electronics Article
Filtering Techniques for Chaotic Signal Processing Denis Butusov 1, * , Timur Karimov 2 , Alexander Voznesenskiy 3 , Dmitry Kaplun 3 , Valery Andreev 2 and Valerii Ostrovskii 2 1 2 3
*
Youth Research Institute, St. Petersburg Electrotechnical University “LETI”, St. Petersburg 197376, Russia Department of Computer-Aided Design, St. Petersburg Electrotechnical University “LETI”, St. Petersburg 197376, Russia;
[email protected] (T.K.);
[email protected] (V.A.);
[email protected] (V.O.) Department of Automation and Control Processes, St. Petersburg Electrotechnical University “LETI”, St. Petersburg 197376, Russia;
[email protected] (A.V.);
[email protected] (D.K.) Correspondence:
[email protected]; Tel.: +7-950-008-71-90
Received: 16 November 2018; Accepted: 17 December 2018; Published: 19 December 2018
Abstract: The vulnerability of chaotic communication systems to noise in transmission channel is a serious obstacle for practical applications. Traditional signal processing techniques provide only limited possibilities for efficient filtering broadband chaotic signals. In this paper, we provide a comparative study of several denoising and filtering approaches: a recursive IIR filter, a median filter, a wavelet-based denoising method, a method based on empirical modes decomposition, and, finally, propose the new filtering algorithm based on the cascade of driven chaotic oscillators. Experimental results show that all the considered methods make it possible to increase the permissible signal-to-noise ratio to provide the possibility of message recognition, while the new proposed method showed the best performance and reliability. Keywords: chaos; communication systems; chaotic synchronization; denoising; secure communications; digital signal processing; filtering; empirical modes decomposition
1. Introduction Chaotic communication has number of advantages over traditional harmonic signal-based schemes, namely, better security and steganographic properties [1], low power spectrum density, robustness to multi-path fading and ease of wideband communication system implementation [2]. Chaotic communications are also of interest due to the wide bandwidth of chaotic signals making them useful in systems where narrowband channels are overloaded [3]. Today a number of practical implementations of chaotic communication systems are known [4]. At the early stage of investigations in the field of chaotic communication, several simple methods have been proposed, including chaotic masking, chaotic regime switching, the nonlinear mixing of the information into the chaotic carrier, etc. Later, it was found that these methods suffer from low robustness to noise in the transmission channel [1,4,5]. To overcome this drawback, a number of techniques have been introduced, which can be divided into two main classes: noise-stable communication schemes and conventional communication schemes with noise reduction; some hybrid types are reported as well. Noise-stable communication approaches include communication with use of generalized synchronization [3,5], the use of analytically-solvable chaotic systems [6], correlation detection with symbolic dynamic encoding [7], etc. Noise reduction techniques for chaotic signals cannot take advantage of the simple signal spectra of conventional multi-harmonic communication systems, forcing them to use linear FIR or IIR filters. Therefore, only relatively complex techniques are feasible for chaotic signal denoising. These approaches include noise reduction methods via a deterministic optimization technique [8], wavelet denoising [9], hidden Markov model approach [10],
Electronics 2018, 7, 450; doi:10.3390/electronics7120450
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Nevertheless, no generally accepted technique is known to date, and practical comparisons of these and the usage of Kalman filter for noisy state variables estimation [11]. Nevertheless, no generally approaches are fragmentary and scarce. accepted technique is known to date, and practical comparisons of these approaches are fragmentary In our work, we study two more filtration techniques which we believe not to have been and scarce. reported yet in the context of chaotic communications: empiric mode decomposition denoising and In our work, we study two more filtration techniques which we believe not to have been reported cascading of driven chaotic oscillators. We compare these techniques with wavelet denoising, yet in the context of chaotic communications: empiric mode decomposition denoising and cascading of median filtering and IIR filtering using the example of a chaotic communicational system model driven chaotic oscillators. We compare these techniques with wavelet denoising, median filtering and with parameter modulation. IIR filtering using the example of a chaotic communicational system model with parameter modulation. Our paper is organized as follows. In section 2, we describe a model of a chaotic communication Our paper is organized as follows. In Section 2, we describe a model of a chaotic system with noisy transmitter-receiver channel, and consider several filtration algorithms. Section 3 communication system with noisy transmitter-receiver channel, and consider several filtration describes the experimental study and presents a comparison of noise reduction methods. The algorithms. Section 3 describes the experimental study and presents a comparison of noise reduction conclusion and a discussion are presented in Section 4. methods. The conclusion and a discussion are presented in Section 4. 2. Materials and Methods 2. Materials and Methods 2.1. Synchronization Synchronization of of Rössler Rössler Chaotic Chaotic Systems Systems 2.1. After the the discovery discovery of of the the chaotic chaotic synchronization synchronization phenomenon, phenomenon, several several chaos-based chaos-based After communication techniques techniques were were invented: invented: chaotic communication chaotic signal signal masking masking [12], [12], chaotic chaotic shift shift keying keying [13], [13], nonlinear mixing of informational signal into chaotic carrier [14], and chaotic parameter nonlinear mixing of informational signal into chaotic carrier [14], and chaotic modulation parameter [15] among [15] others. All these types chaotic techniques suffer from noise the modulation among others. All of these typescommunication of chaotic communication techniques sufferin from transmitting channel, making them hardlythem applicable practice [5]. noise in the transmitting channel, making hardlyin applicable in practice [5]. Let us consider a chaotic communication system with parameter modulation modulation as as presented presented in in Let us consider a chaotic communication system with parameter Figure1. 1. A digital informational (t ) chaotic affectsgenerator the chaotic generator parameter Figure A digital informational signal m(signal t) affectsmthe parameter set P = [ a, b, c . . .], [ ] set P = a , b , c ... , forcing it to take value P (logical 0) or P (logical 1). In a particular case, only one forcing it to take value P1 (logical 0) or P21 (logical 1). In a particular case, only one parameter p can 2 be changed. pSlave’s parameter a pre-defined value When parameter are similar, the can be changed.has Slave’s parameter has Pa1 . pre-defined value values P1 . When parameter parameter synchronization error small; in ideal error cases,isitsmall; converges to zero. values are similar, the is synchronization in ideal cases, Otherwise, it convergesthe to synchronization zero. Otherwise, error is notable; a detector recognizes the input value and matches it with the corresponding logical the synchronization error is notable; a detector recognizes the input value and matches it with the values 0 or 1. corresponding logical values 0 or 1. The The situation situation significantly significantly changes changes when when noise noise appears appears in in the the signal signal path. path. The The synchronization synchronization error becomes notable in both cases, so the detector needs to be very sophisticated error becomes notable in both cases, so the detector needs to be very sophisticated to to distinguish distinguish logical 0 from 1. At the SNR ratio θ > 1, energies of both signals corresponding to 0 and logical 0 from 1. At the SNR ratio θ > 1 , energies of both signals corresponding to 0 and 11 become become close, close, and and communication communication fails. fails.
