55 zyx Signal Processing zyxwvu LsfRor zyxwv Limit Cycles in Residue Number System Filters Gian-Carlo Cardarilli Dipartimento di Elettronica zyxwvut I1 Universita di Roma ((Tor Vergata)) Via 0. Raimondo 8, 00173 Roma zyxwvuts - Italy Martin Hasler Department of Electrical Engineering Swiss Federal Institute of Technology CIRC-CM Ecublens CH-1015 Lausanne, Switzerland Abstract. The zero input limit cycle behavior of canonical second order sections of digital filters using a certain kind of Residue Number System (RNS) arithmetic are investigated. It zyxwv is shown that constant and period two limit cycles are frequent in a large part of the linear stability trian- gle. Therefore, the designer should either modify the overflow and quantization characteristic, or use filter structures that are not pIagued with limit cycles. 1. INTRODUCTION In recent years, much effort has spent in searching for efficient hardware implementations of digital fil- ters in order to increase their speed as much zyxwvu as possi- ble. One of the methods that leads to potentially fast filters is to use Residue Number System (RNS) arith- metic [l]. An approach which uses the RNS arithme- tic in the time domain has been proposed in [2, 31. Whi- le the paper [2] concentrates on the realization of a ro- tator, we consider here, as in 131, a conventional cano- nical second-order filter section. It has been shown in [4, 51 that the multiplication and the division modulo m can be reduced to an addi- tion by means of look-up tables, if zyxwvutsrq rn is a prime num- ber or a power of 2. This leads to very efficient imple- mentations. Furthermore, overflow in the ipternal si- gnals of the filter does not affect the find result, as long as the latter remains within the prescribed dyna- mic range. This property is also shared by the conven- tional 2's complement arithmetic, if quantization is performed only after summation and not already af- ter multiplication. In fact, the procedure of [2, 31 can be considered as a generalization of the 2's comple- ment arithmetic, On the other hand, it is well known that second-order sections with 2's complement arith- metic are plagued with limit cycles [6]. Such limit cy- cles can corrupt the correct functioning of the filter. It is therefore of upmost importance to assess the pos- sibility of limit cycles in these zyxwvutsrq RNS filters. This que- stion is addressed here. 2. SECOND ORDER RNS FILTER SECTION Digital filters are called RNS filters if they use mo- dular arithmetic, i.e. its products and sums are evalua- ted modulo some fixed integer zyx N, the modulus [5]. In this paper we discuss a special kind of RNS implemen- tation of a canonical second order filter, which uses simultaneously two moduli, Nand M. This work deals with zero-input limit cycles in the canonical second or- der filter section (Fig. 1). In the absence of an input signal, such a section is described by the following dif- n l+t----l j l Fig. 1 - Second order digital filter. Vol. 3, NO. 5 Sept.-Oct. 1992