A Closed-Form Solution for the Optimal Time Evolution of an Exponentially Decaying Signal with Thermal Noise
Concepts in Magnetic Resonance Part A
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. (C) 2014 Wiley Periodicals, Inc.
Williams, Brian W. and Rovnyak, David. "A Closed-Form Solution for the Optimal Time Evolution of an Exponentially Decaying Signal with Thermal Noise." Concepts in Magnetic Resonance Part A 43, no. 2 (2014) : 54-56.