Figure 1. 1. Block of communication communication system system based based on on Rössler Rössler oscillator oscillator with with parameters parameters Figure Block diagram diagram of modulation. modulation.
Let us take Rössler chaotic system, a well-known chaotic oscillator [16]:
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Let us take Rössler chaotic system, a well-known chaotic oscillator [16]:
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x = −( y + z ) . y = x + 0.2 y (1) x= −(y + z) . z = 0.2 + z ( x − 5.7) (1) y= x + 0.2y z. = 0.2 + z( x − 5.7) Synchronization of two chaotic systems in the master-slave topology may be performed using different methods such as chaotic Pecora-Carroll [17], Hamiltonian formsbe[18], or the Synchronization of two systems technique in the master-slave topology may performed Open-Plus-Close-Loop approach (OPCL) [19]. For our study, we chose the simplest technique of using different methods such as Pecora-Carroll technique [17], Hamiltonian forms [18], or the linear feedback control approach method. The main [19]. idea isFor to include the we driving the driven Open-Plus-Close-Loop (OPCL) our study, chosestate the variable simplestinto technique of system with a coefficient k ∈ ( 0 ; 1 ] . Thus, we derive the following slave system from the Equation linear feedback control method. The main idea is to include the driving state variable into the driven
system with a coefficient k ∈ (0; 1]. Thus, we derive the following slave system from the Equation (1): (1): . x = −( y + z ) + k ( x − x ) s −(y s+ z s) + k( xm − xs ) s s m s x. s = 0.2sy s , , (2) s +0.2y (2) ys y=s = xsx+ z. z= = 0.+ 2 +zsz(sx( sx s−−5.7 5.7)) s s 0.2
hereindex indexs stands s stands driven state variables, index is driving the driving (“master”) variable, k is here forfor driven state variables, index m ismthe (“master”) variable, and kand is the the coupling constant. Value k = 1 is usually taken. This type of synchronization is available for coupling constant. Value k = 1 is usually taken. This type of synchronization is available for many many known oscillators, as thecircuit, Chua circuit, Sprott chaotic simple chaotic flows, and others. known chaoticchaotic oscillators, such assuch the Chua Sprott simple flows, and others. In our case, the Rössler-driven oscillator has 3 parameters: a = 0 . 2 , b = 0 . 2 , c = 5 . 7 . In our case, the Rössler-driven oscillator has 3 parameters: a = 0.2, b = 0.2, c = We 5.7.can Wemake can logical 1 more or less distinguishable by choosing another parameter set for the master oscillator. To make logical 1 more or less distinguishable by choosing another parameter set for the master oscillator. 0101 obtain the result presented in Figure 2 we suppose a = 0 . 2 , b = 0 . 2 , c = 3 . 5 . Message was sent in To obtain the result presented in Figure 2 we suppose a = 0.2, b = 0.2, c = 3.5. Message 0101 was sent in this example. One synchronization error corresponding to this message in Figure 3a. this example. One cancan seesee thethe synchronization error corresponding to this message in Figure 3a. The The conversion of this signal binary code be performed using envelope detection conversion of this signal intointo binary code cancan be performed using e.g.e.g. envelope detection [20].[20]. Let Letus usconsider consideradditive additivedisturbances disturbancessimulated simulatedwith withaawhite whiteGaussian Gaussiannoise noisesource. source.The Thesame same synchronization synchronizationerror errorininaanoisy noisychannel channelwith withSNR SNR==55dB dBisispresented presentedin inFigure Figure3b. 3b.Figure Figure44presents presents the theconversion conversionofoferror errorsignals signalsinto intodigital digitalform formusing usingthe thealgorithm algorithmdescribed describedbelow. below.
(a)
(b)
Figure Figure2.2.The Thedynamics dynamicsof ofmaster master(1) (1)and andslave slave(2) (2)systems systems(a) (a)and andcorresponding correspondingattractors attractors(b). (b).
