The Karhunen-Loeve Expansion of 1D and 2D OFDM Channel Responses

The discrete Karhunen-Loeve expansion (KLE) of a random vector x uses the orthonormal eigenvectors of the correlation matrix of x as the basis vectors. In the KLE we have uncorrelated random coefficients in the expansion. We can apply the KLE in the feedback and estimation of channel responses (CRs). In this paper, we study the KLE of CRs in the orthogonal frequency-division multiplexing (OFDM) system.We first consider the KLE for one-dimensional (1D) blocks of OFDM CRs across time slots and subchannels, respectively. We further derive the KLE for the two-dimensional (2D) blocks of OFDM CRs. Based on the Kronecker products, we show that the orthonormal matrices, which are served as the basis functions in the 2D KLE, can be obtained from the basis vectors in the KLEs for the two 1D blocks of CRs. We apply the KLE in the pilotassisted CR estimation for the OFDM systems, including both 1D and 2D blocks of CRs. The numerical results show that the meansquares errors (MSEs) of the KLE-based CR estimation approach the performance lower bounds. The KLE-based estimation for the 2D block of CRs has lower MSE than the MSE lower bound for the 1D block of CRs.