Figure 5.1: Double-channel chaotic secure communication scheme based on drive–response synchronization.

where k is a compression ratio constant, which ensures that the useful signal m(t) is much less than the chaotic signal. x1 is the masking signal, and x2 is the synchronous driving signal.

At the receiver, the mathematical model of the demodulator is defined as

Obviously, as long as the synchronization between the response system and drive system is achieved completely, the useful information can be demodulated correctly. In order to save the channel resources, this scheme is improved, and the principle is shown in Figure 5.2. In this scheme, the driving signal consists of useful signal and chaotic signal, that is, the drive signal is the sum of the useful signal and chaotic signal, so only one channel is required to realize the secure communication. Here, taking Lorenz chaotic system as an example, the chaotic system equations for the sender and the receiver are given by, respectively,

Figure 5.2: Single-channel chaotic secure communication scheme based on drive–response synchronization.

In this scheme, the channel resource is saved. However, one of the obvious characteristics is that the drive signal is disturbed by the message signal, and synchronization cannot be accurately realized, which leads to the fact that the information signal cannot be fully recovered. Another notable feature is that the information signal doesn’t have any impact on the dynamic characteristics of the driving system, so the security of the secure communication system is not high enough, and it cannot effectively resist the third-party attack. To improve the security further, an improved scheme was proposed in Ref. [114] as shown in Figure 5.3. In this scheme, the sum of the information signal and the chaotic signal feedbacks to the drive system, so that the drive system and the response system have the same dynamic characteristics, so it can realize the complete synchronization, and the receiving signal can be demodulated completely. As the information signal is integrated into the chaotic system, it is more difficult for the attacker to obtain useful information from the channel. Here, taking Lorenz chaotic system as an example, the chaotic system equations for the sender and the receiver is written as, respectively,

Figure 5.3: Improved single-channel chaotic secure communication scheme based on drive–response synchronization.

To improve the secure performance of chaotic masking communication further, Carroll et al. [115] spread this improved chaotic masking scheme to the high-order-linked synchronization system, and its synchronization principle and secure communication scheme are shown in Figure 5.4. In this scheme, the receiver and the sender are composed of two cascaded chaotic systems. Taking Lorenz chaotic system as an example, the equations for the sender and the receiver are presented as, respectively,

Figure 5.4: Improved single-channel cascade chaotic secure communication based on drive–response synchronization.

5.2Chaos Shift Keying Communication

First, in 1993, Parlitz et al. [116] and Dedieu et al. [41] proposed the digital modulation method of CSK. Soon, Kolumban et al. [117, 118] successively put forward the improved CSK digital communication system, which mainly includes chaotic on–off keying (COOK), differential chaos shift keying (DCSK), frequency modulation differential chaos shift keying (FM-DCSK), quadrature chaos shift keying (QCSK) [119], and so on. Among these schemes, DCSK and FM-DCSK have better antinoise performance [120]. FM-DCSK was listed as a long-term research program by European Commission. In 2000, IEEE Transactions on CAS-I published the 12th special issue about chaos to discuss the performance of a class of digital modulation schemes. Compared with DCSK, QCSK has the same bandwidth and bit error rate (BER) performance, but the data transmission rate is higher. There are two demodulation methods, such as coherent demodulation and noncoherent demodulation.

CSK is a kind of chaotic modulation technology for transmitting digital information. The basic idea is to map the different information code of binary (or multinary) digital signal to the piecewise chaotic waveform. The modulated signal is detected in the receiver based on the correlation reception theory. It needs a chaotic basis function or multiple chaotic basis functions for CSK modulation. A basis function of CSK has two types at least, such as COOK and CSK. The antinoise performance of bipolar coherent demodulation of CSK is better than that of COOK. If the bit energy remains constant and the chaotic basis function is recovered accurately at the receiver, the antinoise performance of the CSK is equivalent to that of Binary Phase Shift Keying (BPSK). The noncoherent detection cannot detect the bipolar CSK, and the antinoise performance of the noncoherent COOK is worse than that of the coherent COOK, but the COOK can achieve the best antinoise performance when the bit energy keeps constant. The decision threshold of the demodulation of reverse-phase CSK and COOK depends on the signal-to-noise ratio (SNR). When two basis functions are adopted, this problem can be solved. When symbols “0” and “1” are mapped to two different chaotic waveforms, the CSK of the two basis functions is formed, and the antinoise performance of coherent demodulation is better than that of the noncoherent demodulation. Especially, through choice of orthonormal basis functions, the decision threshold level becomes zero, and the performance has nothing to do with the SNR, and its performance is similar with that of coherent Frequency Shift Keying (FSK).

