Senkronize kaotik devrelerle haberleşme
Chaotic communication using by synchronized chaotic circuits
- Tez No: 66803
- Danışmanlar: DOÇ. DR. CÜNEYT GÜZELİŞ
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Kaotik devreler, Senkronizasyon, İletişim, Chaotic circuits, Synchronization, Communication
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Elektronik ve Haberleşme Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Devreler ve Sistemler Bilim Dalı
- Sayfa Sayısı: Belirtilmemiş.
Özet
Kaotik sistemlerde alt sistemler arasında senkronizasyonun gözlenmesi ile birlikte, taşıyıcı olarak kaotik işaretleri kullanan yaygın spektrumlu haberleşme sistemlerinin geliştirilmesi günümüzün önemli araştırma alanı oldu. Kaotik işaretlerin haberleşmede kullanılabileceğinin itici gücü; kaotik işaretten gürültü benzeri yay gın bir spektruma sahip olması dolayısıyla haber işaretini gizlemede ve gürültüye bağışık kılmada yararlanılabilir olmasıdır. Elektronik olarak en kolay gerçeklene- bilen kaotik sistem Chua devresi olduğu için, Chua devresine dayalı sistemlerin kaotik haberleşme alanında önemli bir yeri vardır. Bu tezde, Chua devresinin gerçeklenmesini gerilim transfer fonksiyonu sentezine indirgemesinden ve yeni türden kuplaj ile sürüm olanakları sağlanmasından dolayı Güzeliş tarafından önerilen kaotik hücre modeli, kaotik verici ve alıcı olarak gözönüne alınmış böylece bir kaotik durum modülasyonlu haberleşme sistemi geliştirilmiştir, önerilen sistemde alıcı ve verici sıfır mesaj durumu haricinde birbirine senkron olmamakta fakat senkronizasyon hatası mesajın alıcıda yeniden oluşturulabilmesini sağlamaktadır. Sistemin zaman tanım bölgesinde kuramsal analizi yapılmış ve analog benzetim ile sonuçların doğruluğu gözlenmiştir. vııı
Özet (Çeviri)
Synchronization of coupled nonlinear oscillators has long been investigated in a diverse field including communications, power system engineering and biology. By the observation of synchronization in chaotic systems [13], synchronized chaotic systems have received a great deal of attention as offering a way of information transmission specifically spread spectrum communications [11]. In communica tion systems based on chaotic synchronization, either chaotic masking, or para meter modulation or state modulation have been used for mixing a chaotic carrier with a message signal [11]. The message signal has been recovered at the rece iver which is designed to be synchronized with the transmitter by some methods based on circuit theory [15], or Pecora-Carrol's drive-response [12] approaches. The communication system under consideration in this thesis is of chaotic state modulation type and is derived as employing a neural networks approach. The overall system can be viewed as a coupled of neurons: One of the neurons, the transmitter, is excited by an external input, message signal, and its output is fed to the other neuron, the recevier. The neuron models used here are the same with the cells of chaotic cellular neural networks [8]- [9] introduced as a third order special case of generalized cellular neural networks of [8]. Each neuron with unity self-feedback becomes equivalent to a Chua's circuit if it is isolated from the other neuron and from the external input. These neurons are, indeed, obtained from Chua's circuit by decomposing Chua's diode into a linear positive resistor, a nonlinear voltage controlled voltage source and a linear voltage controlled current source. As seen from Fig. l.a, defining the voltages of the dependent sources as port voltages, a neuron can be considered as a two-port nonlinear dynamical circuit element and also as an input-output system so that the input is the current of the first port and the output is the voltage of the second port. Such a neural based treatment of Chua's circuit provides some possibilities two of which are as: i) Obtaining a new hardware realization for Chua's circuit by means of voltage transfer function synthesis [10] as an alternative to the known realizations [7], and ii) Having new ways for excitation and also for coupling of Chua's