SPCTR ANLYS TME SER
SPCTR ANLYS TME SER STAT 520
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This 27 page Class Notes was uploaded by Providenci Mosciski Sr. on Wednesday September 9, 2015. The Class Notes belongs to STAT 520 at University of Washington taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/192509/stat-520-university-of-washington in Statistics at University of Washington.
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Date Created: 09/09/15
Fourier Theory Overview 0 spectral analysis stationary processes Fourier 0 basic idea behind Fourier theory given realcomplex valued function over 00 00 or realcomplex valued sequence gt t O 1 i2 want to write represent synthesize or gt as 44f 077Altfgt Cos lt27rftBf sin 27rft Zf770ltfgt67 27rft Where e E cos i sin and i 1 Four Flavors of Fourier Theory 0 periodic with period T Z is sum over fn nT n O 1 i2 cofntinuous tinie discrete frequency 0 square integrable L Iglttgt2 dt lt oo Z is integral over 00 oo cofntinuous tinie continuous frequency 0 gt square surnniable t Ig 2 lt 00 700 Z is integral over 12 12 discrete time continuous frequency 0 gt t O 1 N 1 a nite sequence 2 issuni over fnnNn01N 1 discrete time discrete frequency 0 all used in spectral analysis 0 task de ne Cf for each avor known as Fourier coe icients Cont TimeDisc Frequency I o assumptions gp periodic with period T gpt T gplttgt gp square integrable over one period T2 7m gp1t2 dt lt oo o de nitions nth Fourier coe icient n O lzl l2 1 T2 2 7T n G E TiT2 9plttgt 2 fnldt fn T interpretation of G covariance between gp and complex exponential if similar Gn large mth order Fourier approximation m gp7mlttgt 2 2 anew n7m least squares approximation 7 see pp 6072 Conh ThneI scFrequencyII 0 can show 772 hm Lm 9plttgt gmltt2 dt 0 m OO o shorthand for above 12s 2 Gne fnt 71700 RHS is Fourier series representation of gp nis equality is not pointwise equality see pp 6273 o notation 9plt39gt a G71 0 Parseval7s theoreni 772 cm 42 lgplttgtl2 dt Tngoo lGnl2 LHS is energy in gp over T2 T2 o corollary Why T2 gplttgt gp7mlttgt2 dt TH Gn2 7772 Cont TimeDisc Frequency III 0 corollary lgplttgtl2 dt 200 Gn2 LHS is power in gp related to variance 0 what are energy amp power over mTQ mTQ 0 can decompose power into pieces associated with fnls 0 de ne discrete power spectrum for gp Sn E Gn2 0 can we recover gp from Sn 0 rst example Figures 6172 27r periodic function of Equation 60a 2 91205 E 22008 related to AR1 process if lt 1 square integrable amp Gm gblnl mth order Fourier approximation gp7mt n in Gne mt 1 2 mi gb cos nt 0 two other examples homework exercise 5 Cont TimeCont Frequency I o assumption OO square integrable LOO Iglttgt2 dt lt oo 0 de nition Fourier transform or analysis of g GU L 9lttgt67 2m dt oo lt f lt oo 0 can recover from Fourier synthesis glttgt 128 L anew df is inverse Fourier transform of 00 T 2 W is my 9 iTQ 6723927rfnu du z 27rfnt LO View T 6722qu 623927rft o motivate above using gp7m Figure 64 pp 6475 905 E N Cont TimeCont Frequency II o shorthand for above 90 gt 00 is Fourier transform of is inverse Fourier transform of amp form a Fourier transform pair many different conventions o Parsevalls theorem L 9lt1 gt2 dt L Gfgt2 df LHS is energy in over 00 00 what would power be 0 de ne energy spectral density function Gltfgt2 0 can write Gfewf Gltf de nes amplitude spectrum 6f de nes phase function Cont TimeCont Frequency III 0 example Figure 67 67mg eiwfg Change of variable yields gU E GUM Where 1 t2 2U2 727T2f2g2 9alttgtW 7 Galtfgt note GUM real valued so can see GU2 easily TimeBandLimited Functions 0 time limited to T T if glttgt 0 for all t gt T for some T lt oo lots of examples 0 band limited to VV W if 0 for all gt W for some W lt oo male speech limited to 8000 HZ cycles second orchestra limited to 20000 HZ has representation M 8 EV Gltfgte 2r df can be differentiated arbitrary number of times very smooth 0 can be both time and band limited o of considerable interest Chapters 6 and 7 time limited sequences that are Close to band limited Similarity Theorem 0 rst of three reciprocity relationships ground levelunderworld p 66 from Bracewell o lt gt implies lallngt Wancw o for a gt 1 a129at formed by contracting horizontally expanding vertically 0 example Figure 71 a 1 2 4 l 9 cal2W Gm Equivalent Width 0 measures concentration of signal in time 0 best if real nonnegative even cont at O 0 de nition widthe 90 E L 905 cit90 0 Width of rectangular signal Whose height is 90 area is area under curve of 0 see Figure 72 0 note area L 9t dt Glt0gt amp 90 700 OO Gltfgtdf 0 implies widthe go Gltogt Lquot Gm cl 1 widthe G 0 product of Widths of signal amp transform unity Fundamental Uncertainty Relationship if real amp nonnegative with unit area then is probability density function pdf consider uniform pdf 71117 WT centered at Mr Width 2Wr7 height l2WT variance measures spread of pdf 00 W2 LOO t my Mt MT WT dt 37 relate natural Width77 2WT amp 03 2WT 20M 3 2 0 if has nonunit area forni Lat same Widthl 5105 E 900 L 9006115 de ne WidthV E 20 3 for general g use Widthv suppose integrates to unity ie is a pdf LO lglttgtl2 dt 1 LO Gltfgtl2 df let 03 02 be variances of g2 can show pp 7374 a X 0 Z ll67r2 note equality holds only in Gaussian case 12 Convolution Theorem 0 brie y convolution in time domain same as multiplication in frequency domain 0 convolution of amp is this function of t L guht u du E g gtllt Mt assumes integral exists reflect and translate77 second function g gtllt notation for function de ned above change of variable shows h same as g o Fourier transform of gh see p 82 Exercise 38 L 9 htgt 2wft dt GUWU 9 M lt gt Glt39gtHlt39gt o variety of convolution theorems in literature stipulate conditions for above to hold Convolution as Smoothing Operation 0 regard as a signal as a smoother lter 0 can regard g as smoothed version of 0 example L a signal gt Z A cos 27rth 251 11 i pig202 a smoother Mt 7 lt27r02gt12 a is adjustable smoothing parameter smoothed version of pp 8374 L g ht Z UQMWQAZ cos 27rth gbl 11 gtllt frequencies phases unchanged gtllt O lt e 02 fz22 lt 1 is attenuation factor gtllt smoother shrinks amplitudes toward 0 gtllt as fl gt O attenuation factor increases to l gtllt as l gt oo attenuation factor decreases to O gtllt reduces amplitudes of high frequency terms Speci c Examples of Smoothing example With f1 16 and f2 3 Figure 83 1 gt 5 cos 27F 05 cos 2W3t 11 attenuation factors f1 1 6 099 097 081 f2 3 017 00 00 a 01 a 025 a 0625 12d 0 smoothed version ofg note sincu E sin7ru7ru