A6523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2013

Transcription

1 A6523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2013 Lecture 16 More on spectral analysis Reading: Chaper 13: Bayesian Revolution in Spectral Analysis (already assigned)

2 Upcoming Topics The remaining lectures will include these topics: Model fitting and statistical inference (frequentist and Bayesian) Model definition Linear and non-linear least squares, maximum likelihood Parameter errors (credible intervals, Fisher matrix) Parameter space exploration (grid, SA, GA, GHS) Markov processes and stochastic resonance MCMC (Hastings-Metropolis) Cholesky decomposition Principal component analysis (PCA) Localization/Matched filtering 1D, 2D problems (time, frequency/wavelength, image) Phase retrieval and Hilbert transforms Radon transform Extreme value statistics SA = simulated annealing, GA = Genetic Algorithm, GHS = Guided Hamiltonian Sampler

3 Chirped Signals A Bayesian Approach to Spectral Analysis Chirped signals are oscillating signals with time variable frequencies, usually with a linear variation of frequency with time. E.g. Examples: plasma wave diagnostic signals f(t) =A cos(ωt + αt 2 + θ). signals propagated through dispersive media (seismic cases, plasmas) gravitational waves from inspiraling binary stars doppler-shifted signals over fractions of an orbit (e.g. acceleration of pulsar in its orbit) Jaynes Approach to Spectral Analysis: cf. Jaynes Bayesian Spectrum and Chirp Analysis in Maximum Entropy and Bayesian Spectral Analysis and Estimation Problems Result: Optimal processing is a nonlinear operation on the data without recourse to smoothing. However, the DFT-based spectrum (the periodogram ) plays a key role in the estimation. 1

4 Start with Bayes theorem p(h/di) posterior prob. = p(h/i) prior prob. new data p(d/hi) p(d/i) As in Ch 1 of Gregory In this context, probabilities represent a simple mapping of degrees of belief onto real numbers. Recall p(d/hi) vs.d for fixed H = sampling distribution p(d/hi) vs.h for fixed D = likelihood function Read H as a statement that a parameter vector lies in a region of parameter space. Measured Quantity: y(t) = f(t)+e(t) Data Model 4 parameters: A, ω, α, θ f(t) = A cos(ωt + αt 2 + θ) e(t) = white gaussian noise,<e>=0,<e 2 >= σ 2 Data Set: D = {y(t), t T }, N =2T +1datapoints. 2

5 Data Probability: The probability of obtaining a data set of N samples is Should be product P (D HI)= P [y(t)] = Π T t= T (2πσ 2 ) 1/2 e X t 1 2σ2[y(t) f(t)]2, (1) which we can rewrite as a likelihood function once we acquire a data set and evaluate the probability for a specific H. Writing out the parameters explicitly, the likelihood function is L(A, ω, α, θ) e 1 2σ 2 T t= T [y(t) A cos(ωt + αt 2 + θ)] 2 For simplicity, assume that ωt 1 so that many cycles of oscillation are summed over. Then t cos 2 (ωt + αt 2 + θ) = 1 t 2 [1 + cos 2(ωt + αt2 + θ)] 2T +1 2 N 2 3

6 Expanding out the argument of the exponential in the likelihood function, we have [y(t) A cos(ωt + αt 2 + θ)] 2 = y 2 (t)+a 2 cos 2 () 2Ay(t)cos() We care only about terms that are functions of the parameters, so we drop the y 2 (t) term to get 1 2σ 2 T t= T The likelihood function becomes [y(t) A cos( )] 2 1 [A 2 cos 2 () 2Ay(t)cos()] 2σ 2 t L(A, ω, α, θ) e Integrating out the phase: A σ 2 t A σ 2 t y(t)cos() NA2 4σ 2 y(t)cos(ωt + αt 2 + θ) NA2 4σ 2 In calculating a power spectrum [in this case, a chirped power spectrum ( chirpogram )], we do not care about the phase of any sinusoid in the data. In Bayesian estimation, such a parameter is called a nuisance parameter. Since we do not know anything about θ, we integrate over its prior distribution, a pdf that is 4

