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Power spectrum measurement and the estimation of power spectral density
8
Transactions of the symposium on noise analysis in nuclear systems
Paper No. 21
POWER SPECTRUM MEASUREMENT ESTIMATION OF POWER SPECTRAL
AND THE DENSITY*
R. R. MOHLER Los Alamos Scientific Laboratory, Los Alamos, New Mexico A SHORT review of correlation functions and power spectral densities establishes a foundation for the main topic. The periodogramme estimate, which is frequently incorrectly used to define power spectral density, is discussed at length. Although this estimate is asymptotically unbiased, if the sample length is assumed to approach infinity first, then the expectation does not yield power spectral density. In fact, the latter estimate is a random variable, and if the sample may be considered as coming from a Gaussian process, it has an exponential distribution with a standard deviation greater than or equal to the actual value of spectral density which is being estimated. The object of power spectrum measurement is to estimate the average power per unit bandwidth (power spectral density) of an ensemble of recordings from a single sample of finite length. Although wide-sense stationarity is all that is required of a process to define a power spectral density, it is seen that the ergodic hypothesis is utilized in the measurement-estimation process. For non-ergodic processes ensemble averaging over a large number of samples by digital computation is probably the only technique available. Measurement of a power spectrum by a direct analogue scheme is basically founded on a Fourier analysis or periodogramme estimate. This assumes a perfectly tuned aperture filter of infinitesimal bandwidth, a squarelaw detector with no dynamics, and a perfect integrator. This would result in a poor estimate as mentioned previously. However, by measurement of the average signal power in a finite frequency band, the error in estimation is decreased. Although an increase in bandwidth of the tuned filter decreases the statistical error, it is seen that a limitation exists due to the variations in the actual spectrum over the bandwidth. The sampling theorem is also seen to constrain the measurement resolution, Several physical variations of the direct analogue technique and data speed-up to obtain better accuracy are presented. Indirect methods of estimating power spectral density by way of correlation functions are economical and are discussed in this paper. Error analysis and methods for reducing error in measurement and estimation are emphasized for both the direct and indirect techniques. The paper also includes mention of reactor applications and an extensive bibliography. * Work performed under the auspices of the U.S. Atomic Energy Commission.
Transactions of the symposium on noise analysis in nuclear systems
Paper No. 21
POWER SPECTRUM MEASUREMENT ESTIMATION OF POWER SPECTRAL
AND THE DENSITY*
R. R. MOHLER Los Alamos Scientific Laboratory, Los Alamos, New Mexico A SHORT review of correlation functions and power spectral densities establishes a foundation for the main topic. The periodogramme estimate, which is frequently incorrectly used to define power spectral density, is discussed at length. Although this estimate is asymptotically unbiased, if the sample length is assumed to approach infinity first, then the expectation does not yield power spectral density. In fact, the latter estimate is a random variable, and if the sample may be considered as coming from a Gaussian process, it has an exponential distribution with a standard deviation greater than or equal to the actual value of spectral density which is being estimated. The object of power spectrum measurement is to estimate the average power per unit bandwidth (power spectral density) of an ensemble of recordings from a single sample of finite length. Although wide-sense stationarity is all that is required of a process to define a power spectral density, it is seen that the ergodic hypothesis is utilized in the measurement-estimation process. For non-ergodic processes ensemble averaging over a large number of samples by digital computation is probably the only technique available. Measurement of a power spectrum by a direct analogue scheme is basically founded on a Fourier analysis or periodogramme estimate. This assumes a perfectly tuned aperture filter of infinitesimal bandwidth, a squarelaw detector with no dynamics, and a perfect integrator. This would result in a poor estimate as mentioned previously. However, by measurement of the average signal power in a finite frequency band, the error in estimation is decreased. Although an increase in bandwidth of the tuned filter decreases the statistical error, it is seen that a limitation exists due to the variations in the actual spectrum over the bandwidth. The sampling theorem is also seen to constrain the measurement resolution, Several physical variations of the direct analogue technique and data speed-up to obtain better accuracy are presented. Indirect methods of estimating power spectral density by way of correlation functions are economical and are discussed in this paper. Error analysis and methods for reducing error in measurement and estimation are emphasized for both the direct and indirect techniques. The paper also includes mention of reactor applications and an extensive bibliography. * Work performed under the auspices of the U.S. Atomic Energy Commission.