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Statistical Processes for Time and Frequency A Tutorial Victor S. Reinhardt 10/17/01

Statistical Processes for Time and Frequency A Tutorial Victor S. Reinhardt 10/17/01. Statistical Processes for Time and Frequency--Agenda. Review of random variables Random processes Linear systems Random walk and flicker noise Oscillator noise. Review of Random Variables. Number of

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Statistical Processes for Time and Frequency A Tutorial Victor S. Reinhardt 10/17/01

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  1. Statistical Processes for Time and FrequencyA TutorialVictor S. Reinhardt10/17/01

  2. Statistical Processes for Time and Frequency--Agenda • Review of random variables • Random processes • Linear systems • Random walk and flicker noise • Oscillator noise

  3. Review of Random Variables

  4. Number of Occurrences Nx Nx-dx Nx+dx x x x+dx Continuous Random Variable Ensemble of N Identical Experiments • Random Variable x • Repeat N identical experiments = Ensemble of experiments • Unpredictable (Variable) Result xn • Nx = Number of of times value xn between x and x+dx • Probability density function(PDF) or distribution p(x) Unpredictable Result xN x3 x2 x1

  5. The expectation value of f(x) is the average of f(x) over the ensemble defined by p(x) s x a m b PDF and Expectation Values • Range of random variable x from a to b • Mean value = [x] • Standard variance = d2[x] • Standard deviation = d[x]

  6. Probability Distributions • Gaussian (Normal) PDF • Range = (-, +) – Mean = m • Standard deviation = sd • Uniform • Range = (-D/2, +D/2) – Mean = 0 • Standard deviation = D/120.5 • Examples: Quantization error, totally random phase error Pgauss(x) x Puniform(x) x -D/2 0 D/2

  7. Statistics • A statistic is an estimate of a parameter like m or s • Repeat experiment N times to get x1, x2, …… xN • Statistic for mean[x] is arithmetic mean • Statistics for standard varianced[x] • Standard Variance(m known a priori) • Standard Variance(with estimate of m) • Good Statistics • Converge to the parameter as N   with zero error • Expectation value = parameter value for any N (Unbaised)

  8. Multiple Random Variables • x1 and x2 two random variables (1 and 2 not ensemble indices but indicate different random variables) • Joint PDF = p(2)(x1,x2) (2) means 2-variable probability • Expectation value • Single Variable PDF • Conditional PDF = p(x1|x2) is PDF of x2 occuring given that x1 occurred • Mean & Covariance matrix • Statistical Independence • p(2)(x1,x2) = p(1)(x1)p’(1)(x2) • Then (k & k’ = 1,2)

  9. Ensembles Revisited Ensemble of N Identical Experiments • The ensemble for x is a set of statistically independent random variables x1, x2, ….. xN with all PDFs the same = p(1)(x) • Thus Each with same PDF p(x) xN Eachstatistically independent x3 x2 x1 (F=2 for normal distribution)

  10. Random Processes

  11. Ensemble Average E[...] uN(t) u2(t) u1(t) t Time Average <…> Random Processes • A random function in time u(t) • Is a random ensemble of functions • That is defined by a hierarchy probability density functions (PDF) • p(1)(u,t) = 1st order PDF • p(2)(u1,t1; u2,t2) = 2nd order joint PDF • etc • One can ensemble average at fixed times • Or time average nth member

  12. A Stationary Non-ergodic Process Op Amp Offset Voltage Ensemble Average uN(t) u2(t) u1(t) t Time Average Ergodic_Theorem: Stationary processes are ergodic only if there are no stationary subsets of the ensemble with nonzero probability Time Averages and Stationarity • Time mean • Autocorrelation function • Wide sense stationarity • Strict Stationarity • All PDFs invariant under tn tn - t’ • Ergodic process • Time and ensemble averages equivalent (= 0 for random processes we will consider)

  13. Types of Random Processes • Strict Stationarity: All PDFs invariant under time translation (no absolute time reference) • Invariant under tn tn - t’ (all n and any t’) • Implies p(1)(x,t) = p (1)(x) = independent of time p(2)(x1,t1; x2,t2) = p(2)(x1,0; x2,t2- t1) = function of t2- t1 • Purely random process: Statistical independence • p(n)(x1,t1; x2,t2; ….xn,tn) = p(1)(x1,t1) p (1)(x2,t2) ….. p (1)(xn,tn) • Markoff Process: Highest structure is 2nd order PDF • p(x1,t1;...xn-1,tn-1 | xn,tn) = p(xn-1,tn-1 | xn,tn) • p(x1,t1 ;...xn-1,tn-1 | xn,tn) is conditional PDF for xn(tn) given thatx1(t1) ;...xn-1,tn-1 have occurred • p(n)(x1,t1; x2,t2; ….xn,tn) = p(1)(x1,t1)p(xk-1,tk-1 | xk,tk)

