Discrete Fourier Transform

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Alexander Blake
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1 Discrete Fourier Trasform 3) Compex Case et s distiguish the three cases + J + > + J + + J + < (35) Ad et s begi treatig the isodetermied case + J +, addig at first the hypothesis that J,. I this case we simpy have u Φ u (36) With observatio equatios y u + (37) Coherety, the oise i (37) sha be compex too, ad we reca that the defiitio of a compex white oise is j ρ j + iη (38) j with { ρ j } { η j } I this case it is ceary ad that are two idepedet white oises. σ σ + σ j ρj ηj (39) ad i ay case { } { + } C E E σ I (4) It is usefu to ote that to produce the covariace matrix of v, we have to use the adjoit + of the vector v ad ot its traspose oy. The product with traspose, i fact, wi ot yied i geera a sef adjoit positive defied matrix. From eqs. (36) ad (37) rememberig eq. (7) it foows that a ubiased estimator of u is give by Φ y (4) 3

2 Actuay it is ( ) Φ Φ u+ u + Φ (4) that meas E{ } u + Φ E{ } u (43) Moreover C σ σ σ. (44) uu ˆˆ Φ Φ I Compoet-wise, eqs (4) ad (44) say that e iπ y,,..., (45) ad that σ { }, ( ) E δ δ ˆ δ δ ˆ ˆ um m u u u (46) Remark Eq.(45) is caed Discrete Fourier Trasform because of the simiarity with the cotiuous formua e i πt u( t) dt ; The term / i eq. (45) pays the roe of the differetia dt. et s suppose to have a isodetermied mode shifted i frequecy: thus u Φ u~ a y u + (47) A typica case is whe M + ad we choose J M, so that M iπt u ( t) e u~ M (48) Recaig eq.(3) we have agai: 4

3 Φ ( M ) y σ C I uu ˆ ˆ (49) or agai e iπ y M,...,,..., M. ad E σ { δuδ m} δ m ˆ. et s ow cosider the overdetermied case, puttig for istace J ad <. I this case we sti have u Φ u u Φ ũ (5) where the matrix Φ is formed by the first coums of matrix Φ Φ iπ e (5) I this case the estimator û have to be computed from a compex east squares pricipe : ( ) ( ) iπ ˆ Mi y Φ Mi y Φ y Φ Mi y e u (5) The soutio of (5) is give by Φ Φ Φ y (53) O the other had it is cear that ΦΦ I (54) so that, agai, oy restrictig to the vaues:,...,, eq.(53) yieds e iπ y (55) 5

4 Ad agai, i the same way, is: C σ ˆˆ σ Φ Φ uu I (56) ow we examie the uderdetermied case. I this case it is coveiet to suppose that J +, that is iπt u ( t) e u~ (57) where we have added Fourier coefficiets equa to zero. I doig so, we wi obviousy suppose that the series is coverget, ad aso that the sum is cotiuous, otherwise it makes o sese to set to zero poitwise vaues of u(t) A obvious sufficiet coditio for uiform covergece of (57) (ad therefore to a cotiuous fuctio) is u ~ < + (58) ow ũ is a vector of ifiite dimesio, ad a proper uderstadig of the iformatio that ca be deduced from y o ũ comes from Sampig Theorem or yquist Theorem 6

5 Theorem (yquist) et be U the space of sigas () u t with form (57), with coefficiets vector, u, satisfyig (58). We maitai that two sigas t (,,..., ), if u() t ad wtare ( ) equivaet, with respect to the sampig ( ) ( ) (,,..., ) u t w t (59) that is u w The equivaece cass of a siga u() t which yieds a give vector u { u }, is the same of the siga that has o zero coefficiets oy for frequecies iπ t wt () e w (6) these coefficiets w, moreover, are give by + w u+ r (6) r (compoud coefficiets). Aso u() t ca be represeted by a siga wt) ( ) with a set of o zero coefficiets, if these represet cosecutive frequecies, a+,..., a+ +. Proof First of a we ote that for a fixed vector u a uique siga of kid (6) exists such that ( ) ( ) wt ut, ad its coefficiets w are, as i isodetermied case, fixed by the reatio w Φ u Moreover we aso saw that vector u itsef coud be obtaied from a siga a+ iπ t a w () t e w a 7

6 computig the coefficiets by a w Φau The we oy have to prove that, for a siga, give i its geera form iπt u( t) e u (6) with eq.(6) a equivaet siga ca be geerated, i.e. with the same samped vaues u( t ).. To do that it is sufficiet to write eq. (6) as partia sum of packets i frequecy u( t) + r + iπt iπt iπt i t e u + e u + e u + e π... u r r (63) ad to ote that r + π π( + ) i i r iπ + r e u e u e u r + r so that eq.(63) computed i t becomes + π i iπ r+ + r r ut ( ) e u e u r It is possibe to commute the two sums, because, uder coditio (58) the series absoutey coverget. u r + r is ceary Accordig to a this, apart from the represetatio (6) of the siga wt ( ) with oy coefficiets, equivaet to M + u() t, it is possibe to use the other oe, that is particuary meaigfu whe ut ()~ wt () e u M + iπ t + r M r (64) i t ote that if we cosider e π ad i π e t as two harmoics with the same frequecy (that meas : i (64) the frequecy is ), from formua (64) the yquist Theorem ca be read directy. 8

7 This theorem says that give equay spaced samped vaues of u( t ), it is possibe to compute Fourier coefficiets of u( t ) up to M ) but ot more. (actuay, i the case just cosidered, up to The effect that makes the higher frequecy coefficiets summed up to coefficiets with frequecy M, is caed aiasig. For a siga u() t i U, the oy way of reducig aiasig is to produce a deser sampig, icreasig ; i fact, i this way, the frequecy packet of w that ca be determied, becomes wider, ad to the coefficiets u ( (58) impies that u u for. Remark : M ), smaer coefficiets are added ( u ± u ±...) as the coditio ote that (6) coud be obtaied directy from () of emma. I fact, puttig iπ w e u( t ) we have w e e u u e π + π + i i m iπ( m) m m m m u δ u δ m m r m m r + r m r r m r u 9

Discrete Fourier Transform

Discrete Fourier Transform Discrete Fourier Trasform ) Purpose The purpose is to represet a determiistic or stochastic siga u( t ) as a fiite Fourier sum, whe observatios of u() t ( ) are give o a reguar grid, each affected by a