Noise-induced Synchronization of Crystal Oscillators
Kauzyoshi Ishimura
Isao T. Tokuda
Experimental study on noise-induced synchronization of two uncoupled crystal oscillators is presented. The circuit consists of two Pierce oscillators that are isolated from each other and receive a common noise input from function generator. As the noise intensity is increased, coherent motions of the Pierce oscillators is enhanced and eventually their in-phase synchronization is achieved. Compared to periodic forcing signals such as square and sinusoidal waveforms, the synchronization has been induced more efficiently by the noise input, which requires less amount of the input power. Our experimental framework should be of practical use for synchronizing clocks of multiple CPU systems, distributed sensor networks, and other engineering devices.
IshimuraTokuda2017.tex
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\begin{document}

\title{Noise-induced Synchronization of Crystal Oscillators}

\author{Kauzyoshi Ishimura}
\email[]{kazu-i@fc.ritsumei.ac.jp}
\author{Isao T. Tokuda}
\email[]{isao@fc.ritsumei.ac.jp}
%\homepage[]{http://www.ritsumei.ac.jp/~isao/tokuda-lab/ja/}
\affiliation{Department of Mechanical Engineering, 
Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, 
Shiga 525-8577, Japan}

\date{\today}

\begin{abstract}
Experimental study on noise-induced synchronization of 
two uncoupled crystal oscillators is presented. 
The circuit consists of two Pierce oscillators that are 
isolated from each other and receive a common noise 
input from function generator.
As the noise intensity is increased, 
coherent motions of the Pierce oscillators is enhanced
and eventually their in-phase synchronization is achieved.
Compared to periodic forcing signals such as
square and sinusoidal waveforms,
the synchronization has been induced more efficiently 
by the noise input, which requires less amount of
the input power.
Our experimental framework should be of practical use for
synchronizing clocks of multiple CPU systems, distributed 
sensor networks, and other engineering devices.
\end{abstract}

% insert suggested PACS numbers in braces on next line
\pacs{XX}
\pacs{05.45.Xt, 05.45.Tp, 82.40.Np}
% insert suggested keywords - APS authors don't need to do this
\keywords{Noise-induced synchronization, Crystal oscillator}

\maketitle
\section{Introduction}
Synchronization is a ubiquitous phenomenon of coupled nonlinear 
oscillators found in a diverse fields of science and engineering.
During the last several decades, remarkable progress has been 
made in both theory and experiment on synchronization of 
limit cycle and chaotic oscillators
\cite{winfree2001,kuramoto1984,pikovsky2003synchronization}.
Although the most common situation presumes that the oscillators 
are directly coupled with each other,
indirect interaction through commonly injected noise sources 
has been also known to synchronize uncoupled oscillators.
It has been proven theoretically that a wide class of 
uncoupled limit-cycle oscillators can be in-phase synchronized 
by common weak white noise \cite{teramae2004robustness}.
The theoretical framework has been extended to random impulses 
\cite{arai2008commonrandomimpulse,nagai2009commontelegraphnoise},
general, colored and non-Gaussian, noise %including a heavy-tailed one.
\cite{goldobin2010generalnoise},
and also in the presence of uncommon noise 
\cite{nakao2007noise}.
Experimental systems that show the noise-induced synchronization 
include neuronal systems 
\cite{mainen1995reliability,de2007correlation,galan2006correlation}
and circuit systems 
\cite{arai2008commonrandomimpulse,nagai2009commontelegraphnoise}.
Despite intensive studies on noise-induced synchronization
in a scientific framework, its application to engineering 
problems remains largely limited.
Natural environmental sounds have been utilized as a noise
source to synchronize simulated network of distributed 
sensors \cite{yasuda2013natural,yasuda2016synchronization}.
Numerical study showed that noise can synchronize spin torque 
oscillators to overcome the problem of low output power 
in their array \cite{nakada2012noise}. 

