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Leading Guard Digits in Finite Precision Redundant Representations
May 2006 (vol. 55 no. 5)
pp. 541-548
Redundant number representations are generally used to allow constant time additions, based on the fact that only bounded carry-ripples take place. But, carries may ripple out into positions which may not be needed to represent the final value of the result and, thus, a certain amount of leading guard digits are needed to correctly determine the result. Also, when cancellation during subtractions occurs, there may be nonzero digits in positions not needed to represent the result of the calculation. It is shown here that, for normal redundant digit sets with radix greater than two, a single guard digit is sufficient to determine the value of such an arbitrary length prefix of leading nonzero digits. This is also the case for the unsigned carry-save representation, whereas two guard digits are sufficient, and may be necessary, for additions in the binary signed-digit and 2's complement carry-save representations. Thus, only the guard digits need to be retained during sequences of additions and subtractions. At suitable points, the guard digits may then be converted into a single digit, representing the complete prefix.

[1] D.E. Atkins, “Higher-Radix Division Using Estimates of the Divisor and Partial Remainders,” IEEE Trans. Computers, vol. 17, pp. 925-934, 1968.
[2] A. Avizienis, “Signed-Digit Number Representations for Fast Parallel Arithmetic,” IRE Trans. Electronic Computers, vol. 10, pp. 389-400, Sept. 1961.
[3] P. Kornerup, “Digit Selection for SRT Division and Square Root,” IEEE Trans. Computers, vol. 54, no. 3, pp. 294-303, Mar. 2005.
[4] J.-M. Muller, Elementary Function Evaluation: Algorithms and Implementation. Boston, Basel, Berlin: Birkhäuser, 1997.
[5] T. Noll and E. De Man, “Anordnung zur Bitparallen Addition von Binärzahlen mit Carry-Save Überlaufkorrektur,” European Patent EP 0 249 132 B1, Aug. 1993.
[6] T.G. Noll, “Carry-Save Architectures for High-Speed Digital Signal Processing,” J. VLSI Signal Processing, vol. 3, pp. 121-140, 1991.
[7] B. Parhami, “On the Implementation of Arithmetic Support Functions for Generalized Signed Digit Number Systems,” IEEE Trans. Computers, vol. 42, no. 3, pp. 379-384, Mar. 1993.
[8] D. Timmermann and B.J. Hosticka, “Overflow Effects in Redundant Binary Number Systems,” IEE Electronics Letters, vol. 29, no. 5, pp. 440-441, Mar. 1993.

Index Terms:
Redundant representations, leading guard digits, multioperand additions, pseudo overflows.
Citation:
Peter Kornerup, Jean-Michel Muller, "Leading Guard Digits in Finite Precision Redundant Representations," IEEE Transactions on Computers, vol. 55, no. 5, pp. 541-548, May 2006, doi:10.1109/TC.2006.79
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