Proceedings of the 38th Southeastern Symposium on System Theory Tennessee Technological University Cookeville, TN, USA, March 5-7, 2006
TA2.4
Symbolic Noise, Signal Processing, and Signal Enhancement by the use of Chaos John E. Gray ∗ and Stephen R. Addison
Abstract In this paper, we discuss the symbolic and visual approach to combinations of noise plus noise. Just as in other application chaos is used to disguise noise, we suggest that their is a dual process where chaos may be used to amplify signals in noise. An example of this is presented and some of the implications are discussed.
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Introduction
Chaos [4] is a catchword for a variety of nonlinear phenomena that can occur in both discrete and continuous dynamic systems. While widely discussed in the physics community in the last twenty years, it has had limited impact on the electrical engineering community. The one exception has been the work by Chua [3] in circuit theory in the early stages, while more recent work has emphasized possible usage in communication and coding [14] using symbolic dynamics. The existence of chaos has suggested to a number of authors that they can use chaos to hide signals (information) so that one appears to be broadcasting noise. The proper decrypting scheme allows the decoder to determine what the broadcast signal was. Thus one has a fairly secure means of communication, at least in theory. Since radar is conceptu∗ J.E. Gray is with the Advanced Science and Technology Division, Code B-23, Naval Surface Warfare Center, Dahlgren Division, Dahlgren, VA 22448-5150, USA.
[email protected]
† S.R. Addison is Chair of the Department of Physics and Astronomy, College of Natural Sciences and Mathematics, University of Central Arkansas, Conway, AR 72035, USA.
[email protected]
0-7803-9457-7/06/$20.00 ©2006 IEEE.
†
ally similar to communications, similar ideas apply to radar waveforms. The problem hiding signals using chaos is dual to the problem of enhancement of signals in noise by the use of chaos, this apparently has not been previously been considered [10]. There are two routes that lead to considering the possibility that chaos be used to enhance detection of a signal in noise as opposed to hiding a signal by making the signal apparent noise. Empirical evidence exists for enhancement of signals in low signal-to-noise-ratio (SNR) environment. For instance this is suggested by the phenomenon of stochastic resonance discovered by Benzi [1]. Any weak signal, which is normally undetectable, when subject to periodic modulation can be exhibit enhancement of signal strength by the phenomena of resonance. A weak signal (low signal strength) can thought of as being trapped in a potential barrier with two deep wells. By driving the signal between the two wells, signal enhancement can be achieved if the maximum of the signal can match the maximum of the well as the signal switches between the two minima [2]. There are a variety of other symbolic models of resonance that exhibit the same type of conceptual model. While noise is traditionally thought of in terms of an analytical model based on differential equation models, an approach that dates back to Einstein’s and led to the modern definition of noise [5], another viewpoint is possible that is more symbolic/pictorial. Noise can be imagined as either a symbolic effect that causes transcription errors in determining the correct underlying symbol or as using a visual model of noise. Noise cannot be destroyed by non-linear transformations, but it can be redistributed so it is possible to conceive of non-linear
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transformation which effect SNR by enhancing SNR within a region for a class of signals, while lowering it in another region. (In quantum mechanics, the concept of a squeezed state is analogous to what we have discussed.) The symbolic viewpoint toward noise leads to at least three possible approaches to the interpretation of noise (randomness): visual (fluid-like), symbol transcription error (replacing one letter with another), measure theory (mathematicians viewpoint). For radar/communications applications, the transcription error viewpoint is useful, while the fluid viewpoint is applicable to the problem of detecting signals in noise. The traditional way to think of noise is the way mathematicians do which is in some abstract topological space with a underlying measure associated with it. It is possible to imagine noise visually as picture that captures the essence of its characteristics. If one imagines noise as a fluid, then the effect of a transformation on noise is not to change the volume but rather the shape. For example, if a fluid were confined to a cylinder, by rapidly rotating the cylinder about its perpendicular axis, then the fluid would redistribute itself around the edge. While the volume remains fixed, the average height of the fluid is lowered for a significant area about the center. If an object is located beneath the fluid before the rotation starts, and the object is located in the region were the fluid is lowered, then the object is exposed when the rotation rate becomes fast enough. Thus if the object remains fixed under these rotation, then the visibility of the object is raised by rotation. The analogy to SNR is obvious. For functions that remained fixed or nearly fixed under nonlinear transformations, one sees how, in principle, the detectability of a “signal” is effectively increased (though obviously there are limits to how much it can be increased). The question is what type of transformations and what type of signal this type of analogy might apply to requires some consideration. Since entropy is fixed under linear transformations and nonlinear transformations that are one-to-one [12], then we are restricted to nonlinear transformations that are not one-to-one (x2 for example.) Local signal SNR could be improved in one region by a local decrease in entropy, while increasing entropy in an-
other. Net entropy could increase, but local entropy about the signal could decrease. Once this concept is understood, then an obvious set of non-linear transformations that may have this local signal SNR enhancement property are those type of nonlinear transformations that exhibit chaos.
