A6523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2013 Lecture 4 For next week be sure to have read Chapter 5 of

Transcription

1 A6523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2013 Lecture 4 For next week be sure to have read Chapter 5 of Gregory (Frequentist Statistical Inference) Today: DFT of complex sinusoid (continued)» Detection issues: threshold and false-alarm probability; ROC curves» Examples: SETI, Continuous wave detection of gravitational waves with LIGO Sampling theorem and interpolation Problem Set 2: look for it on the web page tomorrow (Friday Feb 1)

2 Main Points from Lecture 3 Random walks are nonstationary processes that are not ergodic. CLT works for the DFT but does not for the power-spectrum (PS) estimate. The PS is chi-square distributed with two degrees of freedom, i.e. the PDF is a onesided exponential. Smoothing of the PS increases the number of degrees of freedom, making the PDF tend toward a Gaussian in accord with the CLT. Detection and false-alarm probabilities defined for phasor + noise case.

3 White noise RW 1 RW 2 RW 3

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7 Random Walk Examples Spinning objects: Earth, neutron stars Steps in torque or spin rate Observable = spin phase Scattered photon propagation (diffusion) Step = mean-free path Observable = propagation time Cosmic-ray propagation in the Galaxy Step = scattering off of small-scale magnetic field variations Observable = `grammage of interaction based on isotopic content (typically ~ 5 g cm -2 ) Orbital perturbations Asteroid belt objects! Near Earth Objects Motions of planetesimals in protoplanetary disks Galactic orbits of stars from gravitational potential granularity (molecular clouds, spiral arms)! diffusion of stellar populations

8 Other Random Walk Examples MCMC: random walk in parameter space Brownian motion of a dust particle Molecular diffusion Diffusion of biological populations Options pricing in financial markets Step = transaction Observable = price Black-Scholes equation = Fokker-Planck equation

9 Back to Phasor + Noise

10 DFT + noise N=1024 P = 3.99 F = 256

11 Spectrum

12 SETI Search for Extraterrestrial Intelligence First search of nearby stars: 1960 by Frank Drake Typical assumed signal type (radio): monochromatic carrier signal + noise Other cases: sharp pulses (radio, OIR)

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15 Gravitational Waves LIGO = Laser Interferometer Gravitational Wave Observatory Measure effective distance between pendula using lasers Many possible signal types: continuous wave, chirped, bursts (CWs from rotating neutron stars with mountains ) CW analysis done as part of the Einstein@Home volunteer network

16 Gravitational Waves CW = Continuous Wave See for details and image credits

17 Gravitational Waves Chirped GW

18 Gravitational Waves Burst GW

19 Gravitational Waves Stochastic GW

20 Gravitational Waves

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22 LIGO detector in Hanford, WA; A second detector is in Livingston, Louisiana

23 PDF of Phasor Magnitude (Rice distribution) s = 0, 3, 5, 10 sigma_n = 1

24 PDF of Intensity s = 0, 1, 3, 5 sigma_n = 1

25 ROC Curves Receiver Operating Characteristics In a so-called detection problem, we try to establish whether a signal of some assumed type is present in data that include noise This is a universal problem that applies to many laboratory and observational contexts. In astronomy, ROC curves apply to finding sources/signals in images, spectra, time series, etc. An ROC curve = P d vs P fa (detection vs falsealarm probability)

26 ROC Curve Examples

27 LIGO noise! white noise

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29 Arecibo Pulsar Survey Data Flow Survey analysis + Einstein@Home extended survey analysis 13 Nov '11 Challenges Pulsars and Gravity SP 29

30 Fourier Transforms II This lecture is largely about discreteness effects. DISCRETE IN ONE DOMAIN PERIODIC IN THE CONJUGATE DOMAIN We will consider here periodic extensions and sampled versions of continuous functions f(t). Graphically, we have: f(t) F (f) f p (t) Fp (f) f ps (t) the act of sampling assumes or imposes periodicity in the f domain Fps (f) if the periodic extensions overlap, then there are problems (aliasing) Related Kinds of Fourier Transforms/Series The essence of Fourier analysis is the expansion of a function into a series of sinusoidal functions. The particular form of the expansion depends on the kinds of functions we are expanding. We classify functions according to the DT/CT dichotomy but also according to whether they are periodic/aperiodic. The most general case is the aperiodic/ct signal: We have already motivated the Fourier transform relation for these signals x(t) = 1 2π dw X(ω) e iωt (1) X(ω) = dt x(t) e iωt (2) 1

