1. Cadence Design Systems, Inc. Noise in Mixers, Oscillators, Samplers & Logic An Introduction to Cyclostationary Noise Joel Phillips — Cadence Berkeley Labs Ken Kundert — Office of the CTO

2. 2 Intro to Cyclostationary Noise — Phillips & Kundert Motivation  What is the effect of a time-varying bias point ? –Why does the pulse induce more noise? –How much information can we get from the spectrum? –How to model the noise? Frequency Noise Amplitude Noise with Input Noise without Input

3. 3 Intro to Cyclostationary Noise — Phillips & Kundert What is Noise?  Noise signals are stochastic –Small random variation versus time –Repeated identical trials give slightly different results –A group of trials is an ensemble t v n (t) = v(t) + n(t) n

4. 4 Intro to Cyclostationary Noise — Phillips & Kundert Ensemble Averages  Expectation E{ · } is average over many trials  Mean: E{n(t)} = 0 and E{v n (t)} = v(t)  Variance: var{n(t)} = E{n(t) 2 } is noise power  Autocorrelation: R v (t,  ) = E{v(t)v(t+  )}  Power spectral density: n t v n (t) = v(t) + n(t) E{·}

5. 5 Intro to Cyclostationary Noise — Phillips & Kundert Cyclostationary Noise  Periodically modulated noise –Noise with periodically varying characteristics –Results when large periodic signal is applied to a nonlinear circuit  Has many names –Oscillator phase noise –Jitter –Noise folding or aliasing –AM or PM noise –etc.

6. 6 Intro to Cyclostationary Noise — Phillips & Kundert White Noise  Noise at each time point is independent –Noise is uncorrelated in time –Spectrum is white  Examples: thermal noise, shot noise R()R() S(f ) f Fourier Transform  AutocorrelationSpectrum

7. 7 Intro to Cyclostationary Noise — Phillips & Kundert Colored Noise –Noise is correlated in time because of time constant –Spectrum is shaped by frequency response of circuit –Noise at different frequencies is independent (uncorrelated) Time correlation  Frequency shaping R()R() S(f ) f Fourier Transform  AutocorrelationSpectrum

8. 8 Intro to Cyclostationary Noise — Phillips & Kundert Cyclostationary Noise  Cyclostationary noise is periodically modulated noise –Results when circuits have periodic operating points  Noise is cyclostationary if its autocorrelation is periodic in t –Implies variance is periodic in t –Implies noise is correlated in frequency –More about this later  Cyclostationarity generalizes to non-periodic variations –In particular, multiple periodicities

9. 9 Intro to Cyclostationary Noise — Phillips & Kundert Origins of Cyclostationary Noise  Modulated (time-varying) noise sources –Periodic bias current generating shot noise –Periodic variation in resistance of channel generating thermal noise  Modulated (time-varying) signal path –Modulation of gain by nonlinear devices and periodic operating point Modulated noise source Modulated signal path

10. 10 Intro to Cyclostationary Noise — Phillips & Kundert Cyclostationary Noise vs. Time  Noise transmitted only when switch is closed  Noise is shaped in time Noisy Resistor & Clocked Switch vovo NoisyNoiseless t n

11. 11 Intro to Cyclostationary Noise — Phillips & Kundert Cyclostationary Noise vs. Frequency  No dynamic elements  no memory  no coloring  Noise is uncorrelated in time  Spectrum is white  Cannot see cyclostationarity with time-average spectrum –Time-averaged PSD is measured with spectrum analyzer Noisy Resistor & Clocked Switch vovo f S( f )

12. 12 Intro to Cyclostationary Noise — Phillips & Kundert Cyclostationary Noise vs. Time & Frequency  Sample noise every T seconds –T is the cyclostationarity period –Noise versus sampling phase   Useful for sampling circuits –S/H –SCF n m y Y  Y  f f t t t 

