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1 TRANSFORMS & PDE MA65 Quio Bk wih Awr UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART-A. Oi pri diffri quio imiig rirr co d from z A.U M/Ju Souio: Giv z Diff Pri w.r. d p > - p/ q > q/ Suiuig i, w g p q 4z. Form h pri diffri quio imiig h rirr co & from z A.U Apri/M Souio: Giv z Diff pri w.r. d z z p, q, p, q Suiuig i w g pq 4z. Fid h PDE of p hvig qu ircp o h d i. Souio: Th quio of p i z... Diff. w.r.. d pri rpciv p... q... Sovig & w g, pq 4. Oi h pri diffri quio imiig rirr co d from z. Souio: z - Diffriig pri w.r. d w g zp zp - zq zq - Suiuig & i w g z p z q z i.., z p q S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

2 5. Eimi h rirr fucio f from z f d form h PDE. Nov/Dc Souio: z ' p f... z q f '... From & p q p q 6. Form h pri diffri quio imiig h rirr co & from z Souio: Giv z Diff pri w.r. d z z p, q, Suiuig i w g z pq. 7. Form h pri diffri quio imiig f from z f og. Souio: L z f og - Diffri pri w.r. d z ' p f og z ' q f og - - Eimiig ' f from & p p q p q q 8. From h pri diffri quio imiig g from g z z Souio: W kow h if gu, v h u f v z f z Diffriig pri w.r. d W g,. S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

3 ' zp f z p ' zq f z q.. Divid & zp p zq q q zp zpq p zq zpq z p z q 9. Fid h comp igr of pq pq Souio: Giv pq pq Thi i of h p fp,q L z c h ouio for Pri Diff w.r. d w g p d q > > / Thrfor z / c i h comp ouio.. Fid h igur igr of z p q pq Souio: Thi of Ciru p d h comp ouio i z Diff w.r. d w g z, z! & i igur igr.. Fid h comp igr of pq Souio: Giv p q Thi i of h p fp,q L z c h ouio for Pri Diff w.r. d w g p d q > > Thrfor z c h comp ouio.. Sov p q qz. Souio: p q qz - Thi quio i of h form f z, p, q z g h ouio L u z g u dz dz p q du du rduc o dz dz z dz z dz z dz z dz du du du du du du du z S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

4 Igrig og z u i.., og z i h comp ouio.. Sov p og q Souio: Th quio c rducd o h p F,p F,q! p og q! p og q/ k k! p k ; q 4. Sov! z pd qd! z k d k d k z k c k ' D D D z Nov/Dc Souio: ' D D D z Hr c, c -, m, m Gr ouio z f f 5. Sov p q Souio:. Thi i of h form f p, q. Th ouio i giv z c whr Th comp ouio i 6. Sov Souio: z c DD DD z. Thi i of h form DmD... C DmD C DmD C Z Hc m c m c c Hc h C.F. i z φ φ P.I. DD DD Hc, h comp ouio i φ φ 7. Sov Souio: D DD D Z. S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog Z

5 Suiuig D m, & D Th Auiir quio i m - m m,, - Compimr fucio i φ φ φ i.., z φ φ φ 8. Fid h pricur igr of D Souio: DD' 5D' P.I D DD' 5D' Rpc D d ' D,w g D D' z z z z z 9. Fid h gr ouio of 4 8 Souio: Auiir Equio i m m 4m8! m,, f f Gr ouio z. Fid h Pricur igr of Souio: f D D ' DD' D D ' z P.I D D ' Rpc D d D,w g P.I 4 PART - B. Fid h gr ouio of - z p - z q z - Souio: Th quio i of h form PpQqR whr P - z,q - z,r z - [A.U Apr/ M 8 ] Th uiir quio i d z d dz z z Uig h muipir,, z, w g ch rio i qu o d d zdz z z z i.. d d zdz Igrig w g z c S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog

6 . uig h ohr of muipir,, z d d dz z w g ch of h ov rio i qu o z z i.. d d dz z Igrig w g og z og c Th ouio of h ir p.d. i ϕ u, v i.. ϕ z, z Sov D DD' 6D ' z Souio: Th uiir quio i m m 6 m, C.F φ φ P.I D DD' 6D' D D' D DD' 6D' D 6 D D ' D D' D 6 D D ' Th gr Souio i φ φ Sov z p z q z Thi i Lgrg ir quio of h form Pp Qq R Th uidir quio r d d dz z z z - d d d dz d dz z z z z z z i.., z d d z d z z z z z S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog

7 i.., d d z d z z z z z z - i.., d d z d z z z - og og z og Tkig h fir wo rio, d igrig z - 4 Simir kig h wo rio of w g, z z - 5 Bu d z r o idpd ouio for z z z rciproc of h cod ouio. giv z z which i h Thrfor ouio giv 4 d 5 r o idpd. Hc w hv o rch for ohr idpd ouio. d d zdz Uig muipir,, z i quio ch rio i z z d d dz Uig muipir,, ch rio i z zz d d zdz d d dz z z z zz d z d z z z zz d z z d z Hc Igrig z z k z z k z z z z k z zz S. Joph Cog of Egirig & Pg No. 7 ISO 9:8 S. Joph Iiu of Tchoog

