Canonical signed digit

In computing canonical-signed-digit (CSD) is a special manner for encoding a value in a signed-digit representation, which itself is non-unique representation and allows one number to be represented in many ways. Probability of digit being zero is close to 66% (vs. 50% in two's complement encoding) and leads to efficient implementations of add/subtract networks (e.g. multiplication by a constant) in hardwired digital signal processing.

The representation uses a sequence of one or more of the symbols, -1, 0, +1 (alternatively -, 0 or +) with each position possibly representing the addition or subtraction of a power of 2. For instance 23 is represented as +0-00-, which expands to $+2^5 -2^3 -2^0$ or $32 - 8 - 1 = 23$

Implementation

CSD is obtained by transforming every sequence of zero followed by ones (011...1) into + followed by zeros and the least significant bit by - (+0....0-).

As an example: the number 7 has a two's complement representation 0111

$(7=0\times 2^3+1\times 2^2+1\times 2^1+1\times 2^0 = 4+2+1)$

into +00-

$(7=+1\times 2^3+0\times 2^2+0\times 2^1-1\times 2^0 = 8-1)$