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Symbolic Signals and Chaos
A definition of chaos that applies to a wide variety of examples and easy to verify is [6] Definition 1 Let V be a set. f : V → V is said to be chaotic on V if 1. f has sensitive dependence on initial conditions. 2. f is topological transitive. 3. periodic points are dense in V . The simplest example of non-linear transformation that exhibits chaos is the logistic map (useful for population dynamics modeling) xn+1 = λxn (1 − xn ),
(1)
or we can write it as Fλ (x) = λx(1 − x)
(2)
where λ is a parameter λ ∈ [0, 4]. Starting with an initial seed, x0 , one repeatedly iterates using this rule, so (3) x1 = λx0 (1 − x0 ) x2 = λ2 x0 + 2λ2 x30 − −λ2 x40
(4)
and in general xN
= λxN −1 (1 − xN −1 ) N
= λ x0 − f (higher order terms)
(5) (6)
The stable points are solutions to the logistic map equation are the fixed points x = 0 and x = 1 − λ1 . Stable points are attractive to a fixed point x∗ , if a point close to the fixed point, upon iteration becomes closer to it. Thus if
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xn+1 = x∗ + n+1
(7)
n+1 n < 1,
with
(8)
then by writing the logistic map out as a Taylor series we have xn+1 = x∗ + n f (x∗ ) (9) which is stable provided |f (x∗ )| < 1. This discussion applies not only to points, but also to functions as well. Definition 2 We are led to define a signal as being a fixed point signal function s if s(yn+1 ) = s (y ∗ ) + n s (y ∗ )
∀y ∗ , yn ∈ [a, b] . (10)
some degree of sensitivity on the choice of λ. A value close to the value for the onset of chaos seems to work well for signal-to-noise ratios to −10 dB. Another type of signal that the logistic map works well on is a signal that is composed of superpositions of Bessel functions. Thus one can talk about the combination of symbolic signal and noise under chaotic transformations, and examine how components are transformed. Consider the action of Fλ (x) on the combination of signal plus noise in the frequency domain (Y (ω) = X 2 (ω)), Y (ω) = P (ω) + N (ω) (13) so
Fλ (Y (ω)) = λY (ω) (1 − Y (ω)). Definition 3 A fixed point function f is repelling if |f (x)| > 1 for some x ∈ [a, b] or it is a source funcNow if we iterate m times tion. An example of a fixed point signal function is the sinc function. The simplest non-trivial function that occurs in signal processing is the pulse. When transformed to the Fourier domain, it becomes the sinc function sin(πβx) . (11) sinc(βx) = πx It has the stability property. Many signals that occur in radar, communications and sonar can be decomposed as superpositions of sinc functions in the frequency domain. Thus, typical signals can be represented in the Fourier domain as S(ω) =
M
Ai sinc(β(ω − ωi )) + P (ω)
(m)
Fλ (Fλ (...Fλ (Y (ω)))) = Fλ
(15)
we have iterated the noise spectrum so that the transformed noise has a different probability density function than the original. Consider the case of noise that is uniform distributed between frequencies ±Ωa Pω (ω) =
1 2Ωa 0
ω ∈ [−Ωa , Ωa ] otherwise
(16)
The effect of iteration is equivalent to determining the PDF for Zm where
(12)
i=1
where P (ω) is the power spectral density of the disturbing noise distribution. Without noise, a sinc function repeatedly applying the logistic map produces a term that is linear in the sinc function while the remainder of the terms are highly nonlinear that tend toward zero. Thus the ‘signal’ tends to remain within a narrow band while showing some degree of oscillation. The effect of repeatedly applying the logistic mapping to various noise distributions is to redistribute the noise power away from the signal for a number of different noise distributions. There is
(Y (ω))
(14)
Zˆm = Zˆ1 Zˆm−1
(17)
Zˆ1 = λˆ ω (1 − ω ˆ ).
