Microcycle spectral estimation

Abstract

This thesis attempts to estimate the power spectral density of low frequency semiconductor noise over a range of 10 decades, from a microcycle (10-6 cps) to 10 kilocycles (10+4 cps). It is concluded that the behavior is more complex than a simple inverse proportionality to frequency. The spectrum is approximately 1/f in the region around 100 cps and changes gradually to 1/f2 as the frequency decreases to the microcycle region. These spectra represent the noise properties of the first stage transistors of a grounded input dc differential amplifier. The estimated spectra at very low frequencies still reflect strong temperature influences. In order to obtain these measurements it was necessary to control the temperature environment of the noise source. This was accomplished first by passive attenuation and later by active control. The noise source was placed in a circulating oil bath whose temperature was sensed electrically and controlled to a .001° C range. In conjunction with the temperature control activity the power spectral density of room temperature variations was estimated in the frequency range from .1 cps down to 5 x 10-8 cps. Other spectra of interest estimated over the low frequency range were for line voltage amplitude fluctuations and operational amplifier drift. A brief description of the equipment constructed to obtain sample functions of the noise processes is included. The analytical portion of this work is concerned with the mathematical techniques employed in obtaining power spectral density estimates. The basic scheme employed is that of Blackman and Tukey which consists of estimating the auto-correlation function and Fourier transforming the result. A formula is developed for calculating the variance of the spectral estimator actually employed in the computations. The bias and variability are presented for the estimator when estimating a spectra containing a spectral line. A confidence interval approach to the variability of the spectral estimator is examined. A confidence interval which depends only on the data is constructed around the spectral density estimate. A technique for utilizing the available knowledge concerning the expected variability of the spectral estimate is developed. The result is formulated in terms of a maximum liklihood estimator for the average spectral density when several independent estimates are available. Some possible sources of low frequency bias in the spectral estimate are considered in detail. Among these are the effect of mean removal and certain deterministic disturbances such as steps. Prewhitening for 1/f and 1/f2 spectra is examined and shown to lead to very great improvement in the spectral estimate. Some suggestions as to more efficient methods of spectral estimation data collection and processing are offered.