x(n) = a k x(n k)+w(n) k=1

Transcription

1 Autoregressive Models Overview Direct structures Types of estimators Parametric spectral estimation Parametric time-frequency analysis Order selection criteria Lattice structures? Maximum entropy Excitation with line spectra Direct Structures P x(n) = a k x(n k)+w(n) k= Notation differs (again) from text Essentially all of the techniques that we discussed for FIR filters can be applied Many ways to estimate How to estimate the autocorrelation matrix? Degree of windowing (full, pre-, post-, no/short) Weighted least squares? J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 2 x(n) = Properties P a k x(n k)+w(n) k= If ˆR is positive definite (and Toeplitz?), the model will be Stable Causal Minimum phase Invertible True of all the estimators except the proposed unbiased technique Problem Unification P x(n) = a k x(n k)+w(n) k= AR modeling is approximately equivalent to several other useful problems Estimating the coefficients of a whitening filter One-step ahead prediction Maximum entropy signal modeling In the MSE case if the process is minimum phase, these are exactly equivalent J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 3 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 4

2 Windowing We have discussed three types of windowing Data Windowing: x w (n) =w(n)x(n) Used to reduce spectral leakage of nonparametric PSD estimators Correlation Windowing: r w (l) =ˆr(l)w(l) Used to reduce variance of nonparametric PSD estimators Weighted Least Squares: E e = N n= w2 n y(n) c T x(n) 2 Used to weight the influence of some observations more than others Optimal when error variance is non-constant in the deterministic data matrix case Autoregressive Estimation Windowing Parametric windowing is mostly reserved for nonstationary applications Time-frequency analysis Text seems to implicity suggest using data windowing This is a bad idea! Biases the estimate No obvious gain Is much better to perform a weighted least squares Each row of the data matrix and output still correspond to a specific time Estimate is unbiased Permits you to weight the influence of points near the center of the observation window (block) more heavily Can be used with any non-negative window (need not be positive definite) J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 5 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 6 Frequency Domain Representation of the Error Signal x(n) =h(n) w(n) e(n) =ŵ(n) = ĥi(n) x(n) E(e jω )=Ĥi(e jω )X(e jω )= X(ejω ) Ĥ(e jω ) E = e(n) 2 = π X(e jω ) 2 Ĥ(ejω ) dω 2 n= In the AR case, H(z) = A(z) Note the frequency domain equations only exist if the signals are energy signals Segments of stationary processes This means solving for the coefficients that minimize the error also minimize the integral of the ratio of the ESDs AR Spectral Estimation E = e(n) 2 = π X(e jω ) 2 n= Ĥ(ejω ) dω = π X(e jω ) 2 2 Â(ejω ) 2 dω This means solving for the coefficients that minimize the error also minimize the integral of the ratio of the ESDs Why can t we just make Ĥi(e jω ) large or Â(ejω ) small at all frequencies? Recall the constraint a =in â = [ ] â... â P a k = π A(e jω )e jωk dω a = π A(e jω )dω Thus A(e jω ) is constrained to have unit area J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 7 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 8

3 AR Spectral Estimation Properties E = e(n) 2 = π X(e jω ) 2 n= Ĥ(ejω ) dω = π X(e jω ) 2 2 Â(ejω ) 2 dω The frequency domain representation of the error makes it clear that the impact of the error across the full frequency range in uniform There is no benefit to fitting lower or higher frequency ranges more accurately, in general The regions where X(e jω ) > Ĥ(ejω ) contribute more to the total error The error will be minimized if Ĥ(ejω ) is larger in these regions This is part of the reason the estimate is more accurate near spectral peaks, than valleys Nonparametric estimators were also more accurate near peaks, but for very different reasons (spectral leakage) Bias-Variance Tradeoff The order of the parametric model is the key parameter that controls the tradeoff between bias and variance P too large The variance is manifest differently than in nonparametric estimators Spectrum may contain spurious speaks Also possible for a single frequency component to be split into distinct peaks P too small Insufficient peaks Peaks that are present are too wide or have the wrong shape Can only do so much with a pair of complex-conjugate poles J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 9 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. Unbiased Biased Order Selection Problem ˆ MSE u = N f N i + P ˆ MSE b = N f N i + N f N f n=n i e(n) 2 n=n i e(n) 2 Order Selection Methods Concept There are many order selection criteria All try to obtain an approximately unbiased estimate of the MSE All essentially add a penalty as the unscaled error increases Goal: Select the value of P that minimizes the MSE We have two estimates of the MSE One from Chapter 8 and one from Chapter 9 The one from Chapter 8 was unbiased Why don t we use it to obtain the best value of P? Only holds in the deterministic data matrix case The data matrix is always stochastic with AR models J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 2