Notice that in our study, the communication system is only a computer model, so absolute values of variables have no direct physical meaning. The step size and numerical method for chaotic oscillators’ simulation were taken according to [21,22]. The algorithm of detection block is schematically presented in Figure 5. It consists of a peak detector and threshold block. This design was developed as a sample detector, showing the fundamental possibility of the chaotic communication system to work. In real systems, more advanced detectors can be used, including envelope fitters and other sophisticated components. Figure 6 illustrates the dependence between quality of synchronization and the values of parameter c and SNR. One can observe the tendency of synchronization quality enhancement while the value of c is ascending.
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Figure 3. Master-slave synchronization error with ideal (a) and noisy (b) synchronization signal.
(a)
(b)
(a) synchronization error with ideal (a) and noisy (b) synchronization (b) Figure 3. Master-slave signal. Figure3.3.Master-slave Master-slavesynchronization synchronizationerror errorwith withideal ideal(a) (a)and andnoisy noisy(b) (b)synchronization synchronization signal. signal. Figure
(a)
(b)
Figure 4. Correct message recognition in ideal channel (a) and incorrect recognition in noised channel (SNR = 5dB) (b).
Notice that in our study, the communication system is only a computer model, so absolute values of variables have no direct physical meaning. The step size and numerical method for chaotic oscillators’ simulation were taken according to [21,22]. The algorithm of detection block is schematically presented in Figure 5. It consists of a peak detector and threshold block. This design was developed as a sample detector, showing the fundamental possibility of the chaotic communication system to work. In real systems, more advanced detectors can be used, including envelope fitters and other sophisticated components. Figure 6 illustrates and the values of (a) the dependence between quality of synchronization (b) parameter c and SNR. One can observe the tendency of synchronization quality enhancement (a) (b) Figure 4. Correct Correct message recognition ideal channel (a) incorrect and incorrect recognition in channel noised Figure recognition in in ideal channel (a) and recognition in noised while the value of c message is ascending. (SNR = 4. 5dB) (b). channel (SNR = 5dB) (b). Figure Correct message recognition in ideal channel (a) and incorrect recognition in noised channel (SNR = 5dB) (b).
Notice that in our study, the communication system is only a computer model, so absolute values of variables direct meaning. system The stepissize and numerical model, methodso forabsolute chaotic Notice that inhave our no study, thephysical communication only a computer oscillators’ simulation were taken physical accordingmeaning. to [21,22]. values of variables have no direct The step size and numerical method for chaotic The algorithm of detection block is schematically presented in Figure 5. It consists of a peak oscillators’ simulation were taken according to [21,22]. Electronics 2018, 7, x FOR PEER REVIEW 5 of 14 Figure 5. The scheme of detection block. detector and threshold block. This design was developed as ainsample showing the Figure 5. The scheme of detection block. The algorithm of detection block is schematically presented Figure detector, 5. It consists of a peak fundamental of the This chaotic communication system work. In real systems, detector and possibility threshold block. design was developed as ato sample detector, showingmore the advanced detectors can be used, including envelope fitters and other sophisticated components. fundamental possibility of the chaotic communication system to work. In real systems, more Figuredetectors 6 illustrates dependence of other synchronization and the values of advanced can bethe used, including between envelope quality fitters and sophisticated components. parameter SNR. One can observe between the tendency of of synchronization quality enhancement Figure c6 and illustrates the dependence quality synchronization and the values of while the value of c is ascending. parameter c and SNR. One can observe the tendency of synchronization quality enhancement while the value of c is ascending.
Figure 5. The scheme of detection block. Figure 5. The scheme of detection block. Figure andlogical logical00(E (E00)) transmission transmission Figure6.6. RMS RMSratios ratiosfor forsynchronization synchronizationerror errorin incase caseofoflogical logical11(E (E1 1))and depending / E00 == 1 corresponds corresponds to to the the case case of of depending on on parameter parameter ccvalue. value. Levels Levelsequal equaland andlower lowerthan thanEE1 1/E total message loss. total message loss.
Nevertheless, even for the higher values of parameter c , low SNR still means low difference between logical 0 and 1. This may result in high levels of corruption of the received messages. For many real applications, SNR ≤ 5 dB is common value [20], so some kind of filtration of the noisy message signal could be performed to improve the SNR ratio. The experiments described below were performed with c = 3.5 to make the use of filtering techniques more indicative.
Figure 6. RMS ratios for synchronization error in case of logical 1 (E1) and logical 0 (E0) transmission depending on parameter c value. Levels equal and lower than E1/ E0 = 1 corresponds to the case of5 of 14 Electronics 2018, 7, 450 total message loss.
Nevertheless, even for the the higher higher values values of of parameter parameter c, c , low low SNR SNR still still means means low low difference between logical logical 00 and and1.1.This Thismay may result in high levels of corruption of the received messages. result in high levels of corruption of the received messages. For For many applications, SNR 5 dB is common value [20], somekind kindofoffiltration filtration of of the the noisy many realreal applications, SNR ≤ 5≤dB is common value [20], sososome message to to improve thethe SNRSNR ratio. The The experiments described belowbelow were message signal signalcould couldbebeperformed performed improve ratio. experiments described performed with c = 3.5 to make the use of filtering techniques more indicative. were performed with c = 3.5 to make the use of filtering techniques more indicative. 2.2. Filtration Techniques 2.2. Filtration Techniques Numerous applied to denoise communication signals: recursive and Numerous filtration filtrationtechniques techniquesmay maybebe applied to denoise communication signals: recursive non-recursive filtering [23], median algorithms based on discrete transform [24], and non-recursive filtering [23], filtering median [23], filtering [23], algorithms basedwavelet on discrete wavelet Wiener filters [25], and empirical mode decomposition (EMD) [26,27]. The filter is placed at the first transform [24], Wiener filters [25], and empirical mode decomposition (EMD) [26,27]. The filter is stage of the receiver, as presented in Figure 7. placed at the first stage of the receiver, as presented in Figure 7.