7 uniform over [0, 2π]: f θ (θ) = 1 2π 0 θ 2π 0 otherwise. The marginalized likelihood function becomes 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 t y(t)cos(ωt + αt 2 + θ) NA2 4σ 2 dθ exp A σ 1 t y(t)cos(ωt + αt 2 + θ) Uniform pdf has maximum uncertainty Using the identity we have cos(ωt + αt 2 + θ) =cos(ωt + αt 2 )cosθ sin(ωt + αt 2 )sinθ t y(t)cos(ωt + αt 2 + θ) = cosθ t sin θ t y(t)cos(ωt + αt 2 ) P y(t)sin(ωt + αt 2 ) Q 5

8 P cos θ Q sin θ = P 2 + Q 2 cos[θ +tan 1 (Q/P )]. This result may be used to evaluate the integral over θ in the margninalized likelihood function: A P 1 = 2π σ Q 2 cos[θ + tan 1 (Q/P )] dθ e irrelevant phase shift 2π 0 To evaluate the integral we use the identity, I 0 (x) 1 2π dθ e x cos θ =modifiedbesselfunction 2π 0 This yields A = I0 P σ Q 2 We now simplify P 2 + Q 2 : P 2 + Q 2 = t y(t)cos(ωt + αt 2 ) 2 + t y t sin(ωt + αt 2 ) 2 = y(t)y(t )[cos(ωt + αt 2 ) cos(ωt + αt 2 ) t t + sin(ωt + αt 2 )sin(ωt + αt 2 ) cos[ω(t t )+α(t t ) 2 ] 6 Note issue with []; also (t-t ) 2 should be (t 2 t 2 ) here and in the following

9 Define P 2 + Q 2 = t t y(t)y(t )cos[ω(t t )+α(t t ) 2 ]. C(ω, α) N 1 (P 2 + Q 2 )=N 1 t t y(t)y(t )cos[ω(t t )+α(t t ) 2 ], Then the integral over θ gives I0 A NC(ω, α) and the marginalized likelihood is σ 2 L(A, ω, α) =e NA2 A 4σ 2 NC(ω, α) I 0 σ 2. 7

10 Notes: (1) The data appear only in C(ω, α). (2) C is a sufficient statistic, meaning that it contains all information from the data that are relevant to inference using the likelihood function. (3) How do we read L(A, ω, α)? As the probability distribution of the parameters A, ω, α in terms of the data dependent quantity C(ω, α). (Note that L is not normalized as a PDF). As such, L is a quite different quantity from the Fourier-based power spectrum. (4) What is C(ω, α) N 1 t t y(t)y(t )cos[ω(t t )+α(t t ) 2 ]? For a given data set, ω, α are variables. If we plot C(ω, α), we expect to get a large value when ω = ω signal, α = α signal. (5) For a non-chirped but oscillatory signal (ω = 0, α =0), the quantity C(ω, α) is nothing other than the periodogram (the squared magnitude of the Fourier transform of the data). We then see that, for this case, the likelihood function is a nonlinear function of the Fourier estimate for the power spectrum. 8

11 A Limiting Form: For argument x 1, the Bessel function I 0 (x) ex 2πx. In this case the marginalized likelihood is A NC(ω, α) L(A, ω, α) e NA2 4σ 2 I 0 σ 2 e 2πA NC(ω, α)σ 2 1/2. e NA2 4σ 2 A NC(ω, α) σ 2 Since C(ω, α) is large when ω and α match those of any true signal, we see that it is exponentiated as compared to appearing linearly in the periodogram. 9

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13 Interpretation of the Bayesian and Fourier Approaches We found the marginalized likelihood for the frequency and chirp rate to be L(A, ω, α) =e NA2 4σ 2 I 0 A NC(ω, α) and the limiting form for the Bessel function s argument x 1 is I 0 (x) ex 2πx. σ 2. In this case the marginalized likelihood is A NC(ω, α) L(A, ω, α) e NA2 4σ 2 I 0 σ 2 e 2πA NC(ω, α)σ 2 1/2. e NA2 4σ 2 A NC(ω, α) σ 2 Since C(ω, α) is large when ω and α match those of any true signal, we see that it is exponentiated as compared to appearing linearly in the periodogram. 10

