A Stabilized Parallel Algorithm for Direct–Form Recursive Filters Richard P. Brent and Zhou Bing Bing† Computer Sciences Laboratory Australian National University Canberra, Australia Abstract A stabilized parallel algorithm for direct-form recursive filters is obtained using a new method of derivation in the Z domain. The algorithm is regular and modular, so very efficient VLSI architectures can be constructed to implement it. The degree of parallelism in these implementations can be chosen freely, and is not restricted to be a power of two. 1. Introduction Recursive filtering is one of the most important techniques in digital signal processing. Because recursion is involved, high sampling-rate computation is not straightforward. The look-ahead computation concept may be applied to the imple- mentation of recursive filters to achieve parallel computation. Although this conven- tional method has been used successfully in the parallel realization of recursive filters in state-variable form [1,4,6,7,10], it may cause numerical instability when applied to the direct-form implementation of recursive filters because of the effect of finite wordlength (see the detailed discussion in [2,3,8,9,11]). This paper introduces a new method to obtain a stabilized parallel algorithm for computing direct-form recursive filters. The structure of the algorithm is regular and modular. Thus it is suitable for VLSI implementation. Our algorithm is described in Section 2. Section 3 is concerned with the degree of parallelism, stability and complexity of the algorithm. The degree of parallelism is considered in Section 3.1. In Section 3.2 we show how to reduce the number of multiplications compared to the number required in a naive implementation. Numer- ical stability is considered in Section 3.3. Finally, some conclusions and comments on related work by Moyer [5] and by Parhi and Messerschmitt [8,9] are given in Section 4. † Current address: Department of Radio Engineering, Southeast University, Nanjing, Jiangsu 210018, Peoples Republic of China. Appeared in IEEE Transactions on Computers 40 (1991), 333–336. Copyright c  1991, IEEE. rpb105 typeset using T E X