CHAOS 20, 023123 共2010兲
A matched filter for chaos Ned J. Corron,a兲 Jonathan N. Blakely, and Mark T. Stahl U. S. Army Research, Development and Engineering Command, RDMR-WSS, Redstone Arsenal, Alabama 35898, USA
共Received 5 March 2010; accepted 30 April 2010; published online 17 June 2010兲 A novel chaotic oscillator is shown to admit an exact analytic solution and a simple matched filter. The oscillator is a hybrid dynamical system including both a differential equation and a discrete switching condition. The analytic solution is written as a linear convolution of a symbol sequence and a fixed basis function, similar to that of conventional communication waveforms. Waveform returns at switching times are shown to be conjugate to a chaotic shift map, effectively proving the existence of chaos in the system. A matched filter in the form of a delay differential equation is derived for the basis function. Applying the matched filter to a received waveform, the bit error rate for detecting symbols is derived, and explicit closed-form expressions are presented for special cases. The oscillator and matched filter are realized in a low-frequency electronic circuit. Remarkable agreement between the analytic solution and the measured chaotic waveform is observed. 关doi:10.1063/1.3432557兴 In conventional communication systems, a matched filter provides optimal receiver performance in the presence of noise. As such, matched filters are highly desirable, yet they are practical only when a relatively small number of known basis functions are used to represent information. For communications using chaotic waveforms, the unpredictable and nonrepeating nature of chaos suggests that the basis functions are uncertain and ever changing, which would preclude the use of a simple matched filter. Consequently, it is widely accepted that the performance of chaos communications must be worse than that of conventional, nonchaotic systems. In this paper, we show that this assumption is not necessarily true. We describe a simple, low-dimensional chaotic oscillator that admits an exact analytic solution containing a single fixed basis function. The solution can be written as the linear convolution of a symbol sequence and the fixed basis function, similar to how conventional communication waveforms are usually represented. Despite the linear nature of the solution, waveform returns sampled at regular switching times are conjugate to a shift map, proving the oscillator is chaotic. A matched filter for the basis function can be defined and used to extract symbolic information from the chaotic waveform, and we find that its performance in additive white Gaussian noise (AWGN) is comparable to that of binary phase-shift keying (BPSK). Implemented in radio-frequency electronics, the oscillator and its matched filter have potential application in Hayestype chaos communications where a message signal is encoded in the symbolic dynamics via small perturbation control. The discovery of a practical matched filter for chaos finally provides a coherent receiver to complement this elegant encoding scheme.
a兲
Electronic mail:
[email protected].
1054-1500/2010/20共2兲/023123/10/$30.00
I. INTRODUCTION
Coding information in the symbolic dynamics of a chaotic waveform is an intriguing method proposed for highbandwidth data communications.1 Information is encoded using arbitrarily small perturbations to control a chaotic oscillator and target a desired trajectory within the attractor.2 Since the resulting waveform is consistent with the natural, unperturbed dynamics of the chaotic oscillator, an efficiency in the encoding is expected.3 Experimental demonstrations have proven the efficacy of the encoding method for electronic systems.4 However, the development of practical chaos communication systems based on symbolic dynamics has been hindered by the lack of a receiver that complements the elegance of the transmitter. Noise reduction techniques developed for detecting symbols in the presence of noise are computationally expensive and yield only modest results.5–7 In general, chaotic waveforms exhibit a number of properties that make coherent detection difficult.8 In particular, the lack of a fixed basis function and irregular timing deny the implementation of simple matched filters. It has been suggested that synchronization can enable coherent detection, but maintaining high-quality synchronization is also difficult in the presence of significant noise.9,10 Powerful nonlinear filtering techniques have also been considered.11 In this paper, we present a novel chaotic oscillator with several important and surprising attributes that enable coherent reception. This special chaotic oscillator admits an exact analytic solution, which can be written as a linear convolution of binary symbols and a fixed basis function. We show that the sequence of binary symbols forms an exact symbolic dynamics for the oscillator, completely and uniquely specifying any chaotic trajectory. The linear convolution is similar to a standard communication signal, e.g., BPSK.12 We exploit this representation to find a matched filter for the basis function, which enables near-optimal detection of the binary
20, 023123-1
Chaos 20, 023123 共2010兲
Corron, Blakely, and Stahl
