# Alpha max plus beta min algorithm

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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.

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

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

For the closest approximation, the optimum values for $alpha,!$ and are $alpha_0 = frac\left\{2 cos frac\left\{pi\right\}\left\{8\left\{1 + cos frac\left\{pi\right\}\left\{8 = 0.96043387...$ 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] .

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