In the present study, we apply the method of noise-induced 
synchronization to crystal oscillators
\cite{vittoz1988high,matthys1983crystal}.
The crystal oscillator is an electronic circuit composed of 
a piezoelectric resonator that determines the oscillation 
frequency. 
Because of its highly precise and stable oscillations, 
it has been widely used to provide clock signals to 
a variety of digital circuits and 
to stabilize frequencies of radio transmitters and receivers. 
The clock rate of a central processing unit (CPU) is also
determined by the frequency of the crystal oscillator.
Under these circumstances, synchronized operation of multiple 
clocks should be of significant importance in a future technology. 
For instance, to further accelerate the clock speed in CPUs, 
it is getting extremely difficult to integrate all the circuit 
elements into a single chip, because of the increased number 
of transistors, which should be located within 
a limited-size chip. 
Diving the CPU into multiple chips is inevitable, thereby
synchronizing the clocks of the divided chips should be 
an essential requirement.
As a source of the noise to synchronize the oscillators,
utilization of the internal noise, which exists inherently 
within circuit itself, should be advantageous in terms of
an economical recycling of the oscillation energy. 
A variety of further applications, \textit{e.g.}, synchronizing 
the clocks of distributed sensors and CPUs, should be found.
Towards establishment of a basis for such technologies, 
we present here an experimental study on noise-induced 
synchronization of the crystal oscillators.

\section{Methods}
\subsection{Hardware implementation}
As the piezoelectric crystal oscillator circuits, two Pierce 
oscillators \cite{pierce1923piezoelectric}, 
each of which was composed of a single digital inverter
(Toshiba 74HCU04AP), 
one resistor, two capacitors, and 
one quartz crystal ($3.579$ MHz, Kyocera CXH49SFB03579MOPES), 
were built.
As shown in Fig.~\ref{circuit},
the inverters (${\rm INV_1}$ and ${\rm INV_2}$)
amplified oscillations of the crystals
(${\rm XTAL_1}$ and ${\rm XTAL_2}$)
through feedback resistors ($R_1$ and $R_3$),
while the capacitors ($C_2$, $C_3$, $C_5$, and $C_6$)
adjusted their oscillation frequencies.
The output signals ($V_{1}$ and $V_{2}$) were generated 
through the buffers (${\rm INV_3}$ and ${\rm INV_4}$).

As a common noise input, which was unidirectionally injected 
to the two Pierce oscillators,
the function generator (Agilent 33250A) was used.
Schottkey diodes ($D_1$ and $D_2$) rectified the input and 
isolated the two Pierce oscillators from each other.
The coupling capacitors ($C_1$ and $C_2$) passed the alternating 
input signals, while the resistors ($R_2$ and $R_4$)
discharge them. 
Physical parameters of the electric components used in the
present experiment are summarized in Table~\ref{piercec}. 

\begin{figure}[t]
\includegraphics[scale=0.5]{circuit.eps}
\caption{Circuit diagram. 
A common noise input is injected from function generator
to two isolated Pierce oscillators, 
composed of digital inverters 
(${\rm INV_1}$, ${\rm INV_2}$),
feedback resistors ($R_1$, $R_3$),
capacitors ($C_2$, $C_3$, $C_5$ and $C_6$),
and quartz crystals (${\rm XTAL_1}$, ${\rm XTAL_2}$).}
\label{circuit}
\end{figure}

\begin{table}[t]%[H] add [H] placement to break table across pages
\caption{Parameters of two Pierce circuits.}
\label{piercec}
	\begin{ruledtabular}
		\begin{tabular}{@{\hspace{2em}}c@{\hspace{1.5em}}|l@{\hspace{2em}}}
			\multicolumn{2}{c}{Circuit elements of Pierce oscillators}\\ \hline
${\rm XTAL_1}$ and ${\rm XTAL_2}$ & $3.58$ MHz\\ 
${\rm INV_1}$, ${\rm INV_2}$, ${\rm INV_3}$ and ${\rm INV_4}$ & 74HCU04AP\\
$R_1$ and $R_3$ & $1$ M$\Omega$\\
$C_2$, $C_3$, $C_5$ and $C_6$ & $22$ pF\\ \hline
			\multicolumn{2}{c}{Unidirectional input paths}\\ \hline
${\rm D_1}$ and ${\rm D_2}$ & BAT43\\
$R_3$ and $R_4$ & $10$ k$\Omega$\\
$C_1$ and $C_4$ & $20$ pF\\
		\end{tabular}
	\end{ruledtabular}
\end{table}

\subsection{Experimental procedure}
First, we confirmed that, without any input from the function
generator, the two Pierce circuits gave rise to limit cycle 
oscillations. 
Then, white \textit{Gaussian} noise was injected to the two circuits.
To examine the dependence of the coherence between the two Pierce 
oscillators on the input noise intensity, 
the noise strength was slowly increased. 
As a quantity to evaluate the strength of the noise input, that is
denoted by $V_{0}(t)$, its standard deviation was computed as
${\sigma}_{0} = \sqrt{
{\langle}V_{0}^{2}(t){\rangle}-{\langle}V_{0}(t){\rangle}^{2}
}$, 
where ${\langle}{\cdot}{\rangle}$ represents average over time.
As the main variables to characterize the dynamics of the two 
Pierce oscillators, 
the output voltages, denoted as $V_{1}$ and $V_{2}$, 
were simultaneously measured by using 
an oscilloscope (Keysight InfiniiVision DSOX2014A)
for $N=50,000$ data points with a sampling frequency of 
$f_{s}=1$ GHz.
For each setting, $10$ measurements were repeatedly made and
their average and the standard deviation were computed.
To compare the effect of the noise input with those of periodic
ones, sinusoidal waveform as well as square waveform was
also injected as the common input to the Pierce oscillators.

\section{Results} 
Fig.~\ref{noinput_ts} shows output waveforms measured from the 
two Pierce circuits in the case that no common input was injected. 
The top and middle panels represent the output waveforms 
$V_{1}(t)$, $V_{2}(t)$ from the first and the second oscillators, 
respectively, whereas the bottom panel displays the input signal 
$V_{0}(t)$ from the function generator. 
With respect to the triggered waveform on the top panel,
the waveform on the middle panel was moving in time, because
the phase of the second oscillator was not locked to that 
of the first oscillator.
According to the autocorrelation analysis,
the frequencies were estimated to be $3.579$ MHz and $3.581$ MHz 
for the first and second oscillators, respectively.
Because the frequency resolution in our experimental set-up 
is ${\Delta}f = f_s/N = 0.02$ MHz, the frequency difference
of $0.002$ MHz computed between the two oscillators cannot
be verified. However, the observed phase relationship 
that the first oscillator continued to be delayed from the 
second oscillator implies that there exists a slight 
detuning between the two oscillators.
Fig.~\ref{noinput_pp} shows {\it Lissage} plot of the
two outputs on a two-dimensional plane ($V_{1}$, $V_{2}$).
In the absence of the common input, the {\it Lissage} plot 
shows a scattered structure with no clear locking in the 
phase relationship between the two signals.

We then injected a common noise input to the two oscillators
by varying its input strength.
Figs.~\ref{input1_ts}~{\&}~\ref{input3_ts}
show the time waveforms in the case that the noise intensity is  
${\sigma}_{0} = 0.026$ V and $0.093$ V, 
respectively. 
Figs.~\ref{input1_pp}~{\&}~\ref{input3_pp}
show the corresponding {\it Lissage} plots.
No apparent difference from the case without common input was 
recognized in the {\it Lissage} plots.
As the noise intensity ${\sigma}_{0}$ was increased from 
$0.026$ V to $0.093$ V, 
the speed of the phase slipping was also decreased in the
middle panels from Fig.~\ref{input1_ts} to Fig.~\ref{input3_ts},
implying that the frequency difference became smaller with 
the increased noise intensity. 

Fig.~\ref{input4_ts} displays the case that the noise intensity 
was set to ${\sigma}_{0} = 0.123$ V.
The waveform on the middle panel was not slipping from that
on the top panel, since the phases of the two Pierce oscillators 
were locked to each other. Accordingly, the {\it Lissage} 
plot indicated a closed trajectory associated with a constant 
phase shift established between the first and second oscillators
(Fig.~\ref{input4_pp}).
This is a hallmark of synchronization induced by the common 
noise input.

\begin{figure}[t]
\includegraphics[scale=1.]{image5.eps}
\caption{Time waveforms of two Pierce oscillators 
(top panel: $V_{1}(t)$, middle panel: $V_{2}(t)$)
and noise input (bottom panel) in the case of no input
(${\sigma}_{0} = 0$ V).}
\label{noinput_ts}
\end{figure}

\begin{figure}[t!]
\centering
\includegraphics[scale=0.6]{test2.eps}
\vspace*{5mm}
\caption{{\it Lissage} plot ($V_{1}$, $V_{2}$) of
two Pierce oscillators in the case of no input
(${\sigma}_{0} = 0$ V).}
\label{noinput_pp}
\end{figure}

\begin{figure}[t!]
\includegraphics[scale=1.]{image6_2_2.eps}
\caption{Time waveforms of two Pierce oscillators 
(top panel: $V_{1}(t)$, middle panel: $V_{2}(t)$)
and noise input (bottom panel) in the case of 
${\sigma}_{0} = 0.026$ V.}
\label{input1_ts}
\end{figure}

\begin{figure}[t]
\includegraphics[scale=0.6]{n1.eps}
\caption{{\it Lissage} plot ($V_{1}$, $V_{2}$) of
two Pierce oscillators in the case of 
${\sigma}_{0} = 0.026$ V.}
\label{input1_pp}
\end{figure}