8 i.., z z h gr ouio i φ z z, z Sov Souio: p q z Th giv quio c wri zp zq... Thi i of h p of f, z m pf, z m q Pu z Z d P zp, Q zq P zp & zq Equio c wri P Q 4 Q [A.U Nov/ Dc 7]! P - 4 Q - 4 Th quio i of h form f,p f,q!! P - 4 Q ; P d W kow h dz Pd Qd P Q - 4 dz d d Igrig w g Z ih 4 d Q coh. ih z coh Fid h igur ouio of h quio z p q p q Souio: Th quio i of h form z pqfp,q For comp igr rpc p d q,w g! Comp igr z z Diff w.r. d w g z z Sovig w g...4 &...5 S. Joph Cog of Egirig & Pg No. 8 ISO 9:8 S. Joph Iiu of Tchoog

9 Th quio 4 c wri...6 / / Suiuig 6 i 5 w g...7 d...8 Suiuig 7 & 8 i w g 6z 7 which i igur igr. 4. Sov D 5DD' D' z 5i Souio: Th uiir quio i m 5m m, C.F φ φ P.I 5i D 5DD' D' 5i D 5DD' D' D 5DD' D' 5 i i 4D 5D' 9 4 D 5D' 4D 5D' 5 i 4D 5D' 4D 5D' 9 4 D 5D' 5 i 6D 5D' Th Gr Souio i [ 8 co 5 co ] 9 5 φ 8 co 5 co 9 Sov z z p z q z. φ [ ] Souio: 9 Thi i Lgrg ir quio of h form Pp Qq R Th uidir quio r d d dz z z z z Tkig,, z muipir, w hv ch frcio d d zdz d d zdz S. Joph Cog of Egirig & Pg No. 9 ISO 9:8 S. Joph Iiu of Tchoog

10 Igrig i.., z c z C - Agi, kig h wo mmr, w hv d dz z z i.., d dz z z z d z dz d zd dz zdz d zd dz zdz d d z zdz z Igrig w g z z C - From & h gr ouio i z z z φ, Sov D D' D D' z 7 Nov/ Dc 9 Souio: Th giv quio i D D' D D' z Thi c fcorizd D - D'-D--D'- z I i of h form D m D' c D md' c whr m c,m - c. C.F f f 7 P.I D - D'D D'- D D' D D' 7 D D' D D' D D' DD' 7 D D' 9 7 D D ' D D ' D D' 7 D D' D 7 D' 7 9 DD' S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

11 S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog 9 7 D ' D 9 7 D D' D 9 7 D D' D ' ' D D D D D 9 7 D D ' 9 7 D 9 7 D D d D D d f f z 5. Fid h igur ouio of h quio q pq q p p z Souio: Th quio i of h form z p q fp,q For comp igr rpc p d q,w g z! Comp igr z Diff w.r. d w g z z Sovig w g...4 uiuig i w g...5 Suiuig 5 i 4 w g...6 Suiuig 5 & 6 i w g - - z - which i igur igr

12 Sov D DD' D ' D D ' z 4 Souio: D DD' D ' D D ' z D D' D D' z 4 D D' D D' z 4 C.F φ φ 4 4 P.I D DD' D' D D' 4 D DD' D' D D' D D D' DD ' D ' D D ' UNIT II FOURIER SERIES PART-A. S Dirich codiio for h ic of Fourir ri of f i h irv. Nov./Dc.,Rg 8 A fucio f c pdd Fourir ri i h irv ifid. i f i priodic, ig vud d fii i ii f h fii umr of fii dicoiuii d o ifii dicoiui i iii f h fii umr of mim d miim i. Do po Fourir pio?. do o po Fourir pio cu h fucio dicoiui h poi. If 4 co,dduch... i [, ] 6 Pu h 4 co if h foowig codiio r 4. Th coi ri for f i i < < i giv i co co Dduc h. h h ifii S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

13 Giv i co co Pu, poi of coiui > i co co i.. co i.. 5. Fid h vu of h Fourir ri for f. Nov./Dc.,Rg. f f i poi of dicoiui, vu of Fourir ri of f i. f- im f h im h. h h f. f i.5 6. Fid h co rm of h Fourir ri for h fucio f, < < Sic f i v fucio i < < f d d Co rm 7. To which vu, h hf rg i ri corrpodig o f prd i h irv, 5 covrg 5? A h d poi hf rg i ri covrg o zro. Th hf rg i ri corrpodig o f covrg o zro Wh ou m Hrmoic i? Th proc of fidig h hrmoic i h Fourir ri pio of fucio umric i kow hrmoic i. 9. Fid h Fourir co for i i < <, wh prd Fourir ri. Souio: f i f i i f Hr f i v fucio. Fid h roo m qur vu of h fucio f i. RMS vu RMS vu [ f ] d. Oi h co rm of h Fourir coi ri of f i,6 uig h foowig. : 4 5 : o M vu of Σ/6 4 Co rm 7 Fid h vu of i h coi ri pio of f K i h irv, f co d K co d 5 S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

