Alpha max plus beta min algorithm

The alpha max plus beta min algorithm is a high-speed approximation of the square-root of the sum of two squares. That is to say, it gives the approximate absolute magnitude of a vector given the real and imaginary parts.

$|V|={\sqrt {I^{2}+Q^{2}}}$

The algorithm avoids the necessity of performing the square and square-root operations and instead uses simple operations such as comparison, multiplication and addition. Some choices of the α and β parameters of the algorithm allow the multiplication operation to be reduced to a simple shift of binary digits that is particularly well suited to implementation in high-speed digital circuitry.

The approximation is expressed as:

$|V|=\alpha \,\!\mathbf {Max} +\beta \,\!\mathbf {Min}$

Where $\mathbf {Max}$ is the maximum absolute value of I or Q and $\mathbf {Min}$ is the minimum absolute value of I or Q.

For the closest approximation, the optimum values for $\alpha \,\!$ and $\beta \,\!$ are 0.96043387 and 0.397824735, giving a maximum error of 3.96%.