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Nonuniform Sampling: Bandwidth And Aliasing
 in Maximum Entropy and Bayesian Methods
, 2000
"... . For spectroscopic measurements there are good reasons why one might consider using nonuniformly nonsimultaneously sampled complex data. The primary one is that the eective bandwidth, the largest spectral window free of aliases, can be much wider than with uniformly sampled data. In this paper we d ..."
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. For spectroscopic measurements there are good reasons why one might consider using nonuniformly nonsimultaneously sampled complex data. The primary one is that the eective bandwidth, the largest spectral window free of aliases, can be much wider than with uniformly sampled data. In this paper we discuss nonuniformly nonsimultaneously sampled data, describe how these data are traditionally analyzed, analyze them using probability theory and show how probability theory generalizes the discrete Fourier transform: rst for uniformly sampled data, then for nonuniformly sampled data and nally for nonuniformly nonsimultaneously sampled data. These generalizations demonstrate that aliases are not so much removed by nonuniform nonsimultaneous sampling as they are moved to much higher frequencies. 1. Introduction The problem of estimating the frequency of a sinusoid occurs in many dierent areas of science and engineering. Such data may be sampled in time, space, angle, or a host of other ...
A Bayesian revolution in spectral analysis
 In American Institute of Physical Proceedings
, 2001
"... Abstract. The discrete Fourier transforms (DFT) is ubiquitous in spectral analysis as a result of the introduction of the Fast Fourier transform by Cooley and Tukey in 1965. In 1987, E. T. Jaynes derived the DFT using Bayesian Probability Theory and provided surprising new insights into its role in ..."
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Abstract. The discrete Fourier transforms (DFT) is ubiquitous in spectral analysis as a result of the introduction of the Fast Fourier transform by Cooley and Tukey in 1965. In 1987, E. T. Jaynes derived the DFT using Bayesian Probability Theory and provided surprising new insights into its role in spectral analysis. From this new perspective the spectral resolution achievable is directly dependent on the signal to noise ratio and can be orders of magnitude better than that of a conventional Fourier power spectrum or periodogram. This was the starting point for an ongoing Bayesian revolution in spectral analysis which is reviewed in this paper, with examples taken from physics and astronomy. The revolution is based on a viewpoint of Bayesian Inference as extended logic. 1.
Bayesian Analysis V: Amplitude Estimation, Multiple Well Separated Sinusoids
 J. Magn. Reson
, 1992
"... . Bayesian probability theory is used to estimate the amplitude of a single exponentially decaying sinusoid in NMR free induction decay (FID) data. The posterior probability for the amplitude is derived independent of the phase, frequency, decay rate constant, and variance of the noise. The estimate ..."
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. Bayesian probability theory is used to estimate the amplitude of a single exponentially decaying sinusoid in NMR free induction decay (FID) data. 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. The uncertainty in the estimate is shown to varying inversely with the square root of the sampling rate. Finally, the calculation is applied in two examples. The first example demonstrates probability theory's ability to estimate frequencies and amplitudes in very low signalto noise. The second example illustrates the use of this calculation when the data contain multiple, wellseparated sinusoids. Introduction In NMR the signal frequencies and the amplitudes are of fundamental importance. The freq...
Estimating The Ratio Of Two Amplitudes In Nuclear Magnetic Resonance Data
, 1991
"... . Probability theory is applied to the problem of estimating the ratio of the amplitudes of two sinusoids in nuclear magnetic resonance data. The posterior probabilitydensity for the amplitude ratio is derived independent of the phase, frequencies, decayrate constants, variance of the noise, and t ..."
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. Probability theory is applied to the problem of estimating the ratio of the amplitudes of two sinusoids in nuclear magnetic resonance data. The posterior probabilitydensity for the amplitude ratio is derived independent of the phase, frequencies, decayrate constants, variance of the noise, and the amplitude of the other sinusoid. This probabilitydensity function is then applied in an illustrative example, and the results are contrasted with those obtained by traditional analysis. 1. Introduction Investigating the molecular structure of a compound in a nondestructive manner is difficult. If the nuclei of the compound have a magnetic moment, then one way to investigate the structure is to place the compound in a high magnetic field, excite the system with radiofrequency energy, and listen to it "ring." This type of experiment is called a nuclear magnetic resonance (NMR) experiment. The "ringing" is called a free induction decay (FID). When nuclei of the same type, for example prot...
Amplitude Estimation In Nuclear Magnetic Resonance Data
"... : Probability theory is applied to the problem of estimating the amplitudes of the sinusoids in nuclear magnetic resonance data containing two exponentially decaying sinusoids. The posterior probabilitydensity of one of the amplitudes is derived independent of the phases, frequencies, decayrate co ..."
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: Probability theory is applied to the problem of estimating the amplitudes of the sinusoids in nuclear magnetic resonance data containing two exponentially decaying sinusoids. The posterior probabilitydensity of one of the amplitudes is derived independent of the phases, frequencies, decayrate constants, variance of the noise, and the amplitude of the other sinusoid. This probabilitydensity function is then applied in an illustrative example and the results are contrasted with those obtained by traditional analysis techniques. Introduction Investigating the molecular structure of a compound in a nondestructive manner is difficult. If the nuclei of the compound have a magnetic moments, then one way to investigate the structure is to place the compound in a high magnetic field, excite the system with radiofrequency energy, and listen to it "ring." This type of experiment is called a nuclear magnetic resonance (NMR) experiment. The "ringing" is called a free induction decay (FID). Wh...