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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 12, DECEMBER 2002
Phase Noise and Timing Jitter in Oscillators
With Colored-Noise Sources
Alper Demir, Member, IEEE
SCILLATORS are ubiquitous in physical systems, especially electronic and optical ones. For example, in
radio frequency (RF) communication systems, they are used
for frequency translation of information signals and for channel
selection. Oscillators are also present in digital electronic systems which require a time reference, i.e., a clock signal, in
order to synchronize operations.
Noise is of major concern in oscillators, because introducing
even small noise into an oscillator leads to dramatic changes
in its frequency spectrum and timing properties. This phenomenon, peculiar to oscillators, is known as phase noise or
timing jitter. A perfect oscillator would have localized tones at
discrete frequencies (i.e., harmonics), but any corrupting noise
spreads these perfect tones, resulting in high power levels at
neighboring frequencies. This effect is the major contributor
to undesired phenomena such as interchannel interference,
leading to increased bit-error rates in RF communication systems. Another manifestation of the same phenomenon, jitter, is
important in clocked and sampled-data systems: uncertainties
in switching instants caused by noise lead to synchronization
problems. Characterizing how noise affects oscillators is
therefore crucial for practical applications. The problem is
challenging, since oscillators constitute a special class among
noisy physical systems: their autonomous nature makes them
unique in their response to perturbations. A brief review of
previous work on phase noise is given in [1].
In recent publications [1] and [2], we presented a theory
and numerical methods for practical characterization of phase
noise in oscillators with white-noise sources. In this paper,
we present the theory and practical characterization of phase
noise in oscillators due to colored, as opposed to white, noise
sources. Shot and thermal noise sources in IC devices can be
modeled as white-noise sources for all practical purposes. The
characterization of phase noise in oscillators due to shot and
thermal noise sources is covered by our theory of phase noise
[1] due to white-noise sources. Other types of noise sources in
and
IC devices, which have a colored spectral density (e.g.,
burst noise), are significant in practical applications for the phase
noise performance characterization of oscillators. We first derive
the stochastic characterization of phase noise in oscillators
due to colored-noise sources. Then, we calculate the resulting
spectrum of an oscillator with phase noise characterized as
above. We also extend our results to the case when both white
and colored-noise sources are present. Our treatment of phase
noise due to colored-noise sources is general, i.e., it is not
specific to a particular type of colored-noise source. Hence,
our results are applicable to the characterization of phase noise
and burst noise, but also other types of
due to not only
possibly colored noise, e.g., substrate or power-supply noise.
The theory and numerical methods for phase noise are based
on a novel nonlinear perturbation analysis that is valid for
oscillators, which we summarize in Section II.1 We review
the characterization of phase noise due to white-noise sources
in Section III. In Section IV, we analyze the case of colorednoise perturbations and obtain a stochastic characterization
of the phase deviation. Models for burst (popcorn) and
(flicker) noise, the most significant colored-noise sources in
IC devices, are discussed in Section V. Then, in Section VI,
we calculate the resulting oscillator spectrum with phase noise
as characterized in Section IV. In Section VII, we consider
the presence of white and colored-noise sources together, and
derive the resulting oscillator spectrum. Finally, in Section VIII,
we present simulation results in phase noise characterization
of oscillators.
Manuscript received August 12, 1999; revised June 15, 2001. This paper was
recommended by Associate Editor M. Gilli.
The author is with the Department of Electrical and Electronics Engineering,
Ko University, 9010 Saryer-Istanbul, Turkey (e-mail: aldemir@ku.edu.tr).
Digital Object Identifier 10.1109/TCSI.2002.805707
1Originally in [1], we developed the nonlinear perturbation analysis for
oscillators based on an autonomous ordinary differential equation (ODE)
formulation. In [2], we generalized the nonlinear perturbation analysis theory
to an autonomous index-1 differentialalgebraic equation (DAE) formulation.
Here, we summarize the perturbation analysis using an ODE formulation
for notational simplicity.
