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Senkronize kaotik devrelerle haberleşme

Senkronize kaotik devrelerle haberleşme

##### Dosyalar

##### Tarih

1997

##### Yazarlar

Yalçın, Müştak

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### 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.

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 IX which 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 having a 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) XI Fig. 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. xu ffa) - 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<="" (<-o, a* < 0 V* > <0 and ll*(* » 0)11 ^ ( Û W- ) ' e</t

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 IX which 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 having a 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) XI Fig. 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. xu ffa) - 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<="" (<-o, a* < 0 V* > <0 and ll*(* » 0)11 ^ ( Û W- ) ' e</t

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Sosyal Bilimler Enstitüsü, 1997

##### Anahtar kelimeler

Kaotik devreler,
Senkronizasyon,
İletişim,
Chaotic circuits,
Synchronization,
Communication