Spectra of filtered signals

28 April 1997

PHYSICS

LETTERS

A

Physics Letters A 229 ( 1997) 44-48

ELSEiVIER

Spectra of filtered signals Bruno Eckhardt Fachbereich Physik und ICBM der C. v. Ossietzky Universitiit, Postfach 25 03, D-261 I1 Oldenburg. Germany Received

18 November

1996; revised manuscript

received 3 February

1997; accepted

for publication

*

19 February 1997

Communicated by A.P. Fordy

Abstract Resonances of chaotic signals filtered with linear recursive filters are analyzed within periodic orbit theory. The filtering gives rise to an extended system for which the Fredholm determinant factorizes. The spectra of the combined system thus consist of a superposition of the spectrum of the original system and copies shifted by the powers of the eigenvalues of the linear filter. The results are illustrated for a shifted tent map and a one step filter. @ 1997 Published by Elsevier Science

B.V. PAC.9 05.45.+b

The signals derived from dynamical systems undergo various transformations before they are analyzed. Some are intentional and controlled by the observer, for instance to remove high frequency modulations, background noise and other systematic variations, others are due to limitations in the experimental setup, such as finite response times. This raises the question to what extend the measured signals reflect properties of the underlying system rather than of the filter. Badii et al. [ 1 ] address this problem for the case of a one-pole filter, where they find a phase transition: they compare the relaxation rate of the filter p in V = -,XY + x with the fastest contraction rate on the attractor A, and find a phase transition if p becomes less than A,, i.e. if the relaxation rate of the filter becomes slower than the slowest rate of the attractor. The purpose of this note is to study the combination of the original system and a linear filter within the

periodic orbit theory for chaotic systems [ 2-101. The result will turn out to be surprisingly simple and will help to shed some light on the conditions under which the filtered signal agrees with the original one. Let us assume that the dynamics of the original system can be described by a D-dimensional map *,+I = Rx,).

(1)

Without loss of generality we may assume that the first component is observed and sent to a linear filter producing a signal $ according to N

Yn+l= C

aijn+l-i

+

(2)

By the usual steps, this one-dimensional filter with Nstep memory is represented as a one-step filter of a N-dimensional signal: put Y,=(~n,~n-l,...,.iin+l-N)

’ New address: Fachbereich Physik der Philipps Universtgt, Renthof 6. D-35032 Marburg. Germany.

Xl,n.

i=l

and define the matrix

r)375-9601/97/$17.00 @ 1997 Published by Elsevier Science B.V. All rights reserved, PIISO375-9601(97)00173-4

(3)

B. Eckhardt/Physics

Letters A 229 (1997)

44-48

45

The original system f and the filter define an enlarged dynamical system on a D+N-dimensional phase space with state vector X,, = (x,,y,) and mapping

mapping consists of a particular periodic solution to the inhomogeneous equation (a suitably weighted average of the periodic orbit) and the homogeneous solution. If all eigenvalues of the filter are contracting, the homogeneous solution decays and only the periodic part survives. Thus there is a one-to-one lift of any periodic orbit in the original space to one in the extended space. Similarly, the monodromy matrix also factorizes into a contribution from the original system and the filter. Specifically, the derivative of the map F becomes in a block notation

X n+l = F(X,).

DF=

and the vector ei = (6i,t ). Then the filter becomes y,,+i = AY, + II,,~I.

(5)

(6)

It is this dynamical system to which we will apply the classical periodic orbit formalism in order to describe the spectrum. The resonances [ 1 l-141 of a map can be obtained from the Fredholm determinant Fo (z ) = det( l-z F) : the resonances are the inverses of the zeroes. Using the relation det A = exp tr In A, one can connect FLI to traces of powers of F and thus to periodic points [ 6, lo]. In particular, the partition function n(z) =

= -z$lnFD(z) 00 zll - tr F”, c II

(D;;,:’

;),

(10)

where El.1 is a N x D matrix with zero entries except for a 1 in the upper left corner (it arises from the coupling of the first component of x to the first one of y). The product of two such matrices becomes

0 (7) (8)

il=I

has poles at the resonances. The traces select the periodic points. By the usual rearrangement between sums over periodic points and periodic orbits and multiple traversals (see, e.g. Refs. [ 3,4,10] ), one finds

A2 > ’

(11)

where the elements in the lower left block are of minor interest since they do not influence the determinant. This shows that the structure is preserved, that in the upper left corner the D x D stability matrix of the original system builds up and that in the N x N filter subspace powers of A accumulate. The denominator of the weight in the partition function (9) thus factorizes, Idet(l-M;)I=)det(l-mL)jdet(l-.A”,”),

( 12)

(9) where p runs over all primitive periodic orbits with nP points Xi and stability matrix M, = nDF(Xi). Thus, to apply this to the combined system (6) we need information on the periodic orbits and the monodromy matrices M,. Since one part of the dynamics is given by the original map without any feedback of the filter onto the dynamics, the periodic orbits in this part of the space are the same as without filtering. In the filter space, the dynamics is linear with a driving through the xdynamics. The solution to this inhomogeneous, linear

m,, = HDf (Xi> is the stability matrix in the original system. The filter matrix A may be diagonalized. Its eigenvalues wj are the solutions of N

pN =

C aipN-‘.