AbstractPhase noise or timing jitter in oscillators is of major
concern in wireless and optical communications, being a major
contributor to the bit-error rate of communication systems, and
creating synchronization problems in other clocked and sampled-data systems. This paper presents the theory and practical
characterization of phase noise in oscillators due to colored, as
opposed to white, noise sources. Shot and thermal noise sources in
oscillators can be modeled as white-noise sources for all practical
purposes. The characterization of phase noise in oscillators due to
shot and thermal noise sources is covered by a recently developed
theory of phase noise due to white-noise sources. The extension
of this theory and the practical characterization techniques to
noise sources in oscillators, which have a colored spectral density,
noise, is crucial for practical applications. In this paper,
e.g., 1
we first derive a stochastic characterization of phase noise in
oscillators due to colored-noise sources. This stochastic analysis is
based on a novel nonlinear perturbation analysis for autonomous
systems, and a nonlocal FokkerPlanck equation we derive. Then,
we calculate the resulting spectrum of the oscillator output with
phase noise as characterized. We also extend our results to the
case when both white and colored-noise sources are present. Our
treatment of phase noise due to colored-noise sources is general,
i.e., it is not specific to a particular type of colored-noise source.
Index TermsCircuit simulation, colored-noise sources, nonlinear oscillators, nonlocal FokkerPlanck theory, oscillator noise,
phase noise, stochastic differential equations, timing jitter.
I. INTRODUCTION
1057-7122/02$17.00 2002 IEEE
DEMIR: PHASE NOISE AND TIMING JITTER IN OSCILLATORS WITH COLORED-NOISE SOURCES
II. NONLINEAR PERTURBATION ANALYSIS FOR OSCILLATORS
In general, the dynamics of an oscillator can be described by
a system of differential equations
(1)
and
. We consider systems
where
(with period ) to (1), i.e.,
that have a periodic solution
a stable-limit cycle in the -dimensional solution space. We are
interested in the response of such systems to a small state-dewhere
pendent perturbation of the form
and
. Hence, the perturbed system is described by
(2)
Although our eventual intent is to understand the response of
is random noise, it is useful to consider
the oscillator when
is a known deterministic signal. We carfirst the case when
ried out a rigorous analysis of this case in [1] and obtained the
following results.
is
1) The unperturbed oscillators periodic response
by the perturbation, where
modified to
is a changing time shift, or phase deviation, in
a)
the periodic output of the unperturbed oscillator;
is an additive component, which we term the
b)
orbital deviation, to the phase-shifted oscillator
waveform.
and
are such that
2)
will, in general, keep increasing with time even
a)
is always small, and if the
if the perturbation
will settle to a conperturbation is removed,
stant value;
, on the other hand, will
b) the orbital deviation
always remain small [within a bounded factor of
], and if the perturbation is removed,
will
decay to zero.
and
, in particFurthermore, we derived equations for
ular, a nonlinear differential equation for the phase deviation
:
(3)
is a periodically time-varying vector, which we call
where
the Floquet vector [1], [2]. We will not go into a detailed derivation of (3) and the description of the Floquet vector here, which
is obis covered elsewhere. In short, the Floquet vector
tained as follows.
1) The nonlinear differential equations (1) that describe the
oscillator are linearized around the periodic steady-state
to obtain a homogeneous system of linear
solution
periodically time-varying (LPTV) differential equations,
,
which has a periodic solution, the time derivative of
i.e.,
.
2) The adjoint homogeneous LPTV system [2] is formed,
which also has a periodic solution. Any scaled version of
the periodic solution of the adjoint system is also a soluis obtained by scaling
tion for it. The Floquet vector
this periodic solution so that it satisfies
.
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The Floquet vector plays a crucial role in our analysis. For the
derivations and proofs of the above results, we use the Floquet theory [3] of LPTV ODE systems, and its generalization
to index-1 DAE systems [2]. We developed numerical methods
[1], [2], both in time and frequency domain, for the efficient
computation of the periodic Floquet vector
in (3), which
is sufficient to characterize both spectral spreading and timing
jitter in oscillators.
grows very much like the integral of the perFrom (3),
is apturbation. For a constant perturbation, for example,
proximately a linear ramp. This indicates how the frequency of
the oscillator can change due to perturbations, for a linearly increasing phase error is equivalent to a frequency error. In [1],
we concretized the above observation for white-noise perturbations, which we summarize in Section III.
