Using Chaos to Detect IIR and FIR Filters

Transcription

1 PIERS ONLINE, VOL. 6, NO., Using Chaos to Detect IIR an FIR Filters T. L. Carroll US Naval Research Lab, Coe 66, Washington, DC 07, USA Abstract In many signal processing applications, IIR an FIR filters may be use interchangeably, as in the long elay limit, an FIR filter looks like an. It is known in nonlinear ynamics that a properly tune can change the imension of a chaotic signal. This occurs because the feeback in an makes it a ynamical system, so it may be characterize in terms of ynamical quantities such as Lyapunov exponents. An FIR filter, on the other han, is not a ynamical system, an shoul not change the imension of a chaotic signal. In practice, however, an FIR filter with a long tail mixes parts of the chaotic signal that are uncorrelate in time, so that a finite length chaotic signal filtere with an FIR filter will appear to have a larger imension. I will show in this paper that it is possible to use a recently evelope imension estimation metho to istinguish an e chaotic signal from an unfiltere signal, or a signal filtere by an FIR filter. I can also istinguish ifferent s from each other. I can even etect the presence of an when I use a chaotic signal whose banwith is much smaller than the filter banwith. Possible applications of this filter ID metho coul be in ientifying raar or sonar targets base on resonant structures, or remotely ientifying chemical compouns base on their resonance properties.. INTRODUCTION If one is remotely etecting an object using a signal that reflects off the object (such as raar, sonar, or an optical signal), it may be useful to etermine if the object has resonant properties. One example is an object where reflections off ifferent parts of the object prouce constructive or estructive interference in the signal; this sort of resonance acts as an FIR filter. Detection of FIR resonances is use in raar for target ientification. On the other han, a resonance line use to etect a particular chemical is an, as is a magnetic resonance line []. It has been shown that filtering a chaotic signal with an infinite impulse response (IIR) filter, a type of filter that uses recursion, can increase the imension of a chaotic signal [, ]. A finite impulse response (FIR) filter oes not use recursion, an so shoul not increase the imension of a chaotic signal [], but if the FIR filter inclues a long elay time, the filtere chaotic signal may appear to numerical algorithms to have a higher imension []. While analysis of the reflecte power spectrum can reveal the presence of a resonance, this analysis may not be enough to istinguish if the resonance is cause by an or an FIR filter.. FIR FILTER A simple example of an FIR filter that is foun naturally is the reflection of a soun wave from an infinite plate [6]. If a soun wave propagates in a meium such as water, with an acoustic impeance of R, an strikes a plate at normal incience, then the reflection coefficient an transmission coefficients at the front of the plate are R = R R R + R T = R R + R () where R is the acoustic impeance of the plate. After traveling through the plate thickness of, the soun wave is encounters the rear surface, which has a reflection coefficient R = R R R + R () After a time elay of t = /c, where c is the soun velocity in the plate, the soun wave emerges from the front surface of the plate with a transmission coefficient of T = R R + R ()

