Basics of Phase Noise and Jitter Asad Abidi University of California, Los Angeles Electrical Engineering Department

IEEE SSCS Chapter, Toronto: April 1, 2011

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What is Phase Noise?

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Everyone talks of phase noise ...

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... but how many can define it?

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How is it different from voltage or current noise? What do we measure, when we measure phase noise? Do we have phase meters?

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What is the connection between phase noise and jitter ?

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Are there IEEE standards that define these quantities?

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For all these reasons, we must start with some basics of signal theory

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Phase Noise I

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Point rotates around the circle with non-uniform angular velocity (i.e. varying frequency) Traces unequal phases per unit time Phase and frequency are of course related ω(t ) = ddtφ Phase defines a trajectory versus time (t), whose variance around the noiseless straight line trajectory grows proportionally with R elapsed time. This is because φ(t ) = ω(t )dt Nevertheless φ is stationary, and has a well-defined power spectral density Sφ (f ) In this presentation, we will see how to get to Sφ (f ) ω φ

φ 8π 6π 4π 2π T 2T 3T

Figure: Oscillation as a rotating point on a circle

t

Figure: Phase Noise Trajectory 3 / 27

What Causes Phase Noise in an Oscillator? I

There are multiple sources of voltage and current noise in an oscillator circuit

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Noise sources collectively pull the free-running oscillator’s frequency, through injection locking I

f0

This leads to a Lorentzian spectrum around the oscillation frequency f0 , whose width fB is set by the strength of the total RMS noise (voltage) relative to the amplitude (voltage) of oscillation A S (f ) =

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2fB

fB A2 π fB2 + (f − f0 )2

so that, by definition of spectral density, the integrated power under the Lorentzian is equal to the mean-square voltage of oscillation Z∞ A2 S (f )df = 2 0 We can refer to the spectral linewidth of the oscillation, which depends on fB and is a measure of phase noise

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Specifying Phase Noise I

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In wireless communications, because of how a mixer convolves the skirts of the Lorentzian with a strong unwanted signal (reciprocal mixing ), we concern ourselves with these skirts at offset frequencies fm around f0 , with fm >> fB in almost all practical cases. These skirts are defined in units of dBc/Hz as L(fm ) across a 1 Hz bandwidth, normalized to the mean-square oscillation voltage: L(fm ) = I

2 fB π fm2

If fB < 1 Hz, then at sufficiently small offsets fm , L(fm ) is > 0 dBc/Hz ABL fm

(fm)ABL

f±fLO

fLO

f

Figure: Reciprocal mixing of phase noise by blocking signal

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Since measured spectral densities are single-sided, it follows that L(fm ) = 12 Sφ (fm ) Sφ(fm) L(fm) 0

fm

Figure: Definition of Single-sided Phase Noise

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Phase Noise is Equivalent to a Voltage Noise I

We could look at differences between actual phase φ(t ) and ideal phase 2πf0 t, and from that calculate the spectral density Sφ (f )

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But how do you measure Sφ ? I

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Use phase detector, and compare with a reference phase

Or recognize that phase fluctuations in a sinewave correspond to voltage fluctuations; AM sidebands and PM sidebands at offset frequency fm from average oscillation frequency

½aAM

A φ(t ) '

A

–fm

aPM cos (ωm t ) ½a AM +fm⇒ A

–fm

2 φ '

1 2 2 aPM A2

+fm ½aPM

⇒

Sφ (fm ) '

φ

A

1 Sa (fm ) A2 PM

½aPM 7 / 27

A ½aAM

+fm an

A

+fm

–fm

½aPM

Figure: Phasors showing PM

Ac

1 2

a AM

1 2

a AM

+ fm

+fm –fm ½aAM

- fm

½aPM

Figure: Phasors showing AM

½aPM ½aAM

Figure: Decomposing single added tone into AM and PM



x (t ) = Re

0 −ωm )t Ae j ω0 t + 21 aAM e j (ω0 +ωm )t + aAM e j (ω
1 j (ω0 +ωm )t j (ω0 −ωm )t + 2 aPM e − aPM e




 

 = Re e j ω0 t A + Re aAM e j ωm t + jIm aPM e j ωm t

This resembles the description of modulated carrier in terms of the complex analytical function γ(t ) = u (t ) + jv (t ):   x (t ) = Re γ(t )e j ω0 t 8 / 27

Measuring Phase Noise Noisy oscillation is described by   x (t ) = Re (u (t ) + jv (t )) e j ω0 t u (t ) = A + Re[aAM e j ωm t ] v (t ) = Im[aPM e j ωm t ] Therefore, x (t ) = u (t ) cos(ω0 t ) − v (t ) sin(ω0 t ) This shows that the AM and PM components are modulated on quadrature carriers, and share the same bandwidth. ⇒ PM component, v (t ), can now be separated in a number of ways using synchronous or complex downconversion.