4 Possible Order Selection Methods Let N t N f N i +.Then Model Error ˆσ e 2 N t N f n=n i e(n) 2 Final Prediction Error FPE(P )= N t + P N t P ˆσ2 e Akaike Information Criterion AIC(P )=N t log ˆσ e 2 +2P Minimum Description Length MDL(P )=N t log ˆσ e 2 + P log N t Criterion Autoregressive Transfer CAT(P )= N t P k= N t k N tˆσ 2 e N t P N tˆσ 2 e Order Selection Methods Comments The order selection decision is only critical when P is on the same order as N t,sayn t / P<N t This is the difficult case of too little data to make a good decision None of the order selection criteria work good in this case Otherwise Similar performance will be obtained for a wide range of values of P Can simply pick the value where the estimated parameter vector stops changing with increasing values of P Text suggests looking at diagnostic plots (residuals, ACF, PACS, etc.) and selecting the parameter manually Works well in many applications J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 3 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 4 Maximum Entropy Motivation This discussion based on [] Suppose again that we have N observations of a random process x(n) If we knew the autocorrelation, we could calculate the AR parameters exactly (recall chapters 4 and 6) With only N observations, at most we can estimate r(l) only for l <N Many signals have autocorrelations that are non zero for l N The segmentation may significantly impair the accuracy of the estimated parameters Especially true of narrowband processes Nonparametric PSD estimation methods simply extrapolate the estimate r(l) with zeros: ˆr(l) =for l N Can we do better? Maximum Entropy Problem Suppose we know (i.e., have estimated) the autocorrelation of an WSS process for l P How do we extrapolate the estimated autocorrelation sequence for l>p? Let us denote the extrapolated values by r e (l) Then the estimated PSD is given by ˆR x (e jω = P l= P r x (l)e jlω + l >P r e (l)e jlω We would like ˆR x (e jω to have the same properties as a real PSD Real-valued Nonnegative These conditions are not sufficient for a unique extrapolation Need an additional constraint J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 5 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 6

5 ˆR x (e jω = Maximum Entropy Concept P l= P r x (l)e jlω + l >P r e (l)e jlω Pick r e (l) such that the entropy of the process is maximized Entropy is a measure of randomness or uncertainty Maximizing the entropy is equivalent to Making the (estimated) process x(n) as white as possible Making the (estimated) PSD as flat as possible Places as few constraints as possible on x(n) Minimizes the amount of statistical structure Entropy Defined For a Gaussian random process with PSD R x (e jω ), entropy is given by H(x) = π ln R x (e jω )dω The maximum entropy estimate is the one that maximizes this equation However, we have constraints π ln R x (e jω )e jlω dω = r x (l) l P Can solve by setting the partial derivative of H(x) with respect to re(l) to zero This is a tricky because we re optimizing a function of a complex-valued parameter Details are in the appendix of the text For now assume re(l) is real valued J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 7 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 8 Maximum Entropy Estimate Derived ˆR x (e jω )= P H(x) = l= P π H(x) re(l) == = r x (l)e jlω + π π l >P ln R x (e jω )dω r e (l)e jlω R x (e jω ) R x (e jω ) re(l) dω R x (e jω ) ejlω dω A x (e jω ) R x (e jω ) = π A x (e jω )e jlω dω Maximum Entropy Estimate Properties Q x (e jω ) R x (e jω )= =q(l) = R x (e jω ) = π P l= P P l= P q(l)e jlω Q x (e jω )e jlω dω q(l)e jlω The inverse DTFT of the signal PSD R x (e jω ) is zero for l N Thus the maximum entropy PSD estimate for a Gaussian process is an all-pole power spectrum Can obtain a from the Yule-Walker equations with the know (or estimated) autocorrelation sequence for l N If the autocorrelation sequence is of interest, it can be obtained using the Yule-Walker equation as well (covered in ECE 5/638) J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 9 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 2

