Generalized Geometric Mean Decomposition and DFE Transceiver Design—Part I: Design and Complexity

This paper considers a new matrix decomposition which decomposes a complex matrix as a product of several sets of semi-unitary matrices and upper triangular matrices in an iterative manner. The inner most triangular matrix has its diagonal elements equal to the geometric mean of the singular values of the target complex matrix. The complexity (defined in terms of the number of floating point operations) of the new decomposition, generalized geometric mean decomposition (GGMD), depends on its parameters, but is always less than or equal to that of geometric mean decomposition (GMD). The optimal parameters which yield the minimal complexity are derived. The paper also shows how to use GGMD to design an optimal decision feedback equalizer (DFE) transceiver for multiple-input multiple-output (MIMO) channels without zero-forcing constraint. A novel iterative receiving detection algorithm for the specific receiver is also proposed. For the application to cyclic prefix systems in which the SVD of the equivalent channel matrix can be easily computed, the proposed GGMD transceiver has K/log₂(K) times complexity advantage over the GMD transceiver, where is the number of data symbols per data block and is a power of 2. In a companion paper, performance analyses of the proposed GGMD transceiver in terms of arithmetic mean square error (MSE), symbol error rate (SER) and Gaussian mutual information are performed, and comparisons with well-known transceivers are made. The results show that the proposed transceiver reaches the same optimality that a GMD MMSE transceiver can possibly achieve.