13. 13 Intro to Cyclostationary Noise — Phillips & Kundert Cyclostationary Noise is Modulated Noise  If noise source is n(t) and modulation is m(t), then  In time domain, output y(t) is found with multiplication y(t) = m(t) n(t)  In the frequency domain, use convolution Y(f) =  k M k N(f - kf 0 ) ny m t t t

14. 14 Intro to Cyclostationary Noise — Phillips & Kundert Modulated Noise Spectrum Time shaping  Frequency correlation Stationary Noise Source Periodic Modulation Noise Folding Terms Cyclostationary Noise Replicate & Translate Sum -2012-33 -2012-33 f f f f Convolve

15. 15 Intro to Cyclostationary Noise — Phillips & Kundert Duality of Shape and Correlation  Autocorrelation R(t,)R(t,) f Spectrum S(f ) t n f Correlation in time  Shape in frequency Shape in time  Correlation in frequency

16. 16 Intro to Cyclostationary Noise — Phillips & Kundert f f  Noise is replicated and offset by kf 0 –Noise separated by multiples of f 0 is correlated Correlations in Cyclostationary Noise  With real signals, spectrum is symmetric –Upper and lower sidebands are correlated 0 0

17. 17 Intro to Cyclostationary Noise — Phillips & Kundert  Correlations in sidebands  AM/PM modulation  To separate noise into AM/PM components –Consider noise sidebands separated from carrier by  f –Add sideband phasors to tip of carrier phasor –Relative to carrier, one rotates at  f, the other at  f Sidebands and Modulation f

18. 18 Intro to Cyclostationary Noise — Phillips & Kundert Uncorrelated Sidebands AM/PM Noise AM Correlated Sidebands PM Correlated Sidebands Upper and Lower Sidebands Shown Separately Upper and Lower Sidebands Summed

19. 19 Intro to Cyclostationary Noise — Phillips & Kundert Noise + Compression = Phase Noise  Stationary noise contains equal AM & PM components  With compression or saturation –Carrier causes gain to be periodically modulated –Modulation acts to suppress AM component of noise –Leaving PM component  Examples –Oscillator phase noise –Jitter in logic circuits –Noise at output of limiters

20. 20 Intro to Cyclostationary Noise — Phillips & Kundert Ways of Characterizing Cyclostationary Noise  Time-average power spectral density –Hides cyclostationarity –Useful when cyclostationary nature of noise is not important  Time-domain descriptions (noise vs. phase) –Completely characterizes cyclostationary noise –Thresholds and jitter, sampled data systems  Spectrum with correlations –Noise and correlations versus frequency –Completely characterizes cyclostationary noise –Decomposition into AM/PM components

21. 21 Intro to Cyclostationary Noise — Phillips & Kundert When Can Cyclostationarity be Ignored?  Stage 1 produces cyclostationary noise, but we only know the time-average spectrum. –Can we use it to predict the noise of the system?  If we know the noise figure of both stages, can we compute the noise figure of the system? Stage 1Stage 2

22. 22 Intro to Cyclostationary Noise — Phillips & Kundert  Filtering can remove cyclostationarity –Keeps noise-folding terms, but removes correlated frequencies –Filtering must be single-sided with BW < f 0 /2  Examples: mixer w/ filter, SCF w/ anti-aliasing filter  Can use time-averaged power spectral density  Can use noise figure Removing Cyclostationarity LTI Filter OUTRF LO f

23. 23 Intro to Cyclostationary Noise — Phillips & Kundert Ignoring Cyclostationarity Filtering in Disguise  Subsequent stage is non-synchronous –Different reference oscillator (as with spectrum analyzer)  Subequent stage is synchronous, but over many periods –Differing frequencies f 1 and f 2 with f 1 / f 2 = n/m and n, m large (mixer chain) OUTIN LO 1 LO 2 Rolling phase from period-to-period averages noise

24. 24 Intro to Cyclostationary Noise — Phillips & Kundert When to Use the Time-Averaged PSD  When subsequent stage is non-synchronous –Spectrum analyzer  Subsequent stage runs at a sufficiently different frequency f 1 –f 0 /f 1 = N/M, both M, N large ( > 4 ) with no common factors  When filtering eliminates correlation in the noise –SSB filter with BW < f 0 /2 f