gwlttgt 21 6 6 t amp another smoother rt th o erw1se H6 9Wgt L du Z sinc 2fl6Al cos 27rth 0 11 sinc 216 varies about 0 not monotonic in fl example Figure 86 6 16 eliminates f2 term completely 6 14 causes ripples prefer smoothers With monotonic attenuation 15 Cross and Autocorrelations o variations on convolution idea 0 cross correlation of and h 9 Mt E L guhu t du 0 can show Exercise 37al 9 9lt agt o letting yields autooorrelation 9 906 E L gltugt9ltu t du 39 have 9 90 gt 000 IGHI2 0 leads to another measure of Width widtha 90 widthe 9 90 3 00 9 glttgt dt 9 90 Wny lfgooo glttgt dtl2 000 gt2 dt will prove useful in Chapter 6 Disc TimeCont Frequency I o assumptions g nite energy cont at tAt t O1 samples of g gt E gltt At note At gt O is time interval between samples sequence gt square summable t 00 gt2 lt oo 0 de nition discrete Fourier transform of gt is Gpltfgt E Att in gteiz39QWftAt rst motivation if gt G then am 2 f glttgterWdt mt m9lttmgtewm 120 22 second motivation use Dirac delta functions p 88 723927Tt 0 reason for p subscript note 6 1 for integer t Gpltf i0 gt 7 27rf11 ttAt t7oo At gteiz39watAtei wt Gpltfgt t7oo Gp is periodic with period T 1At deja vul 17 Disc TimeCont Frequency II 0 apply cont timedisc freq theory to Gp o Fourier ooe ioients for Gp are say 1 T2 42wa i n i gn TiTQ Gpte dt With fn 7 T 7 nAt i 12At 7227mm At 7 At 71W 0mg dt 0 Fourier synthesis of Gp is thus 00 oo GP 712700 gne wfnt Z gne mtn At 0 changing n to t and ii t to f yields 12At 7239 W gt 712AtGPltfgte 2 ftAt Gpltfgt t goo gte At o letting gt gntAt yields gt lfjfthpUkWWdf Gpltfgt Att gteiz39watAt 2nd equation is de nitidnjolst gives inverse DFT o notation gt agt Gp o Parseval etc falls out readily 18 Two Questions of Interest 0 given just 9771 gm a nite saniple how well can Gp be approximated 0 how are and Gp related 0 answers to questions involve discussion of leakage convergence factors Windows aliasing Finite Sample Approximation of Gp 0 assuming At 1 can approximate Gp using Gmm t gtemft m 12 239 7T 7239 7T gmwapww we Efngpltf gt emfW df t7m 13 12 27 1 12 GPltfIgtD2mlltf J df D2m1 is Dirichlet7s kernel ie oc FT of rectangular sequence rt example of inverse convolution theorem 9t gtlt 7 lt gt 2m 1gtGp D2m1lt gt approximation best in least squares sense 0 Figure 91 shows D2m1 for m 41664 ideally would like to have Dirac 6 function Why central lobe smears out features Figure 92 loss of resolution due to nite sample of data sidelobes cause leakage and Gibbs Figure 93 note some sidelobes are negative 0 can reduce leakage amp Gibbs using Cesaro sums 20 Cesaro Sums I 0 let u2 111 uo ul U2 be an in nite sequence m 0 form mth partial sum 5m E Z ut 7m 0 form average of partial sums of orders 0 m 1 1 mil m LYLE 257 1 Mt m j O tim m to see this work out What am is for eg m 3 0 above called two sided Cesaro sum 0 theorem if 5m gt 5 then am gt 5 also 0 application let 5m kin gteiz39wat Gimme 0 since Gp7mf gt GPO must also have Gl lm 1 gtem a GM ie is another approximation for Gp Cesaro Sums II claini m 7239 7T Ghomlfl tm1 E gte 2 ft 12 m2 112 GpltfgtDEnltf f df sketch of proof 1 t m oc convolution of n With itself FT of rt oc thus FT of 1 oc FT of 1 gtlt gt convolution of FTs related to Fej rls kernel Chapter 6 comparing D2m1 Fig 91 to