2. Spectral properties of chaos generated by the delayed feedback system with one-dimensional skew tent map

A nonlinear dynamical system with delayed feedback of the ring type is used for broadband chaotic signal generation. In the general case, under the assumption that the whole system is inertial-free, it is described by a difference equation of the form:

$$\mathbf{x}(t) = F[\mathbf{x}(t - T\_0), r, a],\tag{1}$$

where F xð Þ ; r; a is the mapping function of the unit interval 0½ � ; 1 of the x-axis into itself F : ½ � 0; 1 ↦ ½ � 0; 1 ; r and a are the map parameters defined by the function F; T<sup>0</sup> is the delay time.

The solutions of Eq. (1) and their correlation properties for the case when the function F xð Þ ;r; a defines a unimodal piecewise-linear map (skew tent map):

$$F(\mathbf{x}, r, a) = \begin{cases} r\mathbf{x}/a, & \mathbf{x} \in [0, a] \\ r(1 - \mathbf{x})/(1 - a), & \mathbf{x} \in (a, 1] \end{cases} \tag{2}$$

chaotic oscillations decreases, and when a ¼ a<sup>1</sup> ¼ 0:7, a fixed stable point appears and the

Figure 2. Cross-sections of the two-parameter bifurcation diagram of the skew tent map obtained for parameter r fixed

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67

Since two points of the mapping function graph are fixed, the layout of the graph is completely determined by the position of the top. There are three regions with qualitatively different

When the top of the map is in region I (Figure 3(a)), the map has two fixed points, one of which is unstable (z0), and the other is stable (z1). Therefore, the iterative process converges to z1. When the top of the map moves to region II (Figure 3(b)), the fixed point z<sup>1</sup> loses its stability, since the absolute value of the derivative at this point becomes greater than 1. The z<sup>0</sup> point is still unstable. In this case, a region of chaotic oscillations with dense filling appears on the bifurcation diagram. With the further movement of the top of the map in the direction toward the region III, there comes a moment when the top and the fixed point merge (this occurs on the bisector of the first quadrant) and when the top of the map falls into the region III (Figure 3(c)), the oscillations in the system disappear, since there remains one fixed point z<sup>0</sup> that gets stable. Thus, in order to obtain chaotic regimes when choosing bifurcation parameter values r and a, it is necessary to be guided

by the condition that the top of the map belongs to the region II.

Figure 3. Areas with qualitatively different states of the system.

oscillations disappear.

system states (Figure 3) for it.

values: (a) r ¼ 0:7; (b) r ¼ 0:9; (c) r ¼ 1.

are investigated by authors in [29].

The evolution of any dynamical system depends on its parameters. Their change leads to an inevitable change in the trajectories of motions of the dynamical system in time. In this case, a small change in the parameters can lead to both an insignificant change in behavior and a significant rearrangement of the phase trajectories (bifurcations). To exploit the map under consideration as source of chaotic sequences, one needs to make selection of its parameters. It is convenient to use a two-parameter bifurcation diagram in terms of the parameters r and a (Figure 1).

In Figure 1, for the sake of clarity, the step along the parameter a is chosen to be sufficiently large, which, however, does not prevent from following the evolution of the iterative process when a changes. More detailed information is contained in the cross sections of the threedimensional picture made for fixed values of the parameter r (Figure 2).

A characteristic feature that is visible on these cross-sections is the presence of three boundary values of the bifurcation parameter a, under which the picture of the behavior of the system abruptly changes. Therefore, for example, for r ¼ 0:7 (Figure 2(a)), the fixed point for a ¼ a<sup>1</sup> ¼ 0:3 loses stability, that leads to the appearance of cycles of intervals, which merged, when a ¼ a<sup>1</sup> ¼ 0:5. With further increase of the bifurcation parameter, the amplitude of the

Figure 1. Two-parameter bifurcation diagram of the skew tent map.

