Basics of Phase Noise and Jitter
Asad Abidi
University of California, Los Angeles
Electrical Engineering Department
IEEE SSCS Chapter, Toronto: April 1, 2011
1 / 27
What is Phase Noise?
I
Everyone talks of phase noise ...
I
... but how many can define it?
I
How is it different from voltage or current noise? What do we
measure, when we measure phase noise? Do we have phase meters?
I
What is the connection between phase noise and jitter ?
I
Are there IEEE standards that define these quantities?
I
For all these reasons, we must start with some basics of signal theory
2 / 27
Phase Noise
I
I
I
I
I
I
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
I
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 ) =
I
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
4 / 27
Specifying Phase Noise
I
I
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
5 / 27
I
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
6 / 27
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 )
I
But how do you measure Sφ ?
I
I
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.
9 / 27
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 ) + . . .
10 / 27
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
I
On plot of φ vs t, if T is the mean period of oscillation, then
τi =
I
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
I
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
14 / 27
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
15 / 27
I
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.
16 / 27
to output noise, but did not consider correlated sidebands (i.e.
I
I
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
17 / 27
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
I
I
I
I
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.
19 / 27
Is the Ring Oscillator a Viable Substitute for the LC
Oscillator?
I
I
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
I
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
I
Noise on frequency control line causes FM, thus PM
I
Straightforward analysis, using expressions for narrowband FM
∂f0
= κV
∂VC
I
⇒
Sf0 (fm ) = κ2V SVC (fm )
⇒
L(fm ) =
κ2V
4fm2
SVC (fm )
This expression specifies the skirts of a Lorentzian spectral density
21 / 27
Oscillators within Phase-Locked Loops
I
Autonomous oscillators drift, unless corrected periodically by some
stable periodic reference. This is the basis of a phase-locked loop
(PLL).
I
Used for frequency multiplication, or for clock recovery.
I
I
I
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
22 / 27
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
I
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
25 / 27
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
26 / 27
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 0109ṗdf
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