5.2.1COOK Communication

COOK is the most simple CSK communication, and the structure of basic principle is shown in Figure 5.5. {bi} is a binary information signal with “1” and “–1.” When bi = 1, the switch is turned on. When bi = –1, the switch is turned off. At the receiver, the information signal is recovered by employing the correlation algorithm. If the input signal is of equal probability distribution, and the average bit energy for a single symbol to E b, then the code energies of “1” and “–1” are 2Eb and 0, respectively. Obviously, the larger the bit energy gap, the better the antinoise performance of the signal. The structure of this scheme is simple. For example, it only needs a chaotic oscillator, but the security of the system is not high.

Figure 5.5: System principle block diagram of noncoherent COOK modulation/demodulation.

In the receiver, coherent or noncoherent mode is used to recover the information signal, and the principle of coherent demodulation is shown in Figure 5.7. The output of the correlator is

Obviously, if bi = 1, then si(t) = g1(t), ri(t) = g1(t) + n(t). We obtain zi1 > zi2, and the recovered signal is bi = 1. If bi = –1, then si(t) = g2(t), ri(t) = g2(t) + n(t). We have zi1 < zi2, and the recovered signal is bi = –1.

Figure 5.6: Block diagram of CSK modulator.

Figure 5.7: Principle block diagram of CSK coherent receiver.

Figure 5.8: Principle block diagram of CSK noncoherent receiver.

For the CSK noncoherent demodulation, the receiver need not recover the chaotic carrier, and the detector is based on the bit energy of the transmission signal. If the decision threshold is set to the median of the two bit energies, then the received signal can be demodulated correctly. Obviously, the selection of the optimal threshold is closely related to the SNR of the system. So the threshold shift is the main defect of the noncoherent demodulation.

Next, we take CSK digital secure communication system based on the unified chaotic system as an example to illustrate, and the system structure is shown in Figure 5.9. The chaotic systems of the transmitter and receiver are unified chaotic system. The chaotic system 11 in the transmitting end and the chaotic system 21 in the receiving end have the same parameter α1, and the chaotic signal generated by them are x11(t) and x21(t), respectively. The chaotic system 12 in the transmitting end and the chaotic system 22 in the receiving end have the same parameter α2, and the chaotic signals generated by them are x12(t) and x22(t), respectively. By employing the synchronization strategy, the unified chaotic systems 11 and 12 at the sending end are synchronized with the unified chaotic systems 21 and 22 at the receiving end, respectively, then we have x11(t) = x21(t), x12(t) = x22(t). The binary digital signal a(t) = {an} is used to control the chaotic signal. When a(t) is of high voltage, x11(t) is sent. When a(t) is of low voltage, x12(t) is sent. The transmitted signal in the channel is expressed as

Obviously, the signal transmitted in the channel is chaotic. If the noise in the channel is Gaussian white noise n(t), then the signal at the receiving end is expressed as

Figure 5.9: Principle block diagram of CSK digital secure communication system based on unified chaotic system.