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where F xð Þ ; r; a is the mapping function of the unit interval 0½ � ; 1 of the x-axis into itself F : ½ � 0; 1 ↦ ½ � 0; 1 ; r and a are the map parameters defined by the function F; T<sup>0</sup> is the delay time. The solutions of Eq. (1) and their correlation properties for the case when the function F xð Þ ;r; a

The evolution of any dynamical system depends on its parameters. Their change leads to an inevitable change in the trajectories of motions of the dynamical system in time. In this case, a small change in the parameters can lead to both an insignificant change in behavior and a significant rearrangement of the phase trajectories (bifurcations). To exploit the map under consideration as source of chaotic sequences, one needs to make selection of its parameters. It is convenient to use a two-parameter bifurcation diagram in terms of the parameters r and a

In Figure 1, for the sake of clarity, the step along the parameter a is chosen to be sufficiently large, which, however, does not prevent from following the evolution of the iterative process when a changes. More detailed information is contained in the cross sections of the three-

A characteristic feature that is visible on these cross-sections is the presence of three boundary values of the bifurcation parameter a, under which the picture of the behavior of the system abruptly changes. Therefore, for example, for r ¼ 0:7 (Figure 2(a)), the fixed point for a ¼ a<sup>1</sup> ¼ 0:3 loses stability, that leads to the appearance of cycles of intervals, which merged, when a ¼ a<sup>1</sup> ¼ 0:5. With further increase of the bifurcation parameter, the amplitude of the

rx=a, x∈ ½ � 0; a rð Þ 1 � x =ð Þ 1 � a , x∈ ð � a; 1

(2)

defines a unimodal piecewise-linear map (skew tent map):

are investigated by authors in [29].

66 Telecommunication Networks - Trends and Developments

(Figure 1).

F xð Þ¼ ;r; a

dimensional picture made for fixed values of the parameter r (Figure 2).

Figure 1. Two-parameter bifurcation diagram of the skew tent map.

Figure 2. Cross-sections of the two-parameter bifurcation diagram of the skew tent map obtained for parameter r fixed values: (a) r ¼ 0:7; (b) r ¼ 0:9; (c) r ¼ 1.

chaotic oscillations decreases, and when a ¼ a<sup>1</sup> ¼ 0:7, a fixed stable point appears and the oscillations disappear.

Since two points of the mapping function graph are fixed, the layout of the graph is completely determined by the position of the top. There are three regions with qualitatively different system states (Figure 3) for it.

When the top of the map is in region I (Figure 3(a)), the map has two fixed points, one of which is unstable (z0), and the other is stable (z1). Therefore, the iterative process converges to z1. When the top of the map moves to region II (Figure 3(b)), the fixed point z<sup>1</sup> loses its stability, since the absolute value of the derivative at this point becomes greater than 1. The z<sup>0</sup> point is still unstable. In this case, a region of chaotic oscillations with dense filling appears on the bifurcation diagram. With the further movement of the top of the map in the direction toward the region III, there comes a moment when the top and the fixed point merge (this occurs on the bisector of the first quadrant) and when the top of the map falls into the region III (Figure 3(c)), the oscillations in the system disappear, since there remains one fixed point z<sup>0</sup> that gets stable. Thus, in order to obtain chaotic regimes when choosing bifurcation parameter values r and a, it is necessary to be guided by the condition that the top of the map belongs to the region II.

Figure 3. Areas with qualitatively different states of the system.

When choosing oscillation modes in systems with time delay, it is necessary to take into account the stability of oscillations and the influence of external noise [30]. Apparently, under the influence of destabilizing factors, which can be described by the presence of additive or multiplicative noise, the boundary of the chosen region of the bifurcation parameters will be blurred.

Thus, when r ¼ 1 (only this case will be considered below), the sequence of iterations of a skew tent map is completely chaotic. Its interesting property is the independence of the invariant measure on the parameter a. As shown in [31, 32], this map is ergodic on the interval 0½ � ; 1 and has an invariant measure p xð Þ¼ 1. This circumstance allows determining exactly the dependence of the Lyapunov exponent on the abscissa of the top of the map. Since the invariant measure is known, averaging over time can be replaced by averaging over x [31] when calculating the characteristic Lyapunov exponent:

$$
\sigma = \int p(\mathbf{x}) \ln \left| \frac{dF(\mathbf{x})}{d\mathbf{x}} \right| d\mathbf{x}.\tag{3}
$$