Fig 95 has snialler sidelobes hurrayl has nonnegative sidelobes hurrayl has Wider central lobe boo conipare Figures 96 and 93 to see tradeoffs m more generally approximate Gp using 2 ctgte t7m ct7s are convergence factors or Windows 22 723927rft Relating Gp to G Aliasing o assunie general At ie At y 1 necessarily 12At Z W o gt agt GPO iniplies gt 712At GPltfgte 2 ftAtdf 39 9gt and gt gt At imply gt r eavewtmdf n 2k12At mftm I i kOO2k712At Gltfgt df i 00 1mm 2 k AttAt 7 kiOO712AtGltfkAtgt df wheref Ef kAt note 27TfkAttAt 2 27TftAtez3927rkt 2 27TftAt7 1mm 00 z vr 9t nm k ltfkAtgtgt 2 def 700 SO 0 equating integrands CPU k Cf kiAt 0 holds for f g 12At E fltNgt E Nyquist frequency 0 fikAt k 75 O are aliases off see Figures 9879 0 highest f that is not an alias of a lower freq is fN 0 when can we recover perfectly froni Gp 23 DiscCont Concentration Problem 0 assumptions At 1 for convenience gt real valued amp time limited to t O N 1 yr gt 9220 12 N71 2 2 energy gt 12Gpltfgt df 0 how close can Gp be to bandlimited o for O lt W lt 12 consider concentration measure 12 WW 2 leltfgtl2 df12Gpltfgtl2 df 0 reduction of numerator using z2 zz N4 N71 11 IGpltfgt2 df jglt12 gt z27rftgt it0 gte wft df N71 N71 W Z W 7 t2 t2 gtgt LWQ 2 m 0 df i N1N1 sin 27rWt t i 220 tgo gtgt 7Tlttl t g E 907 39 39 39 79N71T o A matrix t tth element sin 27rWt t7rt t 0 WW gTAgng Solution to Concentration Problem 2 o to maximize 52W differentiate wrt g d d ng 0 solution g satis es Ag 62Wg eigenvalueeigenvector problem Ag g solution is eigenvector say v0N W associated with largest eigenvalue say 0N W v0N W is subsequence of 0th order dpss A is positive de nite so all N eigenvalues are positive can show that eigenvalues are distinct so order as O lt AN1N W lt lt 1N W lt 0NW lt1 why must 0N W be less than unity rst 2N W Shannon nuniber eigenvalues close to 1 after which NAN W7s fall off rapidly to O can use eigenvectors to form orthonormal basis 1 J k 0 otherwise vjnv WTvN W see Figures 106710 Disc TimeDisc Frequency I o gt t O N 1 sampled At units apart 0 two possible de nitions for Fourier transform form in nite sequence set gt E O for other t7s oo N71 At 2 gt 7227rftAt At 2 gtei wftAt t7oo t0 useful eg periodogram but in nite of f7s de ne discrete Fourier transform DFT of gt N71 N71 G E At 2 gtei wfntAt At 2 theiz27rntN7 t0 t0 Where fn are N Fourier standard frequencies annNAt n01N 1 o inverse DFT derived as follows Nil Gnet27mtN Nil Nil gt t27mt7tN 710 710 t0 N71 N71 Z gt 2 227mt itN AtgtN t0 710 1 N71 o inverse DFT gt Z Gne mtN 710 0 fast Fourier transform FFT algorithm vs DFT 0 two standard forms At 1 or At 1N 26 Disc TimeDisc Frequency II o DFT de nition implies gt 0 beyond t O to N 1 o notion of zero padding add N N gt O zeros to go gN1 to form 90gN1withgtON t N 1 DFT of padded sequence E nN At Nil 00 r G E At 120 gte mtN Attoo gtei mtN Gplf evaluates Gp over ner grid than fn7s useful to compute oonvolutions or DFT via Chirp transform algorithm p 114 0 can also Claim DFT implies periodic extension 1 N71 2 GR 22mm tlt0 tgtN gt NA 6 or so gt t O l1 l2 has period N 0 can use RHS of DFT to de ne G for all n so 1 Nk71 gt i NAt nk for any integer k G 2 27mtN n o summary of Fourier theory pp 11679 27
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