6 Maximum Entropy Discussion Using maximum entropy for stationary process estimation (PSD, autocorrelation, etc.) has been thoroughly discussed and debated Rationale In the absence of any information, impose the least amount of structure on x(n) More reasonable than setting r(l) =for large lags, as the nonparametric methods assume (?) Counter-argument Maximum entropy approach imposes an all-pole structure If data was not all-pole, how do you know the estimated properties of interest will be accurate? Bottom line: depends on the process (application) Sinusoidal Excitation L x(n) = A k cos(ω k n + φ n ) k= All-pole models can also be fit to signals that consist of sums of sinusoids If L<P, can minimize error criterion E p = d L L k= R x (e jω ) ˆR h (e jω ) Not clear to me what the value of this model is Apparently problematic for nearly periodic signals J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 2 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 22 Example : Autoregressive PSD Estimation of Chirp Use a least-squares approach to autoregressive process estimation to perform a time-frequency analysis of the chirp signal. 5.5 Example : Autoregressive PSD Autoregressive Spectrogram Window:5 s Filter Order: J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 23 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 24

7 Example : Nonparametric PSD Example : Autoregressive PSD.5 Nonparametric Spectrogram Window:5 s Autoregressive Spectrogram Window: s Filter Order: Signal J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 25 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 26 Example : Nonparametric PSD Example : Autoregressive PSD.5 Nonparametric Spectrogram Window: s Autoregressive Spectrogram Window:2 s Filter Order: Signal J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 27 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 28

8 Example : Nonparametric PSD Example : Autoregressive PSD.5 Nonparametric Spectrogram Window:2 s Autoregressive Spectrogram Window:5 s Filter Order: Signal J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 29 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 3 Example : Autoregressive PSD Example : Autoregressive PSD 5.5 Autoregressive Spectrogram Window: s Filter Order: Autoregressive Spectrogram Window:2 s Filter Order: J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 3 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 32

9 .5 Example : Autoregressive PSD Window:5 s Filter Order:5 Estimator:Unbiased/no window Window:Rectangular Example : Autoregressive PSD Window:5 s Filter Order:5 Estimator:Unbiased/no window Window:Blackman J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 33 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 34 Example : Autoregressive PSD Window:5 s Filter Order:5 Estimator:Modified Covariance Window:Rectangular Example : Autoregressive PSD Window:5 s Filter Order:5 Estimator:Modified Covariance Window:Blackman J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 35 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 36

10 Example : MATLAB Code clear; close all; User-Specified Parameters fs = ; Sample rate (Hz) N = ; Number of observations from the process Np = 5; Number of samples to throw away to account for transient k = :N/4; t = (k-.5)/fs; yc = chirp(t,.5,t(end),.45); y = [yc fliplr(yc) yc fliplr(yc)] ; ParametricSpectrograms fid = fopen( ARChirp.tex, w ); if fid==-, error( Could not open ARChirp.tex file for writing ); fo = [2 ]; wl = [5 2]; for c=:fo, for c=:length(wl), AutoregressiveSpectrogram(y,fs,wl(c),fo(c),,); colormap(colorspiral(256)); title(sprintf( Autoregressive Spectrogram Window:d s Filter Order:d,wl(c),fo(c))); FigureSet(, Slides ); AxisSet(8); fn = sprintf( ChirpARO2dL3d,fo(c),wl(c)); print(fn, -depsc ); fprintf(fid, ==========\n ); fprintf(fid, \\newslide\n ); fprintf(fid, \\slideheading{example \\arabic{exc}: Autoregressive PSD}\n ); fprintf(fid, ==========\n ); fprintf(fid, \\hspace{-.5em} \\includegraphics[scale=]{matlab/s}\n\n,fn); if c==, NonparametricSpectrogram(y,fs,wl(c)); colormap(colorspiral(256)); title(sprintf( Nonparametric Spectrogram Window:d s,wl(c))); FigureSet(2, Slides ); AxisSet(8); fn = sprintf( ChirpNPO2dL3d,fo(c),wl(c)); print(fn, -depsc ); fprintf(fid, ==========\n ); fprintf(fid, \\newslide\n ); fprintf(fid, \\slideheading{example \\arabic{exc}: Nonparametric PSD}\n ); fprintf(fid, ==========\n ); fprintf(fid, \\hspace{-.5em} \\includegraphics[scale=]{matlab/s}\n\n,fn); fo = 5; wl = 5; for c=:, for c2=:, et = c; wt = c2; AutoregressiveSpectrogram(y,fs,wl,fo,et,wt); colormap(colorspiral(256)); if et==, estimatortype = Unbiased/no window ; else estimatortype = Modified Covariance ; if wt==, windowtype = Rectangular ; else windowtype = Blackman ; J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 37 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 38 st = sprintf( Window:d s Filter Order:d Estimator:s Window:s,wl,fo,estimatorType,windowType); title(st); FigureSet(, Slides ); AxisSet(8); fn = sprintf( ChirpARO2dL3dEdWd,fo,wl,et,wt); print(fn, -depsc ); fprintf(fid, ==========\n ); fprintf(fid, \\newslide\n ); fprintf(fid, \\slideheading{example \\arabic{exc}: Autoregressive PSD}\n ); fprintf(fid, ==========\n ); fprintf(fid, \\hspace{-.5em} \\includegraphics[scale=]{matlab/s}\n\n,fn); fclose(fid); Example : MATLAB Code function [S,t,f] = AutoregressiveSpectrogram(x,fsa,wla,foa,eta,wta,fra,nfa,nsa,pfa); AutoregressiveSpectrogram: Generates the spectrogram of the signal [S,t,f] = AutoregressiveSpectrogram(x,fs,wl,fo,et,wt,fr,nf,ns,pf); x Input signal. fs Sample rate (Hz). Default = Hz. wl Length of window to use (sec). Default = 24 samples. If a vector, specifies entire window. fo Model order. Default = 3. et Estimator type: =Unbiased/no window, =Modified covariance (default). wt Window type: =Rectangular, =Blackman (default). fr Minimum and maximum frequencies to display (Hz). Default = [ fs/2]. nf Number of frequencies to evaluate (vertical resolution). Default = max(28,round(wl/2)). ns Requested number of times (horizontal pixels) to evaluate Default = min(4,length(x)). pf Plot flag: =none (default), =screen. S Matrix containing the image of the Spectrogram. t Times at which the spectrogram was evaluated (s). f Frequencies at which the spectrogram was evaluated (Hz). Calculates estimates of the spectral content at the specified times using an autoregressive model. The mean of the signal is removed as a preprocessing step. The square root of the power spectral density is calculated and displayed. To limit computation, decimate the signal if necessary to make the upper frequency range approximately equal to half the sampling frequency. If only the window length is specified, the blackman window is J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 39 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 4