Rð Þ¼ τ

(

Wð Þ¼ ω 2

Substituting Eq. (4) into Eq. (6) and changing the order of integration, we have:

Wð Þ¼ ω 2

k¼1

function Eq. (4) has the following form (up to a constant factor):

8 >>><

>>>:

ð Þ �<sup>1</sup> <sup>k</sup>

C tðÞ¼

where B tðÞ¼ C tð Þ, t <sup>¼</sup> nT0,

�

0, t 6¼ nT0,

of the process with autocorrelation function Rð Þτ :

Eq. (7) can be replaced by the series <sup>W</sup>ð Þ¼ <sup>ω</sup> <sup>2</sup> <sup>P</sup><sup>∞</sup>

Wð Þ¼ ω; a

T<sup>0</sup> of the signal in the delay line (Figure 5).

calculating the sum of the series <sup>P</sup><sup>∞</sup>

ð ∞

Chaos-Based Spectral Keying Technique for Secure Communication and Covert Data Transmission between Radar…

B tð Þδ τð Þ � t dt, (4)

http://dx.doi.org/10.5772/intechopen.79027

Rð Þτ cos ωτdτ: (6)

B tð Þ cos ωt dt: (7)

C nT ð Þ<sup>0</sup> cos nωT0, and to calculate the sum of

:

e�ky cos kx, where y > 0. After carrying out all calcu-

, 0:5 < a < 1,

, 0 < a < 0:5

(5)

69

(8)

n∈ N In this case, the dependence of the autocorrelation func-

,

0

<sup>C</sup>0ð Þ �<sup>1</sup> <sup>t</sup>

tion of the solution of Eq. (1) with a nonlinear function Eq. (2) has the following form [33, 34]:

where τ<sup>c</sup> ¼ j j 1= ln 2j j a � 1 . Using the Wiener-Khinchin theorem, we derive the power spectrum

ð ∞

0

ð ∞

0

Taking into account that the function B tð Þ differs from zero only at points t ¼ nT0, the integral

n¼0

this series we need substitute C nT ð Þ<sup>0</sup> by values from Eq. (5). As a result, the problem reduces to

lations, we finally obtain that the power spectrum of chaotic process with the autocorrelation

<sup>e</sup><sup>T</sup>0=τ<sup>c</sup> � cos <sup>ω</sup>T<sup>0</sup> eT0=τ<sup>c</sup> þ e�T0=τ<sup>c</sup> � 2 cos ωT<sup>0</sup>

<sup>e</sup><sup>T</sup>0=τ<sup>c</sup> <sup>þ</sup> cos <sup>ω</sup>T<sup>0</sup> eT0=τ<sup>c</sup> þ e�T0=τ<sup>c</sup> þ 2 cos ωT<sup>0</sup>

Thus, the power spectrum of chaotic auto-oscillations in the dynamical system of ring type with a delay in the deviation of a nonlinear map from a symmetric form is a periodic function of the frequency with a frequency period f <sup>T</sup> ¼ ωT=2π ¼ 1=T0, corresponding to the delay time

For 0:5 < a < 1, the first maximum of the power spectrum is located at zero frequency, while when 0 < a < 0:5 it is shifted to f <sup>T</sup>=2 (the maxima are located at frequencies

C0e�t=τ<sup>c</sup> , a > 1=2

e�t=τ<sup>c</sup> , a < 1=2

Taking into account that in our case, the map is represented by a function that differs from 0 only on the interval 0½ � ; 1 and does not take negative values anywhere, after integration, we get <sup>σ</sup> ¼ � ln <sup>a</sup><sup>a</sup> ð Þ <sup>1</sup> � <sup>a</sup> <sup>1</sup>�<sup>a</sup> h i. As seen from the graph of this function (Figure 4), the Lyapunov exponent is positive everywhere within unit interval and reaches its maximum value for a symmetric case (for a ¼ 0:5). As a criterion for selection the value of parameter a, we choose Lyapunov exponents to be at least 0.5 (white area in Figure 4).

It was shown in [29] that the autocorrelation function of the solution of Eq. (1) can be represented in the form

Figure 4. Dependence of the Lyapunov exponent on the asymmetry parameter of a skew tent map.

