Discrete Fourier Transform (DFT)
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1 Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy umrical Mthods or STEM udrgraduats 8/3/29
2 Discrt Fourir Trasorm Rcalld th xpotial orm o Fourir sris s Eqs. 26, 28 i h.., o gts: t iw T T t t iw t dt I tim t is discrtizd at th Eq. 26 bcoms: t iw t 26, rpatd 28, rpatd t t, t2 2t, t3 3t,..., t t, 2
3 Discrt Fourir Trasorm cot. To simpliy th otatio, di: t Th, Eq. 2 ca b writt as: iw ilw Multiplyig both sids o Eq. 3 by, ad prormig th summatio o, o obtais ot: l = itgr umbr ilw iw ilw 2 i l
4 Discrt Fourir Trasorm cot. Switchig th ordr o summatios o th right-had-sid o Eq.5, o obtais: 2 2 l i il 6 Di: 2 l i A 7 Thr ar 2 possibilitis or to b cosidrd i Eq. 7 l 4
5 Discrt Fourir Trasorm as as: l is a multipl itgr o, such as: l m ; or m whr m,, 2,... Thus, Eq. 7 bcoms: A Hc: A im2 cos m2 i si m
6 Discrt Fourir Trasorm as 2 as2: l is OT a multipl itgr o. I this cas, rom Eq. 7 o has: A Di: a 2 i l i l 2 cos 2 l isi 2 l a ; bcaus l is OT a multipl itgr o Th, Eq. ca b xprssd as: A a 2 6
7 Discrt Fourir Trasorm as 2 From mathmatical hadboos, th right sid o Eq. 2 rprsts th gomtric sris, ad ca b xprssd as: A a ; i a a ; i a a Bcaus o Eq., hc Eq. 4 should b usd to comput A. Thus: a A a i l2 cos i l a S Eq. 5 l2 i si l
8 Discrt Fourir Trasorm as 2 Substitutig Eq. 6 ito Eq. 5, o gts A 7 Thus, combiig th rsults o cas ad cas 2, w gt A Substitutig Eq.8 ito Eq.7, ad th rrrig to Eq.6, o gts: ilw 8 8A Rcall l m whr l, m ar itgr umbrs, ad sic must b i th rag, m=. Thus: l m bcoms l 8
9 Discrt Fourir Trasorm as 2 Eq. 8A ca, thror, b simpliid to l ilw 8B Thus: si cos iw lw i lw 9 whr t ad si cos iw w i w, rpatd 9
10 Aliasig Phomo, yquist sampls, yquist rat Wh a uctio t, which may rprst th sigals rom som ral-li phomo show i Figur, is sampld, it basically covrts that uctio ito a squc at discrt locatios o t. Figur Fuctio to b sampld ad Aliasd sampl problm.
11 Aliasig Phomo, yquist sampls, yquist rat cot. Thus, rprsts th valu o t at t t t, whr t is th locatio o th irst sampl at. I Figur, th sampls hav b ta with a airly larg t. Thus, ths squc o discrt data will ot b abl to rcovr th origial sigal uctio t. For xampl, i all discrt valus o t wr coctd by picwis liar ashio, th a arly horizotal straight li will occur btw through ad through rspctivly S Figur. t 9 t 2 t t 8
12 Aliasig Phomo, yquist sampls, yquist rat cot. Ths picwis liar itrpolatio or othr itrpolatio schms will OT produc a curv which closly rsmbls th origial uctio t. This is th cas whr th data has b ALIASED. 2
13 Widowig phomo Aothr pottial diiculty i samplig th uctio is calld widowig problm. As idicatd i Figur 2, whil t is small ough so that a picwis liar itrpolatio or coctig ths discrt valus will adquatly rsmbl th origial uctio t, howvr, oly a portio o th uctio has b sampld rom t through t 7 rathr tha th tir o. I othr words, o has placd a widow ovr th uctio. 3
14 Widowig phomo cot. Figur 2. Fuctio to b sampld ad widowig sampl problm. 4
15 Widowig phomo cot. Figur 3. Frqucy o samplig rat w S vrsus maximum rqucy cott w. max I ordr to satisy F w or w wmax th rqucy w should b btw poits A ad B o Figur
16 Hc: Widowig phomo cot. w max w w s w max which implis: w s 2w max Physically, th abov quatio stats that o must hav at last 2 sampls pr cycl o th highst rqucy compot prst yquist sampls, yquist rat. 6
17 Widowig phomo cot. Figur 4. orrctly rcostructd sigal. 7
18 Widowig phomo cot. I Figur 4, a siusoidal sigal is sampld at th rat o 6 sampls pr cycl or w s 6w. Sic this samplig rat dos satisy th samplig thorm rquirmt o w s 2w max, th rcostructd sigal dos corrctly rprst th origial sigal. 8
19 Widowig phomo cot. I Figur 5 a siusoidal sigal is sampld at th rat 6 o 6 sampls pr 4 cycls or w s w Sic this samplig rat dos OT satisy th rquirmt w s 2w max, th rcostructd sigal was wrogly rprst th origial sigal! 4 Figur 5. Wrogly rcostructd sigal. 9
20 Discrt Fourir Trasorm cot. Equatios 9 ad ca b rwritt as 2 w i 2 2 w i 2 To avoid computatio with complx umbrs, Equatio 2 ca b xprssd as si cos I R I R i i i 2A whr w 2 2
21 Discrt Fourir Trasorm cot. R I i R I I R cos si i cos si Th abov complx umbr quatio is quivalt to th ollowig 2 ral umbr quatios: R I I R cos cos R I si si 2B 2 2D 2
DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1
![DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1](/thumbs/79/79094742.jpg "DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1")
DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT
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