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PLL Based Measurement

DUT Spectrum Analysis

Ref

Narrowband PLL

Mix with pure VCO output in quadrature, and then lowpass filter:  
x (t ) × sin(ω0 t ) = 4j1 γ(t )e j ω0 t + γ(t )e −j ω0 t e j ω0 t − e −j ω0 t   = 4j1 γ(t ) − γ(t ) + . . . = − 21 v (t ) + . . . = −aPM sin(ωm t ) + . . .

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Delay line discriminator φ

DUT T

Spectrum Analysis

Delay oscillator output by T , and mix with itself. Now both inputs to mixer contain PM, one delayed.  i h   vout = Re (u (t ) + jv (t )) e j ω0 t × Re 1 + j v (tA−T ) e j ω0 (t −T )     = u (t ) + v (t )vA(t −T ) cos ω0 T − v (t ) − u (t )vA(t −T ) sin ω0 T + . . . Now if we set ω0 T = π/2, 3π/2, 5π/2, . . ., then     vout = − v (t ) − u(t )vA(t −T ) ' − v (t ) − Av (tA−T ) = − (v (t ) − v (t − T )) Svout (fm ) = Sv (fm )sin2 ( 21 ωm T ) ' (πfm T )2 Sv (fm ) Since Sv (fm ) ≈ 1/fm2 the larger the T (the longer the delay line), the stronger the PM output. 11 / 27

Jitter I

Effective measurement of phase noise, but in the time-domain

Associated with fluctuations in the times of zero-crossings of a ding Jitter I

nominally DC-free periodic waveform While φ(t ) is a continuous random variable, the times of zero crossings comprise a set of discrete random values {τi } g jitter hasI been determining the performance high-speed This critical discretetovariable is specified by the meanofand variance of its probability density function (PDF), associated with which is a systems. Recently, as internal and external data rates of computers spectral density eased to unprecedented levels, reducing jitter has become an even JitterinFluctuations in the timeand difference between successive zero ring highPeriod reliability high-speed databuses integrated circuits. crossings of a rising (or falling) edge I

f a timing ts ideal a system as a duced by every generate, nals. As a he amount of h element of

Ideal Event Timing

Jitter Histogram (Deviation in Event Timing) 12 / 27

φ

ti ti+T

8π φ(t ) 6π 4π 2π i

T 2T 3T

t

Figure: Measuring Jitter on Phase Trajectory

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On plot of φ vs t, if T is the mean period of oscillation, then τi =

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1 1 (φ(ti + T ) − φ(ti )) = ∆φi 2πf0 2πf0

To relate period jitter to phase noise, transform it into power spectral density, and note that it is the first difference of phase across a delay of T = 1/f0 2 S∆φ (f ) = Sφ (f ) 1 − e −j2πf /f0 = 4Sφ (f )sin2 (πf /f0 ) 13 / 27

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Thus, Sτ (f ) = Sφ (f )

sin2 (πf /f0 ) (πf0 )2

At offsets when Sφ (f ) ≈ 1/f 2 , this spectral density of jitter is almost constant I

To deduce the jitter variance in terms of the oscillator’s phase noise spectral density

2 τ =

Z∞

Z∞

Sτ (f )df = 0

Sφ (f ) 0

sin2 (πf /f0 ) df (πf0 )2

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LC Oscillator

C Oscillators: An Alternate Proof , and an Analysis of Q Degradatio I

In steady-state, energy balance prevails between resonator loss (linear resistor RP ) and active, nonlinear negative resistance (modelled by voltage-dependent controlled source Gm (V )) GM0 − GM2 = −

1 RP

IEEE, Jacob J. GRael, Member, IEEE, and Asad A. Abidi, Fellow,ofIEEE nd harmonic where M0,2 are the Fourier coefficients at DC and 2 the nonlinear conductance, as it is subjected across its two terminals to the steady-state oscillation voltage A cos(2πf0 t )

azzanti has offered a nearly-sinusoidal LC uch oscillators (under pendent of the specific ve circuitry. While the oth rely on Hajimiri’s F). In this work, we d by generalizing the Kouznetsov, and Rael.