III. PHASE NOISE IN OSCILLATORS WITH
WHITE-NOISE SOURCES
is
In [1], we considered the case where the perturbation
a vector of (uncorrelated) stationary, white Gaussian noise processesthis situation is important for determining practical figures of merit like zero-crossing jitter and spectral purity (i.e.,
spreading of the power spectrum) when the thermal and shot
noise sources in IC devices are considered, which can be modeled as modulated stationary white-noise processes. The state. For example, the shot
dependent modulation is in
noise intensity in a -junction depends on the (periodic) largesignal current through the junction. Jitter and spectral spreading
are closely related, and both are determined by the manner in
, now also a random process, spreads with time. In
which
[1], for white-noise excitations, we established the following.
1) The average spread of the jitter (mean-square jitter, or
variance) increases precisely linearly with time, i.e.
(4)
where
(5)
2) The power spectrum of the perturbed oscillator is a
Lorentzian about each harmonic. A Lorentzian is the
shape of the squared magnitude of a one-pole lowpass
filter transfer function.
3) A single scalar constant is sufficient to describe jitter
and spectral spreading in a noisy oscillator with whitenoise sources.
4) The oscillators output with phase noise due to whiteis a stationary stochastic
noise sources, i.e.,
process.
to be the Fourier coefficients of
If we define
then, the spectrum of the stationary oscillator output
with white-noise sources is given by
(6)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 12, DECEMBER 2002
where
is the fundamental frequency. The phase dedoes not change the total power in the periodic
viation
, but it alters the power density in frequency, i.e.,
signal
,
the power spectral density. For the perfect periodic signal
the power spectral density has functions located at discrete frespreads
quencies (i.e., the harmonics). The phase deviation
the power in these functions in the form given in (6), which
can be experimentally observed with a spectrum analyzer.
The above results have important implications. The power
spectral density at the carrier frequency and its harmonics has
a finite value, and that the total carrier power is preserved despite spectral spreading due to noise. Previous analyzes based
on linear time-invariant or linear time-varying concepts erroneously predict infinite noise power density at the carrier, as
well as infinite total integrated power. That the oscillator output
is stationary is surprising at first sight, since oscillators are nonlinear systems with periodic swings, hence it might be expected
that output noise power would change periodically as in forced
systems. However, it must be remembered that while forced systems are supplied with an external time reference (through the
forcing), oscillators are not. Cyclostationarity in the oscillators
output would, by definition, imply a time reference. Hence, the
stationarity result reflects the fundamental fact that noisy autonomous systems cannot provide a perfect time reference.
IV. STOCHASTIC CHARACTERIZATION OF THE PHASE
DEVIATION WITH COLORED NOISE SOURCES
1) We first derive a partial integro-differential equation for
the time-varying marginal probability density function
of
defined as
(PDF)
(11)
denotes the probability measure.
where
2) We then show that the PDF of a Gaussian random
variable, asymptotically with , solves this partial integro-differential equation. A Gaussian PDF is completely
characterized by the mean and the variance. We show
becomes [under some conditions on
that
or
], for large (to be concretized) , a Gaussian
random variable with a constant mean and a variance
that is given by
(12)
The quantity in (12) (excluding the constant factor ) is
, i.e.
the variance of the integral of
(13)
is a stationary, zero-mean, Gaussian
Theorem IV.1: If
, and if
stochastic process with autocovariance function
satisfies (10), then the time-varying marginal PDF
of satisfies
We now find the probabilistic characterization of the phase
[which satisfies the differential equation (3)]
deviation
is a (oneas a stochastic process when the perturbation
, Gaussian
dimensional) stationary, zero-mean
be the autocovariance
colored stochastic process.2 Let
be the power spectral density, of the
function, and
stationary Gaussian stochastic process
(14)
(7)
with the initial/boundary condition for
(8)
Note that
(15)
i.e.
is a real and even function of . Let
(9)
and
are vectors] that is periodic
which is a scalar [both
in with period . Hence, (3) becomes
(10)
Proof: See the Appendix.
