Single Stationary Sinusoid Plus Noise

  • G. Larry Bretthorst
Part of the Lecture Notes in Statistics book series (LNS, volume 48)


We begin the analysis by constructing the direct probability, P(D\H, I). We think of this as the likelihood of the parameters, because it is its dependence on the model parameters which concerns us here. The time series y(t) we are considering is postulated to contain a single stationary harmonic signal f(t) plus noise e(t). The basic model is always: we have recorded a discrete data set D = {d 1, ⋅⋅⋅, d N }; sampled from y(t) at discrete times {t 1, ⋅⋅⋅, t N }; with a model equation
$$ {d_i} = y\left( {{t_i}} \right) = f\left( {{t_i}} \right) + {e_i},{\kern 1pt} \quad \left( {i \leqslant i \leqslant N} \right). $$
As already noted, different models correspond to different choices of the signal f(t). We repeat the analysis originally done by Jaynes [12] using a different, but equivalent, set of model functions. We repeat this analysis for three reasons: first, by using a different formulation of the problem we can see how to generalize to multiple frequencies and more complex models; second, to introduce a different prior probability for the amplitudes, which simplifies the calculation but has almost no effect on the final result; and third, to introduce and discuss the calculation techniques without the complex model functions confusing the issues.


Power Spectral Density Prior Probability Prior Information Discrete Fourier Transform Safety Device 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • G. Larry Bretthorst
    • 1
  1. 1.Department of ChemistryWashington UniversityUSA

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