Fig. 1.

A generic negative-Gm LC oscillator model

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If the spectral density of current noise from the nonlinear conductance is proportional to its instantaneous conductance, i.e. Sin = 4kT γGm , then the total noise current responsible for the oscillator’s phase noise is SiPM = 4kT

1+γ RP

independent of the nonlinearity! This is an important general result that applies to all LC oscillators.

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to output noise, but did not consider correlated sidebands (i.e.

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sidebands). The results of this analysis canAM/PM fill out the noise factor F in Leeson’s The exact approach, however, is a generalized version of expression for phase noise in any oscillator: that laid out in [17]. Indeed, in the limiting case of a “hard switching”  2kT F RPinlinearity, f0 the2above analysis degenerates into that presented [9]. L(fm ) = A2 /2 2Qfm

where Q is the quality factor of the LCRP resonator without the negative resistor Dependence on inverse square offset frequency due to exact balance at steady-state amplitude A between fundamental frequency current in Rp and in Gm

to output noise, but did not consider correlated sidebands (i.e. components. These AM/PM components can then be ap I through unloaded LC AM/PM sidebands). Orthogonal PM noise currents flow directly to (19)an & (20). Consider again the noiseless oscillator shown in Fig. The exact approach,resonator however, is a generalized version of assume the external I Indeed, that laid out in [17]. in the limiting of a “hardIn-phase AM noisecase currents upsetthis theinstance, balance, flowthat through the current source, switching” linearity, the above analysis degenerates into that a cyclostationary white noise source [20] (with respect t parallel impedance of RP and effective Gm = GM0 + GM2 frequency). Wecurrent can model current source (a)oscillation Phase modulating case: (i) PM injected this into oscillator; presented in [9]. (ii) Impedance seen by noise PM current sourcei , modulated by an arb stationary white source, x periodic real-valued waveform, w.t/. Accordingly, in will a time-varying power spectral density equal to ibn2 D ibx2 w 2 .t/ .

The modulation of ix .t/ and w.t/ is shown in Fig. 7

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Fig. 5. Differential current source acting on a “noiseless” oscillator I At offset frequencies of interest, spectral density of PM >> spectral density of AM

I This Fig. 6. looks Squared seen by phase and amplitude modulating likeimpedance a hyperbolic spectral density (asymptote at 0), notcurrents a Lorentzian! Why? I Because analysis assumes fixed oscillation frequency, f0 , which is spectrally concentrated into a delta function. In reality, noise locks theIV. oscillator, causing f0 to spread a certain spectral width D ECOMPOSITION OF Aover R ESONATOR -R EFERRED which forms the Lorentzian. C YCLOSTATIONARY W HITE N OISE S OURCE 18 / 27

Ring Oscillators I

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The ring oscillator is compact in chip area, and easily designed (even by “digital designers”). It remains important in applications where reciprocal mixing with nearby blockers is not an immediate concern. Its principle of operation is fundamentally different than of an LC oscillator; it oscillates because of the delay in a feedback loop, not because of energy exchange between electric and magnetic fields Noise sources modulate the delay in each stage, and thus the total delay White noise in all the FETs of a differential ring oscillator leads to phase noise of the form: " ! #  3 1 1 f0 2 2kT 4 + + γ L(fm ) = I ln 2 Veffd Vefft Vop fm where I is the tail current per stage, Veff is the effective gate voltage at balance for the differential pair in each stage and its tail current FET, and Vop is the differential peak voltage swing (per stage). In a collection of delay stages, correlated modulation of the delays will produce a large phase noise. This happens when the tail currents are driven from a common node that is modulated by flicker noise. Ring oscillators display a large 1/f -induced phase noise.

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Is the Ring Oscillator a Viable Substitute for the LC Oscillator? I

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Ring Oscillator is very compact (consumes a small fraction of the area of a small on-chip inductor) For its phase noise to be equal to that of a well-designed LC oscillator at the same oscillation frequency f0 , the relative bias currents IRO in M delay stages relate to ILC as IRO ≈ M × 8Q 2

VDD I Vefft ||Veffd LC

So a 3-stage ring oscillator biased at a 1V supply and FETs biased 0.2V above threshold will consume, per stage, IRO ≈ 50 Q 2 ILC

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and if the inductor Q is 3, the ring oscillator consumes 450× the current. No, it is not a viable replacement when phase noise is at a premium. But for many applications, it is fine. 20 / 27

Phase Noise in VCOs

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Noise on frequency control line causes FM, thus PM

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Straightforward analysis, using expressions for narrowband FM ∂f0 = κV ∂VC

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⇒

Sf0 (fm ) = κ2V SVC (fm )

⇒

L(fm ) =

κ2V

4fm2

SVC (fm )

This expression specifies the skirts of a Lorentzian spectral density

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Oscillators within Phase-Locked Loops

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Autonomous oscillators drift, unless corrected periodically by some stable periodic reference. This is the basis of a phase-locked loop (PLL).

