Fitting FFT-Derived Spectra: Theory, Tool, and Application to Solar Radio Spike Decomposition

Abstract

Spectra derived from fast Fourier transform (FFT) analysis of time-domain data intrinsically contain statistical fluctuations whose distribution depends on the number of accumulated spectra contributing to a measurement. The tail of this distribution, which is essential for separation of the true signal from the statistical fluctuations, deviates noticeably from the normal distribution for a finite number of the accumulations. In this paper we develop a theory to properly account for the statistical fluctuations when fitting a model to a given accumulated spectrum. The method is implemented in software for the purpose of automatically fitting a large body of such FFT-derived spectra. We apply this tool to analyze a portion of a dense cluster of spikes recorded by our FST instrument ) during a record-breaking event (Cerruti et al. 2006) that occurred on 06 Dec 2006. The outcome of this analysis is briefly discussed. Subject headings:

Particle acceleration by a solar flare termination shock

“…(3) The spike bandwidth tends to have less dispersion at larger brightness. These findings are generally consistent with earlier studies of solar spike bursts based on total-power dynamic spectral data (i.e., without spatial information) (38)(39)(40)(41).…”

Determining the Mode, Frequency, and Azimuthal Wave Number of ULF Waves During a HSS and Moderate Geomagnetic Storm

“…A limitation to the above approach is that neither likelihood nor BIC determines whether the tested models are good choices in absolute terms-all of the chosen models could be poor representations of the observed data. Hence, in order to provide a secondary check and determine if models S 0 and S 1 are consistent with the data we utilize the χ 2 -like statistic for exponentially distributed data described by Nita et al (2014), who showed that an appropriate statistic may be written as…”

“…for a single Fourier power spectral sample, where ρ j = i j /s j is referred to as the sample-to-model estimator, that is, the ratio of the data to the best fit model (see Nita et al, 2014, equation (5)), and ν is the number of degrees of freedom. We determine χ 2 υ for each model-data pair using the approximate expression for the probability density function of χ 2 υ (Nita et al, 2014, Appendix A) and subsequently derive a probability value p of the data for each model. In this work, the main purpose of this statistic is to identify intervals and events where the models are clearly inappropriate for the data.…”

Measurement of duration and signal‐to‐noise ratio of astronomical transients using a Spectral Kurtosis spectrometer

“…In their previous study, Nita et al [] presented a maximum likelihood method that has been employed to resolve the individual spectral contribution of the mostly clustered spikes observed by the FST instrument during the 6 December 2006 solar radio burst, as those illustrated in Figure . Although their method did not include the analysis presented here, we note that the accuracy of their results could not have been affected by the duty cycle effects illustrated in this section because, as noted earlier, the very short (100 μs) accumulation time of the FST instrument guarantees that the much longer lived spikes have a 100% duty cycle, which would not distort the true spectral shape of the observed transient signals.…”