Generalized signed-digit number systems: a unifying framework for redundant number representations

Signed-digit (SD) number representation systems have been defined for any radix r⩾3 with digit values ranging over the set {-α, . . ., -1, 0, 1, . . ., α}, where α is an arbitrary integer in the range 1/2r<α<r. Such number representation systems possess sufficient redundancy to allow for the annihilation of carry or borrow chains and hence result in fast propagation-free addition and subtraction. The author refers to the above as ordinary SD number systems and defines generalized SD number systems which contain them as a special symmetric subclass. It is shown that the generalization not only provides a unified view of all redundant number systems which have proven useful in practice (including stored-carry and stored-borrowed systems), but also leads to new number systems not examined before. Examples of such new number systems are stored-carry-or-borrow systems, stored-double-carry systems, and certain redundant decimal representations