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Used for frequency multiplication, or for clock recovery. I

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In frequency synthesis, the VCO inside the loop is often the main source of phase noise In clock recovery, the data-carrying waveform at the loop input is usually the dominant source of jitter in the recovered clock

In either case, straightforward linear analysis of the frequency response of the loop to jitter enables modelling, and optimized design for low jitter

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118

IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 1, MARCH 2008

Fig. 4. Phase noise contributions for a simple frequency synthesizer.

of 10 dB/decade for the operation frequency of the phase comFig. 5. Frequency parator. As seen in Fig. 13, this fact has been experimentally observed. To get a frequency synthesizer with low phase noise, the phase noise can be analyzed by using a simple frequency therefore, it is necessary to use a high reference frequency at the synthesizer architecture, whichphase cancomparator. be utilized in broadcasting

terminals. A single loop frequency synthesizer including all the noise PSD of f IV. CONCLUSIONS influential building blocks generating the phase noise is shown divider output A phase noise model that predicts an accurate phase noise to the divid in Fig. 4. spectrum of phase-locked loop frequency synthesizertion was proFig. 10. Measured phase noise characteristics of frequency synthesizer for posed in this paper. By using the curve-fitting method, the phase Thus, the pha (a) PLL loop bandwidth of 5By kHz and (b) PLL loop bandwidth of 10 kHz. simply adding the respective phase noise power spectral noise spectra of the reference signal source and a VCO were imated by [13] densities, the output phase noisemodeled PSD as forphase Fig.noise 4 iscomponents rewritten from of an oscillator with resonator. Based on relation between the frequency modulation (4): and the phase noise spectral density, the phase noises due to In case of phase noise model neglecting the resistor noise in the low-pass filter in the phase-locked loop were represented by the low-pass filter, there are discrepancies in the range of high phase transfer functions to VCO input port. offset frequency and in the neighborhood of the loop bandwidth Also, the phase noise spectra of phase comparators and freof the frequency synthesizer. At offset frequencies below the quency divider circuits were modeled in the proposed phase Fig. 12. Phase noisealso, characteristics of frequency synthesizers and their dif- of noise prediction model. loop bandwidth, the previous models neglecting effects ferent phase noise contributions. frequency dividers showPLL low phase noiseModel spectra compared with In validity of the proposed phase noise prediction model, the Figure: Noise

Figure: Measured Phase Noise

23 / 27 Fig. 11. Measured phase noise spectra and prediction spectra for frequency

Sources of Jitter, and Identifying Them by Measurement I I I

Jitter can arise in practical systems from multiple sources Produces unique histograms, which can be used to diagnose sources Jitter can be of two types:

Random Unbounded. Tails in its histogram due to Gaussian PDF. Deterministic Bounded. Periodic, data dependent, duty cycle distortion, intersymbol interference. I

Deterministic jitter arises from coupling on to signal lines from: 1. Electromagnetic interference 2. Crosstalk 3. Reflections

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Since random and deterministic effects are independent, the following relation applies between jitter τR and τD that comprise the total jitter τ PDF (τ) = PDF (τR + τD ) = PDF (τR ) ∗ PDF (τD ) Sτ (f ) = SτR (f ) + SτD (f ) 24 / 27

P

fDJ

fRJ fOJ

Δt Figure: Convolution of Random and Determinstic Jitter PDF 4

There are several metrics used to define PLL performance. Metrics are selected depending on the target application of the PLL. In the following example, the SIA-3000 and VISI software analyze jitter sources in a circuit with a PLL. The analysis starts with a histogram of period measurements to identify the magnitude of the jitter problem. Fig. 7 is an example of a non-Gaussian histogram, indicating that DJ is present. The figure shows a Gaussian tail in red fit to the left of the distribution, and Figure 7. This histogram ofComposite period measurementsHistogram includes RJ and DJ. Figure: Typical a Gaussian tail in blue fit to the right Note that the tails have been matched to a Gaussian distribution. of the distribution. The average of the right and left standard deviation provides