Figure 7. Chaotic system with with filter. filter. Figure 7. Chaotic communication communication system
in in referenced publications [23–27]. In A description description of of abovementioned abovementionedmethods methodscan canbebefound found referenced publications [23–27]. this paper, wewe focus onon two In this paper, focus twofiltration filtrationtechniques: techniques:EMD, EMD,asasaarelatively relativelylittle little known known technique technique whose usage in chaotic communications has never been reported; and a novel approach involving chain of driven oscillators. 2.2.1. Empirical Mode 2.2.1. Empirical Mode Decomposition Decomposition EMD EMD (empirical (empirical mode mode decomposition) decomposition) is is aa signal signal decomposition decomposition method method alternative alternative to to Fourier Fourier transform transform and and wavelet wavelet transform transform [28]. [28]. The The basic basic functions functions of of EMD EMD are are unknown unknown aa priori, priori, and and can can be be found computationally during the decomposition process. Thus, EMD makes it possible to decompose found computationally during the decomposition process. Thus, EMD makes it possible to adecompose signal intoanon-stationary and nonlinearand intrinsic modeintrinsic functions (IMF). Discarding noise-specific signal into non-stationary nonlinear mode functions (IMF). Discarding modes in the resulting decomposition, one can restore original signal, keeping it non-stationary and nonlinear. The last property becomes critical when processing chaotic signals, which are nonlinear by their nature, and non-stationary (considering modulation in communication tasks). Empirical mode (EM) or internal oscillation (IMF) is a function that has the following two quantitative properties: 1. The number of extrema (maxima and minima) and the number of zero crossings should not differ by more than one: Nmax + Nmin = Nzero ± 1, (3) where Nmax is the quantity of maxima; Nmin is the quantity of minima; and Nzero is the quantity of zero-crossings.
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2. The average value determined by two envelopes (upper and lower) must be equal to zero. When processing real-world signals, it should not exceed some small value determined by computer accuracy and errors associated with the data acquisition, transformation and transmission. U (k) + L(k) ≤ ε, k ∈ 1 . . . N, 2
(4)
where U (k ) is the upper envelope; L(k) is the lower envelope; and N is the number of signal samples. These empirical mode properties make it possible to apply Hilbert spectral analysis to them. The EMD algorithm represents the given discrete signal as a set of EMs [29]. The algorithm is based on the fundamental idea that at each stage, the signal (current residue) is represented as a sum of the rapidly oscillating components (extracted EM), and slowly oscillating ones (residue). The latter is further decomposed if it has at least one minimum and one maximum. The algorithm can be written as the following sequence of steps: Step 1. Consider the current residue r p (k) (r1 (k)– the first residue equal to the source signal s(k)). Its extrema are determined and split into sets: { Mi }, i = 1, 2, 3, . . . K, . . . and {mi }, i = 1, 2, 3, . . . K, . . . – set of maxima and minima, respectively. Step 2. Two envelopes are built with the cubic splines interpolation fitting the found extrema: Uj (k) = f U ( Mi , k ),L j (k) = f L (mi , k) – upper and lower envelopes, respectively, j is the iteration of the sifting process (see below). Step 3. After building the envelopes, their half-sum is calculated; the local (instantaneous) average value depending on time: Uj (k) + L j (k) . e j (k) = 2 Step 4. The average value is subtracted from the current residue and the obtained result h j (k ) becomes a “candidate” to be the next mode: h j ( k ) = r p ( k ); j = 1 h j ( k ) = r p ( k ) − e j ( k ); j > 1 where p is the number of the extracted mode. In this step, it is necessary to check conditions (3) and (4) to determine if the empirical function h(k) can be classified as empirical mode or not. If both conditions are met, then proceed to step 5. Otherwise, the algorithm returns to step 1 with h j (k) as the current residue. Such an iterative procedure is called a “sifting process”, which can be written as follows: h 1 ( k ) = r p ( k ) ; h 2 ( k ) = h 1 ( k ) − e1 ( k ) h j+1 (k ) = h j (k) − e j (k ); c p (k) = hiter (k ) here, e j (k) is the average value of the function on the j-th iteration of the sifting process; h j (k) is the current result of the j-th iteration of the sifting process; and iter is the total number of iterations for the given EM. At the iteration with the iter number, the sifting process for the next mode extraction is stopped and the transition to step 5 is performed. Step 5. After the EM is extracted in its final form, it is subtracted from the current residue to form a new residue (to update the residue): r p+1 (k) = r p (k) − c p (k); c p (k) is the obtained EM; r p (k) is the current residue, and r p+1 (k) is the new residue. Step 6. Next, r p+1 (k) is used as a residue to perform the algorithm form step 1. Figure 8 shows the diagram of the algorithm.
Step 5. After the EM is extracted in its final form, it is subtracted from the current residue to form a new residue (to update the residue): r p +1 (k ) = r p (k ) − c p (k ) ; c p (k ) is the obtained EM; r p (k ) is the current residue, and r p +1 (k ) is the new residue. Step 6. Next, r p +1 (k ) is used as a residue to perform the algorithm form step 1. Electronics 2018, 7, 450
Figure 8 shows the diagram of the algorithm.
