Block realization of multidimensional IIR digital filters and its finite word effects

This paper describes the formulation and realization of multidimensional block systems and investigates their stability and numerical performance. The block system can be constructed by the concept of block shift invariance. It possesses the general property that if
(\lambda_1,\lambda_2,\cdots,\lambda_N)
is a pole of the original scalar system, then
(\lambda^{L_1}_{1},\lambda^{L_2}_{2},\cdots,\lambda^{L_N}_{N}
will be a pole of the block system, where
L_i
, is the block length in the
i
th tuple. Thus, a stable scalar system will guarantee that its extended block systems are stable. Two methods of deriving block transfer functions from a given scalar transfer function are proposed. Moreover, it is shown that the scalar transfer function can be derived from its extended block transfer function. Based on Givone-Roesser's model, a unified approach of establishing 1-D to
N
-D block state-space models is presented. It is shown for the proposed block model that the dynamic range constraint in each tuple is invariant under block extension. In addition, the average roundoff noise variance due to the rounding errors in the
i
th tuple is reduced by a factor equal to the block length in this tuple when compared with its scalar counterpart.