circuits. The second possibility is exploited in the proposed communication system to have a nonlinear coupling and to design a receiver which is ensured to be operated in double scroll regime and is synchronized to the chaotic carrier component of the transmitted signal under the channel effect In chaotic state modulation systems developed in the literature [18]-[19], the message signal is injected to Chua's circuit used as the transmitter via either a current source [18] in parallel to, or a voltage source [19] in series to Chua's diode. For message signals with sufficiently small amplitudes, Chua's circuit IXwhich is designed to be operated in double scroll mode under no excitation, still operates in a double scroll mode. But, its states are slightly modified by the message. The transmitter circuit of this paper is identical to the one in [18]; but here the output of the transmitter is a nonlinear function of first state variable, i.e., capacitor voltage instead of itself. This property might bring an extra security by keeping additive channel noise away from the message signal which is already hidden in the chaotic signal th rough state modulation. The main difference of the proposed system from other chaotic state modulation systems [18]-[19] is in the way of excitation of receiver by the output of transmitter. In [18]-[19], a unity gain, dependent voltage source which is controlled by the capacitor voltage of transmitter supplies the signal driving the receiver. This makes the whole circuit degenerate, hence provides the synchronization of first state variables not only for steady-state but for all times. Since the receiver is excited by a dependent current source, then the system pro posed here is not degenerate. As a consequence of this type of coupling, even asymptotic synchronization between the transmitter and receiver is obtained only for zero message signal case. But, at the steady-state the receiver's chaotic states follow the transmitter's by a nonconstant time lag caused by the message. This gives the opportunity of recovering the message from the difference between the outputs of the transmitter and receiver. Next section presents the state equations and circuit structure of the proposed communication system with a comparison of circuit structures of the available chaotic state modulation systems. Chaotic State Modulation-Demodulation System A circuit realization of the proposed communication system with using ideal cir cuit elements is shown in Fig. l.a. The transmitter circuit is equivalent to original Chua's circuit if tranfer characteristic /(.) of nonlinear dependent voltage source is as shown in Fig. l.c, the self-feedback coefficient is chosen to be unity, and the linear, dynamical 2-terminal element Zrlc is defined as in Fig. Lb. This can be seen by combining linear positive resistor E/v, dependent voltage and current sources to obtain Chua's diode which has the driving-point characteristic in Fig. 3.2. The input signal m(t) might be a coded or modulated form of a message signal; but throughout the paper the input will be called as the message. As in the design of any chaotic transmitter, one of the points which should be taken into account is that the spectrum of the input signal could have an appropriate shape ensuring chaotic mode of operation [9]. A second point is that coding function or modulating signal should be chosen as providing that message can be hidden in modulated chaotic signal to be transmitted hence providing security. In synchronization, two dependent current sources which are in parallel at the input port of the receiver can be combined and then the receiver and transmitter Chua's circuits becomes equivalent. Synchronization, however, does not occur for nonzero message signal. By defining appropriate state variables and changing parameters (see [9]), the communication circuit of Fig. l.a can be described by the following state equations. In Fig. l.a, aw is considered to be zero for havinga unidirectional transmission desired, in communication systems. Transmitter's state equations: &i = ot. (-(1 + 8). xi + yx + a00 ? f (xi) + aw- f (x2) + m(t)) (1) yi = «i - yi + z\ zi = -0- yi Receiver's state equations: (2) (3) x2 = a- (-(1 + S). x2 + y2 + a01. f (x±) + an- f (xa)) (4) m = x2 - y2 + z2 (5) z2 = -(3- yi where, the piecewise-linear function /(.) with defining m,Q = °f and mi be given as: Oh. G (6) can f(x) = i. (m0 - roO. (|x + fi| - |x - B|) (7) XIFig. 