HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS

Transcription

1 HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS Po-Ceng Cang National Standard Time & Frequency Lab., TL, Taiwan 1, Lane 551, Min-Tsu Road, Sec. 5, Yang-Mei, Taoyuan, Taiwan 36 Tel: , Fax: , betrand@ct.com.tw Abstract We ave done some work to reveal tat Fast Fourier Transform (FFT) sampling may induce unreasonable Allan deviation (ADEV) values wile te numerical integration is used for te time and frequency (T&F) conversion. Tese ADEV errors occur because parts of te FFT sampling ave no contributions to te ADEV calculation for some τ. For example, wen averaging interval τ = (s) and Fourier frequency f = 50 (Hz), te term sin (πτf), wic plays a key role in te matematical conversion, is zero. If FFT sampling in a specific frequency range is only at multiples of 50 (Hz), spectral density in tis range as no contributions to ADEV calculations for τ = N (s), were N is a positive integer. Our lab as also found suc errors in related commercial software. In order to solve tis problem witout skipping over effects from certain values of τ, we try to cange te original sampling data in several ways, like dividing sampling spaces into narrower ones or sifting te FFT sampling frequency a small amount, etc. Te regenerated data using interposition tecniquess are ten calculated via te T&F conversion. According to our tests, te FFT sampling witin logaritmic frequency space exceeds te oters at reducing ADEV errors. As for spur effects, te spectral density wit spurs is likely to double or triple ADEV values from te same density wit spurs removed in our case, so it is meaningful for laboratories to reduce ac power and oter periodic noises in teir own environment. Te power-law processes can also perform te T&F conversion and identify different noise types in te spectral density. ADEV results calculated from tis way are in good agreement wit tose from te numerical integration. I. INTRODUCTION Spectral density is a measure of frequency stability in te frequency domain because of its functional dependence on Fourier frequency. Allan deviation (ADEV), on te oter and, is an example of a time domain measure. In a strict matematical sense, tese two descriptions are connected by te Fourier transform relationsip. For very sort averaging intervals (τ<0.5 s), it is not easy to measure te frequency stability of a device using a timing measurement instrument (ex: counter) because of its capability limitations. Te existence of a time and frequency (T&F) relationsip provides us a useful access to obtain ADEV via its spectral density. To perform te matematical conversion, te numerical integration, or trapezoidal integration to be precise, is a direct way to proceed. Generally, Fast Fourier Transform (FFT) sampling for measuring spectral density is different in individual Fourier frequency sections. Te iger frequency section is 551

2 sampled wit wider frequency space tan te lower one, wile in eac section sampling is evenly spaced. It is possible for te numerical integration to generate some unreasonable ADEV values due to influences of FFT sampling. For example, wen averaging interval τ = (s) and Fourier frequency f = 50 (Hz), te term sin (πτf), wic plays a key role in te matematical conversion, is zero. If FFT sampling in a specific frequency range are only at multiples of 50 (Hz), te spectral density in tis range as no contributions to te ADEV calculation for τ = N (s), were N is a positive integer. In order to solve te above problem, te FFT sampling data are regenerated witin logaritmic frequency space using an interposition tecnique. Te errors of ADEV are ten improved obviously using te regenerated data. In addition, te power-law processes, eac of wic varies as an integer power of Fourier frequency wit corresponding coefficient α, are frequently used for describing spectral density. By locating eac particular noise process in its dominant range of Fourier frequency wit standard regression tecniques, te coefficient α could be properly determined. ADEV wit different τ could be obtained easily using te Cutler s formula wit tese coefficients. It is almost inevitable tat some spurs, wic are mainly from ac power and oter periodic noises, are observed in spectral density of a measurement. Tose spurs can t be described well by te power-law processes, so only te numerical integration is adopted wile teir influences on te ADEV calculation are estimated. ADEV from te spectral density wit spurs is compared wit one from te above spectral density wit spurs removed. Finally, we find ADEV results using bot te power-law processes and te numerical integration from te same spectral density (spurs removed) are in good agreement wit eac oter. Tat means te evaluation of coefficients α is acceptable. II. CONVERSION BETWEEN T&F DOMAIN Te Fourier transform relation between time and frequency domain is as follows [1-3]: f S sin ( πτf ) σ y ( τ) = y ( f ) df (1) ( πτf ) 0 S y (f) is te spectral density of normalized frequency fluctuations, f is te ig frequency cutoff of a low pass filter and σ y (τ) is ADEV. S y (f) can also be represented by te addition of all te power-law processes wit corresponding coefficients α (α =, 1,0,+1,+): S y ( f ) = + α= For πf τ >> 1, combine formula (1) and () to get Cutler s formula: α 4 α f for 0 < f < f () 0 for f > f σ y ( τ) = τ ln(πf (π) 0 τ + 1 ln (3) 6 (π) τ τ) 3 f (π) τ ADEV can be calculated directly from formula (1) using te numerical integration or from formulas () and (3) using regression tecniques. 55