14 Now let s consider the case with no chirp rate, α =0. Examples in the literature show that the width of the Bayesian PDF is much narrower than the periodogram, C(ω, 0). Does this mean that the uncertainty principle has been avoided? The answer is no! Uncertainty Principle in the Periodogram: For a data set of length T, the frequency resolution implied by the spectral window function is Width of the Bayesian PDF: δω 2πδf 2π T. When the argument of the Bessel function is large the exponentiation causes the PDF to be much narrower than the spectral window for the periodogram. 11

15 Interpretation: The periodogram is the distribution of power (or variance) with frequency for the particular realization of data used to form the periodogram. The spectral window also depicts the distribution of variance for a pure sinusoid in the data (with infinite signal to noise ratio). The Bayesian posterior is the PDF for the frequency of a sinusoid and therefore represents a very different quantity than the periodogram and are thus not directly comparable. 12

16 1. The Bayesian method addresses the question, what is the PDF for the frequency of the sinusoid that is in the data.? 2. The periodogram is the distribution of variance in frequency. 3. If we use the periodogram to estimate the sinusoid s frequency, we get a result that is more comparable: (a) First note that the width of the posterior PDF involves the signal to noise ratio (in the square root of the periodogram) NA/σ while the width of the periodogram s spectral window is independent of the SNR. (b) General result: if a spectral line has width ω, its centroid can be determined to an accuracy δω ω SNR. This result follows from matched filtering, which we will discuss later on. (c) Quantitatively, the periodogram yields the same information about the location of the spectral line as does the posterior PDF. 4. Problem: derive an estimate for the width of the posterior PDF that can be compared with the estimate for the periodogram. 13

17 Figure 1: Left: Time series of sinusoid + white noise with A/σ =1sampled N =500times over an interval of length T =500. Right: Plot of the periodogram (red) and Bayesian PDF of the time series. 14

18 Figure 2: Left: Time series of sinusoid + white noise with A/σ =1/4 sampled N =500times over an interval of length T =500. Right: Plot of the periodogram (red) and Bayesian PDF of the time series. 15

19 Maximum Likelihood Spectra Estimation (MLSE) MLSE is a misnomer; a better name is High Resolution Method because the method is derived by explicitly maximizing the sensitivity to a given frequency while minimizing the effects (i.e. leakage, a.k.a. bias) from other frequencies. The MLSE was developed in the 1960s by Capon to analyze data from arrays of sensors to maximize the response to one particular direction and minimize the response to others. e.g. LASA = Large Aperture Seismic Array (test earthquakes vs. underground nuclear tests). There is a close relationship to beam forming in acoustic arrays and beam forming in radio interferometric arrays. In the original development of the method discussed by Capon 1 the spectral estimator is very closely related to a filter that gives the ML estimate of a signal when it is corrupted by Gaussian noise: S + N A Ŝ 1 see Nonlinear Methods of Spectral Analysis, Haykin, ed. pp

20 This system involves: a) a filter that gives the ML estimate of the signal when corrupted by Gaussian noise is also... b) the filter that generally gives the minimum variance and unbiased estimate of the signal for arbitrary noise and... c) has coefficients that yield an unbiased, high resolution spectral estimate for any signal. The way the filter coefficients are derived (i.e. the constraints applied to the maximization problem) imply that the spectral estimate minimizes leakage. The HRM is sometimes described as a positive constrained reconstruction method which minimizes leakage. Thus, the intent of the MLSE technique is much different from the MESE technique: MLSE minimizes variance and bias (recall how spectral bias was related to resolution) MESE in effect (via its relationship to prediction filters) tries to maximize resolution 2

21 We will derive the ML spectral estimate following the derivation of Lacoss. Method: Construct a linear filter that 1. yields an unbiased estimate of a sinusoidal signal and 2. minimizes the variance of the output with respect to corrupting white noise. Pass a signal y n through a linear filter: y n a k x n x n = n a k y n k+1 k=1 (causal) where the input is of the form of a deterministic sinusoid added to zero mean noise having an arbitrary spectrum: y n = Ae iωn + n n. We will determine the coefficients a k by invoking the above two criteria. 3

22 Goal: We want the filter to pass Ae iωn undistorted but to reject the noise as much as possible. Thus, we require N 1. no bias (in the mean): x n a k y n k+1 k=1 N = a k Ae iω(n k+1) + n n k+1 k=1 N = a k Ae iω(n k+1) k=1 Ae iωn (if no bias) N a k e iω(1 k) = 1 constraint equation k=1 4

