NIST Technical Note 1337: Characterization of Clocks and Oscillators

BL-\5ES .-\.\D \·A.Rl.-\.\CES OF SEVERAL FFT SPECTRAL ESTI~.LHORSAS A FC='iCTION OF :-WISE TYPE A~D ~t:~1BER OF SA.\IPLESF. 1. Walls. Time and Frequency Division.:-lational Institute of Standards and Technology, Boulder. CO 80303D. B. Percival, Applied Physics Laboratory,t.;'niversity of Washington, Seattle, WA 98106 From: Proceedings of theW. R. Irelan, Irelan Electronics. 43rd Annual Symposium on412 Janet, Tahlequah, OK 74464 Frequency Control, 1989.AbstractWe theoretically and experimentally investigate the biasesand the variances of Fast Fourier transform (FFT) spectral estimateswith different windows (data tapers) when used to analyzepower·law noise types JO, 1- 1 , 1- 3 and 1- 4 . There is awide body of literature for white noise but virtually no investigationof biases and variances of spectral estimates for power-lawnoise spectra commonly seen in oscillators. amplifiers, mixers,etc. Biases (errors) in some cases exceed 3{) dB. The experimentaltechniques introduced here permit one to lUlalyze theperformance of virtually any window for any power-law noise.This makes it possible to determine the level of a particularnoise type to a specified statistical accuracy for a particularwindow.I. IntroductionFast Fourier transform (FFT) spectrum analyzers are verycommonly used to estimate the spectral density of noise. Theseinstruments often have several different windows (data tapers)available for analyzing different types of spectra. For example,in some applications spectral resolution is important; in others,the precise amplitude of a widely resolved line is important; andin still other applications, noise analysis is important. Thesediverse applications require different types of windows.We theoretically and experimentally investigate the biasesand variances of FIT spectral estimates with different windowswhen used to analyze a number of common power-law noisetypes. There is a wide body of literature for white noise but virtuallyno investigation of these effects for the types of power-lawnoise spectra commonly seen in oscillators, amplifiers, mixers,etc. Speci£cally, we pTe3ent theoretical results for the biasesassociated with two common windows - the uniform and HIUlningwindows - when applied to power-law spectra varyingas 1 0 , 1- 1 and 1- 4 . We then introduce experimental techniquesfor accurately determining the biases of lUly window anduse them to evaluate the biases of three different windows forpower-law spectra varying as r, 1- 1 , 1- 3 IUld 1- 4 • As an examplewe £nd with I-~ noise thAt the uniform window CIUl haveerrors ranging from a few dB to over 30 dB, depending on thelength of span of the f-t noise.We have also theoreticalJy investigated the varilUlces ofFFT spectral estimates with the uniform IUld Hanning windows(confidence of the estimates) as a function of the power-lawnoise type and as a function of the amount c:i data. We introduceexperimental techniques thAt lIlAke it relatively easy toindependently determine the variance of the spectral estimatefor virtually lUly window on any FFT spectrum analyzer. Thevariance that is realized on a particular instrument depends notonly on the window but on the specific implementation in bothhardware and software. We find that the variance c:i the spectraldensity estimates for white Doise, r, is very similar forthree specific windows available on one instrument and almostidentical to that obtained by standard statistical analysis. Thevariances for spectral density estimates of 1- 4 noise are only4% higher than that of 1° noise for two of the windows studied.The third window - the uniform window - does Dot vieldusable results for either 1- 3 or 1- 4 noise. ­Based on this work it is now possible to determine theminimum number of samples necessary to determine the levelof a particular noise type to a specified statistical accuracy as afunction of the window. To our knowledge this was previouslypossible only for white noise -although the traditional resultsare generally valid for noise that varied as 1- 8 , where a wasequal to or lesll than 4.II.Spectrum Analyzer BuiesThe spectrum analyzer which W!lS U!ed in the experimentalwork reported here is fairly typical of a number c:i such instrumentscurrently aV1l.ilable from various manufacturers. Thebasic measurement process generally consists of taking a stringof N. =1024 digital samples of the input wave form, which werepresent here by XI, X 1 , ..•, XN•. The basic measurementperiod was 4 IDS. This yields a sampling time t:.t = 3.90625 iJS.Associated with the FFT of a time series with S. data points.there are usually (N./2) + 1 = 513 frequencies)I, = N.ut' j =0, 1, ... , .'11./2.The fundamental frequency II is 250 Hz, and the Nyquist frequencyI N.n is 128 kHz. Since the spectrum analyzer usesan anti-aliasing filter which significantly distorts the high frequencyportion of the spectrum, the instrument only displaysthe measured spectrum for the lowest 400 nonzero frequencies,namely, II = 250 Hz, h =500 Hz, . ", 1400 = 100 kHz.The exact details of how the spectrum analyzer estimatesthe spectrum for X I, ..., X N. are unfortunately not providedin the documentation !Upplied by the manufacturer, so the followingmust be regarded only as a reasonable guess on our partas to its operation (see [1] for a good discussion on the basicideas behind a spectrum lUlalyzer; two good general referencesfor spectral lUlalysis are [2] and [4]). The sample mean,_ 1 N.Xa N LXr,• talis subtracted from each ci the samples, and each of these "demeaned"~ples is multipled by a window h, (sometimes calleda data taper) to produceThe spectral estimate,(.) -)X, = h, (X, - X .TN-248j = O. 1. .....V. /2.