Period mapping

File:Alpha Max Beta Min approximation.png

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.

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

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:

$\left|V\right|=\alpha \mathbf{M}\mathbf{a}\mathbf{x}+\beta \mathbf{M}\mathbf{i}\mathbf{n}$

Where $\mathbf{M}\mathbf{a}\mathbf{x}$ is the maximum absolute value of I and Q and $\mathbf{M}\mathbf{i}\mathbf{n}$ is the minimum absolute value of I and Q.

For the closest approximation, the optimum values for $\alpha$ and $\beta$ are ${\alpha }_0=\frac{2\cos \frac{\pi }{8}}{1+\cos \frac{\pi }{8}}=0.96043387...$ and ${\beta }_0=\frac{2\sin \frac{\pi }{8}}{1+\cos \frac{\pi }{8}}=0.39782473...$, giving a maximum error of 3.96%.

$\alpha$	$\beta$	Largest error (%)	Mean error (%)
1/1	1/2	11.80	8.68
1/1	1/4	11.61	0.65
1/1	3/8	6.80	4.01
7/8	7/16	12.5	4.91
15/16	15/32	6.25	1.88
${\alpha }_0$	${\beta }_0$	3.96	1.30

References

• Lyons, Richard G. Understanding Digital Signal Processing, section 13.2. Prentice Hall, 2004 ISBN 0-13-108989-7.
• Griffin, Grant. DSP Trick: Magnitude Estimator.