A closed‐form solution for the optimal time evolution of an exponentially decaying signal with thermal noise. (March 2014)

• Record Type: Journal Article
• Authors: Williams, Brian Wesley; Rovnyak, David
• Is Part Of: Concepts in magnetic resonance. Volume 43:Number 2(2014)
• Page Start: 54
• Page End: 56
• Publication Date: 2014-03
• Subjects: signal‐to‐noise ratio -- Lambert Function -- uniform sampling; Magnetic resonance -- Periodicals; Nuclear magnetic resonance -- Periodicals; 538
• DOI: 10.1002/cmr.a.21298
• Languages: English
• ISSNs: 1546-6086

Abstract

Abstract : The signal‐to‐noise ratio of a monoexponentially decaying signal exhibits a maximum at an evolution time of approximately 1.26 T 2 . It has previously been thought that there is no closed‐form solution to express this maximum. We report in this note that this maximum can be represented in a specific, analytical closed form in terms of the negative real branch of an inverse function known as the Lambert W function. The Lambert function is finding increasing use in the solution of problems in a variety of areas in the physical sciences. © 2014 Wiley Periodicals, Inc. Concepts Magn Reson Part A 43A: 54–56, 2014.