On Modern and Historical Short-Term Frequency Stability Metrics for Frequency Sources

Transcription

1 On Modern and Historical Short-Term Frequency Stability Metrics for Frequency Sources Michael S. McCorquodale, Ph.D. Founder and CTO, Mobius Microsystems, Inc. EFTF-IFCS, Besançon, France Session BL-D: Noise in Systems and Oscillators Wednesday, April, 009 1

2 Overview Historical treatment of short-term frequency stability Modern treatment of short-term frequency stability Goals for this work Theory Experimental results Conclusions and future work Historical Modern Goals Theory Experimental Conclusions of 50

3 HISTORICAL TREATMENT OF SHORT-TERM FREQUENCY STABILITY 3 of 50

4 Noise in oscillators v(t) = V o sin o t v n (t) = (V o + (t))sin( o t + (t)) Ideal oscillator Noisy oscillator v(t) Ideal Period = T o t Period (1-cycle) The standard deviation of the period (1 st jitter difference) v n (t) T = var(t k ) t 0 t t 1 t t 3 t 4 T 0 T 1 T T 3 the time-stamp of a positive-slope transition { t k } { } T k := t k +1 t k the sequence of periods Cycle-to-cycle The standard deviation of the difference between adjacent cycles ( nd jitter difference) T = var(t k +1 T k ) 4 of 50

5 Noise in oscillators v(t) = V o sin o t v n (t) = (V o + (t))sin( o t + (t)) Ideal oscillator Noisy oscillator P v P vn f o f Single sideband (SSB) phase noise power spectral density (PSD) Power spectral density of (t) relative to the fundamental, f o, at offset, f m, where S vn (f) is the twosided spectral density of the oscillator waveform f o f N o P o = 4S ( f vn o + f m ) V f m o 5 of 50

6 Modeling phase noise S ( )= b f m f m 0 =4 = f o f m = h f m S ( f m ) 1 4 f m Random walk of frequency f m -4 S ( f m ) 1 3 f m Flicker of frequency f m -3 S ( f m ) 1 f m White of frequency f m - S ( f m ) 1 1 f m Flicker of phase f m -1 f m 0 S ( f m ) 1 0 f m White of phase f m 6 of 50

7 Fourier to time-domain S f ( f m )= h o 0 f m T = 4 S ( f m )sin f m T o o 0 = 4 o h 0 f o 0 = h 0 T o ( ) ( ) f m sin f m T o Convergence df m df m T = 16 S ( f m )sin 4 f m T o o = 16 f h o 0 o f sin4 m = h 0 T o 0 0 ( ) f m T o Convergence ( ) df m df m Derivation of these Fourier to time domain transformations applies the Wiener-Khinchin theorem where S is assumed to be stationary and integrable. This is valid for white of frequency noise. 7 of 50

8 Fourier to time-domain S f ( f m )= h o 1 3 f m T = 4 S ( f m )sin f m T o o = 4 o h 1 f o = 0 0 ( ) ( ) f m 3 sin f m T o Non-convergence df m df m T = 16 S ( f m )sin 4 f m T o o = 16 o 0 0 f h o 1 3 f sin4 m = 4h 1 ln()t o ( ) f m T o Convergence ( ) df m df m Derivation of these Fourier to time domain transformations applies the Wiener-Khinchin theorem where S is assumed to be stationary and integrable. This is NOT valid for flicker of frequency noise. 8 of 50

9 A mathematical pathology sin ( f ) m 3 f m sin 4 ( f ) m 3 f m Integral non-convergence Integral convergence 9 of 50

10 Allan variance (introduced ca. 1960) y ( )= 1 ( y y ) 1 1 N 1 ( ) N1 k=1 ( y k +1 y ) k Measure frequency consecutively over the interval for samples Compute the fractional frequency error (y) for each sample Compute the average of the square of the difference between each sample Small will have higher variation Large will average variation down to flicker of frequency floor and then diverge for random walk 10 of 50

11 Fourier to time-domain y ( ( ) sin4 f m ) ( ) ( )= S y f m df m = f m 0 8 o S ( ) f m 0 ( )sin 4 f m ( ) df m y ( )= 8 f h o ( o ) 0 f sin4 0 m ( f m ) df m y ( )= 8 o f h o ( ) 1 0 f m 3 sin4 ( f m ) df m = h 0 Convergence = h 1 ln() Convergence 11 of 50

12 Visualizing the Allan variance f f f f y () Figures adapted from: J. R. Vig, Quartz Crystal Resonators and Oscillators For Frequency Control and Timing Applications A Tutorial, Jan of 50

13 Summary of the past Phase noise has been historically modeled by a power-law spectrum With this model, a pathology exists where the classical variance does not converge in the presence of prevalent device-oriented pink noise The Allan variance was introduced to address this mathematical pathology The Allan variance is utilized predominately in the high-precision (e.g., atomic and optical) clock space and for long averaging intervals 13 of 50

