Null-space of block convolution matrix

A novel, efficient, non-iterative algorithm to find the null-space of the block convolution matrix which gives rise to a particular block banded block Toeplitz matrix (BBTM) is developed. This BBTM structure arises in the context of a multiple input multiple output (MIMO) (Nr receivers and Nt transmitters) orthogonal frequency division multiplexing (OFDM) model where the effective L taps channel impulse response length is to be shortened using a N tap channel shortening prefilter. This computationally efficient algorithm to find the null-space is derived from standard Gaussian elimination and exploits the structure of the MIMO-OFDM channel matrix. When compared to standard Gaussian elimination with partial pivoting which has a complexity of O(N³) for a L-tap channel the proposed algorithm has a run-time complexity of only O(N).