Bayesian analysis. V. Amplitude estimation for multiple well-separated sinusoids

Bayesian probability theory is used to estimate the amplitude of multiple well-separated exponentially decaying sinusoids in NMR free-induction-decay data. Specifically the posterior probability for the amplitude is derived independent of the phase, frequency, decay rate constant, and variance of the noise. The estimate is shown to be accurate and precise in the sense that as the noise approaches zero, the estimate approaches the true value of the amplitude and the uncertainty in the estimate approaches zero. For fixed data-acquisition time, the uncertainty in the estimate is shown to vary inversely with the square root of the sampling rate. Finally, the calculation is applied in two examples. The first example demonstrates the ability of probability theory to estimate frequencies and amplitudes in very low signal-to-noise. The second example illustrates the use of this calculation when the data contain multiple, well-separated sinusoids.