid i w t DCT α 5 T ti is t if w is t idtit ti d is cstt w I c T I c T c t ti is t d its ws v t s - b t w bsv wic sws tt t ws f t bt d t v t s -s T t f s

24 5 w I t Ts I I I I I t t t t t t t t t t I I I I I b F Eq 5 t t t t t t t s f Eq 5 t 5 si cs DT w DT si cs si cs si cs Tf DT si cs si cs si cs It c b sw tt twis if if if cs cs twis if if if si si si cs cs si c si cs DT t f s

25 55 DT si cs si cs si cs Rpci b i t fist s d b i t scd s w t DT si cs si cs si cs si cs cs si cs si si cs DT DT b T pit DT f is DT c DT DT - si cs si cs si t tit ppt w bsv tt t bv pdct is q t if d is q t z if c DT 5 cs d si Tf DT t f s

26 5 O Ts DT cs si cs si F t sts f Pb 55 w v DT DT DT cs cs DT DT si si DT DT DT DT DT DT DT DT DT DT DT DT 5 DCFT β β αβ αβ αβ αβ α β αβ δ αβ αβ δ δ αβ αβ αβ αβ δ δ If w qi t t fwi cditis st b stisfid: α β αβ 5- d α β αβ 5- b t α α Svi f β d β i Eqs 5- d 5- w iv t α α β β α α α α T t ivs DCFT is iv b α α DCFT α α c If α * α α αi wit α d α i t pssi f DCFT dcs t t f s

27 si cs i DCFT α α wic is T ivs DCFT is t iv b si cs DCFT i α α α α d It c b si sw tt t disct t tsf DT f Eq 59 is spci cs f t DCFT wit i α α 59 d b F t stct f d it c b s tt d c Tf c T * T T T T w is t i t bit i t bi pstti f i i i b b T b i M5 ipt't v f '; -:; s*; w :*pi55:*pi; fqz w; dft fft; ::*; ptwpibs**bsdft''; b'\\pi'b'apitd'; t f s 5

28 b Rpc t sttt -:; wit :; s*; wit s; t sttt ::*; wit :; d t sttt ptwpibs**bsdft''; wit ptwpibs*bsdft'';i t p f Pt c Add t sttt bs; bw t sttt s*; i t p f Pt d Rpc t sttt s*; wit s* bs; i t p f Pt Rpc t sttt s*; wit cspi**; i t p f Pt T pts td f sw bw w t cics dt t DFT sps b 5 Apitd Apitd c 5 5 d Apitd Apitd t f s

29 Apitd M5 T cd fts t b sd s fws: fft*fft; iift; O T tpt td si t bv cd fts is b w 5O v T tpt td si t bv cd fts is w Cs t 5i -9 i i Cs t 5 9 i - i c 5O T tpt td si t bv cd fts is M5 ; % sqc t 5; :-; p-*; fft; % Ppt fftc; c :-:; % Vif % Ppt c :-:; fft; t f s

30 % Vif c % Ppt ; fft; 5*c :-:; % Vif % Ppt *i; fft; 5*-c :-:; % Vif % Ppt 5 5 5*c :-:; 5 fft5; % Vif 5 % Ppt 5*-c :-:; fft; % Vif *i M5 ; :-; 5; p-*; fft; % Ppt p 5* :-:; p 5*- :-:; p fftp; p fftp; % Vif p d p *i % Ppt :-:; % Vif c; % d i -i % bs bs d - M55 ; % is t f t sqcs 5; :-; p-*; cspi*; fft; fft; % Ppt t f s

31 p5; bt5; p*bt*; fft; % Vif p*bt* % Ppt ; % is t t f sift : :; fft; % Vif p-** % Ppt ; p-**pi***; fft; : :; % Vif % Ppt ; fft; * :-:; % Tis fs * d- % Vif ; % Ppt 5 % T cct cic cvti btw % d s q :-:; T tpitz'; 5 T*'; 5 fft5'; % Vif 5 * % Ppt *; fft; :-:; T tpitz' ; % ' is t ct tsps *T*'; % Vif ' M5 ipt'tp i fist sqc '; ipt'tp i scd sqc '; i*; F fft; Fc cf; t; F zs-'; f :-; F Fcd-; t f s 9

32 d F Fc F; F F F; F F - F; M ; F fft; :9; F p-i**pi*5*f; tpt F F sf sf;; disptpt dispsbsf*bsf T tpt dt td b tis p is --i --i M5 -i* -i* i* -i* i 5; :; Fcd-; F F:; ifftf; :; pi**pi**; tpt s s; disptpt disp* T tpt dt td b tis p is i M59 :55; **p-*; pt;is 55 ; b'';b'apitd'; tit'oii si'; ps z zs5 s5 zs5; *d5-; F z*fft; iv ifftf; s iv; pts;is 55 - ; b'';b'apitd'; tit'is cptd si'; ps zc s5 zs5 s5; SF zc*ffts; ifftsf; pt;is 55 ; b'';b'apitd'; t f s

33 tit'si ft Fi-di fiti'; Apitd Oii si Apitd is cptd si Si ft Fi-di fiti Apitd M5 fcti vpsv t; %t f sqc M t; %t f st sqc fps; if M > %E cditi disp''; d %c tp cim; %Fid t f cic cvti ^tp; %z pddi t st sqc if > M f i M: i ; d d ci--m; wi *-M < if*-m < &*-M > f : *-M; d d if*-m<&*-m> %dfw dstt t f s

34 f : ; d f *-M:*-M if > -*-M ; d d d if*-m > %vfw dstt f : ; d f :-*-M *-M; d d w cicv; %cic cvti si DFT f i :M- i ; d f i M: i wi; d f M: if*-m < M if*-m > O*-M ; d d d ; d %disp'cvti si Ovp Sv:'; O; fcti cicv t; t; if 'Sqcs f q ts' d fft; fft; _RES *; ifft_res; T MATAB p f pfi cvti si t vp-sv td is ; R 5; t f s

35 d dr - 5; ::R-; s **9^; s d; % ; vpsv; :R-; pt'-':r'b--'; b'ti id ';b'apitd'; d'-' 's''b--'''; T tpt pt td b t bv p is sw bw: s Apitd 5 Ti id t f s

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