symbols constituting the symbolic dynamics. Analysis of the matched filter detection in AWGN provides an expected bit error rate 共BER兲, and closed-form analytic expressions are achieved for certain best and worst case scenarios. Importantly, we show that the BER performance of chaos communication using this wide band oscillator is comparable to BPSK. It is commonly assumed that the complexity of chaos denies analytic solution, but this is not necessarily true.13,14 For difference equations, it has been shown that trajectories of the shift map and baker’s map can be written as the convolution of a random process and an acausal basis pulse.15 Continuous-time chaotic waveforms can be directly constructed using linear superposition without an explicit dynamical system.16,17 Chaotic waveforms have been synthesized in reverse time using a randomly driven filter.18–20 Recently, an exactly solvable chaotic differential equation was described.21 The oscillator described here is a hybrid system in that it contains both a continuous differential equation and a discrete switching condition. This system is similar to the class of differential equations with piecewise constant arguments.22–24 In the intervals between switching events, the differential equation is linear and unstable: the instability provides the stretching required for chaos. The instability is contained by the nonlinear switching events, which provide the necessary folding. A similar construction combining linear instability and hysteresis has also been used to obtain chaos.25–27 For these systems, conditions on a return map can be found to assure chaotic dynamics.28,29 For the oscillator described here, returns in the continuous state sampled at regular switching times are conjugate to a shift map, proving the oscillator is chaotic. To explore potential use in technological applications, we demonstrate the oscillator in an audio-frequency electronic circuit. Our circuit contains both analog and digital circuit elements to implement the hybrid dynamics. In operation, the electronic circuit generates waveforms that appear similar to the analytic solution. Sampled at switching times, the circuit generates a shift map consistent with the analytic model. Using symbols extracted from a measured waveform, we construct the corresponding analytic solution and find that it closely matches the observed waveform. Together, these observations confirm that the exactly solvable analytic model provides a good representation of the electronic circuit. We also realize the corresponding matched filter in a circuit and demonstrate its capability for detecting the symbolic content of a chaotic waveform. The relatively simple realization of the oscillator and matched filter suggests that a high-frequency circuit realization may be practical. We now outline the remainder of this paper. In Sec. II, we define the hybrid dynamical system, show it is chaotic, and derive an exact analytic solution, which we write as a linear convolution with a fixed basis function. In Sec. III, a matched filter for the basis function is derived and its theoretical noise performance is examined. In Sec. IV, we present an experimental system where the oscillator and matched filter are realized in electronic circuits. Finally, we conclude
2
u(t), s(t)
023123-2
1 0 -1 -2 0
10
20
t
30
40
50
FIG. 1. 共Color online兲 Typical waveforms u共t兲 共oscillations兲 and s共t兲 共square wave兲 obtained from numerical integration of the hybrid dynamical system for  = ln 2.
in Sec. V by indicating the impact that a matched filter provides in making chaos communications a viable technology. II. CHAOTIC HYBRID SYSTEM
We consider the hybrid system comprising of a continuous state u共t兲 苸 R and a discrete state s共t兲 苸 兵⫾1其. The continuous-time dynamics is described by the differential equation u¨ − 2u˙ + 共2 + 2兲共u − s兲 = 0,
共1兲
where and  are fixed parameters satisfying = 2 and 0 ⬍  ⱕ ln 2. Transitions in the discrete dynamics are defined by the guard condition u˙共t兲 = 0 ⇒ s共t兲 = sgn共u共t兲兲,
共2兲
meaning the discrete state s共t兲 is set to the sign of u共t兲 whenever the derivative of the continuous state vanishes and s共t兲 maintains this value until the guard condition is next met. In this system, we define the signum function as sgn共u兲 =
再
+ 1, u ⱖ 0 − 1, u ⬍ 0,
冎
共3兲
where the choice that sgn共0兲 = +1 is arbitrary and chosen for definiteness. Figure 1 shows a typical waveform obtained via numerical integration of the hybrid dynamical systems 共1兲 and 共2兲 for  = ln 2. Figure 2 shows a corresponding phase-space projection. These solutions are obtained using an adjustable step size Runge–Kutta integrator 共MATLAB’s ODE45兲 to integrate the ordinary differential equation, while the switching condition is implemented as a detectable event in the integrator. These solutions give the appearance of chaotic dynamics, and we show in Sec. II A that these waveforms are indeed chaotic. A. Analytic solution
For the hybrid systems 共1兲 and 共2兲, we can obtain an analytic representation of a general solution. To this end, we first consider the initial conditions u共0兲 = u0, u˙共0兲 = 0, and s共0兲 = s0, where 兩u0兩 ⱕ 1 and s0 = sgn共u0兲. For 兩u0兩 = 1, we immediately see that the fixed points u共t兲 = u0 and s共t兲 = s0 are a solution to the hybrid system for all t ⬎ 0. For 兩u0兩 ⬍ 1, Eq. 共1兲 admits the solution
023123-3
Chaos 20, 023123 共2010兲
A matched filter for chaos
this problem as equivalent to the original initial value problem, allowing for a unit time translation and an increment of the subscripts. Thus, we immediately write the solution as
10
冉
5
du/dt
u共t兲 = s1 + 共u1 − s1兲e共t−1兲 cos t −
冊
 sin t ,
共11兲
which is valid for 1 ⱕ t ⬍ 2. Similarly, at the end of this interval, we find
0
-5
共12兲
u共2兲 = u2 , where
-10 -2
-1
0
1
2
FIG. 2. 共Color online兲 Phase-space projection from numerical integration of the hybrid dynamical system for  = ln 2.
冉
冊
 sin t , w
共4兲
which is valid while s共t兲 remains unchanged. From the derivative u˙共t兲 = −
2 + 2 共u0 − s0兲et sin t,
共5兲
we see that the guard condition in Eq. 共2兲 is first met when t = 1 / 2. Evaluating the solution 共4兲, we find u共
1 2
兲 = s0兵1 + 共1 − 兩u0兩兲e/2其.
共6兲
Since 兩u0兩 ⬍ 1, we see that sgn兵u共 21 兲其 = s0 .
共7兲
Thus, the discrete state remains unchanged by this initial event and solution 共4兲 is valid at least to the second trigger of the guard condition. Continuing the solution for t ⬎ 1 / 2, we find that the guard condition is next met when t = 1. At this event, the continuous state given by Eq. 共4兲 is 共8兲
u共1兲 = u1 , where u1 = eu0 − 共e − 1兲s0 .
