The specification of oscillator characteristics from measurements made in the frequency domain

Abstract

A cross-correlation technique for measuring the very short-term (milliseconds to seconds) properties of stable oscillators is described. Time-dependent functions representing signals from two separate oscillators are led to a function multiplier where the instantaneous product of the functions is made. The oscillators are either set to a given phase relation or allowed a small relative drift so that a slow beat frequency is observed. Short-term fluctuations superimposed upon the slow beat signal from the multiplier output will represent the instantaneous phase difference between the oscillators when the inputs are in quadrature. When the inputs are in and out of phase, the fluctuations represent amplitude fluctuations. The time averaging function is determined by a filter having arectangular pass band from nearly zero frequency to a cutoff frequency vC. The mean square frequency deviation measured in a bandwidth wC is obtained by differentiating, filtering, squaring, and averaging the signal from the function multiplier data being taken when the input signals are in quadrature. Mean square averages of amplitude and phase averaged over various bandwidths wC may be obtained by bypassing the differentiator. Sample data from measurements on hydrogen masers are presented, and the effect of thermal noise is seen to be the major factor limiting the short-term frequency stability of the signals.

Some aspects of the theory and measurement of frequency fluctuations in frequency standards

“…(Again a time delay of r / 2 seconds has been introduced.) This response is described by 6(t) -6(tr ) hb(t) = (8) 7 Because both filters h, and ha are clearly difficult to with limits symmetric about t . However, to insure physical realiza-1 For mathematical symmetry, the integrals have been written bility of the filter, the impulse response has been drawn from 0 to T , changed to t and t + T , respectively.…”

“…, is used as the approximation. I t is certainly perfectly valid to place a sharp filter at the output of the phase detector and measure the rms voltage appearing a t t h e filter output as an indirect measurement of S;(w) (see, for example,[7]). The error appears when one tries to identify this rms voltage with u [ ( $ )~,~] .…”

A collapse of the cross-spectral function in phase noise metrology

“…Cross-spectral analysis has been used to improve the sensitivity of measurements in the field of modulation noise metrology for nearly fifty years. [1][2][3][4][5] The last twenty years have seen extensive use of cross-spectral analysis in the laboratory [6][7][8][9][10][11][12] and it has even made its way into commercial test equipment, at first with small companies 8,13 and now with major test equipment manufacturers. It is generally understood that when a desired signal has some level of correlation with an interferer, occasionally some level of cancellation of signals can be observed.…”

Phase inversion and collapse of cross‐spectral function

“…Cross-spectral analysis has been used in the field of modulation noise metrology for improving the sensitivity of measurements for nearly fifty years [1][2][3][4][5]. The last twenty years has seen extensive use of cross-spectral analysis in the laboratory [6][7][8][9][10][11][12] and it has even made its way into commercial test equipment, at first with small companies [8,13] and now with major mainstream test equipment manufacturers.…”

“…Unfortunately, there is no easy way to determine if a past cross-correlation measurement was affected by the cause given in this paper. (1) where, X(f) and Y(f) are the Fourier transforms of x(t) and y(t). The cross-power spectral density S xy (f) can be determined from the ensemble average of X(f) multiplied by the complex conjugate of Y(f).…”