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On the Implementation of Arithmetic Support Functions for Generalized Signed-Digit Number Systems
March 1993 (vol. 42 no. 3)
pp. 379-384

Ordinary signed-digit (OSD) number representation systems have been defined for any radix r alpha >r. The most important property of OSD number representation systems is the possibility of performing carry-free addition and (by changing all the digit signs in the subtrahend) borrow-free subtraction. Generalized signed-digit (GSD) number systems cover all useful redundant number representations as special cases. Most GSD number systems support carry-free addition and borrow-free subtraction, and even those that do not can be dealt with using limited-carry or limited-borrow algorithms which yield the ith sum or difference digit z/sub i /as a function of the digits x/sub i/, y/sub i/, x/sub i-1/, y/sub i-1/, x/sub i-2/ and y/sub i-2/ of the operands x and y. Additional topics that are important for practical implementation of arithmetic functions using GSD number systems are treated. Because GSD number systems may have asymmetric digit sets, one must consider subtraction (or at least sign change for representations with alpha <0 and beta <0) explicitly. Zero detection, sign detection, and overflow handling are also treated in depth.

[1] A. Avizienis, "Signed-digit number representation for fast parallel arithmetic,"IRE Trans. Electron. Comput., vol. EC-10, pp. 389-400, 1961.
[2] A. Avizienis, "On a flexible implementation of digital computer arithmetic," inInform. Processing '62 (Proc. IFIP Congress), North-Holland, Amsterdam, 1963, pp. 664-670.
[3] A. Avizienis, "Binary-compatible signed-digit arithmetic," inAFIPS Conf. Proc. (1964 Fall Joint Computer Conf.), pp. 663-672.
[4] L. J. Guibas and F. M. Liang, "Systolic stacks, queues, and counters," inProc. Conf. Advanced Res. VLSI, M.I.T., 1982, pp. 155-164.
[5] A. Guyot, B. Hochet, and J.-M. Muller, "A way to build efficient carry-skip adders,"IEEE Trans. Comput., vol. C-36, no. 4, pp. 1144-1151, Oct. 1987.
[6] M. J. Irwin and R. M. Owens, "Digit pipelined arithmetic as illustrated by the paste-up system,"IEEE Comput. Mag., pp. 61-73, Apr. 1987.
[7] M. Lehman and N. Burla, "Skip techniques for high-speed carry propagation in binary arithmetic units,"IRE Trans. Electron. Comput., vol. EC-10, no. 4, pp. 691-698, Dec. 1961.
[8] O. L. MacSorley, "High-speed arithmetic in binary computers,"Proc. IRE, vol. 49, pp. 67-91, Jan. 1961, Reprinted in [16].
[9] G. Metze and J.E. Robertson, "Elimination of carry propagation in digital computers," inInform. Processing '59 (Proc. UNESCO Conf., June 1959), 1960, pp. 389-396.
[10] B. Parhami, "Systolic up/down counters with zero and sign detection," inProc. Symp. Comput. Arithmetic, Como, Italy, May 1987, pp. 174-178.
[11] B. Parhami, "A general theory of carry-free and limited-carry computer arithmetic," inProc. Canadian Conf. VLSI, Winnipeg, Canada, Oct. 1987, pp. 167-172.
[12] B. Parhami, "Carry-free addition of recoded binary signed-digit numbers,"IEEE Trans. Comput., vol. 37, no. 11, pp. 1470-1476, Nov. 1988.
[13] B. Parhami, "Zero, sign, and overflow detection schemes for generalized signed-digit arithmetic," inProc. 22nd Asilomar Conf. Signals, Syst., and Comput., Pacific Grove, CA, Oct./Nov. 1988, pp. 636-639.
[14] B. Parhami, "A new method for designing highly parallel binary multipliers," inProc. 3rd Annu. Parallel Processing Symp., Fullerton, CA, Mar. 1989, pp. 176-185.
[15] B. Parhami, "Generalized signed-digit number systems: A unifying framework for redundant number representations,"IEEE Trans. Comput., vol. 39, no. 1, pp. 89-98, Jan. 1990.
[16] E. E. Swartzlander, Ed.,Computer Design Development, Principal Papers. Hayden, 1976.

Index Terms:
zero detection; arithmetic support functions; generalized signed-digit number systems; OSD number representation; borrow-free subtraction; redundant number representations; carry-free addition; sign detection; overflow handling; digital arithmetic.
B. Parhami, "On the Implementation of Arithmetic Support Functions for Generalized Signed-Digit Number Systems," IEEE Transactions on Computers, vol. 42, no. 3, pp. 379-384, March 1993, doi:10.1109/12.210182
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