14 . If f i prd Fourir ri i h irv -, o which vu hi ri covrg? f f 6 Sic i d poi h f covrg o 4 4. S Prv horm i h irv c,c. If h Fourir ri corrpodig o f covrg uiform o f i c,c c h [ f ] d c 5. Wri dow h comp form of Fourir Sri i c,c i f c whr c 6. If f i < <, fid. Souio: co d c i c f co i { }. 7. Drmi i h Fourir ri pio of f i < <. f i d i d co i 8. Fid h co rm i h prio of Souio: f i < <. Hc how h S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog d co Fourir ri i h irv,. co i co d d Co rm 9. Giv h prio for h Fourir ri coffici for h fucio f dfid i. Souio: f i d. Drmi h vu of & i h Fourir ri pio of f i < < Souio: f f f f i odd fucio. If PART B

15 i.... ii Souio: W kow h f co i d 4 4 d 4 6 α co d 4 co d i co i 4 co i d 4 co i co 4 S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog

16 4 f co. co co... Pu f... i p of dicoiui. f > Pu f i Hr i p of coiui. f. > S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog

17 Fid h Fourir ri for i. Souio: Giv Sic i v fucio S. Joph Cog of Egirig & S. Joph Iiu of Tchoog Pg No. 7 ISO 9:8

18 Suiu i quio w g [ ] f co. Oi h Fourir ri o rpr h fucio f i < < d dduc h Souio: Giv f f f Th giv fucio i v fucio. Hc f co d co d { } i co co if i v 4 if i odd 4 f co, 4 co co... Pu S. Joph Cog of Egirig & Pg No. 8 ISO 9:8 S. Joph Iiu of Tchoog

19 f Hr i p of coiui f Fid h fir wo hrmoic of h Fourir ri of f giv 4 5 f Souio: Hr h gh of h i v i 6, co i co i f co i co i S. Joph Cog of Egirig & Pg No. 9 ISO 9:8 S. Joph Iiu of Tchoog

20 co 8. 6 i co. 6 i f 8.co.co.5i.. Fid h Fourir ri for h fucio Souio: Giv Thrfor i odd fucio. Thrfor Suiu i w g Fid h fir wo hrmoic of h Fourir ri of f. Giv 4 5 f S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

21 Souio: Th vu of i rpiio of h fir; o h fir i vu wi ud Th vu of co, co, i, i ud f co i co i co co.9,.7, i i.7, Oi h Fourir ri for of priod d dfid foow Hc dduc h Souio: Giv Th Fourir ri i S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

22 Dducio i Pu Dducio ii Pu Epd coi ri i S. Joph Cog of Egirig & S. Joph Iiu of Tchoog d dduc h vu of Pg No. ISO 9:8

23 Souio: Giv Th coi ri i if i v 4 if i odd Suiuig i quio w g Dducio i B Prv idi S. Joph Cog of Egirig & S. Joph Iiu of Tchoog Pg No. ISO 9:8

24 Dducio ii 4 S S 6 96 Fid h comp form of h Fourir ri of i Souio: Th comp form of h Fourir ri i i giv Whr Hc com S. Joph Cog of Egirig & S. Joph Iiu of Tchoog Pg No. 4 ISO 9:8

25 Fid h hf rg Fourir i ri for f i h irv, d dduc h... 5 Souio: Th i ri i f i f i d i d co i co 4 f Dducio: [ ] 8,,,... 8 i if if i v i odd Pu i poi of coiui 8 f UNIT- III - APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS PART A. S wo umpio md i h drivio of h o dimio wv quio. i. Th rig i prfc ic d do o offr ric o dig. ii. Dfcio d op d/d vr poi of h rig r m, o h hir highr powr m gcd. S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog

26 Wri dow h pri diffri quio govrig o dimio wv quio? C whr, i h dipcm of h rig.. I h wv quio C, wh do C d for? C T/m, whr T i h io d m i h m of h rig. 4. Wri poi ouio of h rvr virio of h rig i o dimio. i, A p B p C p D p ii, A co p Bi p C co p i p. iii, A B C D 5. Wri dow h ppropri ouio of h virio of rig quio. How i i cho? Th ppropri ouio of h virio of rig quio i, A co p B i p C co p D i p. Wh w d wih h virio of ic rig, rprig h dipcm of h rig poi, i mu priodic i. Hc h ov ouio which coi of priodic fucio i i h propr ouio of h prom o virio of rig. 6. A igh rchd rig wih fid d poi d i iii r i i quiirium poiio. If i i virig givig ch poi voci -, wri dow h oudr d iii codiio of h ov prom., d, for,, d λ 7. S Fourir w of h coducio. for Th r which h fow hrough r i joi proporio o h r d o h mprur grdi orm o h r. 8. Wri dow h pri diffri quio h rpr o dimio h quio. u u α, whr α k ρc 9. Wh r h vriou poi ouio of o dimio diffuio quio? i u, A p B p C α p α p ii u, A co p B i p C iii u, A B. p p α p Wh c u, A B C h corrc ouio i ovig o dimio h quio? A, u. I i o poi.. I d codiio driv h ouio of o dimio h fow quio. S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog

27 u u Th o dimio h fow quio i α. u u u I d codiio,. A u A B.. A iud rod of gh 6 cm h i d A d B miid d rpciv. Fid h d mprur i h rod. Nov. Sd h quio i du d Giv codiio r u u Th ouio i u A B.. Suiuig i w g u, which i h d mprur i h rod.. Wri dow h pri diffri quio h rpr vri h fow i wo dimio? Dduc h quio of d h fow i wo dimio. u u u Th h fow quio i wo dimio Cri co-ordi m i α Wh d codiio prvi i h p h mprur poi of h p do o dpd o u, u dpd o d o i... Thu d mprur diriuio i wo u u dimio p i Lpc quio. 4. Wri wo ouio of h Lpc quio oid h mhod of prio of vri ivovig poi rm i d. u, A p B p C co p i p & u, A co p Bi p C p - p 5. Giv h oudr codiio o qur or rcgur p, how wi ou idif h propr ouio? If h o-zro mprur i prcrid ihr o or o vric dg, h propr ouio wi u, A p B p C co p i p If h o-zro mprur i prcrid ihr o or horizo dg, h propr ouio wi u, A co p Bi p C p - p 6. A ifii og p i oudd wo pr dg d d righ g o hm. Th rh of h dg i d i i miid co mprur u poi d h ohr dg r kp zro mprur. Formu h oudr vu prom o drmi h d mprur. u ujc o u,, u, for u, d u, u for 7. A p i oudd h i,,, d. I fc r iud. Th dg coicidig wih -i i kp. Th dg coicidig wih -i i kp. Th ohr wo S. Joph Cog of Egirig & Pg No. 7 ISO 9:8 S. Joph Iiu of Tchoog

28 dg r kp. Wri h oudr codiio h r dd for ovig wo dimio h fow quio. Nov. Boudr codiio r # u, C; < < i ii # u, C;< <. ii. iv # u, 5 C; < <. # u, C; < <. 8. Th oudr vu prom govrig h d mprur diriuio i f hi u u p i giv, u,, u,, u, d u, 4i. Fid C wh h mo gr ouio i u, c i ih. i i u, c i ih 4i 4 4 c i ih c i ih c i ih... i i c ih, c ih, c c ih ih c c c c Dfi hrm iud d. Th d which h mprur grdi i zro i cd hrm iud d. i. hr i o h fow hrow h d.. Wh h h fow i cd wo dimio? Wh h h fow i og curv id of og righ i, h curv ig i pr p, h h fow i cd wo dimio. PART B. A uiform rig i rchd d fd o wo poi ' ' pr. Moio i rd dipcig h rig io h form of h curv k d h rig i from hi poiio im. Fid h dipcm of h poi of h rig dic from o d im. Souio: O dimio wv quio i Boudr codiio r i, ii, iii, iv, f k S. Joph Cog of Egirig & Pg No. 8 ISO 9:8 S. Joph Iiu of Tchoog

29 Th corrc ouio i Aco p Bi p C co p Di p Pu d pp h oudr codiio i i, A B co p Di p A. Equio com, Bi p C co p Di p Pu d pp h oudr codiio ii i, Bi p C co p Di p i p p p Equio com, Bi C co D i Diffri Pri w. r. o, B i C i D co 4 Pu d pp h oudr codiio iii i 4, B i D D Equio com, Bi C co B i co whr B BC B h uprpoiio pricip, h mo gr ouio i, B i co 5 Pu d pp h oudr codiio iv i 5 S. Joph Cog of Egirig & Pg No. 9 ISO 9:8 S. Joph Iiu of Tchoog

30 S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog. i, f B 6 B hf rg Fourir i ri i f 7 B comprig 6 d 7,w g [ ] [ ] B odd i if k v i if k k k k d k d f co. co i co i i Th quio 5 com,, 5 co i 8, k. If rig of gh i iii r i i quiirium poiio d ch of i poi i giv voci v uch h < < < < c c v ; ;, fid h dipcm im. Souio: O dimio wv quio Th oudr codiio r B

31 i, ii, iii, iv c;, c ; < < < < Th corrc ouio i, Aco p Bi p C co p Di p Pu d pp h oudr codiio i i, A B co p Di p A. Equio com, Bi p C co p Di p Pu d pp h oudr codiio ii i, Bi p C co p Di p i p p p Equio com, Bi C co D i Pu d pp h oudr codiio iii i, B i C C C Equio com, Bi Di B i i whr B BD B h uprpoiio pricip, h mo gr ouio i, B i i 4 Diffri Pri 4 w.r.o S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