\begin{figure}[t!]
\includegraphics[scale=1.]{image10_2.eps}
\caption{Time waveforms of two Pierce oscillators 
(top panel: $V_{1}(t)$, middle panel: $V_{2}(t)$)
and noise input (bottom panel) in the case of 
${\sigma}_{0} = 0.093$ V.}
\label{input3_ts}
\end{figure}

\begin{figure}[t!]
\includegraphics[scale=0.6]{n3.eps}
\caption{{\it Lissage} plot ($V_{1}$, $V_{2}$) of
two Pierce oscillators in the case of 
${\sigma}_{0} = 0.093$ V.}
\label{input3_pp}
\end{figure}

\begin{figure}[t!]
\includegraphics[scale=1.]{image13_2.eps}
\caption{Time waveforms of two Pierce oscillators 
(top panel: $V_{1}(t)$, middle panel: $V_{2}(t)$)
and noise input (bottom panel) in the case of 
${\sigma}_{0} = 0.123$ V.}
\label{input4_ts}
\end{figure}

\begin{figure}[t!]
\centering
\hspace*{10mm}
\includegraphics[scale=0.6]{n4.eps}
\vspace*{5mm}
\caption{{\it Lissage} plot ($V_{1}$, $V_{2}$) of
two Pierce oscillators in the case of 
${\sigma}_{0} = 0.123$ V.}
\label{input4_pp}
\end{figure}

\begin{figure}[t!]
\includegraphics[scale=0.7]{noise_corr_and_timelag2.eps}
\caption{Dependence of the maximal correlation 
$r_{\max}$ (solid line, left y-axis)
between two oscillators on noise intensity ${\sigma}_{0}$.
The corresponding phase shift ${\tau}_{\max}$ is drawn
simultaneously (dotted line, right y-axis).
The error-bars represent standard deviation of $10$ 
different measurements.}
\label{corr_and_time}
\end{figure}

\begin{figure}[t]
\includegraphics[scale=0.65]{corr3.eps}
\caption{Dependence of the maximal correlation $r_{\max}$ 
between two oscillators on noise intensity ${\sigma}_{0}$.
The case of noise input (solid line) is compared with
those of square waveform (dashed line) and sinusoidal 
waveform (dotted line) inputs.}
\label{corr_input}
\end{figure}

To quantify the level of synchrony,
the correlation coefficient between the two outputs
$V_{1}(t)$, $V_{2}(t-{\tau})$ with time lag $\tau$
was computed as
\begin{eqnarray}
r({\tau}) &=& 
\frac{{\langle}V_{1}(t)V_{2}(t-{\tau}){\rangle}-
{\langle}V_{1}(t){\rangle}{\langle}V_{2}(t-{\tau}){\rangle}}{
\sqrt{{\sigma}_{1}^{2} {\sigma}_{2}^{2}}},
\label{eqn:correlation}
\end{eqnarray}
where 
${\sigma}_{1}^{2} =
{\langle}V_{1}^{2}(t){\rangle}-{\langle}V_{1}(t){\rangle}^{2}$,
${\sigma}_{2}^{2} =
{\langle}V_{2}^{2}(t-{\tau}){\rangle}-{\langle}V_{2}(t-{\tau}){\rangle}^{2}$.
Then, the maximal correlation 
\begin{eqnarray}
r_{\max} = {\max}_{0{\leq}{\tau}{\leq}T} r({\tau})
\label{eqn:syncindex}
\end{eqnarray}
provides an index of synchronization, whereas the 
corresponding time lag
${\tau}_{\max} = {\argmax}_{0{\leq}{\tau}{\leq}T} r({\tau})$ 
gives the phase shift.
Here, $T = 2.79 {\cdot} {10}^{-7}$ sec represents the natural
period of the crystal oscillator. 
If the two oscillators are phase-locked with ${\tau}_{\max}$,
the two variables
$V_{1}(t)$, $V_{2}(t-{\tau}_{\max})$
should be highly correlated with each other.
Otherwise, their phase difference would change in time, 
resulting in a low correlation.