共9兲
It is straightforward to show that 兩u1兩 ⱕ 1. Since the sign s1 = sgn共u1兲
共13兲
s2 = sgn共u2兲.
共14兲
and
u
u共t兲 = s0 + 共u0 − s0兲et cos t −
u2 = eu1 − 共e − 1兲s1
共10兲
explicitly depends on u0, a transition in the discrete state may occur at t = 1. Since the state is uncertain in the general case, the initial solution given in Eq. 共4兲 is reliably valid only for the interval 0 ⱕ t ⬍ 1. However, we also note that Eq. 共4兲 subsumes the fixed points for u0 = ⫾ 1, so the solution is valid for all 兩u0兩 ⱕ 1. To continue the solution for t ⱖ 1, we now consider the hybrid system for the initial conditions u共1兲 = u1, u˙共1兲 = 0, and s共1兲 = s1, where 兩u1兩 ⱕ 1 and s1 = sgn共u1兲. We recognize
We also find that 兩u2兩 ⱕ 1, so the solution can be extended again. In general, the solution process can be repeated indefinitely. We find that the general solution is
冉
u共t兲 = sn + 共un − sn兲e共t−n兲 cos t −
冊
 sin t , w
共15兲
which is valid for n ⱕ t ⬍ n + 1. The returns at the transition times satisfy the recurrence relation un+1 = eun − 共e − 1兲sn
共16兲
with sn+1 = sgn共un+1兲.
共17兲
Thus, for a given initial condition 共u0 , s0兲, the response of the hybrid dynamical system can, in principle, be exactly calculated using this solution for all t ⱖ 0. We note that the recurrence relation given by Eqs. 共16兲 and 共17兲 is an iterated shift map. The map is closed on the interval 兩un兩 ⱕ 1, yet it is piecewise linear with the constant slope e ⬎ 1; thus, the iterated map has positive entropy and is chaotic. Since this map comprises returns of the continuous state sampled at regular return times, it forms a Poincaré return map for the continuous-time dynamics. Thus, it follows that the hybrid system is also chaotic, with Lyapunov exponent = . In this iterated map, the discrete values sn = ⫾ 1 correspond to symbols derived from the map iterates un relative to a partition at zero. We show in the following that these symbols form a symbolic dynamics that fully describes the continuous-time dynamics of the hybrid system. It is straightforward to verify that the representation
un = e
n
再
n−1
u0 − 共1 − e 兲 兺 sie−i −
i=0
冎
共18兲
satisfies the initial condition and recurrence relation in Eqs. 共16兲 and 共17兲. It is useful to invert this equation and write the initial condition in terms of the future returns. Thus, we find
023123-4
Chaos 20, 023123 共2010兲
Corron, Blakely, and Stahl 2
n−1
u0 = e−nun + 共1 − e−兲 兺 sie−i ,
共19兲
i=0
P(t)
1
which is valid for all n ⬎ 0. Recognizing un is bounded, we take the limit as n → ⬁ to get
0
⬁
u0 = 共1 − e−兲 兺 sie−i ,
共20兲
-1
i=0
-6
⬁
un = 共1 − e 兲 兺 si+ne−i
共21兲
i=0
so that returns may be written in terms of the current and future symbols. From this representation, we see that the symbolic dynamics is characterized by a one-sided shift. As such, the symbols form a symbolic dynamics for the chaotic map. Returning to the continuous-time state, combining Eqs. 共15兲 and 共21兲 yields u共t兲 = sn +
再
⬁
− sn + 共1 − e 兲 兺 si+ne −
i=0
冉
· e共t−n兲 cos t −
−i
冊
冎
 sin t ,
共22兲
B. Linear representation
Significantly, solution 共22兲 can be written as the linear convolution ⬁
兺
共23兲
sm · P共t − m兲,
m=−⬁
where
P共t兲 =
冦
冉 冉
1 − e共t−1兲
冊 冊
 sin t , 关t兴 ⬍ 0  cos t − sin t , 关t兴 = 0 0, 关t兴 ⬎ 0
共1 − e−兲et cos t −
0
2
FIG. 3. 共Color online兲 Basis function for  = ln 2.
the basis function reveals that the state only depends on the current and future symbols. In addition, we find that the basis function has the spectral properties
冕
⬁
P共t兲cos共nt兲dt =
−⬁
再
1, n = 0 0, n ⫽ 0
冎
共25兲
and
冕
⬁
P共t兲sin共nt兲dt = 0,
共26兲
−⬁
where n = 关t兴 is the integer portion of the continuous time. In this form, the continuous-time waveform is also represented entirely in terms of the current and future symbols. However, solution 共22兲 does not include any explicit dependence on the initial condition at t = 0. By time translation, an initial condition can equally be specified at any integer time, and solution 共22兲 still defines the subsequent waveform. We consider the limiting case with an initial condition at t = −⬁, so solution 共22兲 is valid for all 兩t兩 ⬍ ⬁. In this case, the symbols also exist for all time, forming a bi-infinite sequence 兵si : 兩i兩 ⬍ ⬁其.