(14)

1=I For stability reasons, all eigenvalues are strictly inside the unit circle (an eigenvalue one would introduce a

B. Eckhardt/Physics

46

Letters A 229 (1997) 44-48

continuous family of periodic orbits, see the discussion below EZq. (9) ). Then copying the expansions in geometric series usually applied to the stability matrix [ 61 yields Idet(1 -ML)]-’ =/det(l

-$,)I-‘n(l

-py’)-’

(15) Substitution

into (9) then gives

k,d

k&Go p

r=l

OS

=~...~f2&;‘...&z).

(17)

kN=O

h,=O

In the last line the partition function QJ for the original system without filter has been introduced. From Eq. (7) it follows that the Fredholm determinant factorizes,

kl=O

200

300

400

to lie between Au = 1 and the next to leading order eigenvalue Ai. To illustrate some of the concepts, we study the linear shifted tent map with a one step filter, [ 15,101 Xn+t = 2x/( 1 + 2S), =2(1-x)/(1-2S),

0 6 X < s + l/2, s+1/2<.w<

1,

.v~+I = ay, +x,,

(20) (21)

where - l/2 < s < l/2 and 0 6 a < 1. The resonances of the shifted tent map are given by

kp0

where FD,, is the Fredholm determinant for the original system. Let A; be the resonances of the original system (so that z; = Ai’ are the poles of Q,( z ) and the zeroes of FD,,) . If the system has an attractor, the leading eigenvalue A0 = 1 since the periodic orbits are dense in the attractor. With this one can read off the spectrum of the combined system from ( 18) : they are given by the set &I . . .&A;,

100

Fig. I. Signals for the shifted tent map at s = -0.2. (a) The original signal .r,; (b) the filtered signal for tl = 0.2 and (c) the filtered signal for u = 0.6. Note the smoothing of the large excursions in (c).

( 16) m

0

(19)

for all ki = O,I,..., CQ. Thus even in the case of a single relaxation rate ~1 = al there is an infinite repetition of the spectrum because of multiplication by ,UI and its powers. From this one reads off that significant differences in the decay of correlations will arise if ,uhe = p > A,, i.e. if the filter rate comes

hj

=

[ (1+2S)j+'

+ (-l)j(

1_2S)j+‘]

/2j+‘.

(22)

For ($1 > l/6 the eigenvalue At = 2s is closest to one, for JsJ < l/6 it is A2 = (1 + 12s2)/4 (with a degeneracy at s = l/6 as discussed in more detail in Ref. [ 10 ] ) . Thus for s = -0.2 the correlation properties of the filter signal will to leading order be the same as the original signal for a < 0.4 and dominated by the filter for larger a. The effects on the signal are demonstrated in Fig. 1. Resonances can be extracted from the correlation functions [5,10,11,14]. In particular, in the case of the tent map at s = -0.2, the decay of correlation functions should be dominated by the leading order eigenvalue At = -0.4 for a = 0.2 and by the filter eigenvalue p = 0.6 for a = 0.6. This is shown for the correlation function of yn = yn - (y),

B. EckhardtIPhysics

Letters A 229 (1997) 44-48

10.'

--.

--._ “lx,

a 1o.2

-.._

‘m

0

1.o

--..

CT-..

m

--__ 0

‘. -,._

..

0 g

.

0,

10.3

2

l

0.06

0.5

0

10.' 00 0.00 0.04 :-' L 0 1o.5

0

1

louQee l -4 2

0.0

8 3

4

5

6

7

L

3.0

J 0

1.0

0.5

Y”

Fig. 2. Correlation functions for the filtered signals in Fig. 1, with s = -0.2 and a = 0.2 (full circles) and a = 0.6 (open circles). The inset shows the correlation functions on a linear scale, the full graph the absolute value on a semi-logarithmic scale. The dotted line indicates the decay expected from the tent map eigenvalue A = 0.4, the dashed line the decay expected from the filter eigenvalue fi = 0.6.