The partial integro-differential equation (14) for the timeof
is a generalization of a
varying marginal PDF
partial differential equation known as the FokkerPlanck equasatisfying (10) when
tion [4], [5] derived for the PDF of
is a white-noise process, which is given as
In this section, we will follow the below procedure to find an
adequate probabilistic characterization of the phase deviation
due to the colored-noise source
for our purposes.
(16)
2The extension to the case when b(t) is a vector of uncorrelated white
and colored-noise sources is discussed in Section VII. Noise sources in
electronic devices usually have independent physical origin, and hence
they are modeled as uncorrelated stochastic processes. Hence, we consider
uncorrelated noise sources. However, the generalization of our results to
correlated noise sources is trivial.
depends on the definition of the stochastic
where
integral [4] used to interpret the stochastic differential equation
as a white-noise process. If
is a white-noise
in (10) with
is a Markov process. However, when
process, then
is colored,
, in general, is not Markovian. Equation (16) is
DEMIR: PHASE NOISE AND TIMING JITTER IN OSCILLATORS WITH COLORED-NOISE SOURCES
valid for any initial/boundary condition. On the other hand, (14)
is valid only for initial/boundary conditions of the type
(17)
for some .
We would like to solve (14) for
solving for the characteristic function
defined by
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larger than the oscillation period
. We will further
comment on this condition in Section V, where we discuss the
noise. With (24), (22) and (23) become
models for burst and
(25)
. We do this by first
of
, which is
(26)
From (26)
(27)
(18)
is
series
-periodic, hence we can expand
into its Fourier
follows trivially. Since, the autocovariance
function of , (27) can be rewritten as
is an even
(28)
Lemma IV.1: The characteristic function of
satisfies
Thus, we obtained (12).
of
Lemma IV.2: The variance
rewritten with a single integral as follows:
in (27) can be
(29)
(19)
where denotes complex conjugation.
Proof: Equation (19) is obtained from (14) by using the
definition of the characteristic function in (18).
Theorem IV.2: Equation (19) has a solution that becomes
(with time) the characteristic function of a Gaussian random
variable
(20)
solves (19) for large enough such that
(21)
It can also be expressed in terms of the spectral density of the
as follows:
colored-noise source
(30)
V. MODELS FOR BURST (POPCORN) AND
(FLICKER) NOISE
A. Burst Noise
The source of burst noise is not fully understood, although it
has been shown to be related to the presence of heavy-metal ion
contamination [6]. Gold-doped devices show very high levels of
burst noise. For practical purposes, burst noise is usually modeled with a colored stochastic process with Lorentzian spectrum,
i.e., the spectral density of a burst noise source is given by
where
(31)
(22)
(23)
Assumption IV.1:
is a constant for a particular device, is the current
where
through the device, is a constant in the range 0.5 to 2, and
is the 3-dB bandwidth of the Lorentzian spectrum [6]. Burst
noise often occurs with multiple time constants, i.e., the spectral
density is the summation of several Lorentzian spectra as given
by (31) with different 3-dB bandwidths.
A stationary colored stochastic process with spectral density
(32)
has the autocorrelation function
(24)
(33)
This is satisfied when the bandwidth of the colored-noise source
is much less than the oscillation frequency , or equivalently,
the correlation width of the colored-noise source in time is much
If the 3-dB bandwidth of (32) is much less than the oscillation
of (33)
frequency , or equivalently, the correlation width
, then (24)
is much larger than the oscillation period
is satisfied.
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 12, DECEMBER 2002
The variance of the integral of a stationary stochastic process
with spectral density as in (32) is given by
(34)
(35)
B.
Noise
where the exponential integral
is defined as
The power in a
noise source modeled with a stochastic
process with the spectral density (38) is concentrated at low frequencies, frequencies much less than the oscillation frequency
noise
for practical oscillators. Hence, (24) is satisfied for
sources.