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Figure8.8.The The EMD algorithm. Figure EMD algorithm.
Thealgorithm algorithmhas hasananempirical empiricalnature. nature.Currently, Currently,it ithas hasno nodeeply-studied deeply-studiedtheoretical theoreticalbasis basis The comparedtotoclassical classicalFourier Fourieranalysis analysisororwavelet waveletanalysis. analysis. compared Based on practical results, an empirical formula estimate thethe total number of EMs Based on practical results, empirical formulawas wasderived derivedtoto estimate total number of in the decomposition of an signal [29]:[29]: EMs in the decomposition ofarbitrary an arbitrary signal
M 1 ,1, 2 NN±± M ==log log 2
(5)(5)
here, M is the total EM quantity, and N is number of samples in the signal. here, M is the total EM quantity, and N is number of samples in the signal. Using EMD, the original signal can be restored completely: Using EMD, the original signal can be restored completely: M
s (k ) = Mc p (k ) + r (k ) s(k ) =p =∑ 1 c p (k) + r (k)
(6) (6)
∑ c p ( k ) + r ( k ),
(7)
p =1
or partly: or partly: s(k) =
p∈ P
where P is the set of selected EMs. Considering EMD of a noisy signal, one can easily understand that EMs with small numbers correspond to noise, and with higher numbers correspond to a useful signal. All noise reduction algorithms utilizing EMD are based on this idea. A number of such algorithms have been developed, e.g., ensemble EMD [26], conventional EMD, and iterative EMD [27]. As an example, we used so-called conventional EMD (CEMD). This algorithm is based on the assumption that the energy of “pure noise” EMs decreases according to the law: Eˆ p =
Eˆ 1 2.01− p , p = 2, 3, 4 . . . , 0.719
(8)
algorithms utilizing EMD are based on this idea. A number of such algorithms have been developed, e.g., ensemble EMD [26], conventional EMD, and iterative EMD [27]. As an example, we used so-called conventional EMD (CEMD). This algorithm is based on the assumption that the energy of “pure noise” EMs decreases according to the law: Electronics 2018, 7, 450
Eˆ1 Eˆ p = 2.01− p , p = 2,3,4... , 0.719
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(8)
here, ppisisthe thethe energy of of EM. thenumber numberofofEM; EM;and andEˆ pEˆisp is energy EM. The cleaned signal can be written in the following The cleaned signal can be written in the following form: form: M xe(~ xt()t )== ∑ c pp E
pp==11
+ rr((tt)),, ˆ ((tt)) + Epp>>EEˆpp
i.e. only those of of which exceeds thethe energy of the “pure noise” EM those of ofEM EMare aresummarized, summarized,the theenergy energy which exceeds energy of the “pure noise” due to the of the signal. EM due topresence the presence ofuseful the useful signal. 2.2.2. Nonlinear Filtration 2.2.2. Nonlinear Filtration by by Chain Chain of of Driven Driven Oscillators Oscillators It It is is known known that that chaotic chaotic motion motion can can be be observed observed in in state state variable variable filters filters [30]. [30]. Thus, Thus, the the reverse reverse statement that chaotic systems can work as filters can also be valid. In our study, we propose statement that chaotic systems can work as filters can also be valid. In our study, we propose aa non-linear non-linear filter filter construction construction as as aa cascade cascade of of driven driven oscillators. oscillators. The The number number of of oscillators oscillators in in the the chain chain defines the order of such a filter. Figure 9 presents a chain involving four oscillators. Notice that defines the order of such a filter. Figure 9 presents a chain involving four oscillators. Notice that the the first oscillator works asas a signal amplifier to first driven driven oscillator oscillatorworks worksas asaafilter filterfrom fromFigure Figure7,7,the thesecond second oscillator works a signal amplifier feed thethe nextnext oscillator, and and so on. experiments made made with awith four-oscillator filter have to feed oscillator, soPrimary on. Primary experiments a four-oscillator filtershown have promising performance. shown promising performance.
Figure 9. 9. The oscillators involving involving 44 oscillators. oscillators. Figure The chain chain of of driven driven oscillators
ofsignal signalpropagating propagating from point to x 4 the shows theofprocess of information Analysis of from point x0 toxx04 shows process information extraction and amplification, see Figure 10. extraction and amplification, see Figure 10. Notice that noise superimposed on signal of logical 0 also tends to be amplified, so a more sophisticated filter could be designed which was capable of analyzing the signal not only at point x4 , but also at other points.
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5 0 -5 -10 -15
0
2000
4000
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[n], samples
(a)
(b)
(c)
(d)
Figure 10. Restored Restored message messagein innoisy noisysequence sequence0101 0101(SNR (SNR 0 dB) first stage second stage = 0= dB) at at thethe first stage (a),(a), second stage (b), (b), stage (c) and thirdthird stage (c) and forthforth stagestage (d). (d).