Figure 5.10: Correlation characteristic of chaotic signals generated by the unified chaotic system: (a) autocorrelation and (b) cross-correlation.

where ρ = x11(t)x12(t)dt/Eb is the correlation coefficient of chaotic signals. When T the two signals are completely orthogonal, ρ = 0 is the bit energy of the chaotic signal. Obviously, if the correlator output signals, z1 and z2, are sampled to be decided, then the original signals can be recovered by the decision rules

Obviously, ignoring the effect of noise, the noise tolerance of coherent demodulation in the receiver system is Nc = (1 – ρ)Eb. The output signal of the decision device is b(t). It can be seen that the demodulation performance of the system is related to the characteristics of the chaotic signal. Because the autocorrelation of the chaotic signal generated by the unified chaotic system with different parameters is close to δ(t), and its cross-correlation is very weak, the system has a good correlation demodulation performance and the original signal can be recovered, that is b(t) = a(t).

Next, we discuss the BER performance and security performance of CSK secure communication based on unified chaotic system. Let the two possible deterministic signals arriving at the receiver be x1(t) and x2(t), respectively, and their durations are (0, T), and have the same energy Eb. The noise n(t) at the input of the receiver is Gauss white noise with zero mean, and its unilateral power spectral density is n0. According to the principle of the best reception, in the case of equal priori probabilities, the limit BER performance is

where erf(x) is the error function, and ρ is the cross-correlation coefficient between signals x1(t) and x2(t), and its range is (–1, 1). Because x1(t) and x2(t) are orthogonal signals, the cross-correlation coefficient is zero. The BER performance of the system is rewritten as

The BER simulation result of the CSK system is shown in Figure 5.11. When the SNR is greater than 10 dB, the BER is less than 10–6.

Figure 5.11: BER performance of the CSK system.

Figure 5.12: Simulation model of CSK digital secure communication based on unified chaotic system.

Figure 5.13: Synchronization phase diagram of the variables between the sender and receiver: (a) x11(t)–x21(t) and (b) x11(t)–x22(t).

Figure 5.14: Simulation results of the CSK secure communication system.

Figure 5.15: Principle block diagram of the DCSK modulator.

Figure 5.16: Principle block diagram of the DCSK coherent demodulation.

Because each bit is mapped to two continuous segments with the length T/2, the information signal at the receiver can be recovered by coherent demodulation. The demodulation principle of the DCSK scheme is shown in Figure 5.16. In this demodulation scheme, the output signal of the correlator is

Compared with CSK and COOK, the advantage of the DCSK scheme is that the decision threshold is zero whether there is noise or not. The reference signal and carrying information signal go through the same channel, so it is not sensitive to channel distortion, and it has a good antinoise performance. The shortage of DCSK is that it can only be used in the chaotic digital communication with low bit rate. To solve this problem, in 1998, Kolumban et al. proposed a DCSK scheme based on FM-DCSK [118].

5.2.4FM-DCSK Communication

FM-DCSK is constituted by DCSK and a frequency modulator. The output of FM is a chaotic signal with band-limited and uniform power spectrum density. The structure of FM-DCSK modulator is shown in Figure 5.17. It employs the chaotic analog phase-locked loop (APLL) to generate band-pass chaotic signal, and then FM is applied for chaotic signals. The subsequent parts are consistent with DCSK, which transmitsreference signal in [0, T/2] and transmits same phase chaotic FM signal or inverse phase chaotic FM signal in [T/2, T].

Figure 5.17: Principle block diagram of FM-DCSK modulation.

Figure 5.18: Principle block diagram of FM-DCSK demodulation.

If x(t) is a chaotic reference signal defined in the interval t ∈ [0, τ], and it has a zero mean. In the interval t ∈ [0, τ], x(t) can be expanded as follows by Fourier series (A0 = 0):

where ω = 2π/τ. Correspondingly, the average power of x(t) in the interval t ∈ [0, τ] is

The orthogonal signal y(t), in the interval t ∈[0, τ], can be obtained by each Fourier frequency component of x(t) with the phase shift π/2, namely

Then the signals x(t) and y(t) are orthogonal over the interval t ∈ [0, τ] and their power is equal, i.e.