23 This can be written in matrix form using to denote transpose conjugate: ε a =1 ε 1 e iω e i2ω. e i(n 1)Ω a = a 1 a 2. a N 5

24 2. Minimum variance of the filter output: σ 2 [x n x n ] 2 = = 2 a k y n k+1 Ae iωn k a k Ae iω(n k+1) = k k = a Ca, cancels last term a k n n k+1 2 where C is the covariance matrix of the noise, n. +n n k+1 Ae iωn 2 from 1. k a k n n k+1 n n k +1 a k k 6

25 3. Minimize σ 2 w.r.t. a and subject to the constraint ε a =1. By minimizing σ 2 subject to the constraint, we get the smallest error and no bias. Therefore we minimize L with respect to a: L = σ 2 + λ(ε a 1) = a Ca + λ(ε a 1) We can take L/ Re(a j ) and L/ Im(a j ) separately to derive equations for a, then recombine these equations to get This is the same as we get by taking a C + λε =0. a L L a = a a Ca + λ a (ε a) = a C + λε = 0 for a = a 0. 7

26 The solution for a 0 is a 0 C = λε C a 0 = λ ε a 0 = λ ( C ) 1 ε Now substitute back into the constraint equation ε a 0 =1(the no bias relation) to get ε a 0 = λ ε ( C ) 1 ε =1 or λ = 1 ε ( C ) 1 ε Note denominator is real (quadratic form) a 0 = ( C ) 1 ε ε ( C ) 1 ε ε ( C ) 1 ε = ε C 1 ε 8

27 4. Minimum variance: Substitute a 0 back into the expression for σ 2 to find the minimum variance: σ 2 min a 0 Ca 0 = = = σ 2 min = ( C ) 1 ε ε ( C ) 1 ε ε C 1 ε C (ε C 1 ε )(ε C 1 ε ) 1 ε C 1 ε 1 ε C 1 ε ( C ) 1 ε ε ( C ) 1 ε This is the power in the noise components with the same frequency as the signal Ω. (Note we have used the Hermitian relation C C.) 9

28 Interpretation: 1. σmin 2 = portion of noise that leaks through the filter, which is attempting to estimate a sinusoid corrupted by the noise. 2. Note that the filter coefficients and σmin 2 are functions of Ω and of the noise covariance matrix. But they do not depend on the amplitude of the sinusoid. 3. The trick: now take away the signal but keep the noise. We allow Ω to vary across a range of frequencies we are interested in. Then, σmin 2 (Ω) is a spectral estimate for the noise spectrum (which was left arbitrary) 4. maximum likelihood spectral estimator Ŝ ML (f) = 1 ε C 1 ε with Ω =2πf τ Further comments: 1. As used, the covariance matrix C is an ensemble average quantity. Applications to actual data require use of some estimate for the covariance matrix. 2. The derivation is for equally spaced data. 3. The spectral estimate should work well on processes with steep power-law spectra because the estimator is derived explicitly to minimize bias. 10

29 Data-adaptive aspect of the MLSE spectral estimator: Recall that the Fourier-transform based estimator has a fixed window. The MLSE has a data adaptive spectral window, as we will show. The filter coefficients are a function of the frequency of the sinusoid, Ω: a 0 (Ω) = ( C 1 ) (Ω) (Ω) ( C 1 ) (Ω) As Ω is varied, the coefficients a 0 vary but subject to the normalization constraint a 0 =1. For a given Ω, which labels the frequency component we are attempting to estimate, what is the response to other frequencies, ω? Define the window function W (ω, Ω) =a 0 (Ω) (ω) as the response to frequency ω of a filter designed to pass through the frequency Ω. The window function satisfies (normalization) W (Ω, Ω) 1. The equivalent quantity for a Fourier transform estimator might be sin(ω Ω)T/2 W (ω, Ω) = (ω Ω)T/2. 11

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31 Simulating the HRM Generate a process with specified noise + signal spectrum or just noise with an arbitrary spectrum by passing white noise through a linear filter. white noise h(t) x(t) From one or more realizations of x(t) estimate the autocovariance and put it in the form of a covariance matrix, C. For each frequency of interest (Ω), calculate the MLM/HRM filter coefficients a 0 = Calculate the power-spectrum estimate as Ŝ(Ω) = The window function can be calculated as C 1 ε ε C 1 ε. 1 ε C 1 ε. W (ω, Ω) =a 0 (Ω)ε (ω). 12