The variance of the integral of a stationary stochastic process
with spectral density as in (38) is given by
noise is ubiquitous in all physical systems (as a matter of
noise is varied.
fact, in all kinds of systems). The origins of
In IC devices, it is believed to be caused mainly by traps associated with contamination and crystal defects, which capture and
release charge carriers in a random fashion, and the time constants associated with this process give rise to a noise signal with
energy concentrated at low frequencies. For practical purposes
it is modeled with a stationary and colored stochastic process
with a spectral density given by
(42)
(36)
is a constant for a particular device, is the current
where
through the device, and is a constant in the range 0.5 to 2.
There is a lot of controversy both about the origins and modeling
noise. The spectral density in (36) is not a well-defined
of
spectral density for a stationary stochastic process. It blows up
. Keshner in [7] argues that
noise is really a nonat
stationary process, and when one tries to model it as a stationary
process, this nonphysical artifact arises. We are not going to
dwell into this further, which would fill up pages and would
not be too useful other than creating a lot of confusion. Instead,
we will postulate a well-defined stationary stochastic process
noise: We will introduce a cutoff frequency in
model for
and attains a
(36), below which the spectrum deviates from
. To do this, we use the following integral
finite value at
representation [8]
VI. SPECTRUM OF AN OSCILLATOR WITH PHASE NOISE
DUE TO COLORED NOISE SOURCES
Having obtained the stochastic characterization of
due
to a colored-noise source in Section IV, we now compute the
. We
spectral density of the oscillator output, i.e.,
first obtain an expression for the nonstationary autocovariance
of
. Next, we demonstrate that
function
the autocovariance is independent of for large time. Finally,
by taking the
we calculate the spectral density of
Fourier transform of the stationary autocovariance function for
.
We start by calculating the autocovariance function of
, given by
(43)
Definition VI.1: Define
to be the Fourier coefficients of
(37)
We introduce the cutoff frequency
in (37), and use
(38)
The following simple Lemma establishes the basic form of the
autocovariance:
Lemma VI.1:
(39)
for the spectral density of a stationary stochastic process that
noise. The spectral density in (38) has a finite value
models
at
(40)
The autocorrelation function that corresponds to the spectral
density in (38) is given by
(41)
(44)
is
The expectation in (44), i.e.,
. This expectation
the characteristic function of
is independent of for large time as established by the following
theorem:
Theorem VI.1: If is large enough such that
(45)
DEMIR: PHASE NOISE AND TIMING JITTER IN OSCILLATORS WITH COLORED-NOISE SOURCES
then
is a Gaussian random variable and its
characteristic function, which is independent of , is given by
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Corollary VI.2:
(56)
(46)
is as in (28), (29) or (30). Note that the condition
where
(45) is same as the condition (21) in Theorem IV.2.
We now obtain the stationary autocovariance function from
(44) using (46).
Corollary VI.1:
(47)
To obtain the spectral density of
Fourier transform of (47):
is a finite nonnegative value.
where
noise discussed in Section V,
For the models of burst and
we have
from (32), and
from (40). Thus,
, and hence,
are satisfied for both.
we have
Since
(57)
, we calculate the
and hence
(58)
(48)
where
and
(49)
(50)
We reproduce
in the th
for any colored-noise source. The total power
harmonic of the spectrum is preserved. The distribution of the
.
power in frequency is given by
Lemma VI.3: If
(59)
then
in (27)(30) here for convenience
is nonnegative and finite. If
(60)
(51)
then
(52)
where
is nonnegative and finite, and
(53)
(54)
where
is the autocovariance/spectral density pair for the colored-noise
.
source
The Fourier transform in (50) does not have a simple closed
form. Mullen and Middleton in [9] calculate various limiting
forms for this Fourier transform through approximating series
expansions. We are going to use some of their methods to calculate limiting forms for (50) for different frequency ranges of
interest, but before that, we would like to establish some basic,
,
and
.
general properties for
Lemma VI.2:
(55)
Proof: Equation (55) is obtained using (51).