The above properties are equivalent to

According to definition formula (5.26) of the reference quadrature signal y(t), the x(t) and y(t) are extended into the periodic signal with the periodic τ. It is found that y(t)is the Hilbert transform of x(t), and the Hilbert transform of the given signal can be calculated by the following methods:

1. Frequency domain method: It is basically the same as the method of generating Hilbert transform in Amplitude Modulation - Single Side Band modulation. A fast Fourier transform (FFT) is applied to a chaotic sequence with length k, and then remove the negative frequency (signal energy) corresponding to FFT coefficients. It is obtained by inverse FFT transformation. At present, the algorithm can be realized by a special digital signal processing (DSP) chip.
2. Time domain method: It is achieved by the design of digital Hilbert converter with finite impulse response (FIR) or infinite impulse response (IIR). A simple method for calculating the Hilbert transformation of a given discrete signal is to design an FIR filter which is approximated by Hilbert transform. The FIR filter can be implemented by the ideal impulse response with appropriate width:

In the interval I = [0, τ], a chaotic sampling orthogonal base function is defined as

where E4 = Pxτ is the energy of x or y over the interval I. So we have

The orthogonal condition (5.32) indicates that the energy of cx and cy is constant at each interval I. The constant is one in special case. The chaotic signal can be represented by a linear combination of cx and cy, namely ms(t) = ascx(t) + bscy(t), where s represents a signal space symbol. Equivalently, the symbol s can be expressed in plural form, namely,

Being similar to DCSK and QCSK, the modulator structure is shown in Figure 5.19. It includes a chaotic generator, which generates the signal cx at each interval [0, T/2], and the corresponding quadrature signal cy is generated by the filter. For simplicity, it is assumed that the system has no time delay. According to the symbol s, the encoder generates a linear combination of cx and cy. Information symbol is provided by a bit/symbol converter, and the converter maps each pair of information bits into a symbol. Thus, each symbol period transmits 2 bits information. Then, the QCSK signal is formed by the two-channel analog multiplexer with the switching time T/2. Note that although there is no filter in the figure, the QCSK scheme must contain a low-pass band-limiting filter to limit the bandwidth of transmission signal. The chaotic reference signal segment r(t) = Ebcx(t) is transmitted in the first half symbol period of the transmission symbol s, and the information segment i(t) = Ebms(t) is transmitted in the second half of the symbol period, where s represents the symbol that needs to be transmitted, while Eb represents the energy per bit. The orthogonal condition (5.32) shows that the energy per transmitted bit is a constant. If the symbol period is T = 24, then the QCSK signal is

Figure 5.19: Principle block diagram of QCSK modulator.

where s = 0, 1, 2, 3.

The simplified block diagram of QCSK demodulator is shown in Figure 5.20. QCSK signal may be demodulated by the method of differential coherent detection. During the process of demodulation, the symbol period is assumed to be synchronized. That is, the beginning of each symbol is taken as the starting of the coherent process at the receiver.

Figure 5.20: Principle block diagram of QCSK demodulation.

At the receiver, we can observe noise, filtered reference signal r(t), and information segment i(t). If the signal is only affected by the additive Gaussian white noise, then the input of the correlator A in Figure 5.20 is

The estimated value of the quadrature signal H[r̃] is generated by the disturbed chaotic reference signal, where H is the Hilbert transform operator:

where n′(t) = H [n(t)]. With the ideal Hilbert transform, the statistical properties of n′(t) are the same as that of n(t).

Considering the orthogonality of cx and cy, the output of the correlator is

CSK belongs to chaotic digital secure communication. When the basis function is strictly recovered, the noise performance of the coherent reception is better than that of the noncoherent reception. If the orthogonal basis function is chosen, the chaotic keying and the traditional digital modulation can achieve similar noise performance. However, at present, it is difficult to meet the requirements of the strict recovery basis function, and it is more difficult to regenerate the chaotic basis function than to regenerate the periodic basis function. As a result, the performance of the coherent chaotic keying communication is slightly less than that of the same kind of traditional digital communication. But CSK belongs to broadband communication, and the performance of chaos communication shows more superiority in multipath communication environment. Among CSK communication schemes, QCSK and FM-DCSK have better performance and application prospect.