2 PIERS ONLINE, VOL. 6, NO., 00 9 The result is a sum of elaye versions of the original soun wave. In principle, the sum of reflections is infinite, so the term finite impulse response is not completely accurate, but we will keep the stanar language, where infinite impulse response (IIR) refers to a system with feeback, while finite impulse response (FIR) refers to a system without feeback.. IIR FILTER The use in this work is a moel of a simple ynamical system. A linear ODE moel is use: y t = α f ( βy + y ) + x y t = α f ( y βy ) where x is the riving signal. The Lyapunov exponents for this filter are βα f, corresponing to a complex conjugate pair of eigenvalues.. DIMENSION For a ynamical system with Lyapunov exponents λ λ... λ, the Lyapunov imension of the attractor is j λ k D L = j + () λ j+ where j is the largest inex for which the sum over k is non negative. If the signal is filtere, then the Lyapunov exponents η i of the filter will also be inclue in the sum of Eq. (). The aitional exponents may alter the Lyapunov imension of the signal [], as emonstrate below.. MEASURING DIMENSIONS After embeing a chaotic signal in a phase space by the metho of elays [7], the imension of the chaotic signal coul be measure using the metho of [8]. Applying this metho prouces an average phase space raius ɛ (τ), further escribe in [9]. The change in this raius as the embeing imension changes is relate to the actual imension of the chaotic signal. In orer to calculate the attractor imension, it is necessary to pick a series of multiimensional inex points on the attractor an fin other points on the attractor that are nearby in phase space. This set of phase space neighbors forms a reference, an the inices of the set are store in a reference file. References are create for both filtere an unfiltere signals, an are use below when comparing ɛ (τ) values for ifferent signals. 6. CHAOTIC SYSTEM k= To generate the chaotic signal, we use a map of the form x (n + ) = a c x (n) mo (6) where a c is to be etermine. The initial conition was x(0) =, an the first 00 iterates from the map were iscare. The resulting signal was frequency moulate onto a sinusoial carrier. The frequency of the moulate signal s(i) was etermine by ζ(n) = + 0.[x(n) 0.], so that the n th cycle of the signal s(i) was given by s (i) = sin (πt/ [0ζ (n)]) [i = 0,,..., i < 0ζ (n)] (7) If ζ(n) =, then s(i) will have a perio of 0 points. The signal s(i) has a constant envelope. For the map of Eqs. (6) an (7), the Lyapunov exponent is log( a c )/cycle, where log is the natural logarithm. The chaotic signal s(i) is filtere through the of Eq. (), with a time step of 0. s. The filter time constant α f is set to.6 so that the signal s(i) is at the center of the filter passban. The amping constant β is chosen so that α f β is less than the Lyapunov exponent of the chaotic signal. When a c =., the Lyapunov exponent of the chaotic signal is log(.)/(0 pts 0. s/pt) = 0.08 s. The amping constant β is set to 0.0, so α f β = 0.06 (or the Lyapunov exponents are both 0.06). ()

3 PIERS ONLINE, VOL. 6, NO., 00 9 Figure shows ɛ (τ) vs. imension for the map signal with a c =. moulate onto a sine wave. The reference signal is erive from the e signal. The lowest values of ɛ (τ) occur for the e signal, the highest values for the unfiltere signal, an the sum of signals gives an intermeiate value of ɛ (τ). 7. NOISE MODULATION As the value of α f β in the increases, the Lyapunov exponent of the chaotic signal must also increase if the is to increase the signal imension. As the Lyapunov exponent of the map of Eq. (6) becomes larger, the map signal looks more an more like noise. Taking the limit of this tren, β for the of Eq. () was increase to 0.0, an the signal x(n) in Eqs. (6) an (7) was replace by computer generate uniformly istribute ranom noise. Figure shows that the noise signal, when moulate onto a sinusoial carrier, is also useful for istinguishing an IIR filtere signal from an unfiltere signal. While the noise signal itself is infinite imensional, the sine wave onto which the noise is moulate is imensional, so the resulting signal appears to have a imension between an. Figure shows the normalize power spectrum of the chaotic signal use for Figure, along with the response of the for α f =.6 an β = 0.0. The imension increase of the chaotic signal epens on the relative Lyapunov exponents of the filter an the chaotic signal, not on the relative banwiths. Moulating the chaotic signal onto a carrier allowe the Lyapunov exponent an banwith of the chaotic signal to be varie inepenently, so Figure shows that the chaotic signal use to etect the presence of the occupies only a narrow portion of the filter response function. The full with at half maximum (FWHM) for the chaotic signal is 7. 0 Hz, while the FWHM for the response is. 0 Hz. If the chaotic signal banwith was as large or larger than the filter response, then etecting the coul be one with spectral methos. Since the chaotic signal banwith is much smaller than the filter response, etection using spectral methos woul be ifficult. unfiltere IIR + unfiltere unfiltere + unfiltere Figure : ɛ for an unfiltere signal generate by frequency moulating a chaotic signal onto a sine wave (Eqs. (6) an (7)), ɛ for the same signal filtere by an, an ɛ for a sum of the signals. P response signal power f (Hz) Figure : Frequency response of the of Eq. () with α f =.6 an β = 0.0, an power spectrum of the signal create by using uniformly istribute noise to moulate a sine wave in Eq. (7) Figure : ɛ for a filtere an unfiltere signal when a noise signal is substitute for the chaotic signal x(n) in Eq. (7). filter filter Figure : ɛ (τ) for filters with a 0% ifference in frequency, showing the ability to istinguish between ifferent s. Both filters were riven by a noise moulate sine wave, an the reference file for filter was use.

