Cadence Design Systems, Inc.Cadence Design Systems, Inc.

Noise in Mixers, Oscillators, Samplers & LogicNoise in Mixers, Oscillators, Samplers & LogicAn Introduction to Cyclostationary NoiseAn Introduction to Cyclostationary Noise

Joel Phillips — Cadence Berkeley LabsKen Kundert — Office of the CTO

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Cyclostationary NoiseCyclostationary Noise

Periodically modulated noise–Noise with periodically varying characteristics–Results when large periodic signal is applied to a nonlinear

circuitHas many names

–Oscillator phase noise–Jitter–Noise folding or aliasing–AM or PM noise–etc.

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White Noise White Noise

Noise at each time point is independent –Noise is uncorrelated in time–Spectrum is white

Examples: thermal noise, shot noise

R() S(f )

f

FourierTransform

Autocorrelation Spectrum

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Colored NoiseColored 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() S(f )

f

FourierTransform

Autocorrelation Spectrum

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Cyclostationary NoiseCyclostationary 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

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Origins of Cyclostationary NoiseOrigins 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

Modulatednoise source

Modulatedsignal path

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Cyclostationary Noise vs. TimeCyclostationary Noise vs. Time

Noise transmitted only when switch is closedNoise is shaped in time

Noisy Resistor& Clocked Switch

vo

NoisyNoiseless

tn

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Cyclostationary Noise vs. FrequencyCyclostationary Noise vs. Frequency

No dynamic elements no memory no coloringNoise is uncorrelated in timeSpectrum is whiteCannot see cyclostationarity with time-average spectrum

– Time-averaged PSD is measured with spectrum analyzer

Noisy Resistor& Clocked Switch

vo

f

S( f )

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Cyclostationary Noise vs. Time & FrequencyCyclostationary 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

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Cyclostationary Noise is Modulated NoiseCyclostationary Noise is Modulated Noise

If noise source is n(t) and modulation is m(t), thenIn time domain, output y(t) is found with multiplication

y(t) = m(t) n(t)In the frequency domain, use convolution

Y(f) = k Mk N(f - kf0)

n y

m

t

t

t

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Modulated Noise SpectrumModulated Noise Spectrum

Time shaping Frequency correlation

Stationary NoiseSource

PeriodicModulation

Noise FoldingTerms

Cyclostationary Noise

Replicate &Translate

Sum

-2 -1 0 1 2-3 3

-2 -1 0 1 2-3 3

f

f

f

f

Con

volv

e

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Duality of Shape and CorrelationDuality of Shape and Correlation

Autocorrelation

R(t,)

fSpectrum

S(f )

t

n

f

Correlation in timeCorrelation in timeShape in Shape in frequency frequency

Shape in time Shape in time Correlation in frequencyCorrelation in frequency

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f

f

Noise is replicated and offset by kf0

–Noise separated by multiples of f0 is correlated

Correlations in Cyclostationary NoiseCorrelations in Cyclostationary Noise

With real signals, spectrum is symmetric–Upper and lower sidebands are correlated

0

0

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Correlations in sidebands AM/PM modulationTo separate noise into

AM/PM components–Consider noise

sidebands separatedfrom carrier by f

–Add sideband phasors to tip of carrier phasor– Relative to carrier,

one rotates at f,the other at f

Sidebands and ModulationSidebands and Modulation

f

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Uncorrelated Sidebands

AM/PM NoiseAM/PM NoiseAM Correlated

SidebandsPM Correlated

Sidebands

Upper and Lower Sidebands Shown Separately

Upper and Lower Sidebands Summed

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Noise + Compression = Phase NoiseNoise + Compression = Phase Noise

Stationary noise contains equal AM & PM componentsWith 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

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Oscillator Phase NoiseOscillator Phase NoiseHigh 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

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The Oscillator Limit CycleThe Oscillator Limit CycleSolution trajectory follows a stable orbit, y

–Amplitude is stabilized, but phase is free to driftIf perturbed with an impulse

–Response is y–Decompose into amplitude and phasey(t) = (1 + (t))y(t + (t)/2 fc) y(t)–Amplitude deviation, (t), is resisted by mechanism that controls output level

t as t –Phase deviation, (t), accumulatest as t

t0t1

t2

t3

t4

y1

y2

y(0) t1

t0

t2

t3

t4

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The Oscillator Limit Cycle (cont.)The Oscillator Limit Cycle (cont.)If perturbed with an impulse

–Amplitude deviation dissipates(t) as t –Phase deviation persistt as t – Impulse response for phase is

approximated with a step s(t)

For arbitrary perturbation u(t)

Su( f )

(2f )2Sf

dttudutst )()()()(

y1

y2

y(0)

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1/1/f f 22 Amplification of Noise in Oscillator Amplification of Noise in OscillatorNoise 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

Flicker 1/f 3

White 1/f 2

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Difference Between Difference Between SS and and SSvv Noise NoiseOscillator 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 Sv but does not predict corner

S

Sv

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JitterJitter

Jitter is an undesired fluctuation in the timing of events–Modeled as a “noise in time”

vj(t) = v(t + j(t))–The time-domain equivalent of phase noise

j(t) = (t)T / 2Jitter is caused by phase noise or noise with a threshold

Jitter

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Noise + Threshold = JitterNoise + Threshold = Jitter

v

t

tc

Threshold

Histogram

Jittert

v(tc)SR(tc)

t =

His

togr

am

Noise v

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Cyclostationary Noise from Logic GatesCyclostationary Noise from Logic Gates

t

t

Out

put S

igna

lO

utpu

t Noi

se

Thermal Noise of MP Thermal Noise of MN

Noise from the gateflicker noise, gate

resistanceNoise from preceding stage

outMP

MN

in

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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 GatesNoise in a Chain of Logic Gates

outMP

MN

in

tOut

put N

oise

Thermal noise from last stage

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Characterizing Jitter in Logic GateCharacterizing Jitter in Logic GateIf 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

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k cyclesti ti+k

Jk k-cycle jitter–The deviation in the length of k cycles

For driven circuits jitter is input- or self-referenced– ti is from input signal, ti+1 is from output signal, or– ti and ti+1 are both from output signal

For autonomous circuits jitter is self-referenced– ti and ti+1 both from output signal

)var()( ikik ttiJ

Characterizing JitterCharacterizing Jitter

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Jitter in Simple Driven Circuits (Logic)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–Jk = t for all k

log(Jk)

log(k)

log(S)

log(f)

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Jitter in Autonomous Circuits (Ring Osc, ...)Jitter in Autonomous Circuits (Ring Osc, ...)Assumptions

–Memory of circuit isshorter than cycle period

–Noise is white (NBW >> 1/T )–Self-referenced measurement

Implications– Each transition relative to previous–Jitter accumulates– tkJk log(Jk)

log(k)

1/2

log(S)

log(f)

2

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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– Jk = T where

Jitter in PLLsJitter in PLLs

tkJk

L2 f

tT

log(Jk)

log(k)1/2fLT

T

FD

PFD/CP LPF VCOin out

log(S)

log(f)fL

McNeill, JSSC 6/97

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SummarySummaryCyclostationary 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.

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