5.3Chaos Parameter Modulation

Chaos parameter modulation is also known as wide spectrum transmission. The basic idea is that an information signal is injected into the transmitter, and thus it changes the dynamic characteristics of the original chaotic system. So the information signal is modulated. Compared with CSK and chaotic masking, chaos parameter modulation has the following advantages. First, it employs the whole range of the chaotic signal spectrum to hide information. Second, it increases the sensitivity to parameter variation, thereby enhancing confidentiality.

For CSK technology, no matter when the value of binary information signal changed, there is always a synchronization waiting time because of the difference of both initial conditions. While for chaos parameter modulation, receiver tracks transmitter continuously, and two chaotic system states would surely be the same. Compared with chaotic shift keying, it is expected to get a higher bit rate. In addition, with further development of chaos modulation technology, the direct chaotic spectrum expansion technique is also used as a chaos modulation technique.

Research results show that the main drawback of chaos communication scheme is being susceptible to channel noise and distortion. The traditional masking communication requirements of the power information signal are far lower than that of chaotic signal, which makes chaotic masking communication more difficult to be used in practice. A new method of chaotic pulse position modulation (CPPM) is proposed in Ref. [121], which shows good performance in the case of channel distortion. This scheme is based on the chaotic pulse sequence, and the interval between the pulses is determined by the chaotic characteristics of the nonlinear function. The pulse sequence of chaotic pulse interval modulation is used as a carrier. Binary information is modulated onto a carrier by pulse position modulation (PPM). As a result, each pulse is unchanged or delayed some time depending on the transmission information “0” or “1.” The pulse duration of “0” or “1” is estimated by chaotic pulse sequence synchronization at the receiver, and thus the transmission information is demodulated in this way.

The block diagram of chaotic pulse position modulator is shown in Figure 5.21. The chaotic pulse signal generated by chaotic pulse generator is expressed as

Figure 5.21: Principle block diagram of chaotic pulse position modulator.

where W(t – tj) represents a pulse waveform generated at time tj, and tj is expressed as

where Tn is the time interval between the nth pulse and the (n–1)th pulse. The time interval sequence Ti represents the iteration of the chaotic process. For the design convenience, we consider the chaos is generated by a one-dimensional map Tn = F (Tn–1), where F (⋅) is a nonlinear function.

The information is encoded in the chaotic pulse signal by adding the delay time in the pulse interval Tn. Thus, the resulting pulse sequence can be defined as

where Sn is the information signal. Only binary information is considered here, so Sn is equal to 0 or 1. m represents the modulation degree, and the parameter d is the time delay constant for the modulation and demodulation. When we design chaotic pulse generator, the choice of the nonlinear function F (⋅), and parameters d and m should ensure the chaotic performance of the map.

The transmission signal is a CPPM signal in the channel, which is denoted by

where Tn is generated by formula (5.43), and we assume that the duration of each pulse W(t) of the pulse sequence is much less than the minimum value of the pulse interval Tn. When detecting information at the receiver, the demodulator is triggered by receiving pulse U(t). After measuring successive time intervals Tn and Tn–1, the information signal is recovered as

If the nonlinear function F (⋅), parameters d and m at the receiver are same with those of the sender, then the encrypted information Sn can be easily recovered. When the matching accuracy of nonlinear functions is not sufficient, large decoding error occurs.

The demodulator principle is shown in Figure 5.22. When the demodulator is synchronized with the pulse modulator, to detect a single bit of the transmission information, the demodulator must simply determine whether the desired position of the transmitted pulse is delayed. In the figure, the integrator, sample and hold circuit, and a nonlinear threshold function module are reset or triggered by the pulse from the transmitter, not by the feedback loop within the receiver. To be exact, only when the input signal U(t) from the input channel exceeds the threshold, they are triggered. For each received pulse, the result of V(tn+1) – F(vn) is calculated and used to determine whether the pulse is delayed. Assuming that the difference is less than the reference value β(d + m/2), the data bit Sn+1 is determined to be “1,” otherwise, it is determined to be “0.”

