ANALYSIS IN THE FREQUENCY DOMAIN

ANALYSIS IN THE FREQUENCY DOMAIN

SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind as th Fourir transform F { τ } γ

Proprtis of :. is a ral function of th ral variabl, 0 R Im. is a positiv function, 0 R 3. is a vn function, R 4. is a priodic function with priod ual to π + k π R, k Z Obsrvation: as a consunc of 4, w will plot th spctral dnsit in th intrval [ π, π ]. π π 3

Obsrvation π π is th largst frunc for sinusoidal discrt tim signals. Indd, th minimum priod is T π which corrsponds to π 4

Invrs transform of : F γ τ F { γ τ } + τ + γ τ π π π jτ c bi-univocal rlationship jτ d γ τ and carr th sam information on th proprtis of th procss t th spctral dnsit is an altrnativ rprsntation of th procss nd ordr proprtis Obsrvation: + π γ d d π + π j 0 0 π π i.. th procss varianc is th rscald ara undrling. π π π 5

Spctral dnsit of a whit nois WN Lt us considr t WN µ, λ. Covarianc function: γ λ τ γ τ λ 0 if τ if τ 0 0 τ Spctral dnsit: + τ jτ j 0 j γ τ γ 0 + γ + γ γ 0 λ j + L λ - π Whit Noiss hav constant and ual to λ spctral dnsit. 6

SPECTRAL DENSITY OF S.S.P. GENERATED AS THE OUTPUT OF DIGITAL FILTERS Lt th procss t th stad-stat output of an asmptoticall stabl digital filtr fd b an S.S.P., i.. t F v t v t F t Thn, th following formula for th spctral dnsit of t holds: F j v output spctral dnsit j F suar absolut valu of th filtr transfr function valuat for j filtr frunc rspons input spctral dnsit v If th input v t is a Whit Nois with varianc λ, thn: j F λ 7

Exampl MA procss t t + c t, c R ral cofficint t WN 0, + c γ 0 c γ + c c ± γ τ 0 whn τ..., ± 3, ± 4,... Covarianc function plot τ γ + c c > 0-5 -4-3 - - 3 4 5 τ c < 0 c 8

Spctral dnsit via th dfinition + τ γ τ jτ j onl γ 0, γ ± ar not null j + j γ 0 + γ + γ + 0 j j + c + c + Eulr rprsntation of th xponntial j + + j cos j sin + cos + j sin cos W hav + c + c cos which is: ral 0 c vn priodic with priod π 9

Spctral dnsit computd through th main thorm t + c t, t WN 0, + c j Rcalling that:. a + jb + a + jb a jb a b. + + c c j + j + ccos + + ccos + jsin jsin + ccos jsin j + j + c + c j j + c + c + + c + ccos Plot of + c c < 0 + c c c > 0 - π π π π 0

0 + c + c + c π + c π + c c c Lt us comput γ 0 basd on + π γ 0 π π π + c + c cos d + π + π + c d + c cosd π π π π π + c + csin π + c + c π π π

An altrnativ intrprtation of th spctral dnsit Kinchin-Winr Suppos that an S.S.P. t is filtrd through an idal pass-band filtr: FP B frunc rspons t t FP B F PB pass - band filtr 0 + δ t is th filtrd output procss Thorm Kinchin-Winr: δ 0 limγ 0 Spctral dnsit man nrg of procss raliations frunc b frunc.

Exampl Whit Nois A WN procss raliation is rratic and unprdictabl complt uncorrlation at diffrnt tim instants 8 7 6 5 4 3 0 - - 0 0 0 30 40 50 60 70 80 90 00 Th WN spctral dnsit is constant... λ - π... i.. WN nrg is uall-distributd all ovr th frunc domain 3

Exampl gnral cas t t, WN0, + 0.9 4 3 0 - - -3-4 0 0 40 60 80 00 0 00 80 60 40 0 0-4 - 0 4 5 00 80 60 0 40 0-5 0 0 40 60 80 00 0-4 - 0 4 t t, WN0, 0.9 4

Obsrvation Man diffrnt rprsntations for an ARMA procss. Tim-domain rprsntation: diffrnc uation t a t +... + am t m + c0 t +... + cn t n t WN µ, λ. Opratorial rprsntation: transfr function C t t t WN µ, λ A 3. Probabilistic charactriation: man & covarianc function - m - γ τ τ 0, ±, ±, ± 3, K 4. Frunc domain charactriation: man & spctral dnsit - m - R N.B.: nithr nor γ τ carr an kind of information about th man valu of th procss. Ar all th four rprsntations uivalnt for a wid-sns procss charactriation? Ys! Clarl, 3 4,, 3, 4 What about 3,4,? 5