25. 25 Intro to Cyclostationary Noise — Phillips & Kundert When Knowing Time-Average PSD of a Stage is not Enough  When the subsequent stage varies synchronously with the first –When subsequent stage shares the same LO or clock –Switched-capacitor filter followed by S&H and/or ADC –When output signal causes subsequent stage to respond nonlinearly –Oscillator driving mixer –Chain of logic gates –Large interferer in receiver chain  In these cases, must use complete representation –Noise versus time, or spectrum with correlations

26. 26 Intro to Cyclostationary Noise — Phillips & Kundert Oscillator Phase Noise  High levels of noise near the carrier –Exhibited by all autonomous systems –Noise is predominantly in phase of oscillator –Cannot be eliminated by passing signal through a limiter –Noise is very close to carrier –Cannot be eliminated by filtering  Oscillators have stable limit cycles –Amplitude is stabilized; amplitude variations are suppressed –Phase is free to drift; phase variations accumulate f

27. 27 Intro to Cyclostationary Noise — Phillips & Kundert Importance of Oscillator Phase Noise  In Receivers: Reciprocal Mixing  In Transmitters: Interference f Interferer LO Desired f Interfering IF Desired IF f Nearby Transmitter Distant Transmitter

28. 28 Intro to Cyclostationary Noise — Phillips & Kundert The Oscillator Limit Cycle  Solution trajectory follows a stable orbit, y –Amplitude is stabilized, but phase is free to drift  If perturbed with an impulse –Response is  y –Decompose into amplitude and phase  y(t) = (1 +  (t))y(t +  (t)/2  f c )  y(t) –Amplitude deviation,  (t), is resisted by mechanism that controls output level  t  as t   –Phase deviation,  (t), accumulates  t  as t   t0t0 t1t1 t2t2 t3t3 t4t4 y1y1 y2y2  y(0) t1t1 t0t0 t2t2 t3t3 t4t4  

29. 29 Intro to Cyclostationary Noise — Phillips & Kundert The Oscillator Limit Cycle (cont.)  If perturbed with an impulse –Amplitude deviation dissipates  (t)  as t   –Phase deviation persist  t  as t   –Impulse response for phase is approximated with a step s(t)  For arbitrary perturbation u(t) S u ( f ) (2  f ) 2 S   f  y1y1 y2y2  y(0) 

30. 30 Intro to Cyclostationary Noise — Phillips & Kundert 1/  f 2 Amplification of Noise in Oscillator  Noise from any source –Is amplified by 1/  f 2 in power –Is amplified by 1/  f in voltage –Is converted to phase  Phase noise in oscillators –Flicker phase noise ~ 1/  f 3 –White phase noise ~ 1/  f 2 ff ff Flicker 1/  f 3 White 1/  f 2

31. 31 Intro to Cyclostationary Noise — Phillips & Kundert Difference Between S  and S v Noise  Oscillator phase drifts without bound – S   as   Voltage is bounded, must remain on limit cycle –Total signal power is independent of noise level –Corner frequency is proportional to noise level –PNoise computes S v  but does not predict corner S    S v  

32. 32 Intro to Cyclostationary Noise — Phillips & Kundert Oscillator Phase Noise  Comparing phase over long periods –Phase drifts randomly over long periods –Drift randomizes phase, signal appears stationary –Smeared correlation in frequency –Occurs in radar with long time-of-flight  Comparing phase over short periods –Phase is not randomized, signal appears cyclostationary –Occurs in –RF circuits –Radar with short time-of-flight

33. 33 Intro to Cyclostationary Noise — Phillips & Kundert Jitter  Jitter is an undesired fluctuation in the timing of events –Modeled as a “noise in time” v j (t) = v(t + j(t)) –The time-domain equivalent of phase noise j(t) =  (t)T / 2   Jitter is caused by phase noise or noise with a threshold Jitter