Another important part of the receiver is the window selection module. Once the receiver has correctly received two consecutive pulses, the receiver can predict the fastest time to receive the next pulse. It means that the input of the demodulator can be masked at this time, which is done by the window selection module. In the experiment circuit, the input of the receiver is opened by a window control pulse generated by the comparator at the moment and the input end is turned on until the demodulator is triggered by the first received pulse. So the window selection module greatly reduces the probability of the receiver triggered by the noise, interference, or the pulse of other users.

Figure 5.22: Principle block diagram of the chaotic pulse position demodulator.

where di(n) is the information bit value of the ith user, di(n) ∈ { – 1, 1}. Let xi(n) be the chaotic spread spectrum sequence of the sender of the ith user, and its expression is where gi(n) is the spread spectrum characteristic sequence of

Figure 5.23: Principle block diagram of multiuser chaotic sequence spread spectrum communication system. the ith user.

where Ai(n) is the carrier signal amplitude of the ith user. N (n) is the zero mean Gauss white noise. So, the output of the correlator of the receiver of the ith user is

where is the chaotic spread spectrum sequence of the receiver of the ith user, and the system synchronizes between the sender and the receiver, that is, Let Tb be the time width of each original information bit, then

For formula (5.49), the first item is the desired information component. The second item is the recovery data obtained by the received signal with the kth signal itself, and multiple access interference (MAI) caused by other users. Obviously, the MAI term is related to the user number k, the amplitude of the signal of other user, and the cross-correlation coefficient ρi,k, where The third item is noise interference.

After the integral (correlation operation) of the receiver for the ith user, the signal is decided by the decision threshold, which is zero. The data is demodulated by the ith user at the receiver according to

To sum up, whether the user information is correctly recovered is greatly related with the characteristics of the spread spectrum code.

5.4.2Generation of Chaotic Spread Spectrum Code

Taking three typical chaotic maps, including logistic, Chebyshev, and Tent map as examples, the binary chaotic spread spectrum sequence is generated [123]. In the multiuser chaotic sequence spread spectrum communication system, spread spectrum sequence is a binary sequence with value ±1, namely, the real values of chaotic map sequences are quantized into binary to generate spread spectrum sequence.

5.4.2.1Logistic Chaotic Map

Logistic map equation is

where r is the system parameter, and r ∈ (0, 4]. When 3.5699 . . . < r ≤ 4, the system is chaotic. The mean value of the sequence is 0.5.

5.4.2.2Chebyshev Chaotic Map

The ω-order Chebyshev chaotic map is defined as

where ω is usually taken as an integer power of 2, and the mean value of the sequence is 0.

5.4.2.3Tent Chaotic Map

The Tent chaotic map is defined as

where 0 < a < 1, and the mean value of the sequence is 0.5.

Next, we discuss the balance and correlation of the three kinds of chaotic sequences.

5.4.2.3.1 Balance of the Sequences

The balance of a sequence is determined by the statistical distribution about “1” and “–1” in the sequence. If P and Q denote the number of “1” and “–1” in the chaotic sequence, respectively, then the balance degree of the sequence E is defined as

where N is the total number of symbol sequence. The sequence balance is related with the carrier suppression degree of the DS/SS system. The imbalance of spread spectrum sequence will lead the leakage of carrier information, or be prone to error or lost. Thus, it is important to study the balance of the direct spread spectrum system (DS/SS).

Applying formula (5.54) to analyze the impact of the initial value of three kinds of chaotic maps, the results are shown in Figure 5.24. When r = 4 for the logistic map, the initial value x0 is one of the 9,999 different numbers between 0 and 1. From Figure 5.24(a), we can obtain that the effect of x0 on the balance is small, but there are three peaks, and the initial value x0 = 0.25, 0.5, and 0.75, respectively. When ω = 20 for the Chebyshev map, the initial value x0 belongs to one of the 9,999 different numbers between 0 and 1. From Figure 5.24(b), we found that the effect of x0 on the balance is very small, and there are no peaks. When a = 0.49 for the Tent map, the initial value x0 forms the 9,999 different numbers between 0 and 1. From Figure 5.24(c), we find that the effect of x0 on the balance is small, and there are three peaks, and the initial value x0 = 0.2401, 0.49, and 0.7501, respectively. It is worth pointing out that the peak point of Tent map is related with the initial value selection. If the accuracy of the initial value is not enough, the peak of this map may be missed.