I c(r)

= (%+&)n,

(23)

in Fig. 2. In both cases, the absolute value of the correlation function is dominated by powers of the appropriate leading eigenvalue. The inset clearly shows the suppression of the oscillatory nature of the decay (due to the negative eigenvalue of the map) as the filter eigenvalue (which is positive) takes over. The phase space reconstruction of the original map by plotting x,+1 versus X, reveals the tent map. The phase space reconstruction of the filtered signal y,,+l versus yn is not one-dimensional but smeared out and fractal (Fig. 3). Its Hausdorff dimension can be calculated from the contraction rate u and the expansion rates of the one-dimensional map within periodic orbit theory [ 3,161. The result is Du=min(Z,l-E).

(24)

There is a phase transition for a = l/2, related to the topological entropy. The chaotic l-d map controls the stretching and folding in the unstable direction whereas the filter controls the contraction perpendicular to it. Covering the attractor with boxes one notes that for a < l/2 the contraction in the stable direction is sufficiently fast to separate boxes on neighboring

0.0

L------I

0.0

0.5

1.0

1.5

20

Y"

Fig. 3. Phase space reconstruction of the cases shown in Fig. 1. In both cases s = and (b) a = 0.6. According to FQ. (24) fractal changes from DH z 1.43 in (a) to

filtered signal for the -0.2 and (a) (1 = 0.2 the dimension of the DH =2 in (b).

branches of the attractor (their number increasing like 2”). For a > l/2 the contraction rate is too small and the boxes will start to overlap, indicating a covering of the plane without holes, i.e. DH = 2. This effect is independent of the tent map parameter s and thus of the visibility of the next to leading order decay rate of the system. Thus, while in Fig. 3a the line structure of the original tent map can still be recognized, it is almost completely gone in Fig. 3b. Extensions of this formalism to other maps and more complicated situations are straightforward. In particular, for smooth systems and linear filters the Fredholm determinant also factorizes and the spectra

48

B. Eckhardi/Physics L_eifers A 229 (1997) 44-48

multiply according to the eigenvalues of the filter. For nonhyperbolic systems where the Fredholm determinant can have branch cuts at the origin [ 171, the influence of the filter will extend to even the longest times and cannot as easily be separated from the signal. The effects of nonlinear filters are more difficult to estimate. For once, the nonlinearity of the filter can give rise to a many-to-one relation between orbits in the extended space and in the base space. But even if this does not happen, the contraction rates of the filter will vary from orbit to orbit and hence spoil the factorization: the matrix A in ( 12) and the eigenvalues ,u, in ( 15) will depend on the orbit and cannot be treated separately. However, for weakly nonlinear filters the proliferation of eigenvalues will presumably persist.

References [ I j R. Badii, thesis (1988); R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi and M.A. Rubio, Phys. Rev. Len 60 ( 1988) 979. 12I P. Cvitanovic, Phys. Rev. Lett. 61 ( 1988) 2729.

131 R. Artuso, E. Aurell and P. Cvitanovic, Nonlinearity 3 (1990) 325. 14 1 R. Artuso, E. Awe11 and P Cvitanovic, Nonlinearity 3 (1990) 361. E Christiansen, G. Paladin and H.H. Rugh, Phys. Rev. Lett. 65 (1990) 2087. 161P Cvitanovic and B. Eckhardt, J. Phys. A 24 ( 1991) L237. 171 B. Eckhardt, Acta Phys. Pol. 24 ( 1993) 771, 181 B. Eckhardt and G. Ott, Z. Phys. B 93 (1994) 259. H. Shigematsu and B. Eckhardt, Z. Phys. B 92 191 H. Fuji&a, (1995) 235. [lOI B. Eckhardt and S. Grossmann, Phys. Rev. E 50 (1994) 4571. D. Ruelle, Phys. Rev. Lett. 56 (1986) 405. D. Ruelle, J. Diff. Geo. 25 ( 1987) 99, 117. H.H. Rugh, Nonlinearity 5 (1992) 1237. V. Baladi, J.-P Eckmann and D. Ruelle, Nonlinearity 2( 1989) 119. 1151 S. Grossmann, in: Evolution of chaos and order (Springer, Berlin, 1982); S. Grossmann and S. Thomae, Z. Naturforsch. 32a (1977) 1353; S. Thomae and S. Grossmann, J. Stat. Phys 26 ( 1981) 485. It61 D. Auerbach, P Cvitanovic, JR Eckmann, G. Gunaratne and I. Procaccia, Phys. Rev. Lett. 58 (1987) 2387. I171 P. Dahlqvist, Nonlinearity 8 (1995) II.