31 Note that ω is the radian frequency which is related to cyclic frequency f by ω =2πf. Defining the F.T. in terms of f yields x(t) = df X(f) e 2πift (3) X(f) = dt x(t) e 2πift (4) Note disappearance of (2π) 1 in front. Some definitions of the ω transform place a factor of 1 2π in front. Now, from the aperiodic /CT transform relations, we can derive Fourier series for periodic /CT functions and the discrete Fourier transform for DT functions. Kinds of Fourier Transforms: Fourier transform Fourier Series DFT function f(t) f p (t) f ps (t) periodic extention sampled ( t) of f(t), period = and periodic extention transform F (f) ak = 1 F ( k ) a k continuous? continuous, continuous, discrete, (fn, transform) continuous discrete discrete periodic? aperiodic, periodic, periodic, (fn, transform) aperiodic aperiodic periodic 2

32 Useful Quantities: We will use the following FT relations for delta functions: 1 δ(f) δ(t) 1 where implies a Fourier transform relation. It is useful to consider a periodic train of impluses s(t) = δ(t n ) n= Shift theorem If, f(t) F (f) then, f(t t 0 ) e 2πift0 F (f) This follows by inspection of the definition for F : F.T. {f(t t 0 )} = dt f(t t 0 ) e 2πift ; change variable to t = t t 0 t = t + t 0 : F.T. = dt f(t ) e 2πift e 2πift0 F (f) Similarly, if f(t) F (f) then e +2πif0t f(t) F (f f 0 ) This is the basis for heterodyned receiver systems. 3

33 Sampling Function: Using the first form of the shift theorem we have so, δ(t n ) e 2πifn S(f) = n= e 2πifn. How does this behave? As a series of delta functions in f. Note for f = integer S(f) is the integral of a sinusoid (many cycles) 0. However, for f = k =0, ±1,..., S(f) Thus, we can write (guessing) S(f) = 1 k= δ(f k ) We introduce the factor 1 to keep dimensionally correct, note S has units of s time and s has units of (time 1 ) S is dimensionless. Note that s(t) has a Fourier transform S(f) that is of the same form as s(t) (a sequence of delta functions). The only other function I know of that has this property is a Gaussian function: e πt2 e πf 2 In the following we will use the sampling function several times. For convenience, we will label it with the period : S(t, ) n δ(t n ). 4

34 FOURIER SERIES Consider a function f(t) F (f) which is time limited: f(t) =0outside some range (t a,t b ). Now convolve f(t) with the impulse train and find the resultant fourier transform. This is trivial because of the convolution theorem. f p (t) f(t) S(t, ) = f(t) n n= f(t n ) δ(t n ) F (f) S(f) = 1 = 1 k k δ f k F (f) k F δ f k = periodic function with period ; has discrete Fourier transform Thus, or (simplifying further) f p (t) = f p (t) 1 k k F δ f k df F p (f) e 2πift (5) = = 1 f p (t) = k df 1 k k k F δ f k e 2πift (6) F ( k ) e2πi( k )t (7) a k e 2πi( k )t, a k F ( k ) (8) 5

35 Note that the F.S. is a series in orthonormal basis functions g k (t) =e 2πi( k )t and coefficients a k. We can show that : /2 /2 dt g k g k = = = /2 /2 /2 /2 dt g k (t) g k (t) =δ kk dt e 2πi(k k )t/ 2πi(k k ) [e2πi(k k )/2 e 2πi(k k )/2 ] π(k k ) sin π(k k )= 1 k = k 0 otherwise Thus we have a Fourier series representation of the periodic signal f p (t) in terms of the Fourier transform of the aperiodic function f(t). We can write this in the form of a cos and sin series by expanding the exponential and using a k = b i + ic k f p (t) = k a k e 2πi( k )t = k (b k + ic k )(cos 2π k t + i sin 2π k t) Graphically, we have f(t) F (f) f p (t) Fp (f) periodicity in the time domain discreteness in the frequency domain By reciprocity we can show that discreteness in the time domain periodicity in the f domain 6

