For the matrix-based case, the matrix of observations X ∈ C M × N is mapped to the following centro-Hermitian matrix Z = X Π M · X ∗ · Π N ∈ C M × 2 N. (14).
Here, X ∈ C M × N is the matrix of observations from M channels at N subsequent time instants, A ∈ C M × d is the unknown mixing matrix or the array steering matrix, S ∈ C d × N contains the unknown source symbols, and W represents the additive noise samples.
This model has the following form: (13) where, A and B are the first and second design matrices of dimensions (k × m) and (k × n), respectively, with n ≥ k ≥ m, x is the vector of unknowns, ε is a residual vector, w is the vector of mis-closures, Q is the covariance matrix of observations and E stands for the statistical expectation.
With each new observation vector x ( n ) ∈ C M 1 · M 2 × 1, we obtain a new matrix of observations for the one-space and the two-space of X which is given by X ~ ( n ) ∈ C M 1 × M 2 for X ( 1 ) and X ~ T ( n ) for X ( 2 ).
A quick review of the CAM follows: (5) where B is the coefficients matrix of observations, ε stands for the vector of observation noise, w is the misclousure vector, E stands for the mathematical expectation, Q is the co-factor matrix or variance-covariance matrix of the observations, c′ is the constant vector of the CAM, L is the observation vector and, finally, is an a priori variance factor.
To see this, let 𝒳 be the n × p matrix of observations and S = n−1𝒳⊤𝒳 be the empirical covariance matrix.
The modified dynamic approach, where the observation vector and design matrix of observation equation are simultaneously filtered by empirical parameters, is implemented in this study.
H t) is the matrix of observation at moment t. (bar Rleft (t right) = bar B_{t}^{- 1}) is the covariance matrix of observation noise, in which ({bar B_{t}}) is the equivalent weight matrix.
E[V(i V T (j)]=R i,j δ ij, i,j=1,⋯,n, in which n is the number of observation variables and R(R>0) is the covariance matrix of observation noise.
If we let G be the m × p matrix of observation-specific derivatives then the variance-covariance matrix can be estimated using the equation Var f ^ t m = G V ^ G where V ^ is the estimated variance matrix for the model parameters.
The parameter update for gradient descent PLR regularization is then computed by: (10) where Λ is a diagonal matrix with the regularization parameter λ along the diagonal and W m is a diagonal matrix of observation weights combining information from the IRLS algorithm and the HME architecture.
