Week 2ELE Adaptive Signal Processing 1 STOCHASTIC PROCESSES AND MODELS

ELE Adaptive Signal Processing2 Week 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 {λ i } An eigenvalue of R The eigenvector of R corresponding to an eigenvalue λ

ELE Adaptive Signal Processing3 Week 2 EigenAnalysis Property 1: Let the eigenvalues of R be λ 1, λ 2,..., λ M Then the eigenvalues of R k are λ 1 k, λ 2 k,..., λ M k Then the eigenvalues of R -1 are λ 1 -1, λ 2 -1,..., λ M -1 Proof: Rq=λq, R 2 q=λ(Rq)= λ 2, etc. Property 2: The M eigenvectors {q i } of R are linearly independent. Proof: Let with at least one non-zero v i. Multiply by R, R 2, R 3,..., R M-1 to obtain Always non-singular Must be an all-zero vector → !!! CONTRADICTION !!!

ELE Adaptive Signal Processing4 Week 2 EigenAnalysis Property 3: The eigenvalues {λ i } are real and non-negative.  Proof: Property 4: If the eigenvalues {λ i } are distinct, then the eigenvectors {q i } are orthogonal to eachother.  Proof: ≠0

ELE Adaptive Signal Processing5 Week 2 EigenAnalysis Property 5: Diagonalisation of R, then by stacking and multiplying by Q H from the left we obtain or

ELE Adaptive Signal Processing6 Week 2 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 -1 d, a small pertubation will result in a large perturbation in A -1.)

ELE Adaptive Signal Processing7 Week 2 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 Adaptive Signal Processing8 Week 2 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 Adaptive Signal Processing9 Week 2 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 Adaptive Signal Processing10 Week 2 Mean and Covariance Stationary (strictly/w.s.s) processes lag k=0 is important μ=0 → variance

ELE Adaptive Signal Processing11 Week 2 Ensemble average sample (time) average 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)

ELE Adaptive Signal Processing12 Week 2 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 Adaptive Signal Processing13 Week 2 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 Adaptive Signal Processing14 Week 2 Correlation Matrix Let u(n) be the Mx1 observation vector Define the MxM correlation matrix as

ELE Adaptive Signal Processing15 Week 2 Properties of the Correlation Matrix Property 1: The correlation matrix of a stationary discrete-time stochastic process is Hermitian symmetric.  Then

ELE Adaptive Signal Processing16 Week 2 Properties of the Correlation Matrix Property 2: The correlation matrix of a stationary discrete-time stochastic process is Toeplitz.

ELE Adaptive Signal Processing17 Week 2 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 Adaptive Signal Processing18 Week 2 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 Adaptive Signal Processing19 Week 2 Properties of the Correlation Matrix Property 6: The correlation matrices R M and R M+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 Adaptive Signal Processing20 Week 2 Properties of the Correlation Matrix

ELE Adaptive Signal Processing21 Week 2 Correlation Matrix of a Sine Wave + Noise Let where v(n) is zero-mean additive white noise with Then

ELE Adaptive Signal Processing22 Week 2 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 Adaptive Signal Processing23 Week 2 Stochastic Models v(n): zero-mean, white random variable with variance.  Generally v(n) is assumed to be Gaussian zero-mean Additive 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 Adaptive Signal Processing24 Week 2 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 Adaptive Signal Processing25 Week 2 AR Model Input-output relation of M-th order AR model: a 1, a 2,... a M : AR parameters. Another perspective convolution in time domain, multiplication in z-domain

ELE Adaptive Signal Processing26 Week 2 AR Model Two interpretation of the model (assuming {a k } 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 Adaptive Signal Processing27 Week 2 AR Model

ELE Adaptive Signal Processing28 Week 2 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 Adaptive Signal Processing29 Week 2 MA Model Input-output relation of M-th order MA model: b 1, b 2,... b M : MA parameters. Two interpretation of the model (assuming {b k } 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 Adaptive Signal Processing30 Week 2 MA Model  Process Analyser Given the stochastic process u(n), the white process v(n) is produced IIR impulse response

ELE Adaptive Signal Processing31 Week 2 ARMA Model Input-output relation of M-th order ARMA model: b 1, b 2,... b M : 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 Adaptive Signal Processing32 Week 2 ARMA Model ARMA Generator of order (M, K) assuming that M>K

ELE Adaptive Signal Processing33 Week 2 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 Adaptive Signal Processing34 Week 2 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 Adaptive Signal Processing35 Week 2 Yule-Walker Equations To specify an AR model we need  the model coefficients a 1,a 2,...,a M  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 Adaptive Signal Processing36 Week 2 Yule-Walker Equations Assuming R is non-singular, the coefficients {w k } (hence {a k }) are Noise variance,  for l=0, we have (slide 28)  Therefore Section 1.9: Computer Experiment: Autoregressive Process of Order Two

ELE Adaptive Signal Processing37 Week 2 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 Adaptive Signal Processing38 Week 2 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 Adaptive Signal Processing39 Week 2 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 u N (n) be a window of u(n) with N samples Take DTFT of u N (n) Calculate squared-magnitude of U N (ω) and take expectation

ELE Adaptive Signal Processing40 Week 2 Power Spectral Density But we know that then let l=n-k and after rearranging Finally, PSD is defined as

ELE Adaptive Signal Processing41 Week 2 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 Adaptive Signal Processing42 Week 2 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 Adaptive Signal Processing43 Week 2 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 {λ i } be the eigenvalues of the autocorrelation matrix R of a stationary discrete-time stochastic process, and S(ω) be the corresponding PSD. Then,

ELE Adaptive Signal Processing44 Week 2 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 Adaptive Signal Processing45 Week 2 Power Spectrum Analyser Take a window of bandwidth 2Δω around ω c which can be obtained by a BPF With 2Δω«ω c, we can assume that S(ω) is constant over 2Δω Then, the power in this window is Or, equivalently