5.4.2.3.2 Correlation of the Sequence

From formula (5.49), in the case of same SNR, and without consideration of multipath interference, MAI becomes the major factor in the transmission performance of the system. Root mean square (RMS) of cross-correlation of the sequence characterizes the size of the MAI of the spreading sequence, so RMS of a sequence is important for analyzing the transmission performance. As a result, the actual simulation of the chaotic sequence has a deviation with the theoretical value. Here, we change the length of the chaotic sequence N and maintain the relevant range M, to study the effect of the length of the sequence on the deviation. The RMS 3R12 is defined as

Figure 5.24: Relationship between the balance E and the initial value x0, for different maps: (a) Logistic map; (b) Chebyshev map; and (c) Tent map.

Here, we use formula (5.55) to calculate the RMS values of the cross-correlation of the sequence with length N = 2,000 generated by the above three kinds of chaotic maps. The results are shown in Table 5.2.

From Table 5.2, after the frame processing, RMS value of cross-correlation for each chaotic sequence has been greatly affected, and its change is different. Besides, different frame lengths correspond to different RMS values, which destroy the original relevant characteristics. From Table 5.2, after the frame processing, the crosscorrelation of three chaotic sequences becomes smaller, and it also shows that the chaotic sequence with a certain length has a better cross-correlation compared with the original sequence. But there is no fixed relationship between the frame length and the RMS value, that is to say, after the frame processing, chaotic sequence destroys the original conclusion. That is, the longer the sequence, the smaller the deviations of correlation.

5.4.3Simulation Module Design of Multiuser Chaotic Spread Spectrum System

Multiuser chaotic spread spectrum system involves the design of multiple modules, including the spread spectrum sequence generator module, the correlation receiving module, the multipath channel module, the polarity transformation module, and the frame sampling module. The next sections introduce each of these.

5.4.3.1Spread Spectrum Sequence Generator Module

The chaotic spread spectrum code is generated by Chebyshev, improved logistic, logistic, Tent, and other iterative chaotic maps. In addition, the real value is transformed into the binary sequence by setting a decision threshold ξ, and the real value is converted by the decision function T(x) = [1 + sgn(x – ξ)]/2. Here, two design methods of the chaotic sequence generator module are introduced. One is the design of spread spectrum sequence generator based on Tent sequence, and the other is the design of spread spectrum sequence generator module based on series chaotic map.

Table 5.2: RMS values of three chaotic sequences.

Figure 5.25: Simulation model of chaotic sequence generator based on Tent map.

where u(1) represents xn, and u(2) represents the bifurcation parameter a.

The simulation model of the spread spectrum sequence generator module based on series chaotic map is shown in Figure 5.26. The first chaotic sequence of series sequence adopts improved logistic map, and the second chaotic sequence uses Chebyshev map. Generally, the sampling time of the first chaotic sequence is 1/n times the sampling time of the second chaotic sequence. In this system, we set the sampling time of the first chaotic sequence as 1/192 s, and then the sampling time of the second chaotic sequence is 1/192,000 s. Let the sampling time be 1 s, namely x1 = y1, x2 = y1,000, . . . , x192 = y191,000, that is, the first chaotic sequence should control the second chaotic sequence to generate 1,000 chaotic sequences per second. For simulation, the initial values of the two chaotic sequences are set first. Because x1 = y1, the two chaotic sequences must be set to the same initial value in the series simulation, namely, the initial values of two unit delay modules are the same. Multiple n of the sampling time of the two-stage sequence is controlled by a periodic sequence generator, and the sampling time of the periodic sequence generator can be the same as the sampling time of a chaotic sequence generator. In the multiuser chaotic spread spectrum communication system, when the frames are sampled, it is required that the values of chaotic sequences are discrete. So when the 192nd value controlled by clock arrives, we obtain the chaotic sequence yn, rather than yn+1, and at this moment yn = y192,000. Otherwise chaotic sequence generator cannot work due to the value of the continuous chaotic sequence. Fcn1 in Figure 5.26 is expressed as