4 PIERS ONLINE, VOL. 6, NO., 00 9 P FIR filter f (Hz) Figure : Frequency response of the of Eq. () (ashe line) with α f =.6 an β = 0.0, an the FIR filter from the infinite plate moel with a thickness of.6 wavelengths of the noise moulate sine wave. Figure 6: Comparison of ɛ (τ) for a noise moulate sine wave signal filtere by an an an FIR filter base on the infinite plate moel. The reference file comes from the, so the values of ɛ (τ) resemble the values that woul be seen for an unfiltere signal. 8. SENSITIVITY A ifferent measure of the usefulness of the imension change of the chaotic signal as a etector of ing is the sensitivity of the metho to a change in the ; can we istinguish ifferent s? The time constant of the of Eq. () was increase by 0%, from α f =.6 to.8, while the amping constant β was ecrease by 0%, from 0.0 to 0.06, leaving the prouct α f β constant. The unmoifie filter will be refere to as filter, while the moifie filter is filter. The noise moulate sine wave signal was use to rive both filters. A reference signal was prouce for the e signal from filter an both filtere signals were compare to this reference. Figure shows ɛ (τ) vs imension for filter an filter when compare to a reference from filter. The values for filter are larger for imensions of an above, but the ifference is not large. It is possible to istinguish between ifferent s using the imension increase metho, but the metho is not very sensitive to filter ifferences when using a noise moulate sine wave. The banwith of the noise moulate sine wave is much narrower than the filter response, so one woul not expect great sensitivity. 9. FIR FILTER The FIR filter was erive from the infinite plate moel escribe above, with a thickness of.6 average wavelengths of the moulate signal. The response functions for the IIR an FIR filters are shown in Figure. The FIR filter response is perioic in frequency, but the noise moulate signal is narrow enough in banwith that it will not see the perioicity. The same noise moulate signal was use to rive both filters. Figure 6 shows ɛ (τ) vs. imension for the noise moulate signal filtere by both IIR an FIR filters when compare to a reference generate by the IIR signal. In this case, the values of ɛ (τ) for the FIR filter are very close to the values of ɛ (τ) for an unfiltere signal. 0. CONCLUSION While it is not always possible to istinguish between IIR an FIR filters using conventional methos, this paper shows that these filter types can be istinguishe by taking avantage of the imension increase in filtere chaotic signals. Furthermore, because the banwith an Lyapunov exponent of a chaotic signal may be varie inepenently when moulating it onto a carrier, is is possible to ientify an with a signal that is much narrower than the filter banwith. REFERENCES. Slichter, C. P., Principles of Magnetic Resonance, Springer, New York, Baii, R., G. Broggi, B. Derighetti, M. Ravini, S. Ciliberto, an A. Politi, Dimension increase in filtere chaotic signals, Physical Review Letters, Vol. 60, No., 979, Davies, M. E. an K. M. Campbell, Linear recursive filters an nonlinear ynamics, Nonlinearity, Vol. 9, 87 99, 996.

5 PIERS ONLINE, VOL. 6, NO., Sauer, T. D. an J. A. Yorke, Are the imensions of a set an its image equal uner typical smooth functions? Ergoic Theory an Dynamical Systems, Vol. 7, 9 96, Broomhea, D. S., J. P. Huke, an M. R. Muloon, Linear filters an non-linear systems, Journal of the Royal Stastical Society B, Vol., No., 7 8, Crawfor, F. S., Waves, McGraw-Hill, New York, Abarbanel, H. D. I., R. Brown., J. J. Siorowich, an L. S. Tsmring, The analysis of observe chaotic ata in physical systems, Reviews of Moern Physics, Vol. 6, No., 9, Pecora, L. M., L. Moniz, J. Nichols, an T. L. Carroll, A unifie approach to attractor reconstruction, Chaos, Vol. 7, No., 00-9, Carroll, T. L., submitte to Chaos, 00.