Discrete Fourier Transform

Transcription

1 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 white oise ν y u ν u δ ν,,..., 8 The siga u(t) ad the observed vaues 6 4 u(t) t At first, we wi suppose the siga combiatio of compex fuctios u() t Fig. to be compex; so that the Fourier sum is a fiite iear ( π ) ( π ) iπ t e t i cos si t : L u( t) ~ ue J iπt J, L () We wi therefore cosider the case of a rea siga ad the reatioship with the first case. This time si π t : the Fourier sum is a iear combiatio of rea fuctios cos( π t ) ad ( ) L ( ( π ) ( )) ut () a a cos t bsi π t () Moreover, we start cosiderig the siga u( t) as a determiistic oe ad eave the case of a stochastic u() t at the ed, whe we wi suppose the oise ν idepedet of u() t, thus arrivig at the same resut.

2 We assume t [,) i a fiite iterva [,,, so that we ca choose δ. ote that, as every fiite data set is cotaied ab ) ( a t b ), it is aways possibe to map this iterva o the iterva [, ) t a usig the iear trasformatio t b a i of the idepedet variabe t. We observe aso that u( t ) ca be automaticay exteded as a periodic siga from [ ) whoe rea axis, by puttig u() t u( t h), h Z π the whoe rea axis, because cos( π t ), si ( π t ), i t Z. With these assumptios, formuae () ad () give, to the. I doig this, the reatios () ad () hod o e : are periodic fuctios of period, L i u u( δ ) u e π (5) J or L u a acos π bsi π (6) with,,...,. Caig y y, ν ν, u u, u u, J,..., L { } { } { } { } ( ) { }, { }, (,..., ) a a b b L, the probem is to uderstad which iformatio o u (or o ( ) ca be deduced from y. a, a, b ), that is o the siga u(t), Just for the sake of easiess we suppose the umber of observatios to be odd, M, idicatig, if ecessary, what chages if this umber is eve, M. I order to avoid mistakes, we wi aways use the etters, k, j for time idices (t / etc), ad the etters, m, r for frequecy idexes. ) Lemmas Here foows some usefu agebraic emmas, which aow to defie the orthogoaity of the discrete Fourier fuctios. We wi use Kroecker otatio k δ k (8) k

3 or δ δ (9) Lemma : For each fixed frequecy, we cosider the sum of the vaues assumed by the correspodig compex Fourier fuctio o the grid poits: Γ iπ e Z () This sum is equa to the umber of observatios if the frequecy satisfies the foowig reatio: r r, ±, ±,... otherwise, it is equa to. That is: r Γ δ () r Put the observed vaues of the compex Fourier fuctio of a fixed i a vector: ϕ cos π i si π cos π i si π ;... cos π π i si Γ oes, e iπ is the sum of the eemets of the vector or the scaar product by a rea vector of Γ ϕ ϕ e e e... 3

4 .9.8 Rea ad imagiary part of Fourier compex fuctio of frequecy cos(*pi**/p) cos(*pi**t) si(*pi**/p) si(*pi**t) cos(*pi**t) si(*pi**t) t, ti/p (p5, 3 4) Fig. : Lemma. The sum of the samped vaues of the compex fuctio of frequecy is equa to the umber of poits Rea ad imagiary part of Fourier compex fuctio of frequecy cos(*pi**/p) cos(*pi**t) si(*pi**/p) si(*pi**t) cos(*pi**t) si(*pi**t) t, ti/p (p5, 3 4) Fig.3: Lemma. The sum of the samped vaues of the compex Fourier fuctio of frequecy is equa to. 4

5 Proof: I order to prove the emma, we cosider the compex umber q q i e π () the we have : Γ q if q q q (3) if q ( ) I fact ( q ) Γ ( q ) ( q ) ( q q... q )( q ) so that, if q q Γ ( q) q ow, as it is aways ( ) q q... q q q... q q iπ iπ q e e cos( π) isi( π ) (4) it is cear that Γ if if q q (5) that impies (). Coroary Takig the cojugate of () we have: iπ e δr Z r Γ Γ (6) 5