32 S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog, B co i 5 App h oudr codiio iv i 5, ; ; i f c c B < < < < 6 B hf rg Fourir i ri i f 7 B comprig 6 d 7, w g B B c d d c d f i co i co i i i i 4 i co i co c c i 4 i 4 c c B Th quio 4 com

33 4c, i i i. Th d A d B of rod cm og hv hir mprur kp # C d8 # C, ui d codiio prvi. Th mprur of h d B i udd rducd o 6 # C d h of A i icrd o 4 # C. Fid h d mprur diriuio i h rod fr im. Souio: u u O dimio h quio I h d codiio h ouio i u Th oudr codiio r i u ii u 8 App i i, u Th h quio com u App ii i u Th h quio com u Now coidr h ud codiio. I ud h corrc ouio i u, p Aco p B i p C 4 Th oudr codiio r iii u, 4 iv u, 6 v 5 u, u Sic w hv o zro oudr codiio, w wri h mprur diriuio fucio u, u u, 5 u, u, u S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog

34 To fid u Th ouio i u 6 Th oudr codiio r u 6, u 4 4, u u h quio 6 com u 4 7 To fid, u Giv h oudr codiio r vi u vii u viii u, u, u, u, u, u, u I ud, h corrc ouio i u, p A co p B i p C 8 App h oudr codiio vi i 8 u, AC p hr A h quio 8 com u, p B i p C 9 App h oudr codiio vii i 9 S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog

35 S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog i, p p BC u p p i h quio 9 com C B u i, B BC u i i, whr B BC B h upr poiio pricip, h mo gr ouio i i, B u App h oudr codiio viii i. i, f B u Hf rg Fourir i ri of f i i f From h quio & w g B. co i co i i d d f

36 [ ] B h quio com u, [ ] i Th h rquird mprur diriuio fucio i [ ] u, 4 i 4. A rcgur p i oudd h i,,,. I urfc r iud. Th mprur og d r kp # C d h ohr # C. Fid h d mprur poi of h p. Souio: u u W kow h wo dimio h quio L h gh of h qur p Giv h oudr codiio r i u, ii u, iii u, iv u, W wri h mprur fucio u, u, u, A To fid u, Coidr h oudr codiio v u, vi u, vii u, viii u, Th ui ouio which ifig h giv oudr codiio i S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog

37 p p u, A B C co p Di p App h oudr codiio v i u, A B C co p Di p hr A B B A h h quio com u, A A p p A p p C co p Di p C co p Di p App h oudr codiio vi i hr C p p u, A C h h quio com u p p, A Di p App h oudr codiio vii i p p u, A Di p hr i p p p h h quio 4 com u, A D i ADih i B ih i whr B AD S. Joph Cog of Egirig & Pg No. 7 ISO 9:8 S. Joph Iiu of Tchoog

38 Th mo gr ouio i u B ih i 4 App h oudr codiio viii i 4 u, B ih i f 5 To fid B, pd f i hf rg i ri W kow h hf rg Fourir i ri of f i f i 6 From h quio 5 & 6 w g B ih B ih f i d i d co 4 [ ] co if i v if i odd S. Joph Cog of Egirig & Pg No. 8 ISO 9:8 S. Joph Iiu of Tchoog

39 B 4 if i odd ih h h quio 6 com u, To fid u,,, 5 4 ih i ih Coidr h oudr codiio i u, u, i u, ii u, d h ui ouio i hi c i p p u, E co p F i p G H 7 App h oudr codiio i i 7 hr E p p u, E F H h h quio 8 com p p u, F i p G G 8 App h oudr codiio i 8 u, F i p G H hr G H H G h h quio 9 com u, F i p G FG i p p p G p p 9 S. Joph Cog of Egirig & Pg No. 9 ISO 9:8 S. Joph Iiu of Tchoog

40 App h oudr codiio i i 9 p p u, FG i p hr i p p p h h quio com u, FG i FGih i C ih i whr B FG Th mo gr ouio i u, C ih i App h oudr codiio ii i u, C ih i f To fid C, pd f i hf rg i ri W kow h hf rg Fourir i ri of f i f d i From h quio & w g S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog

41 d d C ih C ih d f i d i d co 4 co [ ] if i v if i odd C 4 if i odd ih h h quio com u,,, 5 4 ih ih i Th h quio A com u, u, u,,, 5 4 ih ih i,, 5 4 ih ih i 5. A rcgur p wih iud urfc i cm wid d o og comprd o i widh h i m coidrd ifii i gh wihou iroducig pprci rror. Th mprur hor dg i giv, 5 u, 5 d h ohr hr dg r kp S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog

42 # C. Fid h d mprur poi i h p. Souio: u u W kow h wo dimio h quio Giv h oudr codiio r i u, ii u,, 5 iv u,, 5 iii u, Th ui ouio which ifig h giv oudr codiio i p p u, Aco p B i p C D App h oudr codiio i i p p D u, A C hr A Th h quio com p p C D u, B i p App h oudr codiio ii i p p u, Bi p C D hr i p p p Th h quio com u, B i C D App h oudr codiio iii i S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog

43 B i C D u, hr C Th h quio com u, B i B i Th mo gr ouio i u, B i 4 App h oudr codiio iv i 4 u, B i.,, 5 5 To fid B, pd f i Hf rg Fourir i ri. W kow h Hf rg Fourir Coi ri f i 5 From 4 d 5 w g B 5 5 f i d i d 5 i d S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog

44 co i 5 5 co i co i co i 8 i Th h quio 4 com u, 8 i i UNIT IV FOURIER TRANSFORMS PART - A. Wri h Fourir rform pir. [Nov/Dc, ] Th Fourir rform pir i dfid F [ f ] i f d F : F F f i F d f. S Fourir igr horm. [Nov/Dc ] If f i picwi coiuou, coiuou diffri d ou igr i, h i f f dd. ; for < Fid h Fourir rform of f ; for > Th Fourir rform of h fucio f i F [ f ] i f d. F[ f ] i i i. i d i i i i i i ii F i i 4. Fid f from h igr quio f co S. Joph Cog of Egirig & Pg No. 44 ISO 9:8 S. Joph Iiu of Tchoog

45 givh f cod, co d f Fc co d cod 5. Dfi Sf-rciproc Fourir rform d giv mp. [ Nov / Dc ] If f i h Fourir rform of f, h f i id o f-rciproc udr Fourir rform. F. 6. Fid h Fourir rform of, if [ Nov / Dc ] [ i F f ] d co ii d co d i i d co d 7. S Covouio horm i Fourir Trform. [Nov / Dc ] Th Fourir rform of h covouio of f d g i h produc of hir Fourir rform i.. F { f g } F{ f } F{ g } 8. S Prv idi o Fourir Trform. [Nov/Dc ] F d f d, whr F[ f ] F 9. Fid F { f } from Ff. d d B propr F[ f ] i F[ f ] F[ ] i F[ ] d d d. i i i d. Fid h Fourir i rform of FS [ f ] f id. FS i d Diffri oh id wih rpc o, whr >. S. Joph Cog of Egirig & Pg No. 45 ISO 9:8 S. Joph Iiu of Tchoog

46 d d F i d i d d d co d co d Now igrig oh id w. r. o, F S d. Fid h Fourir i rform of, >. Hc fid F. [ M / Ju ] F [ f ] f i d. F i d d W kow h propr FS [ f ] FC [ f ] d d. F d. Wri h Fourir i rform pir d Fourir Coi rform pir. Th Fourir i rform pir i dfid F [ f ] f d F i : F F [ ] f i F d Th Fourir coi rform pir i dfid Fc [ f ] f d Fc co : F Fc [ ] f co Fc d. Prov h Fc f F c, F c f co d, [ f ] S. Joph Cog of Egirig & Pg No. 46 ISO 9:8 S. Joph Iiu of Tchoog Pu, d d, d d wh, d, Fc f fco Fc f fco d F c. 4. S Prv idi i Fourir i d coi Trform. F c d f d & F d f d d

47 5. Fid h Fourir Si rform of i i F d d 6. If Fc f Fc d F f F Souio: [Nov/Dc, Nov / Dc 9], prov h F f i [ F F ] c [Nov/Dc, MA] Fc f i f i co d f [i i ] d f [i ] d f [i ] d Fc f i [ F F ]. 7. S d prov h chg of c propr of Fourir Trform. [Apr/M,M/Ju ] Sm: If F [f] F h F f F, Proof: F [ f ] i f d, If > Pu, d d d, d i d wh d F f f F. If < Pu, d d, d d wh d d F f f f i i F From & w g F f F, 8. ; < Fid h Fourir coi rform of < f., ohrwi d. --- S. Joph Cog of Egirig & Pg No. 47 ISO 9:8 S. Joph Iiu of Tchoog

48 F C f co d [ f ] F c [ f ] 9. Fid h Fourir Si Trform of f, > W kow h, F [ ] f i d. co d i co co i i i co [Apr/M, MA] F i d i co i If F i h Fourir rform of f, h how h F{ f } F [ Nov/Dc,,M/Ju ] F {f-} f i d f i d i i f d whr - F i F{f} PART - B Show h h Fourir rform of ; < i co f i ; > >. Hc i co i co dduc h d. Uig Prv Idi, how h 4 d. 5 Th Fourir rform of h fucio f i F[ f ] i f d. Giv i < < f d ohrwi. S. Joph Cog of Egirig & Pg No. 48 ISO 9:8 S. Joph Iiu of Tchoog

