Stability Testing of Two Dimensional Recursive Filters Using Mapping Methods

The stability testing of first quadrant quarter-plane (QP) two dimensional recursive digital filters had been a classical problem for the last two decades. In literature, the two major types of stability testing methods available are algebraic and mapping methods. Even though the algebraic methods can determine the stability in finite number of steps, it has a few practical limitations in its implementations and it consumes very large time to find the exact stability as it requires larger number of calculations. Moreover, the accuracy of the algebraic methods is also affected by the finite word length effects of the computer. The mapping methods, in general cannot guarantee the stability of given recursive digital filter in finite number of steps as it can determine the stability only in infinite number of steps. Out of the mapping methods, Jury’s row and column algorithms have been considered as highly efficient, even though they still run short of accuracy due to the undefined length of the FFT used. Hence, the best mapping method is not yet available, though some researchers like Bistritz have been working on this problem even now. The main aim of this paper is to find the simple and fast solution for testing the stability of first quadrant quarter-plane two dimensional recursive digital filters. In this work, it is assumed that the given transfer function is devoid of non-essential singularities of second kind. A new mapping stability test procedure for two dimensional quarter-plane recursive digital filter (QP) is proposed which is primarily based on the row and column concatenation method. One important advantage of the proposed test procedure is that it requires single 1-D polynomials to be tested for stability for two dimensional recursive digital filters of any order. This method gives a simple and best solution procedure even for barely stable or barely unstable recursive digital filters. Sufficient examples are taken from the literature for two dimensional recursive digital filters and the same accurate results as that found in the literature are also obtained using this method.