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:

:

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

For the closest approximation, the optimum values for $alpha,!$ and are $alpha\_0\; =\; frac\{2\; cos\; frac\{pi\}\{8\{1\; +\; cos\; frac\{pi\}\{8\; =\; 0.96043387...$ and , giving a maximum error of 3.96%.

References

*Lyons, Richard G. "Understanding Digital Signal Processing, section 13.2." Prentice Hall, 2004 ISBN 0-13-108989-7.
* Griffin, Grant. [http://www.dspguru.com/comp.dsp/tricks/alg/mag_est.htm DSP Trick: Magnitude Estimator] .