14 MODERN TREATMENT OF SHORT-TERM FREQUENCY STABILITY Historical Modern Goals Theory Experimental Conclusions 14 of 50

15 Fast-forward to the present Though there exists a pathology, does a practical convergence issue exist? Is it a result of the model? After all, the nd jitter difference converges for pink noise (i.e. is not pathological) and may have bounded relationships to the 1 st jitter difference In modern industry, the 1 st jitter difference (period jitter) and nd jitter difference (c-c jitter) are reported for nearly all frequency control devices for microelectronics Both are the most useful and intuitive to modern electronics system design and particularly in wireline links Lastly, measurement equipment unavailable in the 1960 s is now commonly utilized to report jitter (e.g. GSa/s realtime oscilloscopes) Historical Modern Goals Theory Experimental Conclusions 15 of 50

16 Modern link specifications v(t) t BER BER The estimated BER vs. position along the unit interval is extrapolated from the 1 period jitter in modern measurement systems 0UI 1UI Historical Modern Goals Theory Experimental Conclusions 16 of 50

17 Mainstream frequency control devices Historical Modern Goals Theory Experimental Conclusions 17 of 50

18 Our work in solid-state oscillators 1500μm 1500μm Michael S. McCorquodale, et al., A MHz Self-Referenced CMOS Clock Generator with 90ppm Total Frequency Error and Spread Spectrum Capability, IEEE Int. Solid State Circuits Conf. Dig. of Tech. Papers, San Francisco, CA 008. Self-referenced RF LC oscillators (not MEMS) No external components Accuracy below 50ppm demonstrated over process, voltage and temperature Targeted to quartz replacement in consumer electronics applications Historical Modern Goals Theory Experimental Conclusions 18 of 50

19 Emerging frequency control devices Historical Modern Goals Theory Experimental Conclusions 19 of 50

20 Observations on time-domain metrics Allan variance is not reported for any device Period jitter (1 st jitter difference) is reported but theoretically non-convergent in the presence of flicker of frequency noise Cycle-to-cycle ( nd jitter difference) jitter is reported Peak-to-peak jitter is unbounded, but reported without a bounding cycle count Instrumentation type and measurement BW are not disclosed Despite the elegant historical treatment of frequency stability, historical metrics are not used in this segment of industry Historical Modern Goals Theory Experimental Conclusions 0 of 50

21 GOALS FOR THIS WORK Historical Modern Goals Theory Experimental Conclusions 1 of 50

22 Goals for this work Theory Can we relate T and T is a meaningful manner? If T converges in time and in the presence of flicker of frequency noise, what does that imply? Experimental Can the analytical relation of phase noise to T be confirmed by measurement? Can T and T be reconciled with the two common measurement techniques (sampling and real-time)? Overall Reconcile the historical treatment of stability with that which is common in this segment of industry Historical Modern Goals Theory Experimental Conclusions of 50

23 THEORY 3 of 50

24 T = 4 o T = 16 o 0 0 S S ( f m )sin ( f m T o )df m ( f m )sin 4 ( f m T o )df m { sin( x) } = cos( x) { cos( x) } = 4cos( x) { } cos x { ( )} T = 4 T T 4 of 50

25 T = 4 T T T 4 T T T T 4 T T T T T T T T T T T 5 of 50

26 T = var(t k +1 T k ) = var(t k +1 ) + var(t k ) cov(t k +1,T k ) cov(t k +1,T k ) = 0 T = T If T cov(t k +1,T k ) 0 = T cov(t k +1,T k ) cov(t k +1,T k ) = 1 then T = 4 T T = T T = T 6 of 50

27 T = 4 S ( f m )sin f m T o o 0 = 4 o h 0 f o = h 0 T o 0 ( ) ( ) f m sin f m T o df m df m T = 16 S ( f m )sin 4 f m T o o = 16 f h o 0 o f sin4 m = h 0 T o 0 0 ( ) ( ) f m T o df m df m T T = T = T 7 of 50

28 Observations The nd jitter difference ( T ) is related to the 1 st jitter difference ( T ) by both an upper and lower bound; exceeding these bounds is non-theoretical The lower bound is attained in the presence of white of frequency noise where each period in the sequence {T k } is independent and identically distributed If the nd jitter difference exceeds the lower bound, it is due to memory from the underlying noise process and the covariance between samples is non-zero and negative 8 of 50

29 An aside on RMS and peak jitter Although T and T converge for white noise and T converges for flicker noise, the distribution of the underlying process is unbounded Therefore, peak-to-peak (p-p) jitter can only be specified by considering a bounding cycle count (i.e. number of observations) Though this can be measured in practice, it is far more practical to measure sufficient cycles for T and T to converge and then compute peak jitter for a bounding cycle count 9 of 50