The solid line in Fig.~\ref{corr_and_time} 
shows dependence of the maximal correlation $r_{\max}$
on the noise intensity ${\sigma}_{0}$. 
For each setting of the noise, the maximal correlation was computed
for $10$ time segments (each segment has data points of $N=50,000$) 
and their average was drawn with the standard deviation indicated 
by the error bar.
In the region where the noise intensity ${\sigma}_{0}$ was small, 
the maximal correlation was small.
As the noise intensity ${\sigma}_{0}$ was increased, the
maximal correlation gradually increased and, 
for ${\sigma}_{0}>0.12$, the correlation reached nearly to
unity, indicating the onset of a clear synchronization between 
the two crystal oscillators.
The small error bars discernible in this region also indicate
that the high correlation was independent of the time segment
and the observed synchronization was quite robust. 
On the same figure, the phase shift ${\tau}_{\max}$ was also
drawn by the dashed line. 
Relatively large error-bars in the region of small noise 
intensity indicated that the optimal phase shift was quite
variable from one segment to another, implying that the
phase relationship between the two oscillators has not been
well established.
The scale of the error-bar was comparable to the period length
$T$ of the crystal oscillator. 
For the noise intensity of ${\sigma}_{0}>0.12$, the error-bars 
became much smaller, because of the stable phase relationship 
established between the two oscillators.
In the region of ${\tau}_{\max} {\in} [0.12, 0.16]$, the
phase shift remained relatively large.
As the noise intensity was further increased, the phase shift
became close to zero, indicating that in-phase synchronization 
was achieved.

Finally, to compare the effect of the noise input with those 
of periodic ones, sinusoidal and square waveforms were injected 
to the same Pierce circuits as the common input.
The oscillation frequency of the periodic inputs was set to 
$3.58$ MHz, which is very close to the natural frequencies 
of the crystals. 
Fig.~\ref{corr_input} shows dependence of the maximal correlation
$r_{\max}$ on the input intensity ${\sigma}_{0}$.
Compared to the noise input (solid line), the periodic inputs
gave rise to synchrony associated with high correlation coefficient 
at relatively large intensities, \textit{i.e.}, 
$0.34<{\sigma}_{0}$ for sinusoidal waveform (dotted line) and 
$0.26<{\sigma}_{0}$ for square waveform (dashed line).
These results demonstrate the advantage of the noise input 
over the periodic ones. 
One of the reasons why relatively large intensity was needed to 
achieve synchrony in periodic inputs might be because the periodic 
input should entrain the crystal oscillator to their own frequency, 
which requires a stronger condition of synchrony
\cite{kuramoto1984,pikovsky2003synchronization}.
The noise input, on the other hand, does not entrain the oscillator
to itself, because it has no characteristic frequency. Instead, the 
noise adjusts the phases and timings of the oscillators so as to 
enhance coherent motions of the independent oscillators
\cite{teramae2004robustness}.

\section{Conclusions and Discussions}
To conclude, 
noise-induced synchronization of two uncoupled crystal oscillators 
was experimentally demonstrated.
Compared to periodic input signals such as square and sinusoidal 
waveforms, the synchronization has been achieved more efficiently 
by the noise input.
In general, noise is unwanted in electrical circuits, because 
it produces errors in the output signals and moreover lowers the 
precision of the oscillation frequencies. 
By contrast, the present results highlighted a positive
aspect of the noise for inducing the synchrony. 
As a main source of the noise, instead of the function generator 
used in the present study, a variety of sources such as
thermal noise, electromagnetic wave, chaotic oscillation, or 
others can be a good candidate for triggering the synchronized
oscillations of the crystals. 
Recently, noise-induced synchronization has been applied to
several engineering problems. 
For instance, simulation study examined environmental noise 
as a possible source for synchronizing wireless sensor 
networks \cite{yasuda2013natural,yasuda2016synchronization}.
Noise-induced synchronization has been also utilized
in a simulated array of spin torque oscillators to 
overcome their low output power \cite{nakada2012noise}. 
Along the lines of these works, synchronization of the crystal 
oscillators presented in this study should provide a 
strong technological basis for 
unifying multiple CPUs, sensor networks, and other
distributed clock devices. 

In the present study, white noise has been utilized as 
the simplest form of the noise source. The important future 
step would be to optimize the type of the noise that induce 
synchronization 
with a minimal amount of energy. 
If common noise signals are added to the crystal oscillators 
with optimal timing and strength, the phase synchronization 
might be induced more quickly and robustly.
For periodic stimulus, a methodology has been recently 
developed to design an optimal waveform
\cite{harada2010optimal,zlotnik2013optimal}.
Our future study may focus on such optimization problems
of noise inputs.

\section*{Acknowledgements}
This work was partially supported by the 
Grant-in-Aid for Scientific Research 
(No. 17K14685, No. 16K00343, No. 16K06154, No. 26286086) 
from Japan Society for the Promotion of Science (JSPS). 

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