Matrix Factorizations in Fixed Point on the C 6 xVLIW

1 Summary We investigated matrix factorization in three applications: mimimum mean square error decision feedback equalization (which uses a special form of the Cholesky factorization), code division multiple access (which uses a traditional form of the Cholesky factorization) and blind equalization (which uses the singular value decomposition). The MMSE-DFE application's factorization was both faster and more stable numerically; therefore, we focussed on analyzing and developing this algorithm for TI's C6x DSP. In the MMSE-DFE, the Cholesky factorization of a matrix R is used to compute the feedforward and feedback tap coeecients of the equalizer. Matrix R has the property of displacement structure, and so a special algorithm (attributed to Schur) can be used. This algorithm works on a generating matrix G whose size is much smaller than R, and G can be constructed from easily estimated channel properties. Once the Cholesky factor L of R is computed from G, the tap coeecients can be extracted by a simple backward substitution procedure. Unlike traditional Cholesky factorization, the Schur algorithm uses orthogonal operations to eliminate nonzeros in the matrix. Orthogonal operations are inherently stable numerically; in addition, the G of MMSE-DFE is very well behaved for a generating matrix. The combination of these eeects keeps numerical error in the lowest few bits, even in a xed point implementation. Example runs indicate that the error introduced by xed point computation is insigniicant. The Schur algorithm has two key procedures at each step: nd the orthogonal operation to eliminate nonzeros, and then apply it to the remaining portions of G. The latter operation is very well suited to the C6x's parallel computation units and runs very eeciently. The former operation is poorly suited|it is iterative in nature and requires sequential operations and conditional branching. For channels with short memory the iterative operation can have signiicant