3. Results Notice that noise superimposed on signal of logical 0 also tends to be amplified, so a more sophisticated filter comparison could be designed whichtechniques, was capable analyzing the signal not only First, at point For a relative of filtration weofused the following approach. all x , but also at other points. 4 filtering techniques were implemented as MATLAB functions. Different messages were passed though the model of chaotic communication system based on Rössler oscillators, as described in Section 2. 3. Results Filtered synchronization signals were sent to the driven oscillator and its responses were restored as digital the sameofdetector (Figure 5). After number experiments with the same Formessages a relativeusing comparison filtration techniques, weaused the of following approach. First, all SNR value (100 or 200), we increased the intensity of noise source. The whole experiment was made filtering techniques were implemented as MATLAB functions. Different messages were passed for SNRthe values from 30 to 20communication dB. For SNR = system 20 dB, no filtration was required for stable messages though model of − chaotic based on Rössler oscillators, as described in recognition. For SNR = − 30 dB, no algorithm was able to restore any message. Section 2. Filtered synchronization signals were sent to the driven oscillator and its responses were Experimental results areusing generalized in Figure 11.(Figure One can that nonlinear usingwith the restored as digital messages the same detector 5). see After a number offiltration experiments chain of driven oscillators performance, slightlyofbetter wavelet accepted the same SNR value (100 shows or 200),the webest increased the intensity noisethan source. The technique whole experiment by different researches as suitable for chaotic signals denoising [4,9]. The EMD filtration technique was made for SNR values from −30 to 20 dB. For SNR = 20 dB, no filtration was required for stable shows bestrecognition. performance forSNR low =SNR IIR filteringwas andable median filtering enhance the correct messages For −30 values. dB, no algorithm to restore anyalso message. bits rate, but not soresults efficiently. Experimental are generalized in Figure 11. One can see that nonlinear filtration using the It should be taken into account thatbest these results wereslightly obtainedbetter on anthan idealwavelet system with pure chain of driven oscillators shows the performance, technique white noise in as channel. Notwithstanding, we denoising got a confirmation the validity of accepted byaffecting differentmessages researches suitable for chaotic signals [4,9]. TheofEMD filtration our approach involving chain of driven oscillators. It appears to be more efficient than EMD, wavelet technique shows best performance for low SNR values. IIR filtering and median filtering also and median filtration. good performance, making it practically applicable. enhance the correct bitsEMD rate,also but shows not so efficiently.
It should be taken into account that these results were obtained on an ideal system with pure It should be taken into account that these results were obtained on an ideal system with pure white noise affecting messages in channel. Notwithstanding, we got a confirmation of the validity of white noise affecting messages in channel. Notwithstanding, we got a confirmation of the validity of our approach involving chain of driven oscillators. It appears to be more efficient than EMD, our approach involving chain of driven oscillators. It appears to be more efficient than EMD, wavelet 2018, and median filtration. EMD also shows good performance, making it practically applicable. Electronics 450 10 of 14 wavelet and7,median filtration. EMD also shows good performance, making it practically applicable.
1 1
Correctbits bitsrate rate Correct
0.8 0.8 0.6 0.6
No filter No filter IIR IIR Median Median WLT WLT Chain Chain EMD EMD
0.4 0.4 0.2 0.2 0 0 -30 -30
-25 -25
-20 -20
-15 -15
-10 -10
-5 -5
0 0
5 5
10 10
15 15
20 20
SNR, dB SNR, dB Figure 11. Ratio of correctly recognized bits according to SNR level in communication channel and Figure Figure11. 11.Ratio Ratioofofcorrectly correctlyrecognized recognizedbits bitsaccording accordingtotoSNR SNRlevel levelinincommunication communicationchannel channeland and filtration method: IIR —4-th order low-pass Butterworth filter; MED – median filter; WLT —wavelet filtration filtrationmethod: method:IIR—4-th IIR —4-thorder orderlow-pass low-passButterworth Butterworthfilter; filter;MED— MED – median median filter; filter; WLT—wavelet WLT —wavelet transform filter using external phase Daubechies wavelet with 8 vanishing moments; EMD — transform moments; EMD EMD— transform filter filter using using external external phase phase Daubechies Daubechies wavelet wavelet with 8 vanishing moments; — conventional EMD filter; Chain — chain of driven oscillators technique. conventional oscillators technique. conventionalEMD EMDfilter; filter;Chain— Chain —chain chainofofdriven driven oscillators technique.
Value Value
The first first example is is presented in in Figure 12. 12. The black black waveform represents represents the master-slave master-slave The The firstexample example ispresented presented inFigure Figure 12.The The blackwaveform waveform representsthe the master-slave synchronization error error when transferring message 01010101 a channel noise in channel synchronization transferring the the message 01010101 withoutwithout a noise in synchronization errorwhen when transferring the message 01010101 without a noise in(compare channel (compare to Figure 3a), blue dotted waveform represents the synchronization error with the filtered to Figure 3a), blue dotted waveform represents the synchronization error with the filtered noisy (compare to Figure 3a), blue dotted waveform represents the synchronization error with the filtered noisy =(SNR = −10dB) synchronization signal. Original synchronization error waveform (Figure 12a) (SNR − 10dB) synchronization signal. Original synchronization error waveform (Figure 12a) may be noisy (SNR = −10dB) synchronization signal. Original synchronization error waveform (Figure 12a) may to bebe seen to bethe unlike theinsignal in12b. Figure 12b. seen unlike signal Figure may be seen to be unlike the signal in Figure 12b.
(a) (a)
(b) (b)
Figure 12. 12. Recovering Recovering of of master-slave master-slave synchronization synchronization error error at at SNR SNR== − −10 driven 10 dB using chain of driven Figure 12. Recovering of master-slave synchronization error at SNR = −10 dB using chain of driven oscillators denoising denoising (a) (a) and and EMD EMD filtration filtration (b). (b). oscillators oscillators denoising (a) and EMD filtration (b).