1. a. Proposed chaotic state modulation-demodulation communication cir cuit, b. R,L,C equivalent of linear, dynamical 2-terminal element denoted by Zrlc c. Piecewise-linear transfer characteristic of voltage-controlled voltage so urces. Time-Domain Analysis of the Proposed System In this part, it will be shown that the message signal m(t) can be recovered from the signal m(t) which is defined to be the difference between the input /(#i) and the output f{x-i) of the receiver as m(t) = fM-fM (8) By defining the error vector E to be the difference of the state vectors of the transmitter Xi = [xi yx z-j] T and the receiver X2 = [x2 t/2 22] T as E = Xi - X2 (9) and substracting the state equations of the transmitter from the state equations of the receiver, the following equations are obtained. E = -a -(1 + 6) a 0 1 -1 1 0 -0 0 E + oc- (aQ0 - a0i) 0 0 fM + m(t) (10) Using the lemma in [24], the following can be written. xuffa) - fM = sk ? (x2 ~ Xi) + 0(\x2 ~ Xi\) (11) Where sk is the slope of x - f(x) characteristic in the region where x\ lies, for the time interval tk-i <t<tk, hence sk ? {mo, mi} This residue termö(\x2 - xi\) is nonzero only when x\ and x2 are at different regions and its magnitude goes to zero as \x2 - x\\ goes to zero. If x2{t) (and/or xx(t)) is rapidly changing, then the probability that xx and x2 are placed at different regions will be very high and this nonzero residue term will be frequently occurred. By substituting the relation (11) into (10), the following state equations defined for each time interval Tk = [tk-i tk] are obtained E = -a- [(1 +S) + (a00- an) ? sk] a 0 1 -1 1 0 -0 0 E + at. (aoo - «oi - «il + aio) 0 0 f(x1) + bN-0(\x2-x1\) + b-m{t) (12) Where b=[ a 0 0 ], b^ = I -a. (an - aw) 0 0 ]. By choosing the con nection weights to satisfy the relation oqo - % - an - aio = 0 with a little loss of generality, the equation (12) is further simplified as E = Ak. E + b. m(t) + bN. 0(\x2 - xt\) for t 6 Tk Where, Ak is a time- varying state matrix (13) A* = -a. [(1 + 8) + (a0a - an). sk] a 0 1 -1 1 0 -/? 0 which is equal to strictly Hurwitz constant matrix A(m0), and respectively A(mi), for the times when X\ 6 Pq, and respectively for the times when X\ Ç P+[JP~. The time- varying state equation system of e = Ak. e ter* admits the state transition matrix in (15) (see [25]). (14) *(<, 0) = eA»(*-'»-i). eA»-ıAn-ıgAıAı j^ feTn (15) XlllWhere hj =: tj - tj^ for j e {1, 2,...., n - 1} and, *0 = 0. Since each of Ak matrices is either A(ra0) or A(mi) both of which are strictly Hurwitz matrices, then there exists a K < oo such that ||$(i, 0)|| < K,\/t. This is a consequence of the fact that each ||eAfc (<-<0)|| < Nke«k(t-t0) for some jVA >o, a* < 0 V* > <0 and ll*(* » 0)11 ^ ( Û W- ) ' e<a»<*-'-')+«»-iA»-i++aiAl) ^ ? Tn. (16) When i goes to infinity, the sum an(t - f“_i) + a”_1^n_1 +.... + ct\hi tends to minus infinity, then lim^oo ||$(£, 0)|| = 0. The complete solution of (12) for the times £n_ı < t < tn is E(0 = *(<,0)-E(0)+r*(<,r).b.TO(r)rfr+/**(*,T)-bN.O(|ar3(r)-xj(r)|)rfT Jo Jo (17) The first term will approach to zero when t approaches to infinity. The forced solution due to Ö (\x2 - %i\) can be neglected since |C?