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InfiniBand, SONET, Serial ATA, 3GIO and Firewire components and systems. I Random and deterministic jitter may be deconvolved from In high-speed serial communication The TailFit algorithmhistogram, — a patented WAVECREST innovation — is capable of separating RJ byjitter first fitting tails of distribution to best-fit Gaussian signals, is caused by many from actual measurement distributions by using the Gaussian nature of the tail regions of factors, including: I Then non-Gaussian histograms. The algorithm first identifies a tail region of the histogram, then histogram of deterministic jitter is extracted by deconvolution. fits the data with a Gaussian histogram that best coincides with the tail region. The process repeats for each side of the histogram. The RJ valueseffects for the tails • Bandwidth on are ISIaveraged to represent the RJ for the distribution when calculating TJ. Figure 5 shows a Gaussian tail fit to the left • Optical and electrical (red) and rightI (blue) of the distribution. Chi-squared is used as a gauge to determine the connectors and quality of fit. It is an iterative process, and ends when the cables results converge. To limit the iterative process, an estimate of the parameters made by the algorithm • initial Noisefitting on the PLL’s isreference using the tail portions of the distribution. Most important, you can determine the DJ and frequency signal RJ components, regardless of the shape of the data histogram.

Histogram does not specify frequency of jitter-inducing signal. Spectral density of jitter is useful to isolate frequencies, which appear as discrete lines.

Keep adjusting 1σ, mean and magnitude until tails obtain the best fit with the data.

• • • • •

Power supply noise Internal switching noise Crosstalk Signal reflections Optical laser source

b

Measuring jitter on a high-speed serial device can be done with the SIA-3000 and DataCOM software. Data signals can be analyzed with a repeating pattern or data with a bitclock. We can determine the DCD and ISI components Figure 6. The TailFit algorithm enables the that user to identify Gaussian curve with a coincident tail region in order to Figure 10a shows DDJ as a function of the bit position. can aprovide information about quantify the random or Gaussian component of the distribution. Various curves are fitted against the distribution Figure: Tail fittingcurveto Gaussian Figure: spectrum 10b showsMeasured an FFT with a periodic spike at 52(FFT) MHz that asvalue well asparticular any tail. Figure until an optimal match is found. Then, the bandwidth 1σ of the matched limitations, is used as the RJ for that This is repeated for both sides of the distribution, and the two RJ values are averaged to get the overall RJ value. contributes 38 ps of jitter. Together, these figures illustrate the of jitter PJ component that could be caused DJ components of TJ. by crosstalk or EMI, and RJ (which affects long-term system reliability).

lyzing jitter on h-speed devices

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1. Anon. (2004, May 5, 2010). Clock (CLK) Jitter and Phase Noise Conversion. Maxim Integrated Products (App Note 3359), 8. 2. A. A. Abidi, ”Phase Noise and Jitter in CMOS Ring Oscillators,” IEEE Journal of Solid-State Circuits, vol. 41, no. 8, pp. 1803-1816, 2006. 3. A. Hajimiri, ”Noise in phase-locked loops,” in Southwest Symposium on Mixed-Signal Design, 2001, pp. 1-6. 4. Y. W. Kim and J. D. Yu, ”Phase Noise Model of Single Loop Frequency Synthesizer,” IEEE Transactions on Broadcasting, vol. 54, no. 1, pp. 112-119, 2008. 5. M. Li. (2009, May 5, 2010). Deterministic Jitter (DJ) Definition and Measurement Methods. Available: ˙ www.ieee802.org3bapublicjan09li 01 0109pdf 6. A. Mirzaei and A. A. Abidi, ”The Spectrum of a Noisy Free-Running Oscillator Explained by Random Frequency Pulling,” IEEE Transactions on Circuits and Systems I, vol. 57, no. 3, pp. 642-653, 2010. 7. D. Murphy, J. J. Rael, and A. A. Abidi, ”Phase Noise in LC Oscillators: A Phasor-Based Analysis of a General Result and of Loaded Q,” IEEE Transactions on Circuits and Systems I, 2009. 8. E. Rubiola. (2009, May 5, 2010). Phase Noise. Available: www.rubiola.org 9. D. Scherer. (1985, May 5, 2010). The ”Art” of Phase Noise Measurement. HP Application Notes (Hewlett-Packard), 34. Available: www.hparchive.com 27 / 27