Fourier Series and Parseval s Relation Çağatay Candan Dec. 22, 2013

Transcription

1 Fourir Sris nd Prsvl s Rlion Çğy Cndn Dc., 3 W sudy h m problm EE 3 M, Fll3- in som dil o illusr som conncions bwn Fourir sris, Prsvl s rlion nd RMS vlus. Q. ps h signl sin is h inpu o hlf-wv rcifir circui wih h following inpu-oupu rlionship, hlf., ohr Find h Fourir sris cofficins of hlf. Hin You my us Eulr s rlion o prss sin. sin Hlf-wv rcifid Full-wv rcifid sc. b h signl sin is h inpu o full-wv rcifir circui wih h following, inpu-oupu rlionship full., ohr Eprss h Fourir sris cofficins of full in rms of h Fourir sris cofficins found in pr. No You cn cll cofficins in pr,, nd solv pr b using s. Soluion h soluion is mor dild hn i nds o b. h signl is priodic wih. h Fourir sris cofficins cn b found hrough h rlion ω o d whr ωo rd/sc.

2 sin / / / / d d d d A his poin, i should b nod h ind of h squnc is n ingr nd hrfor,. Subsiuing ino h ls rlion, w cn g h following odd vn I pprs from h ls rlion h for ±, w hv. his is no ru, sinc h quion ** givn bov shows h h drivd vlus r only vlid whn ±. W nd o min h css of ± sprly. From h quion *, givn bov, w cn prss h cofficin s / / d d. Similrly, w cn show h. Givn ll, h FS cofficins cn b wrin s. / / ohr vn

3 Now, w cn wri h Fourir sris pnsion of h hlf-wv rcifid sinusoidl signl Hlf ωo ωo ωo ωo ωo sin ωo sin ωo, vn, vn, vn ωo cos ω o I is lwys usful nd fun o vrify h pnsion wih fw lins of Mlb cod linspc-.5,.5,; FSrms; ; w*pi/; ou /pi /*sinw*; %Firs fw rms for FSrms, %Rmining rms in h sris /pi/-^; ou ou **cosw**; nd; plo,sinw*,'-.'; hold ll plo,ou; hold off; lgnd'sin wv','hlf wv rcifid';

4 .5 Sin wv Hlf wv rcifid Figur h plo gnrd by h givn Mlb cod b If Hlf, h full wv rcifid sinusoid signl cn b prssd s follows. Full So, Hlf Hlf / / / Full vn W should b lil crful in his clculion, sinc h priod of Hlf is scond, whil h priod of full wv rcifid sin signl is /. For /, h corrsponding fundmnl frquncy is ω. Whn wriing Full vn, wih cofficins, for vn c,, ls

5 i is implicily ssumd h Full is lso priodic wih scond. I is indd ru h Full is priodic wih scond, bu his is no h fundmnl priod / nd is no h fundmnl frquncy ω. is qul o hlf of h fundmnl frquncy ; hrfor, c s r no h FS cofficins of Full. By subsiuing in, on cn sily find h FS cofficins b of Full Full vn b, b, [No h for ny priodic signl p wih fundmnl frquncy ω, on cn prss p s h sum of compl sinusoidls ll frquncis h r mulipls of ω / L, for L, 3, ; howvr, corrsponding cofficins, c, r ll zro whnvr m is n ingr. Abov mpl corrsponds o L. ml In ordr o prsrv h uniqunss of h FS rprsnion p, nd o sfly l bou h h hrmonic powr of p looing ; only h FS cofficins in h pnsion wih rspc o compl sinusoidls ll mulipls of frquncis ω, whr ω is h fundmnl priod r clld h FS cofficins of p.] In ordr o obin h FS pnsion in dil, s found in pr cn b subsiud ino. Full b cos I should b nod h h ls rlion is h convnionl pnsion of Full nd / corrsponds o h cofficin of h h hrmonic. W cn us Mlb o vrify our findings.

6 linspc-.5,.5,; FSrms; ou /pi; %DC rm for FSrms, %Rmining rms in h sris /pi/-*^; ou ou *cos*pi**; nd; plo,sinw*,'-.'; hold ll plo,ou; hold off; lgnd'sin wv','full wv rcifid';.8 Sin wv Full wv rcifid Figur h plo gnrd by h givn Mlb cod

7 Prsvl s Rlion, RMS Vlus nd Hrmonic Sris L s rmmbr h dfiniion for h RMS vlu of priodic wvform RMS [ ] In h rminology of lcricl nginring, is considrd o b h priodic wvform rprsning ihr currn or volg of n RΩ rsisor. hn, h vrg [ vrms ] powr dissipd ovr h rsisor bcoms, PAVG or P AVG R[ i RMS ]. R Som of h ypicl priodic wvforms uilizd in circui pplicions r givn in figur blow. his figur is providd o rfrsh our mmory on RMS clculions. d. Squr Wvform Sinousoidl Wvform Swooh Wvform Figur Som priodic wvforms ypiclly uilizd in circui pplicions W now h h squr wv wih h mpliud A hs h RMS vlu of A; A sinusoidl wvform hs h RMS vlu of nd h swooh wvform hs h A vlu of. h cofficins,, scling h mpliud in his dscripion 3 3 r clld h RMS scling fcors. In h rs of his scion, w clcul h RMS scling fcor for h hlf wv rcifid sinusoidl signl nd sblish conncion wih h Prsvl s rlion.

8 W sr wih h clculion of h RMS vlu. I should b clr h Hlf provids hlf h vrg powr of sinusoidl signl; hnc is scling fcor should b. W cn sily vrify his guss wih h following, clculion Hlf RMS [ sin ] cos d d h full wv rcifid signl hs h RMS scling fcor of. Prsvl s rlion ss h d. Vry similrly,. Givn our rfrshd nowldg on h RMS vlus, w cn lso wri h Prsvl s rlion s d RMS. Hlf-wv rcifid signl W hv prviously found h RMS vlu of h uni mpliud hlf-wv rcifid signl s. In ddiion, w hv lso found h FS cofficins of his signl s / / vn. ohr hn Prsvl s rlion d idniy. h summion givn bov cn b rwrin s follows. Subsiuing,, vn RMS givs us h following, w rch. h sm rlion cn 8, vn lso b wrin s 8, vn nd simplifid o h following,

9 , vn prssd s 8 6 nd h summion on h lf hnd sid cn b compcly 8. 6 h nd rsul of his clculion is h summion idniy givn bov. hs idniis involving rciprocl of ingr powrs r difficul o prov wih lmnry mns. hrfor, w cn no provid ny ohr rgumns for h vlidiy of his idniy; bu w cn lwys us Mlb o numriclly min h corrcnss of h his idniy >> vc; >> pril_sum sum./-*vc.^.^ pril_sum.68 >> pi^-8/6 ns.69 I sms h vryhing is in ordr.