32 Comparison of Spectral Estimators Bartlett MLM MEM N 2 ε Cε (ε C 1 ε ) 1 ε C 1 δ 2 100% error f = 1 N τ = 1 T large sidelobes same or better better resolution resolution (up to 2 of Bartlett) lower sidelobes Note all estimators are real because the quadratic form ε t Cε is real for C Toeplitz or, in the MEM case, the estimator is manifestly real. 13

33 Appendices

34 Derivation of the Maximum Entropy Spectrum This section follows Edward and Fitleson IEEE Trans on IT, 19, , 1973 and Mc- Donough in Nonlinear Methods of Spectral Analysis, ed. Haykin, pp Using the expression for the entropy rate in terms of the power spectrum (for a Gaussian process) h = 1 fn df ln S(f) 4f N f N we will derive a spectral estimate given that we know a finite number of values of the ACV. That is, suppose we know C(n) (X k x)(x k+n x), τ = sample interval. n = M, M +1,...,0,...M For now, assume that we actually know C(n) rather than some estimates for C(n), Ĉ(n). Letting S(f) carry the integration limits, we, therefore, have the constraint equations. C(n) = df e 2πifn τ S(f) which we incorporate into the maximization problem by using Lagrange multipliers λ n. Therefore, we maximize M L = h λ n C(n) (minus sign for convenience) n= M 21

35 which can be written as L = df Now we vary S(f) to find δl: δl = dt δs(f) 1 4f N ln S(f) M n= M 1 4f N 1 Ŝ(f) n λ n e 2πifn τ S(f) λ n e 2πifn τ =0 This holds for any δs(f) when S(f) equals the function that extremizes L. Thus, Ŝ(f) = 1 4f N 1 λ n e 2πifn τ Now substitute back into the constraint equations to get equations for the λ n : C(n) = df Ŝ(f) e2πifn τ = 1 e 2πifn τ df, 4f N λ n e 2πifn τ n = M,...,M This is a system of nonlinear equations for λ n. n n 22

36 Following Edward and Fitelson, note that the spectral estimate can be put into the form Ŝ(f) = 1 1 M where A(f) α 4f N A(f) 2 e 2πifl τ, which follows from the positive semi-definiteness of Ŝ(f) and is easy to see by analogy with the Wiener-Khinchin theorem: ACF S(f) FT 2. The coefficients α are related to the Lagrange mulipliers (λ is like a correlation function, α l a time series): M 2 A(f) 2 = α e 2πif τ =0 = α α e 2πif τ( ) Thus, both sides are equal if = M q= M M n= M λ q = M q =0 M q =0 l=0 α α q e 2πif τq λ n e 2πifn τ. α α q. 23

37 Now we can find a solution to the constraint equations. Start with: S(f) = 1 4f N 1 A(f) 2. Multiply Ŝ(f) by A (f) e 2πifn τ and integrate: fn f N df Ŝ(f) A (f) e 2πifn τ = 1 4f N The left-hand side becomes The right-hand size is LHS = = = fn fn f N df A (f) A(f) 2 e2πifn τ df Ŝ(f) M α e 2πif τ f N =0 n fn =0 α f N df Ŝ(f) e2πif τ(n ) C(n ) M α C(n ) =0 RHS = 1 4f N fn 24 f N df e2πifn τ A(f). e 2πifn τ

38 So we have M =0 α C(n ) = 1 4f N fn f N df e2πifn τ A(f). To further reduce the RHS we perform a contour integral in the complex plane for f. Let f = f r + if i and constrain [A(f)] 1 to be analytic 2 in the upper-half plane. Choose the contour ζ: By Cauchy s Integral Theorem 3 the integral around the closed contour vanishes: e 2πifn τ df =0= df [ ]+ df [ ]+ df [ ] + df [ ]. A(f) ζ (1) 2 Analytic functions (Sokolnikoff and Redheffer, p. 540): A function f(z) that has a derivative f (z) at a given point z = z 0 and at every point in the neighborhood of z 0 is analytic at the point z = z 0. The points where f(z) is not analytic are singular points. In order that f(z) =u(x, y)+iv(x, y) be analytic at z 0 = x 0 + iy 0 is that u and v and their partial derivatives be continuous and that the Cauchy-Riemann equations (2) u x = v y, v x = u y be satisfied throughout the neighborhood of x 0,y 0. 3 Cauchy s Integral Theorem If f(z) is continuous in a closed, simply connected region, R+C and analytic within the simple closed curve C, then dz f(z) =0. c (3) (4) 25

