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

( ) ( )s tA xx(t) is white Gaussian with mean(t) is white Gaussian with meanPDF of one measurement vector PDF of one measurement vector xx(t)(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-2VectorVector xx(t) is white, zero-mean Gaussian (t) is white, zero-mean Gaussian random vector with covariance matrixrandom vector with covariance matrix

-log likelihood function (-log likelihood function (llSMLSML) is proportional to) 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

s

U sU

Distance between AT and s

U

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