Figure 5.26: Simulation model of spread spectrum sequence generator based on series chaotic map.

where u(1) represents xn of the Chebyshev map, u(2) is the bifurcation parameter q of the Chebyshev map, u(3) represents xn+1 of improved logistic map which belongs to the first chaotic sequence, and u(4) is the sampling time of multiple n.

5.4.3.2Correlation Receiving Module

The block diagram of traditional correlation detection at receiver is shown in Figure 5.27. The dashed box in the figure corresponds to a match filter.

Integral module is the core part of the spread spectrum system, which consists of unbuffer module, integral sampling module, zero-order hold module, and frame sampling module, as shown in Figure 5.28. It works as follows. Each frame data (N ∗ 1) from In1 module is decomposed to (1 ∗ N) sequences by unbuffer module, and then do sampling integral. The zero-order hold module makes the time duration, hold time, and information code sampling time of each binary symbol to be the same after the integration. Finally, they are converted into frame format by the sampling module. The integrator parameter settings are as follows: Output sampling time is the sampling time of information code, and it is the same with simulation step. The sampling number of the buffer module is N.

Figure 5.27: Principle block diagram of the detector of the traditional DS-Code Division Multiple Access system.

Figure 5.28: Simulation block diagram of correlation receiver.

where the weighted coefficients di(t) are defined as

Obviously, when the correlator (e.g. correlator i) fades, the energy of output ci is small, and the corresponding weighted coefficients di also decrease automatically, and it leads the component in formula (5.59) to decrease.

The simulation model of the Rake receiver is shown in Figure 5.30. Assuming that the data transmitted in channel appears four multipath transmissions, and the time delay is 0, 2, 6, and 7, respectively. At the receiver, these multipath signals are done in the time period corresponding 0, 2, 6, and 7 in advance, and then sum them. Finally, the equal-gain combining will decrease the BER significantly. The equal-gain combining means that the various branch signals are summed in the same phase, and the weighted factors of each branch are equal.

Figure 5.29: Structure block diagram of rake receiver with L channels.

Figure 5.30: Simulation model of Rake receiver.

where L is the number of decomposable paths in the multipath channel. αl, ϕl, τl are the gain, the phase, and the time delay of the lth decomposable channel, respectively. δ(t) is the delta function. Under the condition of Rayleigh channel, αl obeys the Rayleigh distribution, and ϕl obeys the uniform distribution in [0, 20).

The simulation model of multipath channel is shown in Figure 5.31. Through the multipath channel, signal becomes four columns delayed signal, of which the time delay is 0, 2, 6, and 7, respectively, and the frame sampling format is used. Note that the sampling time of multipath channel should be the same as that of information code.

5.4.3.5Polarity Transformation Module

The operator of spreading spectrum and de-spreading spectrum can be realized by the XOR operator of the unipolar binary code, but it is difficult to process when the result is zero sometimes. When the signal is superimposed noise, it is no longer a binary code, and we cannot use XOR operator at this time. Employing multiplication of bipolar binary symbols can also realize spreading and de-spreading operation, and can overcome the shortcomings of the above-mentioned method. During simulation, the polarity conversion is realized by bipolar to unipolar converter module and unipolar to bipolar converter module.

Figure 5.31: Multipath channel simulation model.

Figure 5.32: Simulation model of multiuser chaotic spread spectrum communication system.

Figure 5.33: Waveforms of (a) transmitted signal and (b) received signal.

Figure 5.34: Relationship curves between the BER of spread spectrum system and the initial value x0 of different maps.

Figure 5.35: BER of spread spectrum system with five kinds of spreading codes.

Figure 5.36: BER of the system with three kinds of chaos spreading code for different users.

Figure 5.37: BER of the system for different sequence lengths.