34. 34 Intro to Cyclostationary Noise — Phillips & Kundert Noise + Threshold = Jitter vv tt tctc Threshold Histogram Jitter t v(tc)v(tc) SR(t c )  t = Histogram Noise v

35. 35 Intro to Cyclostationary Noise — Phillips & Kundert Jitter + Sampler = Noise vv tt tctc Histogram Jitter t  v = SR(t c )  t Histogram Noise v

36. 36 Intro to Cyclostationary Noise — Phillips & Kundert Cyclostationary Noise from Logic Gates t t Output Signal Output Noise Thermal Noise of M P Thermal Noise of M N  Noise from the gate flicker noise, gate resistance  Noise from preceding stage out MPMP MNMN in

37. 37 Intro to Cyclostationary Noise — Phillips & Kundert  Thermal noise of last stage often dominates the time- average noise spectrum — but not the jitter! –Is ignored by subsequent stages –Must be removed when characterizing jitter Noise in a Chain of Logic Gates out MPMP MNMN in t Output Noise Thermal noise from last stage

38. 38 Intro to Cyclostationary Noise — Phillips & Kundert Characterizing Jitter in Logic Gate  If noise vs. time can be determined –Find noise at peak –Integrate over all frequencies –Divide total noise by slewrate at peak  If noise contributors can be determined –Measure noise contributions from stage of interest on output of subsequent stage –Integrate over all frequencies –Divide total noise by slewrate at peak –Alternatively, find phase noise contributions, convert to jitter  Otherwise –Build noise-free model of subsequent stage –Apply noise-contributors approach

39. 39 Intro to Cyclostationary Noise — Phillips & Kundert k cycles titi t i+k  J k  k -cycle jitter –The deviation in the length of k cycles  For driven circuits jitter is input- or self-referenced – t i is from input signal, t i+1 is from output signal, or – t i and t i+1 are both from output signal  For autonomous circuits jitter is self-referenced – t i and t i+1 both from output signal Characterizing Jitter

40. 40 Intro to Cyclostationary Noise — Phillips & Kundert Jitter in Simple Driven Circuits (Logic)  Assumptions –Memory of circuit is shorter than cycle period –Noise is white (NBW >> 1/T ) –Input-referenced measurement  Implications –Each transition is independent –No accumulation of jitter –J k =  t for all k log (J k ) log (k) log (S  ) log (f)

41. 41 Intro to Cyclostationary Noise — Phillips & Kundert Jitter in Autonomous Circuits (Ring Osc,...)  Assumptions –Memory of circuit is shorter than cycle period –Noise is white (NBW >> 1/T ) –Self-referenced measurement  Implications –Each transition relative to previous –Jitter accumulates – log (J k ) log (k) 1/2 log (S  ) log (  f) 2

42. 42 Intro to Cyclostationary Noise — Phillips & Kundert  Assumptions –Memory of circuit is longer than cycle period –Noise is white (or NBW >> 1/T ) –Self-referenced measurement  Implications –Jitter accumulates for k small – –No accumulation for k large – J k =  T where Jitter in PLLs log (J k ) log (k) 1/2  f L T TT FD PFD/CPLPFVCO inout log (S  ) log (  f) fLfL McNeill, JSSC 6/97

43. 43 Intro to Cyclostationary Noise — Phillips & Kundert Summary  Cyclostationary noise is modulated noise –Found where ever large periodic signals are present –Mixers, oscillators, sample-holds, SCF, logic, etc.  Cyclostationary noise is correlated versus frequency –Leads to AM and PM components in noise  Several ways of characterizing cyclostationary noise –Time-average spectrum –Incomplete, hides cyclostationarity –Noise versus time and frequency –Useful for sample-holds, SCF, logic, etc. –Noise versus frequency with correlations (AM & PM noise) –Useful for oscillators, mixers, etc.

44. 44 Intro to Cyclostationary Noise — Phillips & Kundert how big can you dream?