49 S. Joph Cog of Egirig & Pg No. 49 ISO 9:8 S. Joph Iiu of Tchoog [ ] [ ] co i i co i co i co i co i co F d d i d d i d f F i B ivr Fourir rform d F f i co co i i co i co co i i co co i co i d f d i d d i d f i Pu &, w g 4 co i. co i 4 d i d Rpc, w g 4 co i d. B Prv Idi,

50 S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog ] [, F f F whr d f d F d d d d d d d co i 8 co i 8 co i 8 co i. 5 d co i 5 d co i, d, wh, d d, Pu 5 d co i d co i Fid h Fourir i rform of whr >. [Nov/Dc, 8] i d f f F S Diffriig oh id w.r.o i co co co d df d d d df

51 Igrig w.r.o d f Pu, f c c. c c f Show h i f-rciproc wih rpc o h Fourir coi Trform. Th Fourir coi rform of h fucio F C f i [ f ] co co d i i R.Pof d R.Pof d i F C f co d. d i i R.Pof d R.Pof d i R.Pof d Pu i, h d d, d R.Pof d R.Pof R.Pof F [ ] C f i f-rciproc wih rpc o Fourir rform d R.P of i d ; if < 4 i Fid h Fourir rform of f. Hc Evu d ; if. 4 Th Fourir rform of h fucio Giv f i < < d ohrwi. f i F[ f ] i f d. S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog

52 F [ f ] F i d co d i i co co co ii d i d i B ivr Fourir rform f F d f f Pu, w g co co co d i i d co co d co i d co d co co co i i d i. i co d d. L, h d d d i d i B Prv idi F d d f d co d d co d d co d d S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog

53 co d d co d i co i [ ] d d L, h d d d d i 4 d i 4 d d Evu uig Fourir Coi Trform of d,, >. Th Fourir coi rform of h fucio f i FC [ f ] f cod. F C [ ] co d d F C [ ] co d L f d g B Covouio horm, F [ f ] F [ g ] d f g i.. d d c d d c d d d S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog

54 4 d d Rpc, w g d Fid h Fourir rform of f dfid, < f d hc fid h vu, > i of d i o prov h d. Th Fourir rform of h fucio Giv f i < < d ohrwi. f i F[ f ] i f d. F [ f ] i. d i i i i F i i i i i B ivr Fourir rform f F d f i i d i co d i i f co d Pu & w g i d B Prv idi i i d i i i co i i d i i co d i i d, Rpc, w g d. F d f d i i S. Joph Cog of Egirig & Pg No. 54 ISO 9:8 S. Joph Iiu of Tchoog

55 i i d d Rpc, w g i d [ ] i d d. i i Fid Fourir Si d Coi rform of d Fourir coi rform of d Th Fourir i rform of h fucio F [ ] i d S. Joph Cog of Egirig & Pg No. 55 ISO 9:8 S. Joph Iiu of Tchoog [] d i d d hc fid h Fourir i rform of. f i [ f ] Th Fourir coi rform of h fucio Fc [ ] co d Fc F d Fc d F S f i d. f i [ f ] co d 5 Fid h fucio f if i i rform i L f F d d F c f co d. [ Nov / Dc ] Th f i d Diffriig oh id wih rpc o w g df d df d co d co i co d

56 Igrig w.r. d f c pu, f from from, w g f c, f Epr, < f Fourir igr, hc vu, > i co d d fid h vu of i d W kow h h Fourir igr formu for f f co d d f co d i i i i co f d d. Thi i fourir igr rprio of f. From Pu, i co d f i co, < d f, > i d d UNIT V Z TRANSFORMS AND DIFFERENCE EQUATIONS PART - A. Wh i h Z- rform of dicr ui p fucio Souio: Dicr Ui p fucio i u,, <... z Z u z z z z z if z > S. Joph Cog of Egirig & Pg No. 56 ISO 9:8 S. Joph Iiu of Tchoog

57 . Fid Z. Souio: Z z z z z. Fid Z u... z 4z 8z z z z, z > z Souio: Z u u z u z z z z z z z z z z z z z z z z z if z > 4. Fid h Z- rform of ui impu fucio Souio: Ui impu fucio i, δ, δ Z δ z z. 5. If Z f U z h fid Z f Souio: Z f f z f / z U z/. 6. k If Z f U z, h how h Z f k z U z Souio: k k Z f k f k z z f k z k r k z f r z z U z S. Joph Cog of Egirig & Pg No. 57 ISO 9:8 S. Joph Iiu of Tchoog

58 f, h Z z U z dz. 7. If z f U z Souio: f f Z z f z dz, ic z z dz z f z dz z f z dz U z dz. 8. If Z f U z, h fid Z f Souio: Z f f z z f z d d d dz dz dz z f z z f z z U z. 9. Fid Z. Souio: z Z z Z z. Fid Z z z z Souio: L z z z U z Th U z A B z z z z z z, B z, A U z z z z S. Joph Cog of Egirig & Pg No. 58 ISO 9:8 S. Joph Iiu of Tchoog

