|
|
| Line 1: |
Line 1: |
| [[File:Alpha Max Beta Min approximation.png|800px|centre]]
| | Irwin Butts is what my wife enjoys to contact me although I don't truly like becoming called like that. For a whilst I've been in South Dakota and my mothers and fathers live close by. For years he's been operating as a receptionist. To gather cash is a thing that I'm totally addicted to.<br><br>My web-site :: home std test kit ([http://www.animecontent.com/blog/436250 click here to read]) |
| | |
| 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 (geometric)|vector]] given the [[Real number|real]] and [[Imaginary number|imaginary]] parts.
| |
| | |
| :<math> |V| = \sqrt{ I^2 + Q^2 } </math>
| |
| | |
| 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:
| |
| | |
| :<math> |V| = \alpha\,\! \mathbf{Max} + \beta\,\! \mathbf{Min} </math>
| |
| | |
| 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.
| |
| | |
| 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%. | |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! <math>\alpha\,\!</math> || <math>\beta\,\!</math> || Largest error (%) || Mean error (%)<br />
| |
| |-
| |
| | align="right" | 1/1 || align="right" | 1/2 || align="right" | 11.80 || align="right" | 8.68
| |
| |-
| |
| | align="right" | 1/1 || align="right" | 1/4 || align="right" | 11.61 || align="right" | 0.65
| |
| |-
| |
| | align="right" | 1/1 || align="right" | 3/8 || align="right" | 6.80 || align="right" | 4.01
| |
| |-
| |
| | align="right" | 7/8 || align="right" | 7/16 || align="right" | 12.5 || align="right" | 4.91
| |
| |-
| |
| | align="right" | 15/16 || align="right" | 15/32 || align="right" | 6.25 || align="right" | 1.88
| |
| |-
| |
| | align="right" | <math>\alpha_0</math> || align="right" | <math>\beta_0</math> || align="right" | 3.96 || align="right" | 1.30
| |
| |-
| |
| |}
| |
| | |
| ==References==
| |
| | |
| *[[Richard Lyons|Lyons, Richard G]]. ''Understanding Digital Signal Processing, section 13.2.'' Prentice Hall, 2004 ISBN 0-13-108989-7.
| |
| * Griffin, Grant. [http://www.dspguru.com/dsp/tricks/magnitude-estimator DSP Trick: Magnitude Estimator].
| |
| | |
| [[Category:Approximation algorithms]]
| |
| [[Category:Root-finding algorithms]]
| |
Irwin Butts is what my wife enjoys to contact me although I don't truly like becoming called like that. For a whilst I've been in South Dakota and my mothers and fathers live close by. For years he's been operating as a receptionist. To gather cash is a thing that I'm totally addicted to.
My web-site :: home std test kit (click here to read)