11 applied as a data window. The specified window length should be odd. If called with a window with an even number of elements, a zero is appended to make the window odd. Example: Generate the parametric spectrogram of an intracranial pressure signal using a Blackman-Harris window that is 45 s in duration. load ICP.mat; x = decimate(icp,5); AutoregressiveSpectrogram(x,fs/5,3,3); D. G. Manolakis, V. K. Ingle, S. M. Kogon, "Statistical and Adaptive Signal Processing," McGraw-Hill, 2. Version. JM See also specgram, window, decimate, and Spectrogram. Error Checking if nargin<, help AutoregressiveSpectrogram; return; if length(x)==, error( Signal is empty. ); Calculate Basic Signal Statistics nx = length(x); xr = x; mx = mean(x); sx = std(x); Process Function Arguments fs = ; if exist( fsa ) & ~isempty(fsa), fs = fsa; wl = min(24,nx); Default window length if exist( wla ) & ~isempty(wla), wl = max(3,round(wla*fs)); if ~rem(wl,2), wl = wl + ; Make odd fo = 3; Default filter oder if exist( foa ) & ~isempty(foa), fo = foa; if var(x)==, error( Signal is constant. ); if exist( fra ) & length(fra)==, error( Frequency range must be a 2-element vector. ); et = ; if exist( eta ) & ~isempty(eta), et = eta; wt = ; Modified covariance estimation type Blackman data matrix window J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 4 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 42 if exist( wta ) & ~isempty(wta), wt = wta; fmin = ; Lowest frequency to display fmax = fs/2; Highest frequency to display if exist( fra ) & ~isempty(fra), fmin = max(fra(),); fmax = min(fra(2),fs/2); nf = max(2^9,round(wl/2)); if exist( nfa ) & ~isempty(nfa), nf = nfa; ns = min(,nx); if exist( nsa ) & ~isempty(nsa), ns = min(nsa,nx); pf = ; Default - no plotting if nargout==, Plot if no output arguments pf = ; if exist( pfa ) & ~isempty(pfa), pf = pfa; More Error Checking if fo>wl, error( Filter order is larger than the window length. ); Preprocessing switch wt, case, wn = ones(wl-fo,); case, wn = blackman(wl-fo+2); wn = wn(2:end-); Trim off the zeros x = x(:). ; Make into a row vector wn = wn(:); Make into a column vector x = x - mx; Remove mean Variable Allocations & Initialization st = (nx-)/(ns-); Step size (samples) st = max(st,); te = :st:nx; Evaluation times (in samples) nt = length(te); No. evaluation times nz = nf*(fs/(fmax-fmin)); No. window points needed in FFT nz = 2^(ceil(log2(nz))); Convert to power of 2 for FFT b = floor(nz*(fmin/fs))+; Lower frequency bin index b = ceil (nz*(fmax/fs))+; Upper frequency bin index fi = b:b; Frequency indices f = fs*(fi-)/nz; Frequencies nf = length(fi); No. of frequencies that PSD is evaluated at Main loop switch et, case, Short/no window X = zeros(wl-fo,fo); y = zeros(wl-fo,); case, Modified Covariance X = zeros(2*(wl-fo),fo); J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 43 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 44

