Time Domain Frequency Stability Estimation Based on Fft Measurements

The standard characterizations of frequency stability are, in the time domain, the Allan (or two-sample) variance and, in the frequency domain, the spectral density function (SDF). The former is mathematically related to the latter by the conversion between time and frequency domain. In this paper, the biases of the Fast Fourier transform (FFT) spectral estimate with Hanning window are checked and the resulting unbiased spectral density are used to calculate the Allan variance. Both the numerical integral and the curve-fitting methods are presented to calculate the variances. The numerical integral is a straightforward method to use, and we can get the integral approximation after eliminating some spike points from SDF, e.g. noise caused by ac power. In addition, a common model for SDF is linear combinations of powerlaw processes, which are distinguished by the integer powers in their functional dependence on Fourier frequency with the appropriate coefficients. Fitting a form of the above model to the resulting SDF using standard regression techniques can estimate these coefficients. Cutler’s formula is adopted to calculate the integral approximation using these coefficients. The approximations of variances from these two methods are compared and analyzed. Finally, we discuss the limitations and possible errors from these two methods. INTRODUCTION FFT spectrum analyzers generally have several different window functions available for analyzing signals. Recent research has shown that the Hanning window provides excellent performance for analyzing noise [1]. Our lab has established a phase noise measurement system including a phase noise standard (1,5,10,100 MHz), a single-channel noise detector, a delay line unit, and one single-channel FFT spectrum analyzer. The signal reference is from a low-noise frequency reference (LNFR-400) with a noise level of about –173 dBc/Hz (5 MHz PM, at Fourier frequency 100 KHz). For passive devices, the system can measure up to –177 dBc/Hz. This year we also have built up a cross-correlation system, which can measure the noise 20 dBc/Hz below the above level. The very short-term stability (τ<0.5 second) by using this phase noise measurement system is a subject of interest to us, since the traditional time interval counter is applicable only when τ is about 1 second. In this paper, we are trying to calculate the Allan variance of the spectral density estimate from experimental results, and possible errors in the spectral density estimate, e.g. biases from window functions, noise from ac power, etc. will be considered. In general, if the spectral density of the normalized frequency fluctuations ) ( f S y is known, its mathematical relation to the Allan variance can be expressed as [2]: Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 00 SEP 2004 2. REPORT TYPE N/A 3. DATES COVERED 4. TITLE AND SUBTITLE Time Domain Frequency Stability Estimation Based On Fft Measure 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER