Closed-form expression for the average weight of signed-digit representations

In radix-r number system, the minimal weight signed-digit (SD) representati on has minimal number of nonzero signed-digits which belong to the set {+/- 1,+/-2,...,+/-(r-1)}. In this article, we derive closed form expressions fo r the average number of nonzero digits in the minimal weight SD representat ion and for the average length of the canonical SD representation, a specia l case of the minimal weight SD form, of a positive integer whose radix-r f orm is of length n, n greater than or equal to 1.