Kalman filtering for matrix estimation

Self-Matching Space-Time Block Codes for Matrix Kalman Estimator-Based ML Detector in MIMO Fading Channels

“…In the succeeding sections, scalars are in italic, vectors are in lowercase bold, MATRICES are in uppercase bold, I is the identity matrix, 0 is the zero matrix, E{·} is the expectation, · * is the conjugate, · H is the Hermitian transpose, rank(·) is the rank, tr(·) is the trace, det(·) is the determinant, diag(·) is the diagonal matrix, vec(·) is the vectorization operation 1 [17], ⊗ is the Kronecker product [22], · F is the Frobenius norm [5], M ∼ CN (M, Σ) denotes a circularly symmetric (proper) complex Gaussian distribution with meanM = E{M} and covariance…”

“…A matrix captures this by designating its row and column dimensions as the time and space dimensions. Thus, the MSSM in this paper allows a more intuitive, insightful, consistent, and compact formulation of the MIMO system and the resulting CE/DD scheme [17]. In addition, it can be used with any number of transmit and receive antennas, different hypermodels, time-variant fading channels, and diversity-achieving matrix modulations.…”

“…Since ordinary KF only works with VSSMs, an MKF is needed. This paper introduces the procedure that was originally developed in [17] to the area of communications and uses the procedure to formulate an MKF specifically to estimate H(l). The procedure does not lose any information or impose any restriction on the MSSM.…”

“…The procedure does not lose any information or impose any restriction on the MSSM. First, using the vectorization operator and the Kronecker product [17], [22], (2) and (5) are converted to the vectorized representation [35]. Since the matrix quantities are now represented as vectors, the ordinary KF recursion is then applied to the vectorized equations.…”

“…An ordinary Kalman filter (KF) only works with vector state-space models (VSSMs). Hence, this paper introduces the technique that was originally developed in [17] and the definition of the covariance of a matrix in [18] to derive the MKF for the MSSM. The DD is shown to be a Mahalanobis (i.e., weighted Euclidean) distance square DD, where the weighting factor represents the effective signal-to-noise ratio (SNR) and is calculated recursively by the MKF.…”