  14. Linear Systems

  15. u(t) v(t) Linear system h(t), H(f) U(f) V(f) Linear Systems • In time domain given by convolution with response function h(t) • Fourier transform to frequency domain • The fourier transform of the output is Time Domain Frequency Domain

  16. log(f/B3) Single-Pole Low Pass Filter C t1= R1C t2= R2C R2 U(f) - V(f) G=-1 R1 + (Causal filter) (3-dB bandwidth)

  17. Important Property of S(f) System H(f) V(f) = H(f)V(f) U(f) Sv(f) = |H(f)|2Su(f) du v(t) = dt 1 Su(f) = Sv(f) w2 Spectral Density of a Random Process • Requires wide sense stationary process • The spectral density is the fourier transform of the autocorrelation function • For linearly related variables given by • The spectral densities have a simple relationship V(f) = jw U(f) Sv(f) = w2Su(f)

  18. Average Power and Variance • Autocorrelation Function back from Spectral Density • Average power (intensity) • Average power in terms of input • For ergodic processes • Where sd2 the standard variance is (Mean is assumed zero)

  19. H(f) for Ideal Bandpass Filter 1 B B f -fo fo 0 White Noise • Uncorrelated (zero mean) process • Generates white spectrum • At output • Bn is noise bandwidth of system • For Single-Pole LP Filter • Bn  B3-dB as number of poles increases • For Thermal (Nyquist) Noise • No = kT

  20. Filter BandLimited Noise White Noise No/2 B3 Dt 0 Band-Limited White Noise& Correlation Time • White noise filtered by single pole filter • t1 = t2 = to • Called Gauss-Markoff Process for gaussian noise • Frequency Domain • Time Domain • Correlation Time = to • Correlation width = Dt = 2to

  21. Model of Specrum Analyzer Det In Res Filter Br Video Filter Bv Out X T/Dt Independent Samples Dt Dt Dt Dt Dt Dt Dt Dt Averaging Time T Dt = Correlation Width = 1/(2Br) Spectrum Analyzers and Spectral Density • Model of Spectrum Analyzer • Downconverts signal to baseband • Resolution Filter: BW = Br • Detector • Video Filter averages for T = 1/(2Bv) • Spectrum Analyzer Measures Periodogram(Br0) • uT(t) = Truncated data from t to t+T • Fourier Trans • Wiener-Khinchine Theorem • When T   • Periodogram  Spectral Density Radiometer Formula (finite Br) Same as

  22. 1/t h1(t’) t t+t Response Function and Standard Variance for Time Averaged Signals • Finite time average over t • Response Fn for average • Variance of with H1 • For • So s12 diverges when y = (f-fo)/fo v(t) = <y(t)>t,t Response Function Sy(f) |H1(f)|2 0 ( for non-stationary noise)

  23. Sy(f)1/f2 |H2(f)|2 f2 0 Response Function for Zero Dead-Time Sample (Allan) Variance y = (f-fo)/fo • Response for difference of time averaged signals • Variance with H2 (Allan variance) • For • So s22 doesn’t diverge for v(t) = <y(t)>t+T,t - <y(t)>t,t Response Function 1/t h2(t’) t t+t t+2t -1/t ( for noise up to random run)

  24. y(t) Satellite Ranging ~ Satellite Round Trip Time T y(t-T) X Average Dy for Time t Meas Error s Dy y = df/f Graphing to Understand System Errors • Can represent system error as • h(t) includes • Response for measurement • Plus rest of system • Graphing h(t) or H(f) helps understanding • Example: Frequency error for satellite ranging • Ranging: sd2(t,T) = s22(T,t) = Allan variance with dead time t and averaging time T reversed • Radar: sd2(t,T) = s22(t,T) = no resversal of T and t (t > T) h(t) T t+T+t 1/t t t -1/t t t+T t+t v(t) = <y(t)>t+T,t - <y(t)>t,t

  25. Random Walk and Flicker Noise

  26. v(t) ±1(t) t Integrated White Noise--Random Walk (Wiener Process) • Let u(t) be white noise • And • Then • where t< = the smaller of t or t’ • Note Rv is not stationary (not function of t-t’) • This is a classic random walk with a start at t=0 • The standard deviation is a function of t Random Walk Increases as t½

  27. White Noise u(t) WienerFilter h(t-t’) Colored Noise v(t) |H(f)|2 Sv(f) Generating Colored Noise from White Noise • A filter described by h(t-t’) is called a Wiener filter • Must know properties of filter for all past times • To generate (stationary) colored noise can Wiener filter white noise • Can turn convolution into differential (difference) equation (Kalman filter) for simulations White Noise Wiener Filter Colored Noise

  28. he He Wiener Filter for Random Walk C t1= R1C e = R1/R2 U(f) R2 - V(f) G=-1 R1 +