Lt us considr an ARMA procss C t t, t WN µ, λ. A t has a rational spctral dnsit: j C λ j, A whr a spctral dnsit is calld rational if it is a rational function of th variabl j, that is p + p + + p + j j 0 j 0 j + L + L Is th rvrs tru? Ys Thorm. Lt t b a S.S.P. with rational spctral dnsit. Thn, thr xists a whit nois procss ξ t with suitabl man and varianc and a rational transfr function W such that: i.. t is an ARMA procss. t W ξ t 6

Qustion: is th ARMA rprsntation i.. th choic of ξ t and of W uniu? NO Th sam procss t W ξ t, ξ t WN µ, λ can b gnratd according to an infinit numbr of diffrnt ARMA modls. I.. t can b gnratd also as t W ξ t, whr W is a rational transfr function diffrnt from W, and ξ t is a whit nois diffrnt from ξ t. Cas. Lt α b an ral numbr Hr, t dos not chang, w hav multiplid W b th idntit!!! α t W ξ t W ξ t W αξ t W ξ t α α whr: Thanks to th ruls for th composition of transfr functions W W it s still a rational transfr function α ξ t αξ t it s still a WN, ξ t WN αµ, α λ * Hnc, t W ξ t is a nw ARMA rprsntation of th procss t not that t is alwas th sam, it nvr changd. 7

* Lt us vrif that ξ t is actuall a whit nois E[ ξ t ] E[ αξ t] αe[ ξ t] αµ γ ξ τ E[ αξ t αµ αξ t τ αµ ] αe[ ξ t µ ξ t τ µ ] αγ τ ξ clarl, ξ t and ξ t ar two diffrnt whit noiss, although strictl corrlatd Cas. Lt n b an intgr numbr. Hr, t dos not chang, w hav multiplid W b th idntit!!! [ ] n W ξ t n W ξ n t W ξ t W ξ t t n Thanks to th ruls for th composition of transfr functions and rcalling th maning of th shift oprator whr: W W it s still a rational transfr function ξ t ξ t n it s still a whit nois* Hnc, t W ξ t is a nw ARMA rprsntation of th procss t not that t is alwas th sam, it nvr changd. 8

* Lt us vrif that ξ t is actuall a whit nois E[ ξ t ] E[ ξ t n] µ stationarit γ τ E[ ξ t n µ ξ t n τ µ ] γ stationarit ξ τ ξ not that ξ t and ξ t ar not th sam procss although th ar wid-sns uivalnt Cas 3. Lt p b an complx numbr such that p <. p t W ξ t W ξ t W ξ t p whr: Hr, t dos not chang, w hav multiplid W b th idntit!!! p W W p it s still a rational transfr function ξ t ξ t plainl, it s a whit nois Hnc, t W ξ t is a nw ARMA rprsntation of th procss t not that t is alwas th sam, it nvr changd. hr ξ t and ξ t ar th sam procss, but W and W ar diffrnt 9

0 Cas 4. Lt a ro of W such that >. That is: W W Thn, t W t W t W t ξ ξ ξ t W t W ξ ξ whr: W W it s still a rational transfr function t t ξ ξ is it a whit nois? Lt us comput th spctral dnsit of t ξ λ λ ξ j j j j j j + + + + + + cos cos λ λ j j j j Hr, t dos not chang, w hav multiplid W b th idntit!!! Thanks to th ruls for th composition of transfr functions

λ + + cos cos λ Th spctral dnsit is constant for all valus of, so that ξ t is a whit nois! Morovr, E [ ξ t] E[ ξ t] µ µ Hnc, ξ t WN µ, λ and t W ξ t is a nw ARMA rprsntation of th procss t not that t is alwas th sam, it nvr changd. not instad that ξ t and ξ t ar two diffrnt whit noiss, although strictl corrlatd

Apart from th four xamind ons, thr ar no othr sourcs of ambiguit in dfining an ARMA procss. This is xprssd b th following thorm which bttr clarifis th prvious on. Thorm Spctral Factoriation Lt t b a S.S.P. with rational spctral dnsit. Thn, thr xists an uniu whit nois procss ξ t with suitabl man and varianc and an uniu rational transfr function W such that: t W ξ t, and, C and A ar th numrator and dnominator of W :. C and A ar monic i.. th cofficints of th maximum dgr trms of C and A ar ual to. C and A hav null rlativ dgr 3. C and A ar coprim i.. th hav no common factors 4. th absolut valu of th pols and th ros of W is lss than or ual to i.. pols and ros ar insid th unit circl Whn all th four conditions abov ar satisfid, w will sa that t W ξ t is a canonical rprsntation of t

Obsrvation. Conditions,, and 3 rmov an ambiguit as du to th procss dscribd in Cas, Cas, and Cas 3, rspctivl. Th first part of Condition 4 assur that W is asmptoticall stabl so that t is wll dfind, whil th scond part rmovs an ambiguit as du to th procss dscribd in Cas 4. 3