3 III. HOW TO DEAL WITH FFT SAMPLING INFLUENCES A. FFT SAMPLING INFLUENCES FFT sampling may cause errors in ADEV calculations wen te numerical integration is adopted for te T&F conversion. For example, if a spectral density in te Fourier frequency range 1500 ~ (Hz) wit a sampling space 50 (Hz) is converted, ADEV is zero wen τ equals multiples of (s), wic is sown in Figure 1a. Tis is no doubt because te sampling frequencies all make te term sin (πτf) in (1) equal to zero. We ave discussed te penomenon in te Introduction of tis paper. Figure 1b is anoter example for FFT sampling influences. If spectral density in Fourier frequency range 1 ~ 1000 (Hz) wit sampling space 1 (Hz) is calculated, ADEV is zero wen τ equals multiples of 1 (s). Figure 1. FFT sampling influences on ADEV calculations using te numerical integration for spectral density in te Fourier frequency range 1500 ~ (Hz) wit a sampling space of 50 (Hz), and 1 ~ 1000 (Hz) wit sampling space of 1 (Hz). Suc errors are also found in commercial software for te T&F conversion. Before running te software, a sampling space of 10.7 (Hz) in a certain frequency section of te experimental data is observed. Te inverse of tis frequency space is about 819. (ms), so 819. (ms), togeter wit two neigboring values (ms) and 80 (ms) are selected for testing te software. Te corresponding ADEV are , , and respectively. It can be seen tat te ADEV wit an averaging interval of 819. (ms) is muc smaller tan tat wit adjacent averaging intervals, as sown in Figure. Te reason for tis is FFT sampling. In oter word, ADEV errors occur because parts of te FFT sampling ave no contributions to te ADEV calculation wen some values of τ are adopted. B. METHODS FOR IMPROVEMENTS Figure 3a sows calculated ADEV from spectral density consisting of several Fourier frequency sections. For some τ, ADEV values obviously fall below te main curve. In order to solve tis problem, we ave to reduce te possibility of sin (πτf) = 0. For tis purpose, let τ equal some logaritmic values and look at te way in wic ADEV beaves. Figure 3b sows tat te falling ADEV is improved wit tese selected τ values. Spur influences are not included in te above calculation. 553

4 Figure. Errors occur in commercial software. Te ADEV wit an averaging interval of 819. (ms) is muc smaller tan tat wit adjacent averaging intervals. Figure 3. ADEV fall obviously below te main curve for some τ using te numerical integration. Te falling ADEV are improved wit τ of logaritmic values. Tis metod seems to work well, but it tactfully skips averaging intervals equal to certain values for wic ADEV are likely to be underestimated. Furtermore, a logaritmic τ is seldom used in te general case. If τ equal to a non-negative integer wit a finite decimal is preferred, canging te original FFT sampling data may be anoter promising way. After using an interposition tecnique, we ave tree kinds of sampling data. Tey are respectively regenerated wit subdivided sampling spaces, sampling points sifted a small amount, and a logaritmic sampling space. Te first two ave no obvious improvements on te falling ADEV, wile te last one works well after our tests. In te following sections, if te numerical integration is used for ADEV calculation, related FFT sampling data are all regenerated using logaritmic sampling space. IV. ANALYSIS OF CALCULATION RESULTS Spectral density of a pase noise measurement is illustrated in Figure 4. It s from a noise floor test of our laboratory s measurement system wit a Fourier frequency range of 0.1 ~ (Hz). Te measure L(f) on te y-axis is te prevailing expression of pase noise among manufacturers and users of frequency standards. Its relation to S y (f) can be expressed as [4]: 554