6 Lemma Km cos π cos πm,m Z (7) it resuts K m ( r δ mr δ mr ) (8) Put the observed vaues of the two cosie fuctios of fixed frequecy ad m i two vectors. c, c m respectivey: c cos π cos πm cos π cos πm, c m ; π cos cos πm K m represets the scaar product betwee these two vectors: Km c c cc cos π cos πm. Proof Usig () ad (6) we have iπ iπ iπm iπm Km e e e e 4 iπ( m) iπ( m) iπ( m) iπ( m) e e e e 4 4 ( δ mr δ ( m ) r δmr δm r ) ( δ mr δmr ) r Coroary r It is sufficiet to et m i eq.(8) to obtai K cos π δ (9) r r ote that (9) meas that K is if some mutipe of, otherwise. 6

7 Agai this ca be cosidered as the scaar product betwee the vectors c e c e Lemma 3 Puttig Hm si π si πm,m Z, () the we have H m ( r δ mr δ mr ) () Put the observed vaues of the two sie fuctios of fixed frequecy ad m i two vectors. s, s m respectivey: s si π si πm si π si πm, s m ; π si si πm H m represets the scaar product betwee these two vectors: Hm s s s s si π si πm. Proof as i Lemma we have: Hm e e e e 4 iπ iπ iπ m iπ m that, usig eqs. () ad (6), yieds (). Lemma 4 Puttig Rm cos π si πm () the we have 7

8 R, m Z R,m.m Z (3) m R m represets the scaar product betwee the two vectors c ad s m : Rm c sm c sm cos π si πm. Proof I fact, usig (6) it is: iπ( m) iπ( m) iπ( m) iπ( m) Rm e e e e 4 Coroary 4 Puttig i () ad (3) we fid si π m (4) Agai this ca be cosidered as the scaar product betwee the vectors s e se. For the sake of further use, we observe that these four Lemmas ca have a particuary meaigfu matrix form, whe reducig the rage of frequecy idexes,m. Puttig for istace m, (5) ad { Φ} e iπ (6) thus, beig Φ a square compex matrix, eq (5) becomes * ΦΦ ΦΦ I I fact { ΦΦ} m π π i m i i π (m) e e e (7) 8

9 π π π( ) e e... e π π π( ) Φ e e... e ( ) π π π e e... e cos π i si π... cos π ( ) i si π ( ) cos π i si π... cos π ( ) i si π ( ) ( ) ( ) ( ) ( cos π i si π... cos π i si π ) 9

10 * ΦΦ π π π π π π( ) e e... e e e... e π π π π π π( ) e e... e e e... e π( ) π( ) π( ) π π π( ) e e... e e e... e π π π π π π( ) e e e e... e e π π π π π π( ) e e e e... e e π( ) π π( ) π π( ) π( ) e e e e... e e π( ) π( ) π( ) e e... e ( ) ( ) π π π( ) e e... e π ( ) ( ) π π( ) e e... e π π π( ) e e... e π π π( ) e e... e ( ) π π( ) π e e... e Remark: it is aso usefu to ote how to make a frequecy shift of α i the matrix Φ, i.e. iπα ( ) Φα e, (,,..., ),(,,..., ) (8) as we ca put Φ D Φ (9) α α

11 iπ α Dα e δ, we wi fid agai ΦΦΦ DDΦΦΦ I (3) * * * * α α Let s ow treat together Lemmas, 3, 4. Suppose at first M ad put M M e, C cos π, S si π (3) with the further positio Λ [ e C S ] (3) as it ca be see, Λ is sti a square matrix. The we have (with the metioed imitatios) Lemma CC I, ec Lemma 3 S S I Lemma 4 C S, es Such reatios ca a be summarized ito the equatio Λ Λ I I (33) I the case M, it is coveiet to chage as foows the defiitio of S i eq. (3): M S si π (34)

12 thus S comes out to be a rectaguar matrix ( M ). Ad actuay, if we use the previous defiitio (3), the ast coum of u, sice i this case it is: si πm si ( π), M I this way the matrix [ e C ] Λ S S resuts to be ideticay is agai a square matrix, with dimesio M M, ad eq (35) is sti vaid. ote that I I M M CC I M, whist S S I M, with idetity matrices of differet dimesios