Chapter Discrete Fourier Transform

Transcription

1 haptr.4 Dscrt Fourr Trasform Itroducto Rcad th xpota form of Fourr srs s Equatos 8 ad from haptr., wt f t 8, h.. T w t f t dt T Wh th abov tgra ca b usd to comput, h.., t s mor prfrab to hav a dscrtzd formua vrso to comput. Furthrmor, th Dscrt Fourr Trasform or DFT [ 5] w aso factat th dvopmt of much mor ffct agorthms for Fast Fourr Trasform or FFT, to b dscussd haptrs.5 ad.6. Drvatos of DFT Formuas If tm t s dscrtzd at t t t t, t 3 t,..., t,, 3 t Th Equato 8, of haptr. bcoms wt f t To smpfy th otato, df t Th, Equatos ca b wrtt as w f 3 I th abov formua, s a tgr coutr. Howvr, f ad t do OT hav to b tgr umbrs. Mutpyg both sds of Equato 3 by obtas ot: tgr umbr w, ad prformg th summato o, o.4.

2 .4. haptr.4 w w w f 4 w 5 6 Swtchg th ordr of summatos o th rght-had-sd of Equato 6, o obtas f 7 Df A 8 Thr ar possbts for to b cosdrd Equato 8 as: - s a mutp tgr of, such as - m; or + m whr,...,, ± ± m Thus, Equato 8 bcoms: m A 9 + s cos m m Hc: A as: - s OT a mutp tgr of I ths cas, from Equato 8 o has A Df: a + s cos ; a bcaus - s OT a mutp tgr of 3 Th, Equato ca b xprssd as a A 4 From mathmatca hadboos, th rght sd of Equato 4 rprsts th gomtrc srs, ad ca b xprssd as

3 Dscrt Fourr Trasform.4.3 A a f a 5 a f a 6 a Bcaus of Equato 3, hc Equato 6 shoud b usd to comput A. Thus a A S Equato 7 a a Sc s st a mutp of, hc cos{ } + s{ } 8 Substtutg Equato 7 to Equato 8, o gts A 9 Thus, combg th rsuts of cas ad cas, o gts s Equatos ad Equato 9 A + Substtutg Equato to Equato 8, ad th rfrrg to Equato 7, o gts w f a Rcad + m whr, m ar tgr umbrs, ad sc must b th rag, thrfor m. Thus: + m bcoms Equato a ca, thrfor, b smpfd to Thus whr ad f b w w f f { cos w s w } Rmars: t w f 3, rpatd { cos w + s w } a osdr th xpota trm Equato. Lt

4 .4.4 haptr.4 w E If o rpacs by or to th abov quato, th o obtas * [ ] E Thus, Equato dcats that th forc corrspodg to frqucs of ordr ad hav th sam vaus. Hc w w for w for > ad th frqucy corrspodg to th dscrt Fourr srs compots abov s th hghst frqucy that ca b cosdrd w s cad th yqust frqucy. If thr ar harmoc forc w th orga fucto, th ths hghr compots w troduc dstortos th owr harmoc compots ow as ALIASIG phomo. Bcaus of th ALIASIG phomo, th umbr of data pots shoud b at ast twc th hghst harmoc compot prsts th forcg fucto, for suffct computatoa accuracy. As a xamp, f th forcg fucto s gv as 6 F t cos t th, th mmum vau of umbr of samp data pots shoud b 3 m. b Th factor, show th DFT Equato, s mry a sca factor. It ca aso b pacd th vrs Fourr Trasform Equato, but ot both. Thus, Equatos ad ca b r wrtt as f w w f 3 To avod computato wth compx umbrs, Equato ca b xprssd as R I R I + f + f { cos θ s θ } a whr θ w b R I R I I R + f cos θ + f s θ + f cos θ f s θ { } { }

5 Dscrt Fourr Trasform.4.5 Th abov compx umbr quato s quvat to th foowg ra umbr quatos R I R I { f cos + f s θ } θ c I R { f cos f s θ } θ d omputr program mpmtato for th DFT quatos c, d ar gv at Dtad Expaato About Aasg Phomo, yqust Samps, yqust Rat. Wh a fucto f t, whch may rprst th sgas from som ra-f phomo show Fgur, s sampd, t bascay covrts that fucto to a squc f at dscrt ocatos of t. Ths dscrt ocatos ar assumd to hav quay spacd ad th dstac btw ay samps s t. Thus, f rprsts th vau of f t, at t t + t, whr t s th ocato of th frst samp at. If th samp ocatos wr do propry, th th orga fucto f t, ca b rcovrd through trpoato procss of ths dscrt samp vaus. Fgur Fucto to b Sampd ad Aasd Samp Probm. I Fgur, th samps hav b ta wth a fary arg t. Thus, ths squc of dscrt data w ot b ab to rcovr th orga sga fucto f t. For xamp, f a dscrt vaus of f t, wr coctd by pcws ar fasho, th a ary horzota straght w occur btw t through t, ad t through t 6, rspctvy S Fgur. Ths pcws ar trpoato or othr trpoato schms w OT produc a curv whch rsmb w wth th orga fucto f t. Ths s th cas whr th data has b ALIASED.

6 .4.6 haptr.4 Fgur Fucto to b sampd ad Wdowg Samp Probm. Aothr potta dffcuty sampg th fucto s cad wdowg probm. As dcatd Fgur, wh t s sma ough so that a pcws ar trpoato for coctg ths dscrt vaus w adquaty rsmb th orga fucto f t, howvr, oy a porto of th fucto f t has b sampd from t through t rathr tha th tr o. I othr words, o has pacd a wdow ovr th fucto. To avod aasd phomo, th samp spac t shoud b sma ough so that th dscrt squc w rcovr bac th orga fucto f t. Th sampg thorm ca b statd as: If th fucto f t s bad-mtd wth badwdth w max, F w Fourr trasform of f t for w w > max th f t s uquy dtrmd by a owdg of ts vaus at uformy spacd trvas t apart, wth t. wmax Th abov sampg thorm ca b oosy xpad through th hp of Fgur 3.

7 Dscrt Fourr Trasform.4.7 Fgur 3 Frqucy of sampg rat w vrsus maxmum frqucy cott w. S To satsfy F w, for w wmax, th frqucy w shoud b btw pots A ad B of Fgur 3. Hc w max w w s w max whch mps w s w max Physcay, th abov quato stats that o must hav at ast samps pr cyc of th hghst frqucy compot prst yqust samps, yqust rat. max

8 .4.8 haptr.4 Fgur 4 orrcty rcostructd sga. Fgur 5 Wrogy rcostructd sga.

9 Dscrt Fourr Trasform.4.9 I Fgur 4, a susoda sga s sampd at th rat of 6 samps pr cyc or w s 6w. Sc ths sampg rat dos satsfy th sampg thorm rqurmt w s w max, th rcostructd sga dos corrcty rprst th orga sga. Howvr, as dcatd 6 Fgur 5 a susoda sga s sampd at th rat of 6 samps pr 4 cycs or w s w. 4 Sc ths sampg rat dos OT satsfy th rqurmt w s w max, th rcostructd sga woud wrogy rprst th orga sga. Rfrcs [] E.Ora Brgham, Th Fast Fourr Trasform, Prtc-Ha, Ic [] S.. hapra, ad R.P. aa, umrca Mthods for Egrs, 4 th Edto, Mc-Graw H. [3] W.H. Prss, B.P. Fary, S.A. Tosy, ad W.T. Vttrg, umrca Rcps, ambrdg Uvrsty Prss 989, haptr. [4] M.T. Hath, Sctfc omputg, Mc-Graw H 997. [5] H. Josph Wavr, Appcatos of Dscrt ad otuous Fourr Aayss, Joh Wy & Sos, Ic FAST FOURIER TRASFORM Topc Dscrt Fourr Trasform Summary Txtboo ots o dscrt Fourr trasform Major Gra Egrg Authors Duc guy Dat Juy 5, Wb St