Alpha max plus beta min algorithm: Difference between revisions
m \cos \sin |
Zogromalvus (talk | contribs) m 'and' may be more pertinent here |
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:<math> |V| = \alpha\,\! \mathbf{Max} + \beta\,\! \mathbf{Min} </math> |
:<math> |V| = \alpha\,\! \mathbf{Max} + \beta\,\! \mathbf{Min} </math> |
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Where <math>\mathbf{Max}</math> is the maximum absolute value of I |
Where <math>\mathbf{Max}</math> is the maximum absolute value of I and Q and <math>\mathbf{Min}</math> is the minimum absolute value of I and Q. |
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For the closest approximation, the optimum values for <math>\alpha\,\!</math> and <math>\beta\,\!</math> are <math>\alpha_0 = \frac{2 \cos \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.96043387...</math> and <math>\beta_0 = \frac{2 \sin \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.39782473...</math>, giving a maximum error of 3.96%. |
For the closest approximation, the optimum values for <math>\alpha\,\!</math> and <math>\beta\,\!</math> are <math>\alpha_0 = \frac{2 \cos \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.96043387...</math> and <math>\beta_0 = \frac{2 \sin \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.39782473...</math>, giving a maximum error of 3.96%. |
Revision as of 10:21, 1 July 2007
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.
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 is the maximum absolute value of I and Q and is the minimum absolute value of I and Q.
For the closest approximation, the optimum values for and are and , giving a maximum error of 3.96%.
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 | 15/16 | 12.5 | 4.91 |
15/16 | 15/32 | 6.25 | 1.88 |
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.