39 Consider first the (2) and (4) integrals: f r = ±f N,f i [0,f im ] df []+ df [] K 24 (2) = e 2πif N n τ (4) fim 0 df i e 2πf in τ A(f N + if i ) + e 2πif N n τ 0 f im df i e 2πifin τ A( f N + if i ) To proceed we specify that f N = Nyquistfrequency = 1 2 τ or 2 τf N =1. Then e ±2πif N n τ = e ±πin =cos(±πn) =cosnπ. Also A(f) is periodic so that A(+f N ) = A( f N ) A(+f N + if i ) = A( f N + if i ) This can be seen from A(f) = M α e 2πif τ 2 τf N =1 =0 M α e πi(f/f N ), =0 26

40 which implies that Therefore A(±f N + if i ) = fim K 24 =cosnπ df i = M α e π(f i/f N ) e ±πi =0 M α e π(f i/f N ) ( 1). =0 e 2πf in τ A(f N + if i ) 0 equal and opposite terms 0 e 2πfin τ + df i f im A(f N + if i ) K 24 =0 27

41 These results combined with the Cauchy Integral Theorem imply df [] = df [] (1) (3) or fn df e2πifn τ f N A(f) desired integral = = + fn f N fn f N df r df r e 2πin τ(fr+ifim) A(f r + if im ) e 2πifrn τ e 2πn τfim M α e 2πi τf r e 2π τf im Taking the limit as f im we get contributions only for n =0and =0: =0 Thus, lim f im fn e 2πifrn τ e 2πn τfim df r = δ n,0 f N α 0 α 0 fn f N df r. fn df e2πifn τ f N A(f) = 2f N δ n,0 α 0 28

42 The constraint equations are satisifed if: M l=0 α l C(n l) = 1 4f N fn df e2πifn τ f N A(f) = 1 δ n,0 2 α 0 Now put in matrix form by defining α 0 α α = 1. α M δ = ε = 1 e 2πif τ e 2πif2 τ. e 2πifM τ The constraint equations become C 0 C 1... C M α 0 C 1 C 0... C M 1 α 1.. C M C 1 M... C 0 α M = 1 2α or (since we ve assumed the process is real in which case α is also real) C α = 1 2α 0 δ 29

43 which has the solution α = 1 2α 0 C 1 δ (real data) 30

44 Now we can solve for the spectral estimate, Ŝ(f) = 1 4f N 1 A(f) 2 = 1 1 4f N M α e 2πif τ 2 =0 = 1 4f N 1 ε t C 1 δ 2 = 1 4f N 1 ε t α 2 = 1 4f N 4α 2 o ε t C 1 δ 2 (real data) Ŝ(f) = 1 f N α 2 0 ε t C 1 δ 2 31

45 Note that the solution for α = 1 2α 0 C 1 δ implies or α 0 = 1 2α 0 (C 1 ) 00 α2 0 = 1 2 (C 1 ) 00 32

46 Data Vectors Vector of random variables Mean X 1 X 2 X =. X N X = X 1 X 2. X N Dot product complex: Covariance matrix X X = X t X = zero mean case complex Hermitian N j=1 X X = X X = X 2 j N X j 2 j=1 X 1 2 X 1 X2 X 1 XN C = XX X 2 X1 X 2 2 X 2 XN =..... X N X1 X N X2 X N 2

47 Consider vectors A, B and matrix C with lengths N 1, N 1, and N N, respectively. We have (a) A A B = B (b) A A 2 =2A (c) A A t CA =(C t + C)A for real A. (d) A A CA = C t A + CA for complex A. (e) (CA) t = A t C t (f) If A is a zero mean stochastic process (e.g. a vector of N measurements of a noiselike signal), its covariance matrix can be written as C = AA. Here the notation is: = conjugate; t = transpose; = transpose conjugate.