In view of the speed of today's hardware components, wirings between IC pins cannot be regarded as ``short circuit jumper+some C'' but should be treated as a piece of transmission line. The central problem in design of any equipment employing a number of IC is thus to predict the extent of gradual deterioration of waveforms as pulses propagate along a length of transmission line loaded with many lumped constant loads at distances. A simulation technique for this problem is introduced in the paper, and the technique is more practical than those previously described.

The authors of the above-mentioned paper have been informed by H. Haskal that the technique for the analysis of waveform propagation used in this paper had been previously used by Haskal and Rabinovici2 as early as 1965. This fact was unknown to the authors at the time of the publication of their paper.

Abstract—A new residue number system algebra has been previously proposed by the author. The algebra has solved an essential theoretical barrier in the residue number system and has enabled one to pursue additive operations in the residue number system to their full extent, overcoming such difficulties as restrictions on the sign or magnitude of numbers in the system. In this paper, basic theorems in the algebra are introduced first, and then, based on the theorems, table look-up oriented solutions for hardware overflow checking, sign detection, and floating-point additive operations are given. The theorems expound the behavior of a quantity treated as a veiled mysterious function in the literature. To the best knowledge of the author, hardware overflow-checking schemes and floatingpoint additive operations in the residue number system have never been reported elsewhere. So far, the upper limit of the magnitude of numbers in the system ever discussed has been the one that is theory-limited, and floating-point operations in the residue number system have never been discussed.

Improved residue expression and new arithmetic algorithms for addition and subtraction are proposed. In the proposed system positive and negative integers of any magnitude can be handled regardless of the particular choice of the set of relatively prime bases. And such problems as difficulties in overflow detection, handling of sign change in subtraction, or complementation of negative numbers do not exist in the proposed system as far as addition and subtraction are concerned. The difficult problem itself of magnitude comparison in the residue system, however, still remains to be attacked. The proposed system assumes existence of an I/O unit, independent of the residue mode computer, for the purpose of the necessary conversions before and after the computation. Basic properties of the proposed arithmetic algorithms are also derived.