Now, we concentrate on the case when (59) is satisfied, i.e.,
when the spectrum takes a finite value at the carrier frequency
(and its harmonics). Next, we proceed as Mullen and Middleton
in [9] and calculate limiting forms to the Fourier transform in
(50) through approximating series:
Theorem VI.2: Let (59) be true. For away from 0, (50) can
be approximated with
(61)
where
with
. For
around 0, (50) can be approximated
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 12, DECEMBER 2002
where
is the periodically time-varying Floquet vector [1],
[2] of Section II. Let
(65)
and
(66)
(62)
denotes convolution. The first term in (62) is a
where
Lorentzian with corner frequency
and can be used as an approximation for (50) around
by
ignoring the higher order second term. Equation (62) contains
the first two terms of a series expansion for (50).
Proof: Equation (61) is obtained using the representation
in (54), and (62) is obtained using (53).
of
From (61), we observe that the frequency dependence of
is as
multiplied with the spectral density
of
the noise source. This result matches with measurement results
noise sources.
for phase noise spectrum due to
that satisfies (64) becomes a Gaussian
Lemma VII.1:
random variable with constant mean, and variance given by
for large enough such that
(67)
characterized as above, the oscilTheorem VII.1: With
is a stationary process, and its spectral
lator output
density is given by
VII. PHASE NOISE AND SPECTRUM OF AN OSCILLATOR DUE
TO WHITE AND COLORED-NOISE SOURCES
We now consider the case when both white and colored-noise
sources are present. Let there be white-noise sources and
colored-noise sources
(68)
where
are the Fourier series coefficients of
, and
(63)
,
where
,
is a vector of (uncorrelated)
stationary, normalized,3 white Gaussian noise processes, and
are zero-mean, Gaussian,
stationary colored stochastic processes [uncorrelated with each
] with autocorrelation function/spectral
other, and with
density pairs
where
and
this case, the phase error
equation
are assumed to satisfy (24). In
satisfies the nonlinear differential
(64)
3The
spectrum is normalized to 1 (for all frequency), and the amplitude for
is set by the multiplicative modulating factor B in (63).
(69)
The full spectrum of the oscillator with white and colored-noise
sources has the shape of a Lorentzian around the carrier, and
away from the carrier, the white-noise sources contribute a term
frequency dependence, and the colored-noise
that has a
sources contribute terms that have a frequency dependence as
multiplied with the spectral density of the colored-noise
source.
VIII. EXAMPLES
We have derived an analytical expression, given by (68) and
(69), for the spectrum of the oscillator output with phase noise
due to white and colored-noise sources. The analytical expression in (68) and (69) contains some parameters to be computed.
: The Fourier series coefficients of the large-signal
of the oscillator output.
noiseless periodic waveform
:
Scalar that characterizes the contributions of the
white-noise sources.
DEMIR: PHASE NOISE AND TIMING JITTER IN OSCILLATORS WITH COLORED-NOISE SOURCES
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(a)
Fig. 2. Colpitts oscillator.
(b)
(c)
Fig. 1.
Simple oscillator with parallel RLC-nonlinearity.
:
Scalars that characterize the contributions of the colorednoise sources.
We have developed efficient numerical methods, which were
implemented in a circuit simulator, for the computation of the
, and hence for the computation of
periodic Floquet vector
of
the parameters above. Once the periodic steady state
and
are
the oscillator and the scalars
computed, we have an analytical expression that gives us the
spectrum of the oscillator at any frequency . The computation
of the spectrum is not performed separately for every frequency
of interest. We compute the whole spectrum as a function of
frequency at once, which makes our technique very efficient.
Moreover, as will be illustrated below, one can perform a detailed analysis of the mechanism of phase noise generation using
our techniques. In particular, one can calculate the contribution
of the noise sources separately.
