# ω (argument or phase)

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1 Imagiary uit: i ( i Complx umbr: z x+ i y Cartsia coordiats: x (ral part y (imagiary part Complx cougat: z x i y Absolut valu: r z x + y Polar coordiats: r (absolut valu or modulus ω (argumt or phas x cos(ω r x r cos(ω, si(ω y r si(ω si( ω y y x ta( ω ω ata( cos( ω x r y z z ( x+ i y( x i y x i y x + y z aalogous if z ot i st quadrat,.g., x<, y> ω ata( y/x

2 Exrcis: Show that th st C of -dimsioal complx vctors togthr with vctor additio dfid by z w z + w z+w M + M M z w z + w is a commutativ group. FG Exrcis: Show that th st C togthr with vctor additio dfid as abov ad scalar multiplicatio dfid by z λz λ zλ M M z λz is a complx vctor spac, i.., λ(µ z(λ µz, zz, λ(z+w(λ z+(λ w, (λ+µz(λ z+(µ z. FV Exrcis: Show that th ir product of two complx vctors z ad w dfid by z z,w w*z ( w,..., w M z t w t, z satisfis (i v + w, z v, z + w, z, (ii (iii (iv (v (vi (vii z, v+ w z, v + z, w, λ z, w λ z, w, v, λw λ v, w, z, w w, z, z, z, z, z z. FI Two complx vctors z ad w ar said to b orthogoal, if z, w.

3 Exrcis: Show that th orm of a complx vctor z dfid by satisfis z (i z, z zt z t (ii λ z λ z, z t z+ w z + w. FN Hit: Us th Cauchy-Schwarz Iquality z, w z w. Exrcis: Prov th Pythagora thorm z, w z+ w z + w. FP Siusoid: g ( t R si( ωt + φ Paramtrs: R (amplitud ω (frqucy φ (phas A siusoid is priodic with priod p bcaus g ω ( t+ R si( ω( t+ + φ ω ω R si( ωt+φ + R si( ωt+ φ g(t. Th st Fourir frqucy ω implis a priod of p. ω Th d Fourir frqucy ω implis a priod of p. ω M 3

4 Exrcis: Us th Eulr rlatio FE to show that i ω + i ( ω+ cos( ω i si( ω iω, iω iω i(ω ω,. Th vctors FB iω M,,, iω costitut a orthoormal basis for C bcaus,, iω t iω t i( ω ω if. iωt iωt - ( iω t iωt iω t iωt, ( i(ω ω t i( ω ω i(ω ω t i(ω -ω i(ω -ω Thus, ay x C has a rprstatio of th form x λ. Taig ir products of ach sid w obtai x,, λ λ, λ, λ, λ * x, x x x, x λ, t iωt x, λ λλ, λ λ λ. FS Exrcis: Show that ( x t x λ. FF Not: Th siz of λ thrfor idicats how much of th sampl variac ca b attributd to frqucy ω. 4

5 Suppos that x R. If <, th AZ iω-t ( i t i ( t i t iωt iωt iω - t iω t iω t xt xt x t λ, λ iω t λ + λ iω t λ iω t + λ iω t ( a + ib (cos( ω t i si( ω t + + ( a ib (cos( ω t i si( ω t a cos( ω t b si( ω t R si( φ cos( ω t + R R si( ω t+ φ, cos( φ si( ω t whr R ad φ ar th polar coordiats of b +a i, i.., a si( φ, b cos( φ. R si(α+βsi(αcos(β+cos(αsi(β R, If, th ω iω t it, cos( t + si( t si( t+, 3 t λ xt cos(t xt (- λ iω t R si( t R t+ R, cos( R si( ωt+ φ. If, th iω ω, t i t, λ λ iω t x. xt xt x R, Thus, [ / ] iωt xt λ + λ x+ R si( ωt + φ. 5