12 y = zeros(2*(wl-fo),); S = zeros(nf,nt); Power Spectral Density for c = :nt, ic = te(c); Sample index of segment center i = round(ic-wl/2); Index of first segment element i = i + (wl-); Index of last segment element k = (i:i); Collection of indices a = max(-i+,); Number of zeros to append at front a = max(i-nx,); Number of zeros to append at end k = k(+a:wl-a); Subset of valid indice xs = [x()*ones(,a) x(k) x(nx)*ones(,a)]. ; Data segment Create the Target Vector and Data Matrix (full windowing) y(:wl-fo) = wn.*xs(fo+:wl); for c2=:fo, X(:wl-fo,c2) = wn.*xs(fo-(c2-):wl-c2); if et==, y(wl-fo+(:wl-fo)) = wn.*xs(:wl-fo); for c2=:fo, X(wl-fo+(:wl-fo),c2) = wn.*xs(c2+:wl-fo+c2); ah = -pinv(x)*y; s2w = mean(abs(y + X*ah).^2); ah = [;ah]; dft = s2w./(abs(fft(ah,nz)).^2); Calculate FFT S(:,c) = dft(fi). ; drawnow; t = (te-)/fs; Postprocessing x = xr; t = t(:); Convert to column vector f = f(:); Convert to column vector Plot Results if pf>=, td = ; Time Divider if max(t)>2, td = 6; Use minutes Tmax = (nx-)/fs; if pf~=2, figure; FigureSet(); colormap(jet(256)); fmax = max(f); k = :length(x); tx = (k-)/fs; tp = t; tmax = max(t); tw = [(wl/2-)/fs,(tmax-(wl/2)/fs)]; xl = ; X-axis label if max(t)>2, Convert to minutes tx = tx/6; tp = tp/6; tmax = tmax/6; tw = tw/6; xl = Time (min) ; J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 45 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 46 end Parametric Spectrogram ha = axes( Position,[ ]); s = reshape(abs(s),nf*nt,); p = [ prctile(s,98)]; imagesc(tp,f,abs(s),p);,[smin Smax]); xlim([ tmax]); ylim([fmin fmax]); set(ha, YTickLabel,[]); set(ha, XAxisLocation, Top ); set(ha, YDir, normal ); hold on; h = plot([;]*tw,[ fs], k ); set(h, LineWidth,2.); h = plot([;]*tw,[ fs], w ); set(h, LineWidth,.); hold off; Colorbar ha2 = axes( Position,[ ]); colorbar(ha2); set(ha2, Box, Off ) set(ha2, YTick,[]); Power Spectral Density ha3 = axes( Position,[ ]); psd = mean((abs(s).^2). ).^(/2); plot(psd,f, r );,[Smin Smax]); ylim([fmin fmax]); xlim([ max(psd)]); ylabel( ); set(gca, XAxisLocation, Top ); Signal ha4 = axes( Position,[.5..8.]); h = plot(tx,x); set(h, LineWidth,.5); ymin = min(x); ymax = max(x); yrng = ymax-ymin; ymin = ymin -.2*yrng; ymax = ymax +.2*yrng; xlim([ tmax]); ylim([ymin ymax]); xlabel(xl); axes(ha); AxisSet; Process Return Arguments if nargout==, clear( S, t, f ); J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 47 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 48

13 Summary AR models of stochastic processes have many advantages Least squares estimates reduce to solving a linear set of equations Properties of estimators known Several good choices for estimating properties Can weight observations They are also used for many nearly equivalent problems Whitening One-step ahead linear prediction Parametric process modeling Maximum entropy process/psd/autocorrelation estimation References [] Monson H. Hayes. Statistical Digital Signal Processing and Modeling. John Wiley & Sons, Inc., 996. J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 49 J. McNames Portland State University ECE 539/639 Autoregressive Models Ver.. 5