  29. R R C C Wiener Filter for Flicker Noise Heavyside Model of Diffusive Line • Impedance of diffusive line • White current noise generates flicker voltage noise • Ni = Current noise density R R Z C C Impedance Analysis R Z Z C Flicker Voltage Noise White Current Noise v(t) i(t)

  30. Sv 0 f Sf fo f Multiplicative Flicker of Phase Noise • Nonlinearities in RF amplifier produce AM/PM • Low frequency amplitude flicker processes modulatesphase around carrier through AM/PM • Modulation noise or multiplicative noise is what appears around every carrier AM/PM converts low frequency amplitude fluctuations into phase fluctuations about carrier

  31. An Alternative Wiener Filter for Flicker Noise • Single-Pole Filters T. C. = t • Independent current sources • Integrate outputs over t N Independent White Current Noise Sources C t = RC = -1 White Current Source I(f) R = Constant R V(f) - G=-1 + SI(f)=I2 Filter t10 Filter t2 Filter tN Sum (Integrate) Over Outputs Flicker Noise

  32. Results for m = 0 to 8 Sf(f) S (f) vm A Practical Wiener Filter for Flicker Noise • Single-pole every decade • With independent white noise inputs • Spectrum • For time domain simulation turn convolutions into difference equations for each filter and sum Error in dB from 1/f

  33. Oscillator Noise

  34. Df |YR|2 df/dy = 2Q Properties of a Resonator • High frequency approximation (single pole) f = 3-dB full width • Phase shift near fo

  35. Ga, fa Simple Model of an Oscillator Resonator Near Resonance fR = -2QLy • Amplifier and resonator in positive feedback loop • Amplifier • Amp phase noise Sf-amp (f) = FkT/Pin (1+ ff/f) • Thermal noise + flicker noise • Resonator (Near Resonance) fR = -2QLy [ y = (f - fo)/fo ] • Oscillation Conditions • Loop Gain =|GaGL|  1 • Phase shift around loop = 0 fR + famp = 0 Loss =GR = YR Loaded Q = QL Flicker of Phase Thermal Oscillation Conditions |GaGR| = Loop Gain  1 Sf Around Loop = 0 Pin Amp Noise Gain = Ga Phase Shift =famp Noise Figure = F Flicker Knee = ff White Noise Density = FkT

  36. fR = -fa y Converted Noise Original Amp Noise Leeson’s Equation Resonator Phase vs df/f Response • Phase Shift Around Loop = 0 famp= 2QLy = - fR • Thus the oscillator fractional frequency y must change in response to amplifier phase disturbances famp • Amp Phase Noise is Converted to Oscillator Frequency Noise Sy-osc(f) = 1/(2QL)2 Sf-amp(f) • But y = wo-1df/t so Sf-osc(f)= (fo2/f2) Sy-osc(f) • And thus we obtain Leeson’s Equation Sf-osc(f) = ((fo/(2QLf))2+1)(FkT/Pin)(1+ ff/f) The Oscillator df/f must shift to compensate for the amp phase disturbances

  37. Sf(f) K3/f3 Converted Noise QL K2/f2 Amp Noise K1/f K0 f Oscillator Noise Spectrum • Oscillator noise Spectrum Sf(f) =K3/f3 + K2/f2+ K1/f + K0 • Some components may mask others • Converted noise • K2 = FkT/Pin (fo/(2QLf))2 • K3 = FkT/Pin(ff/f) (fo/(2QLf))2 • Varies with (fo/(2QL)2andFkT/Pin • Original amp noise • Ko= FkT/Pin • K1= FkT/Pin(ff/f) • Only function of FkT/Pin • and flicker knee Oscillator Noise Spectrum Leeson’s Equation Sf-osc(f) =(fo/(2QLf))2+1)(FkT/Pin)(1+ ff/f)

  38. R. G. Brown, Introduction to Random Signal Analysis and Kalman Filtering, Wiley, 1983. D. Middleton, An Introduction to Statistical Communication Theory, McGraw-Hill, 1960. W. B, Davenport, Jr. and W. L. Root, An Introduction to the Theory of Random Signals and Noise, Mc-Graw-Hill, 1958. A. Van der Ziel, Noise Sources, Characterization, Measurement, Prentice-Hall, 1970. D. B. Sullivan, D. W. Allan, D. A. Howe, F. L. Walls, Eds, Characterization of Clocks and Oscillators, NIST Technical Note 1337, U. S. Govt. Printing office, 1990 (CODEN:NTNOEF). B. E. Blair, Ed, Time and Frequency Fundamentals, NBS Monograph 140, U. S. Govt. Printing office, 1974 (CODEN:NBSMA6). D. B. Leeson, “A Simple Model of Feedback Oscillator Noise Spectrum,” Proc, IEEE, v54, Feb., 1966, p329-335. References

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