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STOCHASTIC PROCESSES AND MODELS

STOCHASTIC PROCESSES AND MODELS. EigenAnalysis. Read Appendices E, B, C and others! Let R be an M x M autocorrelation matrix corresponding to u (n): M x1 The eigenvalue problem is There are M eigenvalues and M eigenvectors of R. Rewriting or →eigenvalues { λ i }.

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STOCHASTIC PROCESSES AND MODELS

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  1. STOCHASTIC PROCESSES AND MODELS ELE 774 - Adaptive Signal Processing

  2. EigenAnalysis Read Appendices E, B, C and others! • Let R be an MxM autocorrelation matrix corresponding to u(n):Mx1 • The eigenvalue problem is • There are M eigenvalues and M eigenvectors of R. • Rewriting • or →eigenvalues {λi} Theeigenvector of R corresponding to an eigenvalue λ Aneigenvalue of R ELE 774 - Adaptive Signal Processing

  3. EigenAnalysis • Property 1: Let the eigenvalues of R be λ1, λ2, ..., λM Then the eigenvalues of Rk are λ1k, λ2k, ..., λMk Then the eigenvalues of R-1 are λ1-1, λ2-1, ..., λM-1 Proof: Rq=λq, R2q=λ(Rq)= λ2, etc. • Property 2: The M eigenvectors {qi} of R are linearly independent. Proof: Let with at least one non-zero vi. Multiply by R, R2, R3, ..., RM-1 to obtain !!! CONTRADICTION !!! Always non-singular Must be an all-zero vector → ELE 774 - Adaptive Signal Processing

  4. EigenAnalysis • Property 3: The eigenvalues {λi} are real and non-negative. • Proof: • Property 4: If the eigenvalues {λi} are distinct, then the eigenvectors {qi} are orthogonal to eachother. • Proof: ≠0 ELE 774 - Adaptive Signal Processing

  5. EigenAnalysis • Property 5: Diagonalisation of R , then by stacking and multiplying by QH from the left we obtain or ELE 774 - Adaptive Signal Processing

  6. EigenAnalysis • Property 6: Spectral Theorem • Property 7: • Property 8: Condition number Small (~1) is good, large (R is ill-conditioned) (→∞) is bad. (Aw=d, → w=A-1d, a small pertubation will result in a large perturbation in A-1.) ELE 774 - Adaptive Signal Processing

  7. Stochastic Processes • Definition: The term stochastic process (random process) is used to describe the time evolution of a statistical phenomenon according to probabilistic laws. • Computer data, radar signal, measurements, data • A stochastic process is not just a single function of time • It represents an infinite number of different realizations. • One particular realization is called a time series. • u(n), u(n-1), ... , u(n-M) ELE 774 - Adaptive Signal Processing

  8. Stochastic Processes • A stochastic process is strictly stationary, if its statistical properties are invariant to a time shift • Joint pdf of {u(n), u(n-1), ... , u(n-M)} remain the same regardless of n. • Joint pdf is not easy to obtain, • First and second moments are used frequently. ELE 774 - Adaptive Signal Processing

  9. Mean and Covariance • Mean (Expected) Value of u(n) (1st order) • Autocorrelation Function of u(n) (2nd order) • Autocovariance Function of u(n) (2nd order) • u(n): zero mean -> r(n, n-k)=c(n, n-k) ELE 774 - Adaptive Signal Processing

  10. Mean and Covariance • Stationary (strictly/w.s.s) processes • lag k=0 is important • μ=0 → variance ELE 774 - Adaptive Signal Processing

  11. Ergodic Processes • Ensemble averages (expectations) are across the process -> can be obtained analytically (actual expec.) • Sample (time) averages are along the process -> can be obtained emprically (from realizations of a process) (estimated expectation) sample (time)average Ensembleaverage ELE 774 - Adaptive Signal Processing

  12. Ergodic Processes • If sample averages converge to ensemble averages in, e.g. mean square error sense, • We call the process u(n) as ergodic. • We can estimate the mean value of the process as • If the process is ergodic, i.e. then ELE 774 - Adaptive Signal Processing

  13. Ergodic Processes • For a w.s.s. process, the autocorrelation can be estimated if the process is correlation ergodic, e.g. in MMSE sense. • We will generally assume that ELE 774 - Adaptive Signal Processing

  14. Correlation Matrix • Let u(n) be the Mx1 observation vector • Define the MxMcorrelation matrix as ELE 774 - Adaptive Signal Processing