Figure 13 reveals the details of the experiment. experiment. The Thesame samemessage message 01010101 01010101 passes passes through through Figure 13 reveals the details of the experiment. The same message 01010101 passes through the chaotic chaotic communication communication channel. The upper waveform is the master-slave synchronization error the chaotic communication channel. The upper waveform is the master-slave synchronization error without envelope digital message encoded in without noise noisein inthe thechannel channel(same (sameasasininFigure Figure3a), 3a),and anda asquare square envelope digital message encoded without noise in the channel (same as in Figure 3a), and a square envelope digital message encoded synchronization errorerror signal. Other analog waveforms were obtained from thefrom detector. In this example, in synchronization signal. Other analog waveforms were obtained the detector. In this in synchronization error signal. Other analog waveforms were obtained from the detector. In this only the oscillators’ chain method allows us toallows recognize originalthe message. After EMD, After the 3rd and example, only the oscillators’ chain method us tothe recognize original message. EMD, example, only the oscillators’ chain method allows us to recognize the original message. After EMD, 4th the missing, median filter has lostfilter the second 1, andlogical other methods are the bits 3rd are andmissing, 4th bits are the median has lostand thefourth secondlogical and fourth 1, and other the 3rd and 4th bits are missing, the median filter has lost the second and fourth logical 1, and other total loss. are Thistotal example of the detector used, butdetector demonstrates methods loss. also Thisillustrates example the alsoimperfection illustrates the imperfection of the used, the but methods are total loss. This example also illustrates the imperfection of the detector used, but superiority of the chain of driven oscillators well. approach well. demonstrates superiority of the chain of approach driven oscillators demonstrates the superiority of the chain of driven oscillators approach well.
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Figure13. 13.Example Exampleof ofaamessage messagerecognition, recognition,SNR SNR==− −10 filtering; Figure 10 dB. Noisy analog indicates without filtering; IIR indicates indicates 4-th 4-th order order low-pass low-pass Butterworth Butterworth filter; filter; MED MED indicates indicates median median filter; filter; WLT WLT indicates indicates IIR maximal overlap overlap discrete discrete wavelet transform filter maximal filter using using external external phase phaseDaubechies Daubechieswavelet waveletwith with8 driven oscillators oscillators 8vanishing vanishingmoments; moments;EMD EMDindicates indicates conventional conventional EMD EMD filter; filter; Chain indicates 44 driven − Xss indicates noise in in channel; channel; m(t) m(t) chain filter; filter; XXmm − chain indicates master-slave master-slave synchronization synchronization error with no noise indicatesoriginal originaldigital digitalmessage messagewaveform. waveform. indicates
To Tomake makean animportant importantstep steptowards towardspractical practicalimplementation implementationof ofthe thechaotic chaoticcommunicational communicational system, we should take into account that the relative performance of filtration methods system, we should take into account that the relative performance of filtration methodspresented presentedin in Figure Figure11 11isisindependent independentto tothe thedisturbance disturbancetype. type.Often, Often,disturbance disturbanceisismodeled modeledby bynoise, noise,including including colored [31], though otherother signals also may be considered as disturbances [32]. Noise classification colorednoise noise [31], though signals also may be considered as disturbances [32]. Noise involves conception of the noise color, i.e.noise dependency noise powerof from frequency. Colored noise classification involves conception of the color, i.e.ofdependency noise power from frequency. α , where α ∈ [−2;α2]. White noise has flat spectrum has power spectral density of the form S ( f ) ∝ 1/ f Colored noise has power spectral density of the form S ( f ) ∝ 1 / f , where α ∈ [−2; 2] . White noise (α = 0), while the spectral density of a pink noise compared to white noise is attenuated by 3 decibels has flat spectrum ( α = 0 ), while the spectral density of a pink noise compared to white noise is per octave (α = 1), and the spectral density of a blue noise increases by 3 decibels per octave (α = −1). attenuated by 3 decibels per octave ( α = 1 ), and the spectral density of a blue noise increases by 3 White noise is an equivalent of uniformly-disturbed media; pink noise can be naturally generated by decibels per octave ( α = −1 ). White noise is an equivalent of uniformly-disturbed media; pink noise great amount of physical processes; and blue noise may be a good analog of radio waves propagating can be naturally generated by great amount of physical processes; and blue noise may be a good media where higher frequencies are occupied by more powerful sources. Real world ambient noises analog of radio waves propagating media where higher frequencies are occupied by more powerful can often be approximated with fractional α values [31]. sources. Real world ambient noises can often be approximated with fractional α values [31]. Figure 14 shows a comparison of studied filtering techniques in the presence of colored noises. Figure 14 shows a comparison of studied filtering techniques in the presence of colored noises. The problem with pink noise is that chaotic systems have spectra of the same type, i.e., decaying The problem with pink noise is that chaotic systems have spectra of the same type, i.e., decaying in in higher frequencies, so the presence of pink noise masks the chaotic signal and, consequently, higher frequencies, so the presence of pink noise masks the chaotic signal and, consequently, weakens the performance of the filtration by the chain of chaotic oscillators. Instead, blue noise can weakens the performance of the filtration by the chain of chaotic oscillators. Instead, blue noise can be called ‘chaos friendly’, but EMD needs to be properly adjusted when higher harmonics appear. be called ‘chaos friendly’, but EMD needs to be properly adjusted when higher harmonics appear. In In general, this experiment confirms the abovementioned results of relative performance of the general, this experiment confirms the abovementioned results of relative performance of the filtration techniques. filtration techniques.
