Approximation of 2-D weakly causal filters

The use of strictly causal filters in the approximation of two-dimensional asymmetric half-plane filters

“…For this example, the causality cone is Clo,ll, i.e., p = q = t = 1 and r=O. Equation From Table II it is clear that if the number of parameters in the filter is nearly equal to that in [2], the resulting filter has better approximation than that in [2].…”

“…The 2-D digital filter, (l), is said to be strictly causal since eJ is assumed to be null for all i, j < 0. Let the filter to approximate the impulse response (1) be specified as ?j) + B u(i j) ,j) HI 0 ' I> i,j>O (2) where xh(i, j) is the n ~1 horizontal state vector, x"(i, j) is the m x 1 vertical state vector, u(i, j) is the y x 1 input vector, v(i, j) is the X ~1 output vector, and A,, A,, A,, B, C are real constant matrices of appropriate dimensions. The filter (2) is a special form of the Roesser state-space model [6] and is simpler than one used in [5].…”

“…The order of the approximating filter can be reasonably chosen by checking the singular values and since the rational transfer function has separable denominator in the transformed domain (open first quadrant), the stability test is trivial. Two illustrative examples have demonstrated that the approximation using a causal separable denominator filter [2], [5] is better than the one using a strictly causal separable denominator filter provided they have the same order. This is owing to the more restrictive structure, i.e., less number of independent parameters in a strictly causal filter.…”

“…Two examples are presented to illustrate the utility of the proposed technique. [2]. The concept "weakly causal" is a generalization of the concept "causal."…”

“…On the other hand, a mapping technique from the closed first quadrant to ti asymmetric half-plane has been investigated [3], [4]. This mapping technique enables one to use an algorithm for designing quarter-plane causal 2-D digital filters for the purpose of designing asymmetric half-plane weakly causal 2-D digital filters [2]. Lashgari et al [5] has presented a technique based on singular value decomposition to approximate a quarter-plane causal 2-D digital filter by one being separable in the denominator.…”