u共t兲 =
-2
t
which expresses the initial condition exclusively in terms of the future symbols 兵si : 0 ⱕ i ⬍ ⬁其. Combining Eqs. 共18兲 and 共20兲 yields −
-4
冧
where n is any integer. Thus, the power spectrum for the chaotic waveform exhibits zeros at the bit rate and its harmonics, similar to a conventional digital communication waveform.12 Although solution 共23兲 reveals a linear characteristic of the chaotic waveform, we emphasize that the hybrid systems 共1兲 and 共2兲 are nonlinear and an arbitrary linear superposition of basis functions is not, in general, a solution. In fact, the basis function 共24兲 by itself is not a solution. Only the linear convolution 共23兲 for a complete sequence of symbols 兵sm = ⫾ 1 : −⬁ ⬍ m ⬍ ⬁其 constitutes an exact solution to the nonlinear hybrid systems 共1兲 and 共2兲. III. MATCHED FILTER
A matched filter for a given waveform is the optimal linear filter for detecting that waveform in AWGN. Practically, a matched filter can be realized as a filter with an impulse response that is the time reverse of the waveform to be detected. In particular, the matched filter for a real pulse P共t兲 is the linear operator L such that L ⴰ ␦共t兲 = P共− t兲,
共27兲
where ␦共t兲 is the Dirac delta function. A. Matched filter for basis function
共24兲
is the basis function. Figure 3 shows the basis function for  = ln 2. We note that solution 共23兲 assumes a bi-infinite symbol sequence and is valid for all 兩t兩 ⬍ ⬁; however, the form of
It is possible to develop a matched filter for the basis function 共24兲. To this end, we note that the time-reversed basis function P−共t兲 = P共−t兲 satisfies the ordinary differential equation P¨− + 2 P˙− + 共2 + 2兲P− = 共2 + 2兲h共t兲, where
共28兲
023123-5
Chaos 20, 023123 共2010兲
A matched filter for chaos
再
h共t兲 =
1, − 1 ⱕ t ⬍ 0 0,
otherwise
冎
共29兲
is a square pulse with unit amplitude and duration.18 By differentiating Eq. 共29兲, we see that h˙ = ␦共t + 1兲 − ␦共t兲,
共30兲
where the impulse ␦共t兲 has unit area. From these observations, we deduce that the time-reversed basis function is the impulse response of the linear filter
˙ = v共t + 1兲 − v共t兲, 共31兲
¨ + 2˙ + 共2 + 2兲 = 共2 + 2兲共t兲,
B. Detecting symbols
We now consider the action of the matched filter on a received waveform 共32兲
where u共t兲 is the transmitted waveform given by Eq. 共22兲 or Eq. 共23兲, and w共t兲 is the Gaussian white noise. The output of the matched filter is
共t兲 = L ⴰ 共t兲,
共33兲
as defined by Eq. 共31兲. Using Eq. 共27兲 we find that ⬁
兺
共t兲 =
sm
冕
P共兲P共 − t + m兲d
−⬁
m=−⬁
+
冕
⬁
⬁
w共兲P共 − t兲d
共34兲
−⬁
is the continuous-time output of the matched filter. To detect a symbol sn using the matched filter, the output 共t兲 is sampled at t = n and compared to a threshold. Here, we assume the timing problem has been solved and the output can be sampled at the precise time. Without loss of generality, we consider detecting the bit s0. As such, the filter output is sampled at t = 0 to give ⬁
兺
共0兲 =
smIm + N,
共35兲
m=−⬁
where Im =
冕
⬁
P共兲P共 + m兲d
共36兲
−⬁
is the pulse autocorrelation at lag m and N=
冕
⬁
w共兲P共兲d
2 − 32 2  共 2 +  2兲
共38兲
which is necessarily positive. For m ⫽ 0, the autocorrelation can also be evaluated to get Im = e共1−兩m兩兲共1 − e−兲
1 − I0 , 2
共39兲
which is negative for all m. To analyze the effectiveness of detecting the bit s0, we write Eq. 共35兲 as
共0兲 = s0I0 + ⌺ + N,
共40兲
where
where v共t兲 is the filter input, 共t兲 is an intermediate state, and 共t兲 is the filter output. Consequently, Eq. 共31兲 is a matched filter for the basis function 共24兲.
共t兲 = u共t兲 + w共t兲,
I0 = 1 + 共1 − e−兲
共37兲
−⬁
is the filtered noise. For m = 0, the autocorrelation is proportional to the energy contained in the pulse and can be evaluated to get
⬁
⌺=
兺
共41兲
s mI m
m=−⬁ m⫽0
is recognized as intersymbol interference. An estimate for the detected bit is ˜s0 = sgn共共0兲 − 兲,
共42兲
where is the threshold. The probability of detecting an incorrect bit is PBE = Pr共共0兲 ⬍ 兩s0 = 1兲 · Pr共s0 = 1兲 + Pr共共0兲 ⱖ 兩s0 = − 1兲 · Pr共s0 = − 1兲,
共43兲
which is the BER. In general, the threshold can be adjusted to minimize the probability of a bit error. In particular, an optimal threshold will exploit a priori information about the current symbol s0, as well as any knowledge of past and future symbols in ⌺. C. BPSK
As a first case, we assume we have no prior knowledge on the current symbol, but we know all the other symbols with certainty. That is, we have Pr共s0 = 1兲 = Pr共s0 = − 1兲 =
1 2
共44兲
and ⌺ is fixed. In this case, we find PBE = 21 兵Pr共N ⬍ − ⌺ − I0兲 + Pr共N ⱖ − ⌺ + I0兲其.