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$$R(\tau) = \bigcap\_{t=0}^{\infty} B(t)\delta(\tau - t) \, dt,\tag{4}$$

where B tðÞ¼ C tð Þ, t <sup>¼</sup> nT0, 0, t 6¼ nT0, � n∈ N In this case, the dependence of the autocorrelation function of the solution of Eq. (1) with a nonlinear function Eq. (2) has the following form [33, 34]:

When choosing oscillation modes in systems with time delay, it is necessary to take into account the stability of oscillations and the influence of external noise [30]. Apparently, under the influence of destabilizing factors, which can be described by the presence of additive or multiplicative noise, the boundary of the chosen region of the bifurcation parameters will be

Thus, when r ¼ 1 (only this case will be considered below), the sequence of iterations of a skew tent map is completely chaotic. Its interesting property is the independence of the invariant measure on the parameter a. As shown in [31, 32], this map is ergodic on the interval 0½ � ; 1 and has an invariant measure p xð Þ¼ 1. This circumstance allows determining exactly the dependence of the Lyapunov exponent on the abscissa of the top of the map. Since the invariant measure is known, averaging over time can be replaced by averaging over x [31] when

> p xð Þln dF xð Þ dx � � � �

Taking into account that in our case, the map is represented by a function that differs from 0 only on the interval 0½ � ; 1 and does not take negative values anywhere, after integration, we get

exponent is positive everywhere within unit interval and reaches its maximum value for a symmetric case (for a ¼ 0:5). As a criterion for selection the value of parameter a, we choose

It was shown in [29] that the autocorrelation function of the solution of Eq. (1) can be

� � � �

. As seen from the graph of this function (Figure 4), the Lyapunov

dx: (3)

blurred.

<sup>σ</sup> ¼ � ln <sup>a</sup><sup>a</sup>

ð Þ <sup>1</sup> � <sup>a</sup> <sup>1</sup>�<sup>a</sup> h i

represented in the form

calculating the characteristic Lyapunov exponent:

68 Telecommunication Networks - Trends and Developments

σ ¼ ð

Figure 4. Dependence of the Lyapunov exponent on the asymmetry parameter of a skew tent map.

Lyapunov exponents to be at least 0.5 (white area in Figure 4).

$$\mathbf{C}(t) = \begin{cases} \mathbf{C}\_0 e^{-t/\tau\_c}, & a > 1/2 \\ \mathbf{C}\_0 (-1)^t e^{-t/\tau\_c}, & a < 1/2 \end{cases} \tag{5}$$

where τ<sup>c</sup> ¼ j j 1= ln 2j j a � 1 . Using the Wiener-Khinchin theorem, we derive the power spectrum of the process with autocorrelation function Rð Þτ :

$$\mathcal{W}(\omega) = 2 \iint\_0 \mathbf{R}(\tau) \cos \omega \tau \, d\tau. \tag{6}$$

Substituting Eq. (4) into Eq. (6) and changing the order of integration, we have:

$$\mathcal{W}(\omega) = 2 \int\_0^\infty \mathcal{B}(t) \cos \omega t \, dt. \tag{7}$$

Taking into account that the function B tð Þ differs from zero only at points t ¼ nT0, the integral Eq. (7) can be replaced by the series <sup>W</sup>ð Þ¼ <sup>ω</sup> <sup>2</sup> <sup>P</sup><sup>∞</sup> n¼0 C nT ð Þ<sup>0</sup> cos nωT0, and to calculate the sum of this series we need substitute C nT ð Þ<sup>0</sup> by values from Eq. (5). As a result, the problem reduces to calculating the sum of the series <sup>P</sup><sup>∞</sup> k¼1 ð Þ �<sup>1</sup> <sup>k</sup> e�ky cos kx, where y > 0. After carrying out all calculations, we finally obtain that the power spectrum of chaotic process with the autocorrelation function Eq. (4) has the following form (up to a constant factor):

$$\mathcal{W}(\omega, a) = \begin{cases} \frac{e^{T\_0/\tau\_\varepsilon} - \cos\omega T\_0}{e^{T\_0/\tau\_\varepsilon} + e^{-T\_0/\tau\_\varepsilon} - 2\cos\omega T\_0}, & 0.5 < a < 1, \\\\ \frac{e^{T\_0/\tau\_\varepsilon} + \cos\omega T\_0}{e^{T\_0/\tau\_\varepsilon} + e^{-T\_0/\tau\_\varepsilon} + 2\cos\omega T\_0}, & 0 < a < 0.5 \end{cases} \tag{8}$$

