Advanced Parametric Methods in Signal Processing: DML and SML Approaches

Parametric Methods 指導教授：黃文傑 W.J. Huang 學生：蔡漢成 H.C. Tsai

Outline • DML (Deterministic Maximum Likelihood) • SML (Stochastic Maximum Likelihood) • Subspace-Based Approximations

DML (Deterministic Maximum Likelihood)-1 • Performance of spectral-… is not sufficient • Coherent signal increase the difficulties • Noise independent • Noise as a Gaussian white, whereas the signal …deterministic and unknown

DML-2 • Skew-symmetric cross-covariance • x(t) is white Gaussian with meanPDF of one measurement vector x(t)

DML-3 • Likelihood function is obtained as • Unknown parameters • Solved by

DML-4 • By solving the following minimization

DML-5 • X(t) are projected onto subspace orthogonal to all signal components • Power measurement • Remove all true signal on projected subspace ， energy ↓

SML (Stochastic Maximum Likelihood) -1 • Signal as Gaussian processes • Signal waveforms be zero-mean with second-order property

SML-2 Vectorx(t) is white, zero-mean Gaussian random vector with covariance matrix -log likelihood function (lSML) is proportional to

SML-3 • For fixed ,minima lSML to find the

SML-4 • SML have a better large sample accuracy than the corresponding DML estimates ,in low SNR and highly correlated signals • SML attain the Cramer-Rao lower bound (CRB)

Subspace-Based Approximations • MUSIC estimates with a large-sample accuracy as DML • Spectral-based method exhibit a large bias in finite samples, leading to resolution problems,especially for high source correlation • Parametric subspace-based methods have the same statistical performance as the ML methods • Subspace Fitting methods

Subspace Fitting-1 • The number of signal eigenvector is M’ • Us will span an M’–dimentional subspace of A

Subspace Fitting-2 Form the basis for the Signal Subspace Fitting (SSF)

Subspace Fitting-3 • and T are unknown , solve Us=AT • T is “nuisance parameter ” • instead Distance between AT and

Subspace Fitting-4 • For fix unknown A , • concentrated Introduce a weighting of the eigenvectors

WSF (Weighting SF)-1 • Projected eigenvectors • W should be a diagonal matrix containing the inverse of the covariance matrix of

WSF -2 • WFS and SML methods also exhibit similar small sample behaviors • Another method,

NSF(Noise SF)-1 V is some positive define weighting matrix

NSF-2 • For V =I NSF method can reduce to the MUSIC • is a quadratic function of the steering matrix A

Advanced Parametric Methods in Signal Processing: DML and SML Approaches