共45兲
Assuming the filtered noise N is normally distributed with zero mean and standard deviation , we find optimal detection yields
冉 冊
1 I0 PBE = erfc 2 冑2
共46兲
using the threshold = ⌺. We note that Eq. 共46兲 is equivalent to the standard result for BPSK encoding, where I2 Eb = 02 N0 2
共47兲
is the effective ratio of bit energy to noise power spectral density.12 That is, if we can subtract the known intersymbol interference, detecting individual symbols in the received waveform is as efficient as the theoretical limit achieved by
023123-6
Chaos 20, 023123 共2010兲
Corron, Blakely, and Stahl 0
BPSK. In a sense, this knowledge represents a best case scenario, although detection can be improved if a priori information about s0 is available.
10
-1
10
-2
10
D. Intersymbol interference
-3
PBE = Pr共s0 = 1兲 ·
冕
PBE
10
More generally, we must allow for uncertainty in the intersymbol interference. In this case, we write
-4
10
Chaos Matched Filter: known intersymbol interference unknown intersymbol interference Reference: BPSK coherent DCSK
-5
10
Pr共共0兲 ⬍ 兩s0 = 1,⌺兲+共⌺兲d⌺
+ Pr共s0 = − 1兲 ·
冕
-6
10
Pr共共0兲 ⱖ 兩s0 = − 1,⌺兲−共⌺兲d⌺,
-7
10
-5
共48兲 where ⫾ designates a density function for the intersymbol interference 兺 given s0 = ⫾ 1, respectively. In Eq. 共48兲, the integrals are evaluated over the range of all possible values of the intersymbol interference given by Eq. 共41兲. As such, both density functions ⫾ reflect the relatively likelihood of the past and future symbol sequences 兵sm其. Assuming the expected sequence is obtained from a free-running oscillator, this information encompasses the symbolic grammar of the one-dimensional map 共16兲, including disallowed sequences, as well as the relative frequency of occurrence derived from the map’s invariant measure. For the general case, this information cannot be analytically derived, but rather it must be numerically estimated. If the sequence is generated by a controlled oscillator, as it would in a communication system, the densities are not only properties of the oscillator but encompass information about the encoded message as well. However, the special case of a free-running oscillator with  = ln共2兲 does allow analytic treatment. In this case, the return map 共16兲 simplifies to 共49兲
un+1 = 2un − sn and its solution 共21兲 is ⬁
1 un = 兺 si+n2−i , 2 i=0
共50兲
which is a binary representation that depends only on future symbols. The map 共49兲 is a full shift, implying that the grammar is unrestricted and the symbols are independent. Hence, the a priori probability for each symbol in the free running oscillator is Pr共sm = 1兲 = Pr共sm = − 1兲 =
1 2
共51兲
for all m. From Eqs. 共39兲 and 共41兲, the intersymbol interference while detecting s0 is 1 − I0 ⌺= 兺 共s−m + sm兲2−m , 2 m=1
共52兲
which also reveals binary representations of the future and past symbols. Using the independent a priori probability 共51兲, one can show that the sum 共52兲 uniformly spans the continuous interval 兩⌺兩 ⱕ I0 − 1, with the exception of a countable set corresponding to the rational numbers. However, the rational subset of the interval has zero measure and
5
10
Eb / N0 (dB) FIG. 4. 共Color online兲 Analytic BERs for  = ln 2 with complete 共bottom curve兲 and no knowledge 共middle curve兲 about intersymbol interference. For comparison, the bottom curve also represents BPSK, while the top curve shows the BER for coherent differential chaos shift keying 共DCSK兲 共Ref. 29兲.
thus is negligible for computing the probability of a bit error. As such, the density functions are
−共⌺兲 = +共⌺兲 =
再
1, 兩⌺兩 ⱕ I0 − 1 1 · 2共I0 − 1兲 0, otherwise
冎
共53兲
and the probability of a bit error 共48兲 is PBE =
1 4共I0 − 1兲
冕
I0−1
兵Pr共N ⬍ − ⌺ − I0兲
−共I0−1兲
+ Pr共N ⱖ − ⌺ + I0兲其d⌺.
共54兲
Assuming the filtered noise N is normally distributed with zero mean and standard deviation , we find optimal detection yields PBE =
1 4共I0 − 1兲
冕
I0−1
冉冑冊
erfc
−共I0−1兲
I0 + ⌺
2
d⌺
共55兲
using the threshold = 0. Using the indefinite integral
冕
erfc共z兲dz = z erfc共z兲 −
1
冑 e
−z2
,
共56兲
expression 共55兲 is evaluated to give PBE =
再
冉 冊 冉 冊冎 冑再 冉 冊 冉 冊冎
1 2I0 − 1 1 − erfc 共2I0 − 1兲erfc 冑 4共I0 − 1兲 2 冑2 −
m=⬁
0
4共I0 − 1兲
1 共2I0 − 1兲2 2 − exp − 2 exp − 2 2 2
,
共57兲 which provides an exact analytic expression for the BER in this case. We compare this result with the BPSK result in Eq. 共46兲 for  = ln 2 in Fig. 4. In this figure, we show the horizontal axis as the effective ratio of bit energy to noise power spectral density using Eq. 共47兲. Since the matched filter provides a coherent receiver for detecting the symbols, the per-
023123-7
Chaos 20, 023123 共2010兲
A matched filter for chaos
v
-R
1K
+ -
220
C
+ -
1.3µF
220
1K 10K
vd
+5V
10K
0.01µF
+
i
100K
Rd 8.2K
+ -
+
CLK _ Q
4.7K
3K
D
1K
1K
5.6K
+ +
L
CLR
54LS74
4.7K
1K
3K
PRE
10K
3K
0.1µF
vs
+
-15V 100K
FIG. 5. Chaotic oscillator circuit.
