A Bayesian Approach to Spectral Analysis

Transcription

1 Chirped Signals A Bayesian Approach o Specral Analysis Chirped signals are oscillaing signals wih ime variable frequencies, usually wih a linear variaion of frequency wih ime. E.g. f() = A cos(ω + α 2 + θ). Examples: plasma wave diagnosic signals Signals propagaed hrough dispersive media (seismic cases, plasmas) Graviaional waves from inspiraling binary sars Doppler-shifed signals over fracions of an orbi (e.g. acceleraion of pulsar in is orbi) Jaynes Approach o Specral Analysis: cf. Jaynes Bayesian Specrum and Chirp Analysis in Maximum Enropy and Bayesian Specral Analysis and Esimaion Problems Cied by Brehors in Bayesian Specrum Analysis and Parameer Esimaion, (1987) Briefly by Gregory in Chaper 13 of Bayesian Logical Daa Analysis for he Physical Sciences Resul: Opimal processing is a nonlinear operaion on he daa wihou recourse o smoohing. However, he DFT-based specrum (he periodogram ) plays a key role in he esimaion. 1

2 Sar wih Bayes heorem p(h/di) }{{} poserior prob. = p(h/i) }{{} prior prob. new daa {}}{ p(d/hi) p(d/i) In his conex, probabiliies represen a simple mapping of degrees of belief ono real numbers. Recall p(d/hi) vs.d for fixed H = sampling disribuion p(d/hi) vs.h for fixed D = likelihood funcion Read H as a saemen ha a parameer vecor lies in a region of parameer space. 2

3 Daa model: y() = f() + e() f() = A cos(ω + α 2 + θ) wih ω = ω 0 and α = α 0 for he daa e() = whie gaussian noise, e = 0, e 2 = σ 2 Daa Se: D = {y(), T }, N = 2T + 1 daa poins. 3

4 Daa Probabiliy: The probabiliy of obaining a daa se of N samples is P (D HI) = P [y()] = T = T (2πσ 2 ) 1/2 e 1 2σ2[y() f()]2, (1) which we can rewrie as a likelihood funcion once we acquire a daa se and evaluae he probabiliy for a specific H. Wriing ou he parameers explicily, he likelihood funcion is L(A, ω, α, θ) e 1 2σ 2 T = T [y() A cos(ω + α 2 + θ)] 2 For simpliciy, assume ha ωt 1 so ha many cycles of oscillaion are summed over. Then cos 2 (ω + α 2 + θ) = 1 2 [1 + cos 2(ω + α2 + θ)] 2T N 2 4

5 Expand he argumen of he exponenial in he likelihood funcion, we have [ y() A cos(ω + α 2 + θ) ] 2 = y 2 () + A 2 cos 2 (ω + α 2 + θ) 2Ay() cos(ω + α 2 + θ) We care only abou erms ha are funcions of he parameers, so we drop he y 2 () erm o ge 1 2σ 2 T = T [y() A cos(ω + α 2 + θ)] 2 1 [A 2 cos 2 (ω + α 2 + θ) 2Ay() cos(ω + α 2 + 2σ 2 The likelihood funcion becomes L(A, ω, α, θ) e A σ 2 A σ 2 y() cos(ω + α 2 + θ) NA2 4σ 2 y() cos(ω + α 2 + θ) NA2 4σ 2 Inegraing ou he phase: In calculaing a power specrum [in his case, a chirped power specrum ( chirpogram )], we do no care abou he phase of any sinusoid in he daa. In Bayesian esimaion, such a parameer is called a nuisance parameer. Since we do no know anyhing abou θ, we inegrae over is prior disribuion, a pdf ha is 5

6 uniform over [0, 2π]: f θ (θ) = 1 2π 0 θ 2π 0 oherwise. The marginalized likelihood funcion becomes Using he ideniy we have L(A, ω, α) 1 2π dθ L(A, ω, α, θ) 2π 0 = 1 2π A dθ exp 2π 0 σ 2 = exp NA2 4σ 2 1 2π 2π 0 y() cos(ω + α 2 + θ) NA2 4σ 2 dθ exp A σ y() cos(ω + α 2 + θ) cos(ω + α 2 + θ) = cos(ω + α 2 ) cos θ sin(ω + α 2 ) sin θ y() cos(ω + α 2 + θ) = cos θ y() cos(ω + α 2 ) } {{ } P y() sin(ω + α 2 ) }{{} Q P cos θ Q sin θ = P 2 + Q 2 cos[θ + an 1 (Q/P )]. 6

7 This resul may be used o evaluae he inegral over θ in he marginalized likelihood funcion: 1 2π 2π 0 dθ exp A σ y() cos(ω + α 2 + θ) To evaluae he inegral we use he ideniy, This yields 1 2π 2π 0 I 0 (x) 1 2π dθ exp A σ 2π 0 = 1 2π 2π 0 dθ e A σ 2 dθ e x cos θ = modified Bessel funcion y() cos(ω + α 2 + θ) P 2 + Q 2 cos[θ + an 1 (Q/P )] }{{} irrelevan phase shif = I A 0 P σ Q 2 We now simplify P 2 + Q 2 : P 2 + Q 2 = [ y() cos(ω + α 2 ) ] 2 + [ y sin(ω + α 2 ) ] 2 P 2 + Q 2 = = y()y( ) [cos(ω + α 2 ) cos(ω + α 2 ) + sin(ω + α 2 ) sin(ω + α 2 )] }{{} cos[ω( ) + α( 2 2 )] y()y( ) cos[ω( ) + α( 2 2 )]. 7