(18)
and Now if we let u = λˆ ω (1 − ω ˆ ), what is P (u) given we know that the PDF of ω is Pω (ω)? Answer: Solve for zeros 1 1 4u 1+ ω1 = + 2 2 λ and
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1 1 ω2 = − 2 2
1+
4u λ
2
du note dω = √λ 4u , then the PDF becomes 1−
P (u) =
=
References
λ
1 √ Pu (u)|ω1 ,ω2 2 u 2 1 1 P + 1− λ 2 2
[1] R. Benzi, A. Sutera , A. Vulpiani, J. Phys. A14, L453, (1981). 4u λ
+P 1 − 4u λ
1 2
− 1− 1 2
4u λ
[2] A. D. Bulsara and L Gammaitoni, “Tuning in to Noise”, Physics Today, March 1996.
[3] L. O. Chua, “Chaos in Digital Filters”, IEEE Transactions on Circuits and Systems, Vol. 35, (19) No. 6, June 1988.
From the form of this equation, it is clear that the logistic map has a tendency to concentrate the randomness toward the edge of the interval the uniform distribution was originally defined upon. Repeating the process of iteration continues this procedure and sharpens it up. Thus we see the fluid analogy again, where the noise surface becomes more and more concave under iteration
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[4] Chua, L. O. and Madan, R. N., “Sights and Sounds of Chaos”, IEEE Circuits and Devices Magazine, Volume: 4 Issue: 1 , Jan. 1988. [5] L Cohen, “The history of noise on the 100th anniversary of its birth”, IEEE Signal Processing Magazine, Nov. 2005 Volume: 22, Issue: 6, pp 20 - 45. [6] R. L. Devaney, An Introduction to Chaotic Synamical Systems, Addison-Wesley, 1987. [7] C. S. Daw, C. E. A. Finney and E. R. Tracy , “Symbolic analysis of experimental data”, http: www-chaos.engr.utk.edu/abs/abs-rsi2002.html
Conclusions
[8] D. F. Delchamps, “Nonlinear Dynamics of overThis is but one example of a signal detection methodsampling A-to-D Converters”, Proceedings of ology that has broader applications to the usage the 32nd Conference on Decision and Control, chaotic transformations to signal processing. AuSan Antonio, Texas, December 1993. thors have been researching applying chaos to the problem of secure communication problem by trying [9] Gentle, J. E., Random Number Generation and to modulate signal through a circuit that is chaotic in Monte Carlo Methods, Springer-Verlag, 1998. an attempt to disguise the signal in the background noise. Trying to disguise a signal is the dual prob- [10] Gray J. E. and McCole, B. S., “Application of Non-Linear Transformation to Signal Prolem to detecting a signal in a high noise environment. cessing”, Proceedings of Thirtieth Annual AllerApplying transformations that exhibit chaos to noisy ton Conference on Communication, Control, signals has the potential to “demodulate” the signal and Computing, Sept. 30-Oct.2, 1992, Allerton from the noise. This observation has not been made House, Monticello, Illinois. by authors who are trying to apply chaos to communication. Singer [15] has noted that the objective of [11] Madan, R. N. “Observing and Learning Chaotic detection is not to improve SNR, but rather to imPhenomena from Chua’s Circuit”, Proceedings prove detection performance. The symbolic/visual of the 35th Midwest Symposium on Circuits and approach to noise to manipulate noise via a series of Systems, 1992, 736 -745, vol.1 chaotic transformations. If the signal remains invariant under transformation, while the noise is redis- [12] Papoulis, A., 1991: Probability, Random Variables, and Stochastic Processes, Third Edition, tributed, then it is possible to improve signal detecMcGraw-Hill Book Company. tion performance.
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[13] M. R. Schroeder, Number Theory in Science and Communication, Second Enlarged Edition, Springer-Verlag, 1986. [14] J. Schweizer and T. Schimming, “Symbolic Dynamics for Processing Chaotic Signals—II: Communication and Coding”, IEEE Tran. on Cir. and Syst.–1 Fundamental Theory and Applications, Vol. 48, No. 11, Nov. 2001. [15] P. F. Singer, “The Optimal Detector”, SPIE Small Targets Conference, Orlando, Fl, 2002. [16] W. Wang and D. H. Johnson, “Computing Linear Transforms of Symbolic Signals”, IEEE Trans. Signal Processing, March, 2002.
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