Frequency Stability Characterization from the Filtered Signal of a Precision Oscillator

This paper presents the evaluation of the frequency stability characterization of an oscillator when its signal is directly filtered instead of its phase as is usually considered in the development of the standard theory. The amplitude and the phase fluctuations of the filtered signal are expressed as functions of filter impulse response and input signals characteristics. The general expressions for the power spectral density of the amplitude and phase fluctuations of the filtered signal are then calculated. The results obtained for the phase fluctuations of the filtered signal are used to characterize the frequency stability of the oscillator. Every filter will have a particular effect on the measured phase fluctuations and we study some commonly used filters. For each filter, we give the contribution of white additive noise to the Allan variance and to the modified Allan variance (white phase noise). An experimental verification for the Allan variance in presence of additive noise shows an excellent agreement with the theoretical predictions. Introduction Barnes et al. [l], in their paper entitled "Characterization of Frequency Stability", have brought the major policies guiding the field of frequency stability measurement for about twenty-five years. Among other things, they wrote about time domain frequency stability measurement: "Good practice, however, dictates that the system noise bandwidth fh should be specified with any results." Until now no means were provided to help the xperimentalist performing such a measurement o evaluate this bandwidth. One was obliged to suppose a rectangular low-pass [ 2 ] , [ 3 ] or a first-order low-pass [4]-[61 phase filtering in order to specify the experimental conditions and to fully characterize the oscillator under test. These two types of phase filtering are different from most of the experimental setups used to measure the frequency stability. This work presents an evaluation of the effects encountered in frequency stability characterization when a filter is directly applied to the signal of a precision oscillator. We consider the case where the oscillator signal is perturbed by internal noise and by additive noise and where the resulting signal is passed through a linear filter. The amplitude and the phase fluctuations of the filtered signal are expressed as functions of filter impulse response and input signals characteristics. The general expressions for the power spectral density of the amplitude and phase fluctuations of the filtered signal are then calculated. A preliminary version of this work has been presented to the 1984 CPEM in Del:t, The Netherlands [ 7 ] . The results obtained for the p ase fluctuations of the filtered signal are used to characterize the frequency stability of the oscillator. This characterization is done either in the frequency domain by a power spectral density measurement or in the time domain by a variance measurement. A given filter will have a particular effect on the measured phase fluctuations and we study two low-pass filters; a first-order and a Nth-order Butterworth, two band-pass filters; a first-order and a Nth-order Butterworth and a second-order resonant bandpass filter. For each filter, we give the contribution of white additive noise to the Allan variance and to the modified Allan variance. However, contributions from other types of amplitude and phase fluctuations such as flicker noise or random walk could be calculated using the equations provided through this paper. These calculations will be given in ref. [81. Experimental verification for the Allan variance is given when the oscillator signal is in presence of additive noise and when the resulting signal is filtered by a first-order and a eighth-order band-pass filter. Filtering of a Noisy Sinusoidal Signal Description of the Model The model used and the calculations done in this paper will be presented extensively in reference [S]. Fig. 1 gives the block diagram representing our model. The oscillator signal is expressed by the following relation: so(t) = A, [l+co(t)J cos[2n~ot+rpo(t)+00], (1) where A, is the oscillator mean amplitude, ~ ~ ( t ) is its relative amplitude fluctuations, d o is its mean frequency, cpo(t) is its phase fluctuations and Qo(t) is its initial phase, which is uniformly distributed. The relative amplitude and phase fluctuations are two zero-mean wide-sense stationary random processes which are mean square continuous. Moreover the oscillator signal is perturbed by an additive noise which is decomposed as a sum of two parts: an component. wide-sense continuous in phase component and an in quadrature The additive noise, n(t), is a zero-mean stationary random process and is mean square This noise is then expressed as: cos [2nd0t+0,j q(t) sin [2ndot+00], ( 2 ) where the amplitudes of the two components, p(t) and q(t), are also zero-mean wide-sense stationary random processes, they are mean square continuous and uncorrelated with the oscillator relative amplitude and phase fluctuations. These two amplitudes have the same power spectral density, which is the symmetrical part 119 ~~~~8~-0 /85 /0000-0119$1 .00Q19851EEE around the oscillator mean frequency of the power spectral density of the additive noise: These amplitudes are also correlated and their cross-Dower suectral densities are related to the antisymmetrical part around the oscillator frequency of the noise power spectral density: OSCILLATOR n (t) ADDITIVE NOISE