59 . Fid Z. Souio: Z z z U z z z u z z { } z... z z z. z z z z z z z z. Fid Z-Trform of. Souio: Z z... z z z. Fid Z-Trform of Souio: z z og og og z z z. Z { } z z z z 4. Fid Z-Trform of Souio: Z.! Z z... z!!!! z z 5. Sov Souio:, giv h. S. Joph Cog of Egirig & Pg No. 59 ISO 9:8 S. Joph Iiu of Tchoog

60 { } { } Z Z z z, { } zy z Y z Y z Z z z Y z Y z z z z z z z z Y z z z z z z Z z z 6. Uig Covouio horm, vu Z z z Souio: W kow h Z z z z d Z z z Now Z z z. z Z z z z z m m. m m m /. m / 7. Fid Z{ i } Souio: { } { } z i T Z i Z i T z z z z co T z z T z T i T z 4T z T cot S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog

61 8. Fid Z{ i } Souio: Z{ i } i T z ziθ z i T Z i θ θ T z zco z z co T θ θ T 9. S iii d fi vu horm of Z - rform. Souio: Iii vu Thorm, h f f U z If Z f U z Fi vu Thorm Limi Limi z, h f z U z If Z f U z Limi. Form h diffrc quio from A. B. Souio: Limi z Giv A. B., A. B., 4 A. 9 B. Eimiio of A d B form h diffrc quio Sov Souio: PART - B 5 6 5,, uig Z-rform Tkig Z-rform o oh id z z z Y z z z Y z Y z S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog

62 z z 5z 6 Y z z 5 Y z A B C z z5 z 5z 6 z5 z z z5 z z A z z B z5 z C z5 z z 5, 6 A 6 z, C C / z, B B / Y z z 6 z5 z z Y z z z z 6 z5 z z 5. 6 S d Prov Scod Shifig Propr of Z-Trform. Sm: L Z u U z Proof d k k k k Z u u z z u z k k k > h Z u Z U z u u z u z... u z k k k z u z u z k k k... k z U z u uz uz uk z. Sov, giv h Souio:. Tkig z-rform o oh id [ ] Z Z Z S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog 4,, 8 Uig Z-rform.

63 z z z z z z Y z z Y z z z Y z Y z 48 4 Y z z z 8z 4z z 4z 4z Y z z z z z z z z z z z z Y z 4z 4 A B C z z z z z z 4z 4 A z z B z C z z, B B z 9c c /9 4/ z, 4 A C A 4 4/ 8/ 8 4 Y z z z z 8 4 Y z z z If U z z z 4, z fid u d u. Souio: u imi imi z z U z 4 z z z u z U z u z z z imi imi z z 4 S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog

64 im i u z U z u u z z i z z z z im z 4 im i u z U z u u z u z z i z z z z z im z imi z z z z 4z 6z 4z z 4 z z z im i z 4 z z z 8z.. Fid Z 8z covouio horm. z 4z Souio: Z Z z 4z z z 4 8z z z z Z. Z CT z z 4 4 r r r r r 4 r 4 r r 4... r r S. Joph Cog of Egirig & Pg No. 64 ISO 9:8 S. Joph Iiu of Tchoog

65 4 Fid Z Souio: Z{ } Z { } { } { } Z Z Z z z z 4z z z z z z 4 z z z 4z z z z z z z 6z Fid Z r co Souio: θ d hc dduc Z co iθ iθ z Z Z z iθ iθ z z iθ z Z z z z iθ iθ iθ iθ iθ z z θ z z coθ iiθ z zcoθ iziθ z z co coθ z iθ θ θ z z z co θ ii θ i z zco z zco quig r d imgir pr, S. Joph Cog of Egirig & Pg No. 65 ISO 9:8 S. Joph Iiu of Tchoog

66 Z co θ coθ z z z zcoθ rcoθ z z Z{ r co θ} Z{ co θ} z z r z rzcoθ r Z co [ Z{ co θ} ] co z z coθ z z z θ z z coθ θ z z co z Fid h ivr Z-Trform of z z z ridu mhod. Souio: z z F z z Z - z z Fz z Hr z i po of ordr d z z [ RES] L z z z L! dz z! z d dz z [ z z ] L [ z z ] [ ] d L [ z z ] z! dz S. Joph Cog of Egirig & Pg No. 66 ISO 9:8 S. Joph Iiu of Tchoog [ ] 5. Form h diffrc quio corrpodig o h fmi of curv. Souio:

67 from, u i u i. or i Fid Z z z z uig h mhod of pri frcio. Souio: L F z z z z F z A B C z z z z z z z z A z B z z C z A z B z z C z Puig z d z w g A d 4 c Equig h coffici of o oh id AB B A 4 F z 4 4 z z z z S. Joph Cog of Egirig & Pg No. 67 ISO 9:8 S. Joph Iiu of Tchoog

68 F z z z z 4 z 4 z z z z z Z F z Z Z Z 4 z 4 z z 4 4 S. Joph Cog of Egirig & Pg No. 68 ISO 9:8 S. Joph Iiu of Tchoog