Asymmetric high-radix signed-digit number systems for carry-free addition

We propose an asymmetric high-radix signed-digit (AHSD) number system for fast binary addition, and show that the AHSD number system supports carry-free (CF) addition. The CF additions in AHSD, to be classified as quasi-closed and subclosed additions, use only one redundant digit for any radix r greater than or equal to 2. Novel algorithms for constructing the three-stage CF adder (CFA) based on the AHSD number system are also presented. Moreover, if the radix is specified as r = 2(m), where m is any positive integer, the binary-to-AHSD conversion can be done in constant time regardless of the word-length. Hence, the AHSD-to-binary conversion dominates the performance of an AHSD-based arithmetic system. We also propose two efficient algorithms for converting AHSD numbers to binary format. The first uses a novel structure to achieve high speed, while the second uses simple transformations and conventional additions to provide hardware reusability. These results are important since the conversion from AHSD numbers to binary format has been considered the performance bottleneck of AHSD-based arithmetic systems.