TY - JOUR

T1 - Bayesian analysis. V. Amplitude estimation for multiple well-separated sinusoids

AU - Bretthorst, G. Larry

N1 - Funding Information:
The encouragement of Professor J. J. H. Ackerman is greatly appreciated as are extensive conversations with Professor E. T. Jaynes. This work was partially supported by a gift from the Monsanto Company and NIH Grant GM3033 I, J. J. H. Ackerman principal investigator.

PY - 1992/7

Y1 - 1992/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0141903764&partnerID=8YFLogxK

U2 - 10.1016/0022-2364(92)90004-Q

DO - 10.1016/0022-2364(92)90004-Q

M3 - Article

AN - SCOPUS:0141903764

SN - 0022-2364

VL - 98

SP - 501

EP - 523

JO - Journal of Magnetic Resonance (1969)

JF - Journal of Magnetic Resonance (1969)

IS - 3

ER -