5 1 v0 L( f ) = ( S ( f )) y (4) f were v 0 is te carrier frequency. Te x-axis stands for Fourier frequency. L(f) is usually reported in a db format: db C = 10log( L( f )) (5) H Z Te blue line is spectral density wit spurs, wile te red line is tat wit spurs removed. Tose spurs are mainly from ac power and oter periodic noises, wic usually sneak into measurement results. Te spur effects on te ADEV calculation will be evaluated later on. Besides, te power-law processes are anoter way for performing te T&F conversion. Calculation results from bot te numerical integration and te power-law processes will also be compared. Figure 4. Spectral density of a pase noise measurement. Te spurs are mainly from ac power and oter periodic noises. A. EFFECTS OF SPURS Since spurs in spectral density are beyond te model of power-law processes, numerical integration is te only way applicable for te matematical conversion. Bot spectral densities in Figure 4 are converted wit τ ranging from to 10 (s) and its increment equal to (s). We can see in Figure 5a tat te blue line (spurs-included ADEV) varies up and down irregularly above te red one (spurs-removed ADEV), depending on τ. Tat means te spurs ave nonnegligible influences on te ADEV calculations. Figure 5b sows relative biases of te results, and some biases may reac 00%. In oter words, te spectral density wit spurs may triple ADEV values from te one wit spurs removed in tis case. 555

6 Figure 5. Spurs obviously affect te ADEV calculations. Te blue line (spursincluded ADEV) varies up and down irregularly above te red line (spurs-removed one), depending on τ. Relative biases of te ADEV results may reac 00%. B. COMPARISONS BETWEEN CONVERSION METHODS According to te power-law processes, te five noise types are Random Walk FM, Flicker FM, Wite FM, Flicker PM, and Wite PM, wit α equal to, 1, 0, +1, and + respectively. In Figure 4, we observe te spurs-removed spectral density and find tat wen Fourier frequency increases by one decade, L(f) also goes down by one decade in te range of 0.1 ~ 1000 (Hz). Tis indicates tat te dominant noise process ere is Flicker PM. In te range of 10 ~ (khz), L(f) is almost te same, so te dominant noise process sould be Wite PM. Wit standard regression tecniques, te coefficients +1 = and + = could be obtained. From Cutler s formula, te generated ADEV could be compared wit te one using te numerical integration. Figure 6a sows tat ADEV results from te two metods matc eac oter quite well wen τ is from to 10 (s) wit an increment of (s). Furtermore, teir relative biases are all below 10%, as sown in Figure 6b. Figure 6. ADEV from te numerical integration and te power-law processes matc eac oter quite well. Relative biases of te ADEV results are all below 10%. 556

7 V. CONCLUSIONS Underestimated ADEV may occur wen numerical integration is used for te T&F conversion. Tis is because parts of te FFT sampling ave no contributions to te ADEV calculation for some τ. Te penomenon is illustrated by several examples, including results from certain commercial software. In order to solve tis problem, we ave to reduce te possibility tat sin (πτf) = 0. After testing a number of possible ways, FFT sampling wit logaritmic frequency space exceeds te oters at improving te ADEV errors wile τ as values of a non-negative integer wit a finite decimal. As for spur effects, te spectral density wit spurs is likely to double or triple te ADEV from te density wit spurs removed, so it is important and meaningful for laboratories to reduce ac power and oter periodic noises in te environment. Te power-law processes can also perform te T&F conversion wit te advantage of identifying different noise types in te spectral density. In Figure 4, two noise types including Flicker PM and Wite PM are identified. Finally, we compare te generated ADEV from te numerical integration and te power-law processes, and results sow tat tey can matc eac oter quite well. REFERENCES [1] S. R. Stein, 1985, Frequency and Time Teir Measurement and Caracterization, Precision Frequency Control (Academic Press, New York), Vol., cap. 1, pp [] Caracterization of Frequency and Pase Noise, Report 580 of te International Radio Consultative Committee (C.C.I.R), pp , [3] D Allan, H. Hellwig, P. Kartascoff, J. Vanier, J. Vig, G. M. R. Winkler, and N. F. Yannoni, 1988, Standard Terminology for Fundamental Frequency and Time Metrology, in Proceedings of te 4 nd Annual Symposium on Frequency Control, 1-3 June 1988, Baltimore, Maryland, USA (IEEE Publication 88CH588-), pp [4] Femtosecond Systems, Inc., FSS1000E Pase Noise Detector FSS1011A Delay Line Operation Manual, cap.. 557