formance shown in Fig. 4 compares favorably to other correlation-based chaos communication schemes.30
IV. EXPERIMENTAL SYSTEM
In order to demonstrate the potential of this system for practical chaos communications, we designed and built audio-frequency circuits that implement the oscillator and corresponding matched filter. Measurements show the oscillator is chaotic and in good agreement with analytic results, while the corresponding matched filter circuit successfully reveals the information content of the oscillator waveform.
A. Electronic oscillator circuit
A hybrid electronic oscillator, containing both analog and digital components, is shown in Fig. 5. This circuit was constructed using commercially available, discrete components on a solderless breadboard. The analog operational amplifiers are all type TL082, which are powered using ⫾15 V. The diodes are all type 1N4148. The digital integrated circuit is a dual positive-edge-triggered D flip flop 共SN54LS74AJ兲, which is powered with +5 V. The digital and analog components share a common ground. On the left side of the circuit, certain analog components are grouped by dashed boxes. The first grouping, labeled −R, is an active circuit realizing a negative resistor. The second grouping, labeled L, is an impedance converter that provides a nearly ideal inductor. Also significant is the capacitor labeled C, which connects −R and L to a virtual ground provided by an operational amplifier. For the nominal circuit values shown in the figure, we have C = 1.3 F, L = 2.7 H, and −R is tunable by a variable resistor. We recognize the left side of this circuit as a standard RLC oscillator, except that the resistance is negative. Thus, this part of the circuit is modeled by the equations dv v C − +i=0 dt R and
共58兲
L
di = v − vs , dt
共59兲
where v is the tank voltage, i is the current through the inductor, and vs is a feedback voltage applied to the inductor. We introduce the dimensionless time t = , T where T = 2RC
共60兲
冑
L 4R C − L 2
共61兲
and = 2. As we show below, the period T is the return time for the oscillator. Equations 共58兲 and 共59兲 are then written as dv d 2v + 共2 + 2兲共v − vs兲 = 0, 2 − 2 d d
共62兲
where the parameter
=
T 2RC
共63兲
is the dimensionless negative damping. For the circuit, we only consider 0 ⬍  ⱕ ln 2. We now examine the function of the right side of the circuit, which contains the digital circuitry. In the top trace, an operational amplifier is configured as a comparator, which detects the sign of the tank voltage v. The subsequent diode and voltage divider convert the saturated amplifier output to digital logic levels. The middle trace uses a current-tovoltage converter to give v d = − R dC
dv , dt
共64兲
where Rd is the feedback capacitor. A second comparator then detects the sign of this voltage. The following capacitor, diodes, and difference amplifier generate a short trigger pulse for any transition in the comparator output. Thus, the middle trace generates a trigger pulse whenever the derivative of the
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Chaos 20, 023123 共2010兲
Corron, Blakely, and Stahl
2
0
dv/dt (V/s)
v, vs (V)
500
-2 0.0
0.2
0.4
0
0.6
t (s) FIG. 6. 共Color online兲 Typical time series for v 共oscillations兲 and vs 共square wave兲 measured from the oscillator circuit.
-500
-2
tank voltage changes sign. This trigger signal is also scaled to digital logic levels. The digital logic signals from the top two traces control a single flip flop in a 54LS74 digital logic circuit. The flip flop is configured here so that its output encodes and holds the sign of the tank voltage at the last transition in the capacitor current. This output is fed back to the oscillator via the bottom trace. A summing circuit with fixed gain shifts the digital signal to the symmetric levels ⫾V, and the feedback signal vs is applied to the tank inductor. Therefore, the feedback circuit is modeled as dv = 0 ⇒ vs = V sgn共v兲 + V0 , dt
共65兲
meaning that, whenever the derivative of the tank voltage passes through zero, the feedback voltage vs is set to the sign of the tank voltage times the fixed magnitude V. Furthermore, the feedback voltage is held constant until the next trigger event when the derivative transitions. An offset voltage V0 is included to account for a small, yet unavoidable asymmetry in the electronic circuit. Defining the dimensionless states u=
v − V0 V
共66兲
s=
vs − V0 , V
共67兲
-1
0
1
2
v (V) FIG. 7. 共Color online兲 Phase-space projection.
which alternates between the two fixed points. After each switching event, the tank voltage exhibits growing oscillations about the fixed point. Eventually, the oscillations get large and a new switching event is triggered. Figure 7 shows a phase space projection of v and vd for a four-second trajectory. From analysis of the measured switching signal vs, it was determined that the amplitude scale is V = 0.81 V and the offset is V0 = −0.02 V. Using the nominal circuit values, we estimate T = 0.012 s and  = 0.7⬃ log共2兲. However, more precise estimates for T and  are derived from analysis of the measured waveform. Using transitions detected in the measured waveform vs, we determine the times tn for successive returns such that 共dv / dt兲共tn兲 = 0 and 兩v共tn兲 − V0兩 ⬍ V. The average return time is then given by T = 具tn+1 − tn典, which gives T = 0.0119 s. A return map for the measured successive returns v共tn兲 is shown in Fig. 8. This map is in agreement with the analytic solution given by Eqs. 共16兲 and 共17兲, which is a shift map with slope e ⬎ 1. Thus, we can refine our estimate of the negative damping in the circuit using the slope of the
and
1.0
we obtain the dimensionless system given by Eqs. 共1兲 and 共2兲. Thus, we see that the electronic oscillator is a physical realization of the exactly solvable, chaotic hybrid system.