36 DISCRETE FOURIER TRANSFORM A periodic (in t) function can be described by a set of discrete coefficients in a Fourier series. By discretizing the time domain function (still periodic) we can show that a discrete set of time-domain sample describes the resultant periodic and sampled function: 1. Start with f(t) =CT/aperiodic function. 2. Extend it periodically, as before: and write as a Fourier series: f p (t) =f(t) s(t, ) f p (t) = k now CT/periodic a k e 2πi( k )t 3. Now sample f p (t) to get a DT/periodic function: f ps (t) f p (t) S(t, t) =f p (t) n δ(t n t) = k a k e 2πi k t n δ(t n t) k 2πi( 4. Now multiply f ps by e )t and integrate over one period i.e. apply 0 k dt e 2πi( )t to the equation in 3.: 0 k 2πi dt e t f p (t) δ(t n k 2πi t) = dt e t k 2πi a k e t δ(t n t), n 0 k n We integrate over one period so f p (t) =f(t) in this interval. Therefore, LHS = f(n t) e 2πi k n t n :0 n t<. Now RHS = k a k dt e 2πi (k k )t δ(t n t) = n 0 k =e 2πi (k k )n t e 2πi(k k )n t n a k 7

37 Note: {n :0 n t< } or {n :0 n <N = no. samples in interval (assume an integer)} t thus {n :0 n N 1} 0 k = k sum of integer N 1 But e 2πi (k k )n t cycles of sinusoids = n =0 1=N k = k n Nδ kk Kronecker delta Therefore RHS = k a k N δ kk = Na k Thus (using N t ) Na k = n :0 n t< f(n t) e 2πi k n N N 1 a k = N 1 f(n t) e 2πi k n N, k =0,...,N 1 ( ) n 0 In words: The Fourier series coefficients a k for a periodic and sampled function are represented as discrete sums of samples over one period. Note that a k is manifestly periodic: N 1 a k +N N 1 f(n t) e 2πin (k +N)/N n =0 e 2πin k /N e 2πin 1 8

38 N 1 5. Now find the inverse transform: Apply e 2πi k n k =0 N to eqn. : N 1 k =0 a k e 2πi k n N N 1 = N 1 f(n t) n =0 N 1 k =0 e 2πik (n n ) N N δ nn N 1 n f(n t) N δ nn = f(n t) or f(n t) = N 1 k =0 a k e 2πi k n N Thus the samples of a periodic, sampled function can be written as a discrete finite Fourier series of the F.S. coefficients. Eqn. and constitute the discrete Fourier transform. Notation: can drop t, rename a k so that we have DFT : f n = k F k e 2πink/N F k = N 1 n f n e 2πink/N Note that the function is periodic in both domains: f n+n = f n, F k+n = F k 9

39 SAMPLING THEOREM Consider a function that has finite frequency extent, e.g. a function that is band limited in the interval [ f, f]. Then the function can be determined totally by sampling it in the time domain at the rate 2 f. Samples must occur at intervals t 1 2 f Plausibility: consider a sine wave of frequency f f(t) = A cos 2π ft A 2 (e2πi ft + e 2πi ft ) (9) F (f) = A 2 [δ(f f)+δ(f + f)] (10) The sampling theorem says to sample this a rate of at least This requires at least two samples per period. f s =2 f or at intervals t = 1 2 f. One might be unlucky if the samples occur at the zero crossings, but this is ok because, by definition, there is no energy in the sine wave (A =0)for a bandlimited function. 10

40 Logic for derivation of sampling theorem: I. First impose periodicity in the f domain and then derive interpolation formula for critically sampled data. II. Then sample in the t domain at arbitrary rate and derive the resultant interpolation formula. Demonstration of sampling theorem (applies to bandlimited signals) Consider a bandlimited function f(t) F (f), F (f) =0for f f f(t) is CT/aperiodic f(t) = df F (f) e 2πift = f f df F (f) e 2πift our choice here with W =2 f critical sampling Now define a periodic version of F (f) called F p (f): Since F p (f) is periodic, by the duality theorem [if f(t) F ( f)], then F (t) f( f)] we can write it as a Fourier series: period W =2 f (11) F p (f) = C m e 2πi m W f (12) m= where the Fourier coefficients are (as usual) C m = 1 W = 1 2 f W/2 W/2 f f df F p (f) e +2πi( m W )f (13) df F (f) e +2πi m 2 f f (14) 11

41 By comparison with the expression for f(t) (and from our knowledge of how F.S. coefficients relate to the F.T.), we have C m = 1 2 f f(t) t= m = 1 m 2 f 2 f f 2 f Thus, F p (f) = 1 2 f m= m f 2 f e 2πi m 2 f f Consequently, since F (f) = F p (f) in the range f f this result implies that F (f) can be determined by sampling f(t) at intervals 1 2 f. Then, from the fact that f(t) F (f) it follows that the t domain function itself can be determined at all times from a set of discrete samples. The interpolation formula shows that the continuous frequency-domain function can be calculated from discrete data. 12

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