Oscillator With Parallel RLC and a Nonlinear Current Source
We now present simulation results in the phase noise characterization of a simple oscillator in Fig. 1(a). The resistor is
assumed to be noiseless, but we insert a stationary external current noise source across the capacitor. The Floquet vector
is a two-dimensional vector, since the oscillator has two state
variables, namely the capacitor voltage and the inductor current.
corresponding to the capacitor
Fig. 1(b) shows the entry of
voltage.
Now, let us assume that we have two current noise sources
across the capacitor, one of them a white stationary noise source,
and the other a colored stationary noise source with bandwidth
much smaller than the oscillation frequency. To calculate the
spectrum of the capacitor voltage given by (68) and (69), we
need to compute in (65) for the white-noise source, and in
(66) for the colored-noise source. For stationary noise sources,
in (65) and
in (66) are constant functions of time .
which is a periodic function
Fig. 1(c) shows
is the time-average of this quantity.
of time . Note that
.
From (66), we observe that is the time-average of
in Fig. 1(b) for the capacitor voltage
The time-average of
is 0! Thus, we conclude that any stationary [with the modua constant function of time] colored-noise source
lation
(with bandwidth much smaller than the oscillation frequency)
connected across the capacitor has no contribution to the oscilfor this noise
lator spectrum due to phase noise, because
source.
Colpitts Oscillator
The oscillator for this example is a standard Colpitts oscillator
with a single bipolar transistor as the active device, shown in
,
nH,
K ,
Fig. 2 (
pF,
pF,
pF). Fig. 3(a) and (b)
show the difference of the entries of the Floquet vector
for the baseemitter nodes, and collectoremitter nodes of the
transistor, respectively.
for collectoremitter is multiplied by the
The waveform
periodic modulation
for the collector current shot noise of the transistor, and then the
in (65) is calculated as
contribution of this noise source to
the time-average of the square of this periodic quantity. Fig. 3(c)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 12, DECEMBER 2002
(a)
Fig. 4. Single-sideband phase noise spectrum with white and 1=f noise
sources.
IX. CONCLUSION
((b)
We have presented the theory and practical characterization
of phase noise in oscillators due to colored, as opposed to white,
noise sources, based on a nonlocal FokkerPlanck equation that
we derived and solved. The results of this paper together with
the numerical methods described in [2] have been implemented
in a proprietary circuit simulator. It has been used by oscillator
designers and proved to be a very useful tool in the search for
novel oscillator circuit architectures with improved phase noise.
APPENDIX
Proof of Theorem IV.1
be the time-varying conditional probability denLet
given
:
sity function of
(70)
(c)
Fig. 3. Colpitts oscillator phase noise analysis.
shows
for all of the white-noise sources, i.e.,
the collector current shot noise, the base current shot noise,
and the thermal noise sources of the collector, base and emitter
is the time-average of this quantity.
resistances.
for baseemitter voltage is used in calThe waveform
and the burst noise source
culating the contribution of the
connected between the base and the emitter of the transistor in
its noise model [6]. As observed in Fig. 3(a), this waveform has a
nonzero time-average. Any colored-noise source connected between the base and the emitter will have an influence on the
spectrum of the oscillator due to phase noise.
Fig. 4 shows the computed single-sideband phase noise spectrum (that can be measured by a specialized equipment) for an
oscillator which has both white and flicker noise sources. This
.
spectrum is computed using the expression in (69) for
denotes the conditional probability measure
where
. Let
denote the conditional expectation
given
operator associated with the conditional probability measure
. Let
be an arbitrary but smooth function. From
(10) we obtain
(71)
Now, we apply the conditional expectation operator
both sides of the above equation
to
(72)
DEMIR: PHASE NOISE AND TIMING JITTER IN OSCILLATORS WITH COLORED-NOISE SOURCES
which can be rewritten using the conditional PDF
as follows:
1791
of
follows using the form of the characteristic function for a
Gaussian random variable.
is the autocorrelation funcgiven in (7). If we take the time derivative of both
tion of
sides of the above, we obtain
(73)
(82)
We use integration by parts to rewrite the integral on the
right-hand side to obtain
We substitute the operator
from (77) into the above
(83)
(74)
where we assumed that the conditional PDF
and
at
vanishes
(75)
follows immediately, since (74) is satisfied for an arbitrary func. Now, we use techniques from [10] to proceed further.
tion
We rewrite (75) as follows:
(76)
with the operator
(77)
Equation (76) has the formal solution
(78)
Next, we will take the expectation of both sides of the above
solution over all realizations of the noise source
Equation (14) follows from the above equation through the
chain rule for derivatives.