  15. Properties of the Correlation Matrix • Property 1: The correlation matrix of a stationary discrete-time stochastic process is Hermitian symmetric. • Then ELE 774 - Adaptive Signal Processing

  16. Properties of the Correlation Matrix • Property 2: The correlation matrix of a stationary discrete-time stochastic process is Toeplitz. ELE 774 - Adaptive Signal Processing

  17. Properties of the Correlation Matrix • Property 3: The correlation matrix of a discrete-time stochastic process is always non-negative definite and almost always positive definite. • Let a be an arbitrary Mx1 vector and let Then • Since , ELE 774 - Adaptive Signal Processing

  18. Properties of the Correlation Matrix • Property 4: The correlation matrix of a w.s.s. process is nonsingular due to the unavoidable presence of additive noise. • None of the eigenvalues of R is zero due to noise, i.e. • Property 5: If the order of the elements of the vector u(n) is (time)-reversed, the effect is the transposition of the autocorrelation matrix. ELE 774 - Adaptive Signal Processing

  19. Properties of the Correlation Matrix • Property 6: The correlation matrices RM and RM+1 of a stationary discrete-time stochastic process, pertaining to M and M+1observations of the process, respectively are related by or equivalently where r(0) is the autocorrelation of u(n) for lag zero, ELE 774 - Adaptive Signal Processing

  20. Properties of the Correlation Matrix ELE 774 - Adaptive Signal Processing

  21. Correlation Matrix of a Sine Wave + Noise • Let where v(n) is zero-mean additive white noise with • Then ELE 774 - Adaptive Signal Processing

  22. Correlation Matrix of a Sine Wave + Noise • Given noise power , can be obtained from r(0). • Given , the angular frequency can be obtained from r(k), k>0. • Magnitude, and angular frequency can be found from the autocorrelation function. • Autocorrelation lacks the phase information. ELE 774 - Adaptive Signal Processing

  23. Stochastic Models • v(n): zero-mean, white random variable with variance . • Generally v(n) is assumed to be Gaussian • zero-meanAdditive White Gaussian Noise (AWGN) with variance • N(0, ) • Input-output relation of the linear filter (transversal) • A stochastic process described in this way is called a linear process. ELE 774 - Adaptive Signal Processing

  24. Stochastic Models • 1. AR (AutoRegressive) Model • Only the past values of the model output, and the present value of the model input are used. • 2. MA (Moving-Average) Model • Only the past values of the model input, no past value of the model output are used. • 3. ARMA (Auto-Regressive Moving-Average) Model • The past values of both the model input and output are used. (causal models, we are interested in present and past values not future) ELE 774 - Adaptive Signal Processing

  25. AR Model • Input-output relation of M-th order AR model: • a1, a2, ... aM: AR parameters. • Another perspective • convolution in time domain, multiplication in z-domain ELE 774 - Adaptive Signal Processing

  26. AR Model • Two interpretation of the model (assuming {ak} is given) • Process Generator • Given the white process (noise) v(n), the stochastic process u(n) is generated: • IIR impulse response AR Generator (all-pole filter) ELE 774 - Adaptive Signal Processing

  27. AR Model ELE 774 - Adaptive Signal Processing

  28. AR Model • Process Analyser • Given the stochastic process u(n), the white process v(n) is produced • FIR impulse response AR Analyser (all-zero filter) ELE 774 - Adaptive Signal Processing

  29. MA Model • Input-output relation of M-th order MA model: b1, b2, ... bM: MA parameters. • Two interpretation of the model (assuming {bk} is given) • Process Generator • Given the white process (noise) v(n), the stochastic process u(n) is generated: • FIR impulse response MA Generator (all-zero filter) ELE 774 - Adaptive Signal Processing

  30. MA Model • Process Analyser • Given the stochastic process u(n), the white process v(n) is produced • IIR impulse response ELE 774 - Adaptive Signal Processing

  31. ARMA Model • Input-output relation of M-th order ARMA model: b1, b2, ... bM: MA parameters. • Process Generator • Given the white process (noise) v(n), the stochastic process u(n) is generated: • FIR impulse response • Process Analyser • Given the stochastic process u(n), the white process v(n) is produced • IIR impulse response ELE 774 - Adaptive Signal Processing