Thus, the power spectrum of chaotic auto-oscillations in the dynamical system of ring type with a delay in the deviation of a nonlinear map from a symmetric form is a periodic function of the frequency with a frequency period f <sup>T</sup> ¼ ωT=2π ¼ 1=T0, corresponding to the delay time T<sup>0</sup> of the signal in the delay line (Figure 5).

For 0:5 < a < 1, the first maximum of the power spectrum is located at zero frequency, while when 0 < a < 0:5 it is shifted to f <sup>T</sup>=2 (the maxima are located at frequencies

xn <sup>¼</sup> <sup>2</sup>xn�<sup>M</sup> <sup>þ</sup> sn�<sup>M</sup> � <sup>1</sup>

Chaos-Based Spectral Keying Technique for Secure Communication and Covert Data Transmission between Radar…

Following this formula, the problem of generating a chaotic signal in discrete time domain using a dynamical system with delayed feedback and a nonlinear function Eq. (2) reduces to computing the samples of the sequence exploiting the algorithm given by formula Eq. (10). Entering information into a chaotic signal is accomplished by changing the parameter a value. In this case, the computational algorithm consists of a block of delay for M samples and a set of functional blocks performing elementary arithmetic operations. This allows implementation of a digital synthesis module based on FPGA technology, for example, using standard elements of the "Xilinx System Generator for DSP" and "ISE Foundation" libraries [36]. The simulation

Signal is generated in the transmitter as follows. We choose a binary sequence whose elements si take the values "0" or "1" as a test information signal S(t). The information signal controls the switching element, which changes the parameter of the nonlinear function (Figure 7), herewith the parameter values a<sup>0</sup> ¼ 0:25 and a<sup>1</sup> ¼ 0:75 correspond to transmission of the

Figure 7. The dynamical system of ring type with a variable parameter of a nonlinear element as a driver of a chaotic

model of proposed chaos communication system is presented in Figure 6.

Figure 6. Simulation model of proposed chaos-based communication system.

symbols "0" and "1", respectively.

signal in the transmitter.

<sup>2</sup><sup>a</sup> <sup>þ</sup> sn�<sup>M</sup> � <sup>1</sup> : (10)

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71

Figure 5. Power spectrum of chaotic oscillations in a ring dynamical system with delay for different values of the parameter a.

1=ð Þ 2T<sup>0</sup> ,3=ð Þ 2T<sup>0</sup> , …). Therefore, assigning the control parameter value a from the interval ð Þ 0; 0:5 or 0ð Þ :5; 1 allows manipulating the positions of the spectrum maxima, thereby entering information message into a chaotic carrier [35]. To transmit a binary sequence, it is sufficient to use two fixed values of a, for example, following the formula a ¼ 0:5 þ λsign ð Þ si � 0:5 , where si is the information bit that takes the value "0" or "1". Here parameter λ ∈ 0; 1 determines the ratio between maxima and minima (the "depth" of the irregularity) in the formed spectrum. If the signal generated in this way arrives in the receiver at the input of the comb filter, the frequency response shape of which is matched to the signal spectrum, then the response at the filter output will be maximal in comparison with the case, when the signal with a spectrum that does not match to the filter frequency response acts at the filter input. An analysis of the magnitude of the response allows making decision whether "0" or "1" was transmitted and thus restoring the original information.