30 An aside on RMS and peak Jitter Tails extend to T o Solve for a given a bit error rate (BER) representing the inverse of the number of frequency cycles BER = 1 erfc erfc( x)= x 1 e x Choose a bounding cycle count & compute the peak jitter from 1 (requires convergence) J pp = T No. cycles (BER -1 ) (scale factor) of 50

31 EXPERIMENTAL RESULTS 31 of 50

32 Converting phase noise to T T = 8 o 0 N o P o sin ( f m T o ) df m f m 8 o f h 0 N o P o sin ( f m T o ) df m f m Instrumentation does not measure to infinite f m Similarly, the bandwidth of any system is finite Measure over the maximum bandwidth for the instrument, export and pad with far-from-carrier noise Numerically compute T in a math CAD program for various upper limits on f h 3 of 50

33 Phase noise (4MHz solid-state oscillator) Power spectral density (dbc/hz) Frequency offset from carrier (Hz) 33 of 50

34 Jitter filter (4MHz solid-state oscillator) Power spectral density (dbc/hz) Frequency offset from carrier (Hz) 34 of 50

35 Projection (4MHz solid-state oscillator) Power spectral density (dbc/hz) Frequency offset from carrier (Hz) 35 of 50

36 Phase noise (4MHz XO) Power spectral density (dbc/hz) Frequency offset from carrier (Hz) 36 of 50

37 Jitter filter (4MHz XO) Power spectral density (dbc/hz) Frequency offset from carrier (Hz) 37 of 50

38 Projection (4MHz XO) Power spectral density (dbc/hz) Frequency offset from carrier (Hz) 38 of 50

39 Summary of experimental results Integrated phase noise f h = 8f o f h = 6f o f h = 4f o f h = f o f h = f o f h = f o / T (ps) The majority of the period jitter is captured by integrating from the fundamental to f h = f o Higher integration limits merely add white noise which is likely below the noise floor of instrument 39 of 50

40 Real-time oscilloscope (RTO) GSa/s rates with hi-speed A/D 0GSa/s; 50ps spacing Real-time events are captured Sensitive to interpolation Any statistic can be computed Sensitive to dynamic range v(t) v(t) t v samp. (t) t t 40 of 50

41 RTO measurements 41 of 50

42 RTO measurement convergence 4MHz XO T =.47ps T = 4.4ps 4MHz SSO T =.9ps T = 3.97ps 4MHz XO T =.47ps T = 4.4ps 4MHz SSO T =.9ps T = 3.97ps 4 of 50

43 Digital sampling oscilloscope (DSO) Low-frequency (00kHz) sampling Self-triggering technique Waveform reconstruction Split DUT signal for trigger Real-time events not captured Can measure N-cycles away v(t) DELAY t DSO TRIG SPLITTER DUT v samp. (t) v samp. (t) t t Trigger T 43 of 50

44 DSO measurements 100kSa captured for both measurements For 10min. measurement T did not diverge Measurements are made on identical scales 44 of 50

45 Summary of experimental results Integrated SSB phase noise PSD from f m = 0 to f m = f h f h = 8f o f h = 6f o f h = 4f o f h = f o f h = f o f h = f o / T (ps) Sampling oscilloscope 100kSa T (ps).61.7 Real-time oscilloscope 1MSa T (ps) T (ps) of 50

46 Comments on results Integration of phase noise measurements Order of magnitude is consistent with time-domain Integrated jitter for SSO is higher than in time-domain, but is because SSO did not reach its noise floor Most integrated jitter noise is near the fundamental; then integration is highly dependent of noise floor Time-domain measurements Good agreement was found between DSO and RTO T did not violate predicted bounds, but was closer to upper bound, thus indicating presence of flicker noise Convergence was observed because sample interval is small with these measurement techniques; effectively, the pathology is filtered by a HPF 46 of 50

47 CONCLUSIONS Historical Modern Goals Theory Experimental Conclusions 47 of 50

48 Conclusions A review of the historical treatment of short-term frequency stability was presented Short-term frequency stability metrics common to industry were motivated and presented Bounds relating T and T were presented Experimental results were presented relating phase noise to T and for T and T using two different measurement techniques for two different free-running oscillators (XO and SSO) Reasonable agreement was found across all approaches Historical Modern Goals Theory Experimental Conclusions 48 of 50

49 Open questions With these measurement approaches, T converges because the observation interval is short for each capture; but T is still theoretically divergent Does it matter? For most commercial applications, the interface is reset often; long timekeeping is not necessary; but then measured jitter becomes instrumentation-dependent How can we relate the historical work with what is common practice in industry now? How can the mathematical pathology be reconciled? This segment of industry needs to standardize on metrics and measurement techniques; currently, there is no standard metrology Historical Modern Goals Theory Experimental Conclusions 49 of 50

50 QUESTIONS WELCOME 50 of 50