8 Define C(ω, α) N 1 (P 2 +Q 2 ) = N 1 y()y( ) cos[ω( )+α( 2 2 )], Then he inegral over θ gives 2π 0 dθ L(A, ω, α, θ) I 0 A NC(ω, α) σ 2 and he marginalized likelihood is L(A, ω, α) = e NA2 4σ 2 I 0 A NC(ω, α) σ 2. 8

9 Noes: (1) The daa appear only in C(ω, α). (2) C is a sufficien saisic, meaning ha i conains all informaion from he daa ha are relevan o inference using he likelihood funcion. (3) How do we read L(A, ω, α)? As he probabiliy disribuion of he parameers A, ω, α in erms of he daa dependen quaniy C(ω, α). (Noe ha L is no normalized as a PDF). As such, L is a quie differen quaniy from he Fourier-based power specrum. (4) Wha is he quaniy C(ω, α) N 1 y()y( ) cos[ω( ) + α( 2 2 )]? For a given daa se, ω, α are variables. If we plo C(ω, α), we expec o ge a large value when ω = ω signal, α = α signal. (5) For a non-chirped bu oscillaory signal (ω 0, α = 0), he quaniy C(ω, α) is nohing oher han he periodogram (he squared magniude of he Fourier ransform of he daa). We hen see ha, for his case, he likelihood funcion is a nonlinear funcion of he Fourier esimae for he power specrum. 9

10 Inerpreaion of he Bayesian and Fourier Approaches We found he marginalized likelihood for he frequency and chirp rae o be L(A, ω, α) = e NA2 4σ 2 I 0 A NC(ω, α) and he limiing form for he Bessel funcion s argumen x 1 is I 0 (x) ex 2πx. σ 2. In his case he marginalized likelihood is L(A, ω, α) e NA2 4σ 2 I 0 A N C(ω, α) σ 2 e ( 2πA N C(ω, α)/σ 2 ) 1/2. e NA2 4σ 2 A N C(ω, α) σ 2 Since C(ω, α) is large when ω and α mach hose of any rue signal, we see ha i is exponeniaed as compared o appearing linearly in he periodogram. 10

11 Now le s consider he case wih no chirp rae, α = 0. Examples in he lieraure show ha he widh of he Bayesian PDF is much narrower han he periodogram, C(ω, 0). Does his mean ha he uncerainy principle has been avoided? The answer is no! Uncerainy Principle in he Periodogram: For a daa se of lengh T, he frequency resoluion implied by he specral window funcion is Widh of he Bayesian PDF: δω 2πδf 2π T. When he argumen of he Bessel funcion is large he exponeniaion causes he PDF o be much narrower han he specral window for he periodogram. 11

12 Inerpreaion: The periodogram is he disribuion of power (or variance) wih frequency for he paricular realizaion of daa used o form he periodogram. The specral window also depics he disribuion of variance for a pure sinusoid in he daa (wih infinie signal o noise raio). The Bayesian poserior is he PDF for he frequency of a sinusoid and herefore represens a very differen quaniy han he periodogram and are hus no direcly comparable. 12

13 1. The Bayesian mehod addresses he quesion, wha is he PDF for he frequency of he sinusoid ha is in he daa.? 2. The periodogram is he disribuion of variance in frequency. 3. If we use he periodogram o esimae he sinusoid s frequency, we ge a resul ha is more comparable: (a) Firs noe ha he widh of he poserior PDF involves he signal o noise raio (in he square roo of he periodogram) NA/σ while he widh of he periodogram s specral window is independen of he SNR. (b) General resul: if a specral line has widh ω, is cenroid can be deermined o an accuracy δω ω SNR. This resul follows from mached filering, which we will discuss laer on. (c) Quaniaively, he periodogram yields he same informaion abou he locaion of he specral line as does he poserior PDF. 4. Problem: derive an esimae for he widh of he poserior PDF ha can be compared wih he esimae for he periodogram. 13

14 Comparison of Specral Line Localizaion Properies Claim: While he periodogram gives a specral line ha is much broader han he widh of he poserior PDF for frequency, he abiliy o localize he specral line in frequency is he same for boh approaches. Periodogram: The signal-o-noise raio (S/N) of he line is NA/σ (as in DFT of complex exponenial). The specral resoluion is δω res 2π/(2T + 1) (since our ime inerval is [ T, T ]. The widh of he line (e.g. FWHM) is of order he specral resoluion. Assume S/N is large. Poserior PDF: The PDF for ω is dominaed by he exponenial facor E(ω) = exp{a NC(ω, 0)/σ 2 } From he expression for C we have C max = C(ω = ω 0 ) = NA 2 /4 so E max = e A NC max /σ 2 = e (N/2)(A/σ)2 For offse frequencies ω = ω 0 + δω we can expand various hings o show ha E(ω 0 + δω) E max e (N/2)(A/σ)2 [δω(2t +1)/ 12] 2 This funcion has a widh when he exponenial = 1/2 or δω = 12 In erms of resoluion unis his is δω = ( ) σ 12 δω res NA = σ N(2T +1) NA 1 S/N of line in periodogram 14

15 Figure 1: Lef: Time series of sinusoid + whie noise wih A/σ = 1 sampled N = 500 imes over an inerval of lengh T = 500. Righ: Plo of he periodogram (red) and Bayesian PDF of he ime series. 15

16 Figure 2: Lef: Time series of sinusoid + whie noise wih A/σ = 1/4 sampled N = 500 imes over an inerval of lengh T = 500. Righ: Plo of he periodogram (red) and Bayesian PDF of he ime series. 16