### 3. Simulation modeling of the data transmission system

The efficiency of the proposed method of information transmission was tested by means of simulation using the Simulink environment of the MATLAB software package. We used Eq. (1), written in discrete time domain at r ¼ 1 as a source of chaos, controlled by a discrete information sequence:

$$\mathbf{x}\_n = F(\mathbf{x}\_{n-M}, a). \tag{9}$$

where x1, x2, …xM is the vector of initial values by dimension M. If a discrete sequence xn is associated with the binary sequence sn ¼ signð Þ a � xn , then discrete Eq. (9) with the nonlinear function Eq. (2) has the following form:

Chaos-Based Spectral Keying Technique for Secure Communication and Covert Data Transmission between Radar… http://dx.doi.org/10.5772/intechopen.79027 71

$$\infty\_{\mathfrak{n}} = \frac{2\mathbf{x}\_{\mathfrak{n}-M} + \mathbf{s}\_{\mathfrak{n}-M} - 1}{2\mathbf{a} + \mathbf{s}\_{\mathfrak{n}-M} - 1}. \tag{10}$$

Following this formula, the problem of generating a chaotic signal in discrete time domain using a dynamical system with delayed feedback and a nonlinear function Eq. (2) reduces to computing the samples of the sequence exploiting the algorithm given by formula Eq. (10). Entering information into a chaotic signal is accomplished by changing the parameter a value. In this case, the computational algorithm consists of a block of delay for M samples and a set of functional blocks performing elementary arithmetic operations. This allows implementation of a digital synthesis module based on FPGA technology, for example, using standard elements of the "Xilinx System Generator for DSP" and "ISE Foundation" libraries [36]. The simulation model of proposed chaos communication system is presented in Figure 6.

Signal is generated in the transmitter as follows. We choose a binary sequence whose elements si take the values "0" or "1" as a test information signal S(t). The information signal controls the switching element, which changes the parameter of the nonlinear function (Figure 7), herewith the parameter values a<sup>0</sup> ¼ 0:25 and a<sup>1</sup> ¼ 0:75 correspond to transmission of the symbols "0" and "1", respectively.

Figure 6. Simulation model of proposed chaos-based communication system.

1=ð Þ 2T<sup>0</sup> ,3=ð Þ 2T<sup>0</sup> , …). Therefore, assigning the control parameter value a from the interval ð Þ 0; 0:5 or 0ð Þ :5; 1 allows manipulating the positions of the spectrum maxima, thereby entering information message into a chaotic carrier [35]. To transmit a binary sequence, it is sufficient to use two fixed values of a, for example, following the formula a ¼ 0:5 þ λsign ð Þ si � 0:5 , where si is the information bit that takes the value "0" or "1". Here parameter λ ∈ 0; 1 determines the ratio between maxima and minima (the "depth" of the irregularity) in the formed spectrum. If the signal generated in this way arrives in the receiver at the input of the comb filter, the frequency response shape of which is matched to the signal spectrum, then the response at the filter output will be maximal in comparison with the case, when the signal with a spectrum that does not match to the filter frequency response acts at the filter input. An analysis of the magnitude of the response allows making decision whether "0" or "1" was

Figure 5. Power spectrum of chaotic oscillations in a ring dynamical system with delay for different values of the

The efficiency of the proposed method of information transmission was tested by means of simulation using the Simulink environment of the MATLAB software package. We used Eq. (1), written in discrete time domain at r ¼ 1 as a source of chaos, controlled by a discrete

where x1, x2, …xM is the vector of initial values by dimension M. If a discrete sequence xn is associated with the binary sequence sn ¼ signð Þ a � xn , then discrete Eq. (9) with the nonlinear

xn ¼ F xð Þ <sup>n</sup>�<sup>M</sup>; a : (9)

transmitted and thus restoring the original information.

70 Telecommunication Networks - Trends and Developments

information sequence:

parameter a.

function Eq. (2) has the following form:

3. Simulation modeling of the data transmission system

Figure 7. The dynamical system of ring type with a variable parameter of a nonlinear element as a driver of a chaotic signal in the transmitter.