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No filter IIR Median WLT Chain EMD
0.8 0.6 0.4 0.2 0 -30
-20
-10
0
10
20
SNR, dB
(a)
(b)
Figure 14. 14. Ratio recognized bitsbits according to SNR levellevel in communication channel, noise Figure Ratioofofcorrectly correctly recognized according to SNR in communication channel, color indicates pink (a) and blue (b), and filtration method: IIR indicates 4-th order low-pass noise color indicates pink (a) and blue (b), and filtration method: IIR indicates 4-th low-pass Butterworthfilter; filter;MED MED indicates median filter; indicates a wavelet transform filter using Butterworth indicates median filter; WLTWLT indicates a wavelet transform filter using external external phase Daubechies wavelet with 8 moments; vanishing EMD moments; EMD the conventional phase Daubechies wavelet with 8 vanishing indicates theindicates conventional EMD filter; EMD filter; Chain the chain of driven oscillators technique. Chain indicates theindicates chain of driven oscillators technique.
Table Table11summarizes summarizesstudied studieddenoising denoisingmethods methodsapplied appliedfor forchaotic chaoticcommunications. communications. Table Table1.1.Brief Briefcomparison comparisonof ofnonlinear nonlineardenoising denoisingtechniques techniquesfor forchaotic chaoticsignals. signals. Denoising technique Denoising technique
Median filtering Median filtering Wavelet denoising Wavelet denoising EMD denoising
EMD denoising Driven oscillators chain filtering
Driven oscillators chain filtering
Pros Pros Easy to to implement; implement; Easy adjustable parameter parameter one adjustable Fine performance Fine performance Fine performance
Fine performance; performance Fine easy to implement; Fine performance; no parameters to adjust easy to implement; no parameters to adjust
Cons Cons
Not efficient Not efficient Difficult Difficulttotoadjust adjust Difficult to adjust; parameters on SNR Difficult depend to adjust; parameters depend on SNR -
4. Conclusion and Discussion
4. Conclusion Discussion Following and previous research, we demonstrate the possibility of chaotic communication signals to be effectively denoised when messages affected bythe disturbances communication channel. signals As for Following previous research, weare demonstrate possibilityinofa chaotic communication conventional communication systems, the use of proper denoising algorithms significantly enhances to be effectively denoised when messages are affected by disturbances in a communication channel. the resistivity ofcommunication communication. Denoising can denoising determinealgorithms the operation ability of As noise for conventional systems, the techniques use of proper significantly the investigated system in a case of low signal-to-noise ratio. Using white noise as a disturbance, we enhances the noise resistivity of communication. Denoising techniques can determine the operation obtained the correct bit-rate approximately 50% when signal-to-noise ratio was -20 dB using our novel ability of the investigated system in a case of low signal-to-noise ratio. Using white noise as a nonlinear filtration method based on bit-rate chain ofapproximately driven oscillators. This method and empirical mode disturbance, we obtained the correct 50% when signal-to-noise ratio was -20 decomposition denoising were more effective and easier to adjust than the wavelet technique used for dB using our novel nonlinear filtration method based on chain of driven oscillators. This method denoising in chaotic other researchers. and empirical modecommunications decomposition by denoising were more effective and easier to adjust than the Moving towards practical system implementation, we studied the expected behavior of the system wavelet technique used for denoising in chaotic communications by other researchers. in presence of different noise types. Numerical experiments with pink and blue noises show that Moving towards practical system implementation, we studied the expected behavior of the the performance of the whole communication is influenced by noise denoising system in presence of different noise types.system Numerical experiments withtype, pinkbut andrelative blue noises show methods effectiveness is was found that basic chaotic oscillatorby parameters variation may that the performance ofnot. the Also, wholeit communication system is influenced noise type, but relative result in notable improvement of noise resistance. denoising methods effectiveness is not. Also, it was found that basic chaotic oscillator parameters The obtained results can be improvement applied to other chaos-based communication schemes based variation may result in notable of types noise of resistance. on chaotic synchronization principle. The obtained results can be applied to other types of chaos-based communication schemes Our will include enhancements of the detection algorithm, improving based on further chaotic investigations synchronization principle. the filtration algorithm based on EMD, and comprehensive study of the nonlinear filtering method
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based on the slave generators cascading. In particular, we will try to determine optimal cascade number and develop the theory of cascaded chaotic filters. We will also implement the chaotic communication system utilizing state-of-art message encoding and perform some experiments with hardware prototypes, including hydroacoustic and underwater communications systems. Author Contributions: Conceptualization, D.B. and D.K.; Data curation, T.K. and V.A.; Formal analysis, D.B. and A.V.; Investigation, T.K., A.V., V.A. and V.O.; Methodology, D.B. and A.V.; Project administration, D.B. and D.K.; Resources, T.K. and D.K.; Software, T.K., A.V., D.K. and V.A.; Supervision, D.B. and D.K.; Validation, D.B., T.K., V.A. and V.O.; Visualization, V.A.; Writing—original draft, D.B., T.K. and V.O.; Writing—review & editing, D.K. and V.O. Funding: This research was funded by Russian Science Foundation (Project NO 17-71-20077). Acknowledgments: Special thanks to our young colleagues from CAD department of Saint-Petersburg Electrotechnical University “LETI” for their participation. Conflicts of Interest: The authors declare no conflict of interest.
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