Probability Theory and Multiexponential Signals, How Accurately Can the Parameters be Determined?

Abstract.

Estimating the amplitudes and decay rate constants of exponentially decaying signals is an important problem in science. Understanding how the uncertainty in the parameter estimates depends on the experimental parameters is important as an aid in understanding how to improve the reliability of the parameter estimates. In this paper, probability theory has been applied to this problem with the intent of understanding the relevant experimental parameters. In the case of a single exponential, the uncertainty in the estimated decay rate depends directly on the three halves power of the true decay rate constant, inversely on the signal-to-noise ratio, and inversely on the square root of the number of data values. The uncertainty in the amplitude estimate depends directly on the square root of the true decay rate constant, directly on the noise level, and inversely on the square root of the number of data values. The case of two exponentials has also been analyzed with similar results. However, here the presence of the second signal introduces interference effects which make the estimate more uncertain.