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

In radix-r number system, the minimal weight signed-digit (SD) representation has minimal number of nonzero signed-digits which belong to the set {/spl plusmn/1, /spl plusmn/2, ..., /spl plusmn/(r-1)}. In this article, we derive closed form expressions for the average number of nonzero digits in the minimal weight SD representation and for the average length of the canonical SD representation, a special case of the minimal weight SD form, of a positive integer whose radix-r form is of length n, n/spl ges/1.