The operation of the circuit was confirmed using measurements of the oscillator waveforms. To obtain a chaotic output, the negative resistor was adjusted. For R ⬃ 6.5 k⍀, the circuit oscillates chaotically with a fundamental frequency near 84 Hz. Waveforms v, vd, and vs were sampled at 100 kHz using a data acquisition device and a PC. To reduce sampling noise, the oversampled data were smoothed using a running average over a window of ten samples. Figure 6 shows a typical observed time series for the chaotic tank voltage v. Also shown is the switching signal vs,
vn+1 (V)
B. Oscillator measurements
0.5
0.0
-0.5
-1.0 -1.0
-0.5
0.0
0.5
vn (V) FIG. 8. 共Color online兲 Successive return map with fit lines.
1.0
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Chaos 20, 023123 共2010兲
A matched filter for chaos
C. Electronic matched filter circuit
v (V)
2
A matched filter corresponding to the oscillator circuit is shown in Fig. 10. This circuit was also built using discrete components on a solderless breadboard. Operational amplifiers are all type TL082, and the power is ⫾15 V. The analog signal delay is implemented using a separate programmable delay unit, employing analog/digital converters and digital memory. The signal delay is set to ⌬t = 0.0119 s to match the oscillator return time. The left part of the circuit approximates the first equation of the matched filter 共31兲, where
0
-2
Measured Waveform Analytic Solution
∆v (V)
0.1
0.0
=
v V
共68兲
is the filter input and
-0.1 1.0
1.1
=
1.2
t (s) FIG. 9. 共Color online兲 Comparison of measured waveform and analytic solution.
− V共t − ⌬t兲 V
is the intermediate state. The time shift is required to make the filter causal. The variable resistor is tuned as R =
measured return map. Using a linear least-squares fit to each segment of the measured return map, also shown in Fig. 8, we obtain e = 1.92 or  = 0.65. Finally, we directly confirm the agreement of the measured oscillator and the analytic solution from the model. From the switching signal vs, we extract the symbol sequence sn. For the 4-s waveform, a total of 336 symbols is extracted. Using these symbols, the corresponding analytic waveform is constructed using Eqs. 共23兲 and 共66兲. We note that the sum in the analytic model is evaluated for just the known symbols so that the solution is not theoretically exact; however, the exponential nature of the neglected terms implies that the error in the truncated solution is negligible. A typical portion of the measured waveform and the analytic solution are shown in Fig. 9. At the top plot, the two waveforms overlap and are indistinguishable. At the bottom plot, the magnified difference ⌬v between the measured waveform and the analytic solution is shown. The close agreement between the two waveforms confirms the reliability of the circuit model.
共69兲
T , 0.1 F
共70兲
which gives R ⬃ 120 k⍀. Although there is no damping in the first equation of the matched filter 共31兲, a 10 M⍀ feedback resistor is included for stability and to suppress drift in the intermediate state. The right-hand side of the circuit realizes the second term of Eq. 共31兲, where
=
v V
共71兲
is the filter output. To achieve a matched filter, the positive resistance is tuned to R ⬃ 6.5 k⍀, matching the negative resistance used in the oscillator circuit. Figure 11 shows a typical output from the matched filter. In the figure, the top plot shows the continuous and discrete states generated by the chaotic oscillator circuit. The continuous state is transmitted to the matched filter via a direct wire connection; thus, there is effectively no noise in the received signal. The bottom plot shows the corresponding output from the matched filter circuit. We note that the matched filter does not recover the original oscillator waveform; rather, its function is to expose the information content
vξ
3K + -
10M 1K
vν
SIGNAL DELAY
∆t = T
1K
+
1K
3K
1K 100K
Rη
+
0.1µF +
3K
vη
1K
FIG. 10. Matched filter circuit.
0.1µF
L
100K
1.3µF
R
C
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Chaos 20, 023123 共2010兲
Corron, Blakely, and Stahl
matched filter finally provides a coherent receiver to complement the elegant encoding in such systems. By considering coding and long symbol sequences, this oscillator and matched filter can also provide an engine for direct sequence-spread spectrum communications and next generation ultra-wide-band radars. Until now, it has been assumed that a performance penalty must be paid for using chaos. The existence of a practical matched filter as presented here invalidates this assumption so that the oft-cited potential benefits of chaos communications—noiselike obscurity, efficient encoding, resilience to multipath, yet simple implementations—may be favorably compared to conventional, nonchaotic communication systems.