REFERENCES
[1] A. Demir, A. Mehrotra, and J. Roychowdhury, Phase noise in oscillators: A unifying theory and numerical methods for characterization,
IEEE Trans. Circuits Syst. I, vol. 47, pp. 655674, May 2000.
[2] A. Demir, Floquet theory and nonlinear perturbation analysis for oscillators with differentialalgebraic equations, Int. J. Circuit Theory Applicat., pp. 163185, 2000.
[3] M. Farkas, Periodic Motions. New York: Springer-Verlag, 1994.
[4] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry
and the Natural Sciences. New York: Springer-Verlag, 1983.
[5] H. Risken, The FokkerPlanck Equation. New York: Springer-Verlag,
1989.
[6] P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated
Circuits, 2nd ed. New York: Wiley, 1984.
[7] M. S. Keshner, 1=f noise, Proc. IEEE, vol. 70, p. 212, Mar. 1982.
[8] F. X. Kaertner, Analysis of white and f
noise in oscillators, Int. J.
Circuit Theory Applicat., vol. 18, pp. 485519, 1990.
[9] J. A. Mullen and D. Middleton, Limiting forms of FM noise spectra,
Proc. IRE, vol. 45, no. 6, pp. 874877, 1957.
[10] P. Jung, Colored noise in dynamical systems: Some exact solutions, in
Stochastic Dynamics, L. Schimansky-Geier and T. Pschel, Eds. New
York: Springer-Verlag, 1997, pp. 2331.
(79)
From the definitions of
we have
and
, in (11) and (70),
where the expectation is over all realizations of
. We would
like to obtain a closed equation for the marginal unconditional
of
. We can accomplish this if we can exPDF
press the expectation on the right-hand-side of (79) in terms
. In the general case, this is not possible, because
of
and
are correlated. However, if
we restrict ourselves to the case when
, then, we obtain
(80)
The expectation in (80) can be evaluated exactly, using the fact
hence
is a Gaussian process
that
(81)
Alper Demir (M95) received the B.S. degree
in electrical engineering from Bilkent University,
Turkey, the and M.S. and the Ph.D. degrees in
electrical engineering and computer sciences, from
the University of California, Berkeley, in 1991, 1994
and 1997 respectively.
From May 1992 to January 1997 he worked as a
Research and Teaching Assistant in the Electronics
Research Laboratory and the EECS Department at
the University of California, Berkeley. He was with
Motorola, Austin, TX, during Summer 1995, and
with Cadence Design Systems, San Jos, CA, during Summer 1996. He joined
the Research Division of Bell Laboratories, Lucent Technologies, Murray Hill,
NJ, as Member of the Technical Staff in January 1997, where he spent four
years. He was with CeLight, Iselin, NJ, a start-up in optical communications,
from November 2000 until February 2002, where he was the Manager for
Optical Telecommunications Systems Design. He is now an Assistant Professor
in the Electrical and Electronics Engineering Department, Ko University,
Istanbul, Turkey. His research work is on the fundamental theory and algorithms for the design analysis, verification and design automation of electronic
and opto-electronic discrete/integrated circuits and systems, with emphasis on
analog/mixed-signal circuits; electromagnetic, wave propagation, nonlinear,
time-varying and noise phenomena in RF/wireless/high-speed/optical communications. The work he has done at Bell Labs and CeLight is the subject of
eight patents (one issued and seven pending). He co-authored two books in the
areas of nonlinear noise analysis and analog design methodologies.
Dr. Demir received the Regents Fellowship from the University of California
at Berkeley in 1991, and was selected to be an Honorary Fellow of the Scientific
and Technical Research Council of Turkey (Tbitak).
