Handbook of Frequency Stability Analysis

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This handbook includes detailed information about these (and other) stability measures.References for Frequency Stability Analysis1. Proc. of the IEEE-NASA Symposium on the Definition and Measurement of Short-Term FrequencyStability, NASA SP-80, (Nov. 1964).2. D.W. Allan, "The Statistics of Atomic Frequency Standards,” Proc. IEEE, 54(2): 221-230(Feb. 1966).3. Special Issue on Frequency Stability, Proc. IEEE, 54(2)(Feb. 1966).4
3 Definitions and TerminologyThe field of frequency stability analysis, like most others, has its own specializeddefinitions and terminology. The basis of a time domain stability analysis is an arrayof equally spaced phase (really time error) or fractional frequency deviation dataarrays, x i and y i , respectively, where the index i refers to data points in time. Thesedata are equivalent, and conversions between them are possible. The x values haveunits of time in seconds, and the y values are (dimensionless) fractional frequency, Δf/f. The x(t) time fluctuations arerelated to the phase fluctuations by φ(t) = x(t) ⋅ 2πν 0 , where ν 0 is the carrier frequency in hertz. Both are commonlycalled “phase” to distinguish them from the independent time variable, t. The data sampling or measurement interval,τ 0 , has units of seconds. The analysis or averaging time, τ, may be a multiple of τ 0 (τ = mτ 0 , where m is the averagingfactor). Phase noise is fundamental to a frequency stability analysis, and the type and magnitude of the noise, alongwith other factors such as aging and environmental sensitivity, determine the stability of the frequency source.3.1. Noise ModelA frequency source has a sine wave output signal given by [1]Vt ( ) = [ V+ ε ( t)]sin[2 πν t+ φ( t)], (1)0 0Specialized definitions andterminology are used forfrequency stabilityanalysis.where V 0= nominal peak output voltageε(t) = amplitude deviationν 0 = nominal frequencyφ(t) = phase deviation.For the analysis of frequency stability, we are concerned primarily with the φ(t) term. The instantaneous frequency isthe derivative of the total phase:1 dφν () t = ν0+ . (2)2πdtFor precision oscillators, we define the fractional frequency asΔf ν () t −ν 01 dφdxyt () = = = = , (3)f ν0 2πν0dt dtwherext () = φ()/2t πν . (4)03.2. Power Law NoiseIt has been found that the instability of most frequency sources can be modeled by a combination of power-law noiseshaving a spectral density of their fractional frequency fluctuations of the form Sy(f) ∝ f α , where f is the Fourier orsideband frequency in hertz, and α is the power law exponent.Noise TypeαWhite PM (W PM) 2Flicker PM (F PM) 1White FM (W FM) 0Flicker FM (F FM) –1Random Walk FM (RW FM) –25
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Page 6 and 7: PrefaceI have had the great privile
Page 8 and 9: 5.12. MTOT BIAS FUNCTION ..........
Page 11 and 12: 1 IntroductionThis handbook describ
Page 13: Original Allan (a)Overlapping Allan
Page 17 and 18: 15000Random Run Noise50Random Walk
Page 19 and 20: 5 Time Domain StabilityThe stabilit
Page 21 and 22: step) are the sample period τ 0 an
Page 23 and 24: (a)(c)(b)Figure 5. (a) Simulated fr
Page 25 and 26: In terms of phase data, the Allan v
Page 27 and 28: 22 df ⋅ sχ = , (12)2σwhere χ²
Page 29 and 30: References for Time Variance1. D.W.
Page 31 and 32: M− 3m+1 j+ m−121⎧⎫Hσy( τ)
Page 33 and 34: References for Hadamard Variance1.
Page 35 and 36: 5.2.12. Modified Total VarianceThe
Page 37 and 38: Fractional Frequency Data y i, i =
Page 39 and 40: It is necessary to identify the dom
Page 41 and 42: NoiseRW FMTable 3. Thêo1 EDF Formu
Page 43 and 44: References for Thêo1, NewThêo1, T
Page 45 and 46: where S φ is the spectral density
Page 47 and 48: References for Dynamic Stability1.
Page 49 and 50: 1000100N=500250100Full EDF Algorith
Page 51 and 52: The edf for the modified Allan vari
Page 53 and 54: References for Confidence Intervals
Page 55 and 56: 5.5.4. The Lag 1 AutocorrelationThe
Page 57 and 58: The input data should be for the pa
Page 59 and 60: 5.9. B 3 Bias FunctionThe B 3 bias
Page 61 and 62: References for Bias Functions1. “
Page 63 and 64: Figure 24. Frequency stability plot
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include points at all possible tau
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x(t) = a + bt, where slope = y(t) =
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Linear frequency drift can be estim
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References for Environmental Sensit
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AllandeviationHadamarddeviationTota
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For πτf
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S(f)yS(f)=x, (59)(2π f)2where S x
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preferred, either because of measur
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References for Spectral Analysis1.
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where the h α terms define the lev
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The expected PSD values that corres
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8.1. White Noise GenerationWhite no
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9 Measuring SystemsFrequency measur
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factor. For example, if two 5 MHz s
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Figure 37. Random telegraph signal
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10 Analysis ProcedureA frequency st
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5 Remove the frequencydrift, leavin
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10.4. Gap HandlingGaps should be in
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References for Gaps, Jumps, and Out
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Table 23. Stability data for unit u
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B1 ⎡ ⎤∑∑− , (72)⎣ ⎦M
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The total deviation canprovide bett
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The Hadamard variance isinsensitive
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Table 28. Detection of periodic com
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11.6. White FM Noise of a Frequency
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12 SoftwareSoftware is necessary to
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This expression produces a series o
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13 GlossaryThe following terms are
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14 Bibliography• NotesA. These re
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44. P. Lesage and T. Ayi, “Charac
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81. J. McGee and D.A. Howe, “Thê
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• Noise Identification122. J.A. B
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162. W.J. Riley, “Addendum to a T
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OOutlier . 41, 75, 106, 108, 109, 1
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NIST Technical PublicationsPeriodic
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Handbook of Frequency Stability Analysis

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