# Canonical signed digit

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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.[1]

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 ${\displaystyle +2^{5}-2^{3}-2^{0}}$ or ${\displaystyle 32-8-1=23}$

## 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

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

into +00-

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

## References

1. ^ Hewlitt, R.M. "Canonical signed digit representation for FIR digital filters". Signal Processing Systems, 2000. SiPS 2000. 2000 IEEE Workshop on: 416–426. doi:10.1109/SIPS.2000.886740.