  32. ARMA Model ARMA Generator of order (M, K) assuming that M>K ELE 774 - Adaptive Signal Processing

  33. Wold Decomposition • Theorem: Any stationary discrete-time stochastic process, x(n) may be decomposed into the sum of a general linear process, u(n) and a predictable process, s(n) 1. u(n) and s(n) are uncorrelated processes, 2. u(n) is a general linear process represented by a MA model with white noise v(t) as input. 3. s(n) can be predicted from its own past values with zero prediction variance. ELE 774 - Adaptive Signal Processing

  34. Stationarity of AR Processes • For asymptotic stationarity of a discrete-time stochastic process, all the poles of the filter in the AR model must lie inside the unit-circle in the z-plane. • Assuming stationarity, an important recursive relation for the autocorrelation of an AR process • Observe that , then multiply both sides by and take expectation to get • Autocorrelation function must satisfy the characteristic equation of the AR model. ELE 774 - Adaptive Signal Processing

  35. Yule-Walker Equations • To specify an AR model we need • the model coefficients a1,a2,...,aM • and the variance of the input noise, • Given the autocorrelation function of the AR process, how can we find these parameters? • Rewrite • Stacking the eqn.s for each lag l, write the Yule-Walker Equations ELE 774 - Adaptive Signal Processing

  36. Yule-Walker Equations • Assuming R is non-singular, the coefficients {wk} (hence {ak}) are • Noise variance, • for l=0, we have (slide 28) • Therefore Section 1.9: Computer Experiment: Autoregressive Process of Order Two ELE 774 - Adaptive Signal Processing

  37. Complex Gaussian Processes • (Complex valued) Gaussian processes are frequently encountered. • Consider a complex Gaussian process u(n) consisting of N samples. • First-order statistics (zero-mean) • Second-order statistics (Autocorrelation function) • The NxN autocorrelation matrix R of u(n) can be constructed from r(k). • With this definition of first and second order stat.s the process is w.s.s. • Shorthand notation for this process is N(0,R) ELE 774 - Adaptive Signal Processing

  38. Complex Gaussian Processes • Probability density function (pdf) of u(n) is totally defined by mean and correlation matrix where Hence, the process u(n) is 1. strictly stationary (totally defined by w.s.s. parameters) 2. circularly complex, i.e. 3. ELE 774 - Adaptive Signal Processing

  39. Power Spectral Density • Autocorrelation function is a time-domain description and • Power Spectral Density (PSD) is a frequency-domain description of second-order statistics. • PSD analysis requires w.s.s. processes. (non-stationary processes can be analysed by e.g. wavelet transform) • Let uN(n) be a window of u(n) with N samples • Take DTFT of uN(n) • Calculate squared-magnitude of UN(ω) and take expectation ELE 774 - Adaptive Signal Processing

  40. Power Spectral Density • But we know that then let l=n-k and after rearranging • Finally, PSD is defined as ELE 774 - Adaptive Signal Processing

  41. Properties of PSD • Property 1: The autocorrelation function and power spectral density of a w.s.s. stochastic process form a (discrete-time) Fourier transform pair. • Property 2: The frequency support of the PSD is the Nyquist interval, i.e. • Outside this interval PSD is periodic ELE 774 - Adaptive Signal Processing

  42. Properties of PSD • Property 3: The PSD of a stationary discrete-time stochastic process is real. • Property 4: The PSD of a real-valued stationary discrete-time stochastic process is even, if the process is complex-valued, it is not even. • u(n): real → r(-l)=r(l) → S(-ω)=S(ω) • u(n): complex → r(l)=r*(l) → S(-ω)≠S(ω) ELE 774 - Adaptive Signal Processing

  43. Properties of PSD • Property 5: The mean-square value (variance, power) of a stationary discrete-time stochastic process equals the area under S(ω). • Property 6: The PSD of a stationary discrete-time stochastic process is non-negative. • Property 7: Let {λi} be the eigenvalues of the autocorrelation matrix R of a stationary discrete-time stochastic process, and S(ω) be the corresponding PSD. Then, ELE 774 - Adaptive Signal Processing

  44. Transmission of a Stationary Process Through a Linear Filter • Filter is LTI, input process is stationary • Impulse response of the filter is h(n), then its output is and the aucorrelation of the output is taking the DTFT we obtain the PSD of the output ELE 774 - Adaptive Signal Processing

  45. Power Spectrum Analyser • Take a window of bandwidth 2Δω around ωc which can be obtained by a BPF • With 2Δω«ωc,we can assume that S(ω) is constant over 2Δω • Then, the power in this window is • Or, equivalently ELE 774 - Adaptive Signal Processing

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