v, vs (V)
2
0
-2
vξ (V)
2
0
ACKNOWLEDGMENTS -2
0.0
0.1
0.2
0.3
0.4
0.5
t (s) FIG. 11. 共Color online兲 Measured oscillator waveforms 共top兲 and corresponding matched filter output 共bottom兲. The switching state vs is repeated in the bottom plot for reference.
corresponding to the discrete switching state. The switching state is also shown at the bottom plot for reference. We note that a time shift is apparent in the matched filter output, which is necessary for a causal realization of the matched filter 共31兲. V. CONCLUSIONS
In this paper, we presented a novel chaotic oscillator described by a hybrid dynamical system, i.e., a system containing both continuous and discrete-valued states. We showed that this system has several surprising and important characteristics. The system admits an exact analytic solution that can be written as the linear convolution of a symbol sequence and a fixed basis function, similar to conventional communication waveforms. The oscillator is provably chaotic since returns sampled at switching times are conjugate to a shift map. The symbols form a symbolic dynamics, uniquely specifying any and all trajectories of the chaotic oscillator. A matched filter for the basis function can be defined and used to detect the symbolic dynamics of the oscillator waveform. Analytic BERs for symbol detection in noise can be obtained and show performance is comparable to BPSK. The wealth of analytic results that can be obtained for a chaotic system is both unprecedented and remarkable. To demonstrate the potential for practical application, we designed and built electronic circuits that physically realize the oscillator and its corresponding matched filter. Although this implementation operates at low frequencies 共⬃84 Hz兲, a higher frequency version may prove useful for technological applications. In particular, we anticipate a radio-frequency version of the oscillator can enable high-bandwidth chaos communications using symbolic dynamics, where a message signal is efficiently encoded in the symbolic dynamics via small perturbation control.1,2 The discovery of a practical
The authors would like to acknowledge Robert J. Berinato, Erik M. Bollt, Daniel W. Hahs, Scott T. Hayes, Lucas Illing, and Shawn D. Pethel for many helpful discussions while completing this work. S. Hayes, C. Grebogi, and E. Ott, Phys. Rev. Lett. 70, 3031 共1993兲. E. M. Bollt, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 269 共2003兲. 3 S. Hayes and C. Grebogi, IEEE MTT-S Int. Microwave Symp. Dig. 3, 1879 共1996兲. 4 S. Hayes, C. Grebogi, E. Ott, and A. Mark, Phys. Rev. Lett. 73, 1781 共1994兲. 5 E. Rosa, Jr., S. Hayes, and C. Grebogi, Phys. Rev. Lett. 78, 1247 共1997兲. 6 E. Bollt, Y.-C. Lai, and C. Grebogi, Phys. Rev. Lett. 79, 3787 共1997兲. 7 N. Sharma and E. Ott, Phys. Rev. E 58, 8005 共1998兲. 8 G. Kolumbán, M. P. Kennedy, and L. O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 45, 1129 共1998兲. 9 R. Brown, N. F. Rulkov, and N. B. Tufillaro, Phys. Rev. E 50, 4488 共1994兲. 10 C.-C. Chen and K. Yao, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 47, 1663 共2000兲. 11 J. Bröcker and U. Parlitz, Chaos 13, 195 共2003兲. 12 M. B. Pursley, Introduction to Digital Communications 共Pearson Prentice Hall, New Jersey, 2005兲. 13 S. Katsura and W. Fukuda, Physica A 130, 597 共1985兲. 14 K. Umeno, Phys. Rev. E 55, 5280 共1997兲. 15 D. F. Drake and D. B. Williams, IEEE Trans. Signal Process. 55, 1379 共2007兲. 16 S. T. Hayes, J. Phys.: Conf. Ser. 23, 215 共2005兲. 17 Y. Hirata and K. Judd, Chaos 15, 033102 共2005兲. 18 N. J. Corron, S. T. Hayes, S. D. Pethel, and J. N. Blakely, Phys. Rev. Lett. 97, 024101 共2006兲. 19 N. J. Corron, S. T. Hayes, S. D. Pethel, and J. N. Blakely, Phys. Rev. E 75, 045201共R兲 共2007兲. 20 N. J. Corron, J. N. Blakely, S. T. Hayes, and S. D. Pethel, Phys. Rev. E 77, 037201 共2008兲. 21 N. J. Corron, Dyn. Contin. Discrete Impulsive Syst.: Ser. A - Math. Anal. 16, 777 共2009兲. 22 K. L. Cooke and J. Weiner, J. Math. Anal. Appl. 99, 265 共1984兲. 23 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics 共Kluwer, Dordrecht, 1992兲. 24 M. U. Akhmet, Nonlinear Anal. 66, 367 共2007兲. 25 R. W. Newcombe and N. El-Leithy, Proceedings of the IEEE International Symposium on Circuits and Systems 共ISCAS ’84兲, 1984, Vol. 2, p. 856. 26 T. Saito, Proceedings of the IEEE International Symposium on Circuits and Systems 共ISCAS ’85兲, 1985, Vol. 2, p. 847. 27 R. W. Newcombe and N. El-Leithy, Circuits Syst. Signal Process. 5, 321 共1986兲. 28 T. Saito and S. Nakagawa, Philos. Trans. R. Soc. London, Ser. A 353, 47 共1995兲. 29 S. Nakagawa and T. Saito, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 43, 1019 共1996兲. 30 G. Kolumbán, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 47, 1692 共2000兲. 1 2
