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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

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parametric methods

Parametric Methods

指導教授:黃文傑 W.J. Huang

學生:蔡漢成 H.C. Tsai

  • DML (Deterministic Maximum Likelihood)
  • SML (Stochastic Maximum Likelihood)
  • Subspace-Based Approximations
dml deterministic maximum likelihood 1
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
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
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
Subspace Fitting-1
  • The number of signal eigenvector is M’
  • Us will span an M’–dimentional subspace of A
subspace fitting 2
Subspace Fitting-2

Form the basis for the Signal Subspace Fitting (SSF)

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

Distance between AT and

subspace fitting 4
Subspace Fitting-4
  • For fix unknown A ,
  • concentrated

Introduce a weighting of the eigenvectors

wsf weighting sf 1
WSF (Weighting SF)-1
  • Projected eigenvectors
  • W should be a diagonal matrix containing the inverse of the covariance matrix of
wsf 2
WSF -2
  • WFS and SML methods also exhibit similar small sample behaviors
  • Another method,
nsf noise sf 1
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