Alpha max plus beta min algorithm


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} + eta,! mathbf{Min}

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 eta,! are alpha_0 = frac{2 cos frac{pi}{8{1 + cos frac{pi}{8 = 0.96043387... and eta_0 = frac{2 sin frac{pi}{8{1 + cos frac{pi}{8 = 0.39782473..., 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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