To transmit one information symbol of duration TS, it is necessary to fulfill the condition TS > T<sup>0</sup> >> Δt, where T<sup>0</sup> is the delay time equal to the analysis time of the signal spectrum at the receiver, Δt is the duration of one sample of the chaotic carrier (one information symbol is transmitted for a sequence of M ¼ TS=Δt samples of the chaotic signal). During transmission of the symbol "0" a continuous chaotic signal enters the communication channel, in the spectrum of which the positions of maxima are determined from the condition Ω<sup>0</sup> <sup>n</sup> ¼ ð Þ 2n � 1 π=T0, n∈ N. When symbol "1" is transmitted, the maxima in the spectrum of the transmitted signal correspond to the condition Ω<sup>1</sup> <sup>n</sup> ¼ 2ð Þ n � 1 π=T0, n∈ N.

described in [26], where the procedure for forming a spectrum with alternating maxima and minima applies to a pre-generated noise signal. In our approach, a special feature of the spectral characteristics of oscillations in a system with delayed feedback of ring type was used for this, which eliminated the need to include additional signal conversion units to obtain the

Chaos-Based Spectral Keying Technique for Secure Communication and Covert Data Transmission between Radar…

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73

Figure 8. Results of simulation modeling: (a) initial information impulse sequence corresponding to the transmitted message; (b) a fragment of the time series of the signal in the communication channel during the sequential transmission of "1" and "0"; (c) signal at the output in the absence of interference in the communication channel; (d) output signal under the influence of additive Gaussian white noise in the communication channel with a level of þ6 dBrelative to the signal level; (e) output signal under the influence of additive Gaussian white noise in the communication channel with a

level of þ12 dB relative to the signal level; (f) restored information sequence.

spectrum of the required shape.

In the receiver, to derive the information from a chaotic signal, its analysis in the spectral domain is used. A signal with a structural feature of the spectrum in the form of equidistantly located spectral density maxima can be efficiently distinguished with a comb filter, the transmission coefficient modulus of which is j j <sup>K</sup>ð Þ <sup>ω</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>G</sup><sup>2</sup> � <sup>2</sup><sup>G</sup> cos <sup>ω</sup>T<sup>0</sup> <sup>p</sup> , where <sup>G</sup> and <sup>T</sup><sup>0</sup> are the amplifier gain and the delay time in the feedback loop, respectively. For a positive value G, the filter frequency response is matched to the chaotic signal spectrum at 0:5 < a < 1 (transmission of information bit "1"), whereas for its negative value the filter frequency response is matched to the chaotic signal spectrum at 0 < a < 0:5 (transmission of information bit "0").

The received signal is simultaneously feed two comb filters, one of which has a positive gain in the feedback loop (G<sup>1</sup> ¼ 0:9) and the other has negative one (G<sup>2</sup> ¼ �0:9). A detector is connected at the output of each filter, which estimates the dispersion of the incoming signal. The signals from both detectors come to the inputs of a comparator and then to a decision device, at the output of which the logical "0" is formed if the signal at the output of the first channel exceeds the signal at the output of the second one and the logical "1" in the opposite case.

As follows from the simulation results, presented in Figure 8, the binary sequence at the output of the receiver (Figure 8(f)) repeats the modulating sequence in the transmitter (Figure 8(a)). In this case, the signal of the transmitter in the communication channel looks like noise waveform, the moments of changing the information bits are not detected from the observable time series (Figure 8(b)). The fragments corresponding to transmission of information bits "0" and "1" are visually indistinguishable, which allows making conclusion about covert operation of the proposed method of information transmission.

To study noise immunity, an AGWN channel was modeled by adding a Gaussian white noise to the transmitted waveform. Figure 8(d) and (e) show the output signal coming from the comparator output for the case when the signal at the receiver input is completely hidden by noise (signal-to-noise ratio is S=N ¼ �6 dB and �12 dB, correspondingly). Simulation results show that with the system parameters selected, that provide the time-bandwidth product B ≈ 500, information recovery occurs correctly at the signal-to-noise ratios of at least �14 dB. With a further increase in the power of additive interference, the envelopes at the output in each channel have an unacceptably large dispersion, resulting in false triggering of the key circuit in the decoder at the receiver output leading errors in the information bit sequence recovering.

The use of a delayed feedback system as a source of chaotic carrier allows simplifying the scheme of a transmitter with switching chaotic modes compared to, for example, the device described in [26], where the procedure for forming a spectrum with alternating maxima and minima applies to a pre-generated noise signal. In our approach, a special feature of the spectral characteristics of oscillations in a system with delayed feedback of ring type was used for this, which eliminated the need to include additional signal conversion units to obtain the spectrum of the required shape.