6 Th priodogram of x,,x is dfid by For <, I(ω λ I(ω iωt x t. ( a + b ((a + ( b R 8 8. It will b show latr that for ay ozro Fourir frqucy ω, I(ω ca b writt as whr I(ω ˆ γ ( -iω ( γ( ˆ, ( xt x( xt+ x is th sampl autocovariac at lag. If th obsrvatios x,,x com from a statioary procss x, th priodogram I(ω may b rgardd as a sampl aalogu of th fuctio f(ω γ( -iω which is calld th spctral dsity of th procss x. Th statioarity of th procss x implis that all x t hav th sam ma ad th sam variac ad th autocovariacs γ(cov(x t,x t- dpd oly o but ot o t., 6

7 Exrcis: Show that - iω dω if. A Assumig that th itrchag of summatio ad itgratio is ustifid w ca driv th spctral rprstatio of th autocovariac fuctio γ of a statioary procss x with spctral dsity f as follows: iω f(ω dω - - iω γ( γ( γ( -iω γ( - - iω( iω( dω dω dω Rmar: Lt If th f (ω γ( cos(ω. g(ω γ( <, - g(ω dω < ad, by th domiat covrgc thorm, bcaus - lim - lim f (ω dω f (ω dω, (ω f γ(cos(ω γ( g(ω. AR 7

8 Rmar: It follows from f(ω ad γ( - γ( -iω iω f(ω dω that th spctral dsity f ad th autocovariac fuctio γ cotai th idtical iformatio. Giv obsrvatios x,,x, th priodogram I(ω ˆ ( γ( -iω is a vry rratic stimator for th spctral dsity, bcaus th sampl autocovariacs ˆ γ ( (xt x(xt+ x cotai vry fw products ( x x( x + x if is larg. t t A obvious improvmt is to giv lss wight to th mor variabl sampl autocovariacs. A stimator of th typ f ˆ ( ω ˆ ( w γ( -iω is calld a wightd covariac stimator. A widly usd stimator is th Bartltt stimator which uss th triagl wights M if < M, w ls. Th trucatio poit M is a importat paramtr for cotrollig th smoothss of th stimator. A altrativ mthod of smoothig th priodogram is to ta wightd avrags ovr ighborig frqucis. A widly usd smoothd priodogram stimator is th modifid Daill smoothr, which diffrs from a simpl movig avrag of th priodogram oly i that th first ad th last wight ar oly half as larg as th othrs. 8

9 Exrcis: Spctral aalysis of th postwar US GDP Crat a worig dirctory, say C:\GDPq, for th aalysis of th quartrly US GDP. Dowload th ral Gross Domstic Product (quartrly, sasoally adustd as a txt fil (GDPC.txt from th wbsit of th Fdral Rsrv Ba of St. Louis ito your worig dirctory. Th first part (lis - of GDPC.txt cotais oly txt, th scod part cosists of two colums (dats ad GDP valus. Import th data ito R ad plot th GDP, th log GDP, th diffrcd log GDP, ad th priodogram of th diffrcs. stwd("c:/gdpq" # commt: st worig dirctory D <- rad.tabl("gdpc.txt",sip # import data d <- as.dat(d[,] # first colum of D: dats v <-D[,] # scod colum of D: GDP valus N <- lgth(v # N o. of quartrs lgth of vctor v par(mfrowc(, # subsqut plots i x array par(marc(,,, # st arrow margis for plots plot(d,v,pch # plot GDP valus agaist dats y <- log(v; plot(d,y,pch # plot charactr: solid circl r <- y[:n]-y[:(n-]; <- N- # o. of diffrcs plot(d[:n],r,pch,typ"o" # ovrplot poits&lis h <- spc.pgram(r,tapr,dtrdf,fasf,plof c <- *pi; f <- c*h\$frq # Fourir fr. btw ad pi pg <- h\$spc/c; plot(f,pg,typ"o",pch # priodogr. 9

10 Smooth th priodogram with th modifid Daill smoothr. par(mfrowc(, # sigl plot plot(f,pg,typ"l" # oly lis, o poits h <- spc.pgram(r,tapr,dtrdf,fasf,plof, spas3 lis(c*h\$frq,h\$spc/c,col"gr",lwd # add li to xistig plot with li width twic as wid h <- spc.pgram(r,tapr,dtrdf,fasf,plof, spas lis(c*h\$frq,h\$spc/c,col"rd",lwd Th highr th spa (th total umbr of trms i th movig avrag, th smoothr th stimat.

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