(|a?2 - a?i|)| is very small for small amplitude message signals and it occurs only if xi and x2 are in different regions. The forced solution term of (17) due to m(t) is approximately equal to the forced solution of (19) which can be expressed as EforcedW = EPn(t) - eA«((-(»-')eA-'A»-'eAlhi ? EPl(0) +eA“(t-«”_1)eAn_1fcn_1eA2h2. [EP1(^) - Ep,(<i)] +eA"(*-*»-ı)cAB-j&»_ıeAsh3. [Epa(i2) _ Ep,(<a)] +... + ^(«-«-i). [Ep..,^!) - EPn(^_0] (18) The expression (18) is derived by interval-wise application of the relation which gives the forced solution to a linear system in terms of its particular solution and state transition matrix, and by imposing the final value at the present interval on the linear state equations of the next interval as the initial condition. Where, EPk(2) is the particular solution during the time interval J*, for the truncated state equation system of Epk = Ak. Epk + b. m(t) (19) due to a sinusoidal message m(t) - M. Cos (w. t). This particular solution is an approximate particular solution to (13) since the correct particular solution EpCk should satisfy xivEpck = A(sk). EpCk + b. m(t) + O (\x2pCk - xlpCk |) (20) The approximate particular solution is found as EPk(*) = ei(w,sk) -Cos(wt + 9i(w,sk)) £-z{w, sk). Cos (wt + 62(w, sk)) £3(1», sk). Cos (wt + 03(w, sk)) t ? Tk (21) The forced solution in (18) is approximately equal to EPn(t) at the steady state since the vectors [EPn_1 (ira-i) - EPn(£n_i)l are almost equal to zero vector which can be seen from the closeness of the characteristics in the figures (5.10)-(5.11) obtained for different slopes mo, mi, and since matrix exponentials multiplying these small constant vectors are decreasing with time t. The recovered signal m[t) defined in (8) at the time interval Tk is m{t) = sk. CT. E (t) + O (\x2 - Xl\) with C = [1 0 0]T teTk (22) and at the steady-state rh(t) = sk. CT. EPk(t) since, as mentioned before, the forced solution due to m(t) is approximately equal to the particular solution Epk(*) at the steady-state. Hardware Realization In this part, the time domain analysis has been confirmed by experimental ob servations done on a hardware realization. In the realization, off-the-shelf R,L,C components with Cx = 1 nF, C2 = 22 nF, R = 10 kQ, L = 68 mH, RN = 10 kti and the op amps TLC-082 for the synthesis of dependent sources based on the voltage controlled voltage source approach [10] have been used for both of the transmitter and receiver Chua's circuits. For having a large /(#) signal to be transferred and also for having a voltage transfer characteristic x - f(x) which is easy to implement by op amps, firstly voltage dependent sources have been realized for the slopes Ga = y and Gb - y, and then the resulting scaled f(x) has been applied to voltage controlled current sources with a gain factor 10-4; resulting in a characteristic having the slopes Ga - y. 10~4 and Gb = y. 10~4 to obtain double scroll regime. This indicates an implementation advantage of the neural based treatment of Chua's circuit. The current m(t) driving the transmitter has been obtained as the current of a voltage controlled current source with a gain factor 10~5 mho such that the peak to peak value of the control voltage Vm is in the interval of [2V, 11V]. It has been observed that excitation by such a message signal m(t) having the frequency range [4kHz, 100k Hz] does not destroy the double scroll regime of the transmitter. The waveforms of the source Vm (t) with the abovementioned amplitude and fre quency, the transmitted signal, i.e., 104. f(Vci(t)), and the recovered signal m(t) are given in Fig. 2.a-b. xvV LuHfciA.. (a) (b) Fig. 2. a. The transmitted signal 104. f(Vcı(t)) {hV/div - 2msn/div). b. The message control voltage Vm(t) (the above signal) (5 V/div-50/xsn/div), the recovered signal m(t) (lV/div - 50(xsn/div). Conclusion An analysis of the proposed communication system in the time domain has yielded expressions for the magnitude and phase characteristics which can be used to find the recovered signal. This approach does not give the necessary magnitude and frequency range for the message signal to preserve the transmitter's and receiver's chaotic regimes. But it might be said that if the occurrence of the residue term Ö(\x2 - X\\) is not too often, then the magnitude of the forced response due to ö{\xı - X\\) could be low, hence the similarity between m(t) and rh(t) would be high. The results obtained can also be usefull for understanding the behaviours of coupled biological neurons and for the analysis of chaotic cellular neural networks [12H13J. XVI
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