To transmit one information symbol of duration TS, it is necessary to fulfill the condition TS > T<sup>0</sup> >> Δt, where T<sup>0</sup> is the delay time equal to the analysis time of the signal spectrum at the receiver, Δt is the duration of one sample of the chaotic carrier (one information symbol is transmitted for a sequence of M ¼ TS=Δt samples of the chaotic signal). During transmission of the symbol "0" a continuous chaotic signal enters the communication channel, in the

π=T0, n∈ N. When symbol "1" is transmitted, the maxima in the spectrum of the transmitted

In the receiver, to derive the information from a chaotic signal, its analysis in the spectral domain is used. A signal with a structural feature of the spectrum in the form of equidistantly located spectral density maxima can be efficiently distinguished with a comb filter, the trans-

the amplifier gain and the delay time in the feedback loop, respectively. For a positive value G, the filter frequency response is matched to the chaotic signal spectrum at 0:5 < a < 1 (transmission of information bit "1"), whereas for its negative value the filter frequency response is matched to the chaotic signal spectrum at 0 < a < 0:5 (transmission of information bit "0").

The received signal is simultaneously feed two comb filters, one of which has a positive gain in the feedback loop (G<sup>1</sup> ¼ 0:9) and the other has negative one (G<sup>2</sup> ¼ �0:9). A detector is connected at the output of each filter, which estimates the dispersion of the incoming signal. The signals from both detectors come to the inputs of a comparator and then to a decision device, at the output of which the logical "0" is formed if the signal at the output of the first channel exceeds

As follows from the simulation results, presented in Figure 8, the binary sequence at the output of the receiver (Figure 8(f)) repeats the modulating sequence in the transmitter (Figure 8(a)). In this case, the signal of the transmitter in the communication channel looks like noise waveform, the moments of changing the information bits are not detected from the observable time series (Figure 8(b)). The fragments corresponding to transmission of information bits "0" and "1" are visually indistinguishable, which allows making conclusion about covert operation of the pro-

To study noise immunity, an AGWN channel was modeled by adding a Gaussian white noise to the transmitted waveform. Figure 8(d) and (e) show the output signal coming from the comparator output for the case when the signal at the receiver input is completely hidden by noise (signal-to-noise ratio is S=N ¼ �6 dB and �12 dB, correspondingly). Simulation results show that with the system parameters selected, that provide the time-bandwidth product B ≈ 500, information recovery occurs correctly at the signal-to-noise ratios of at least �14 dB. With a further increase in the power of additive interference, the envelopes at the output in each channel have an unacceptably large dispersion, resulting in false triggering of the key circuit in the

decoder at the receiver output leading errors in the information bit sequence recovering.

The use of a delayed feedback system as a source of chaotic carrier allows simplifying the scheme of a transmitter with switching chaotic modes compared to, for example, the device

<sup>n</sup> ¼ 2ð Þ n � 1 π=T0, n∈ N.

<sup>1</sup> <sup>þ</sup> <sup>G</sup><sup>2</sup> � <sup>2</sup><sup>G</sup> cos <sup>ω</sup>T<sup>0</sup>

<sup>p</sup> , where <sup>G</sup> and <sup>T</sup><sup>0</sup> are

<sup>n</sup> ¼ ð Þ 2n � 1

spectrum of which the positions of maxima are determined from the condition Ω<sup>0</sup>

mission coefficient modulus of which is j j <sup>K</sup>ð Þ <sup>ω</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

the signal at the output of the second one and the logical "1" in the opposite case.

signal correspond to the condition Ω<sup>1</sup>

72 Telecommunication Networks - Trends and Developments

posed method of information transmission.

Figure 8. Results of simulation modeling: (a) initial information impulse sequence corresponding to the transmitted message; (b) a fragment of the time series of the signal in the communication channel during the sequential transmission of "1" and "0"; (c) signal at the output in the absence of interference in the communication channel; (d) output signal under the influence of additive Gaussian white noise in the communication channel with a level of þ6 dBrelative to the signal level; (e) output signal under the influence of additive Gaussian white noise in the communication channel with a level of þ12 dB relative to the signal level; (f) restored information sequence.
