VLSI Digital Signal Processing Systems: Design and Implementation

14.3    CARRY-FREE RADIX-2 ADDITION AND SUBTRACTION

Redundant number representations limit the carry propagation to a few bit-positions, which is usually independent of the wordlength W. This carry-propagation-free feature enables fast addition [1]. The algorithm to carry out signed binary digit addition is not unique [5, 12],[15], and therefore its logic implementation can be diverse.

A radix-2 signed-digit number is coded using 2 unsigned binary numbers, 1 positive and 1 negative, as X = X ⁺ – X ⁻. Hence, each signed digit is represented using 2 bits as where and x_i ∈ {,0,1} [4],[13]. This section considers radix-2 hybrid and signed-digit additions and subtractions. In a hybrid operation, 1 input operand and the output operand are in redundant signed-digit representation, and the 2nd input operand is a conventional unsigned number. A signed-digit addition can be viewed as a concatenation of one hybrid addition and one hybrid subtraction.

14.3.1    Hybrid Radix-2 Addition

Consider the addition of a radix-2 signed-digit number X_<2.1> and an unsigned conventional number Y

where X_{<r.α ...}