# Possibilities In Using Chaotic Modulation Computer Science Essay

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Chaotic modulation uses chaotic signals instead of the traditional sinusoids. The various modulation techniques like chaotic modulation (CM), chaotic masking (CMS), or chaotic shift keying (CSK), differential chaos shift keying(DCSK),modified differential chaos shift keying (M-DCSK) , FM-Differential Chaos Shift Keying (FM-DCSK), trellis-based chaotic modulation (called as chaotic-coded modulation: CCM), and their advantages are discussed.

The major concentration of the project is on the comparing the Bit Error Rate (BER) performance of DCSK and M-DCSK in AWGN channels.

Chaotic modulation has gained the attention because of its properties such as non-linearity, noise-like signal and pseudo linearity.

The idea of chaotic modulation is driven by two main facts. First would be due to its efficient hiding of information. Second, due to the broad-band spectrum of chaos a narrow-band information signal can be modulated over a wide-band carrier, thereby spread-spectrum communication can be used.

In terms of channel noise, it is treated as a secure transmission scheme rather than an efficient scheme. Secure transmission is important in wireless field, but usually encryption is done at a higher layer, which leaves the lower layers unsecured i.e. the raw data at these layers can be read. When chaotic modulation is used the information bits to be transmitted are encoded by the chaotic signal instead of the sinusoids so the information bits are not readable even at the physical layer. Hence communication scheme based on chaotic modulation increases security.

Transmission performance is limited by noise in conventional chaotic communication schemes as synchronization of chaos is difficult in presence of noise.

The paper is organized as follows. Section II introduces the various modulations. Comparison between various modulations in done in Section III, concluding it in Section IV.

Chaotic systems generate signals which are purely deterministic, although they show features typical of random signals due to the numerical instability of the system (i.e. sensitivity to initial conditions). In a signal processing context, chaotic signals and systems have been proposed for a wide range of applications: communications, watermarking, cryptography, time series modeling, etc. Many different chaotic communications systems have been proposed: chaotic modulation, chaotic masking, chaos shift keying (CSK) and its variants, spread spectrum techniques, etc.

Chaotic modulation is a way to transmit digital informations by means of chaotic signals instead of sinusoids. Currently proposed schemes are obtained from traditional transmission systems by replacing sinusoids with chaotic signals. The major drawback of such systems is represented by their poor performances when compared with traditional transmission systems.

In Chaotic modulation, the message modifies the state or the parameters of the chaotic generator through an invertible procedure, thus the generated chaotic signal inherently contains the information on the transmitted message. Chaotic modulation with multiplication and feedback (CMMF) has been proposed by the authors. In this method, the message modulates by multiplication the chaotic output signal, the resultant signal

is fed back to the transmitter system and simultaneously sent to the receiver system. In order to synchronize with the transmitter, the receiver is merely a copy of the transmitter which is driven by the received signal.

A dual chaotic modulation scheme, which modulates the delay coefficient and the switching frequency of switching control signal with chaotic signal. The new modulation scheme is more efficient in dispersing the harmonic power than single chaotic modulation scheme, so it is advantageous in improving power converters' EMC.

OPTICAL communications based on chaotic carriers have attracted great interest in recent years. Taking advantage of the undesirable, for the conventional optical communication systems, coherence collapse regime of the semiconductor lasers, they provide a competent solution for secure practical applications. One of the most commonly used techniques for chaos generation is a semiconductor laser subjected

to external optical feedback or to injection by other semiconductor laser sources. However, other structures employing fiber lasers or using optoelectronic conversion have been theoretically developed and experimentally realized by several researchers as well. Focusing on the all-optical semiconductor laser approach, a plethora of work has

been presented concerning the synchronization properties of such chaotic systems forming either an open- or a closed-loop setup for a variety of laser sources including cases where certain parameter mismatch appears between transmitter and receiver lasers.

The last years have witnessed intensive studies of Ultra Wide Band (UWB) communication systems. The interest devoted to such systems is motivated by the various potential advantages the UWB technology can bring to the wireless industry.

The basic idea of synchronization was to adjust the frequencies of weakly interacting periodic oscillators. Nowadays several types of synchronization representing different degrees of correlation between the coupled systems have been identified:

a)- Complete (or full) synchronization,

b)- generalized synchronization and ,

c)- phase synchronization. Full synchronization is achieved when the states of coupled systems coincide, while the dynamics in time remains chaotic.

Generalized synchronization, as introduced for drive-response systems, is defined as the presence of some functional relationship between the states of the response and the drive. Phase synchronization is achieved when the phases of coupled systems lock to each other, while their amplitudes remain uncorrelated and sustain an irregular motion

of their own.

In Chaotic Masking the message to be transmitted is added to a much stronger chaotic signal in order to hide the information, the overall signal is then transmitted to the receiver. This method has been soon forsaken since it has shown vulnerability with respect to channel noise.

In Chaos Shift Keying (CSK) the transmitted signal is obtained by switching between two chaotic generators according to the information level of a binary message. In chaotic modulation the message modifies the state or the parameters of the chaotic generator through an invertible procedure, thus the generated chaotic signal inherently contains the information on the transmitted message.

Chaotic modulation with multiplication and feedback (CMMF) has been proposed. In this method, the message modulates by multiplication the chaotic output signal, the resultant signal is fed back to the transmitter system and simultaneously sent to the receiver system. In order to synchronize with the transmitter, the receiver is merely a copy of the transmitter which is driven by the received signal. The main advantage of this method is that it can be applied to a large class of chaotic systems. Herein, we present a conmparison study to investigate the effect of different nonlinear functions on the behavior of the system from two viewpoints, namely sensitivity to chalmel noise and parameter variations.

The advantages lays in its compatibility with a large class of nonlinear functions gives a wide choice to a transmitter-receiver system to strengthen their security against an intruder.

Chaotic digital modulation is concerned with mapping symbols to analog chaotic waveforms. In CSK, information is carried in the weights of a combination of basis functions, which are derived from chaotic signals. Differential chaos shift keying

is a variant of CSK, where the basis functions have a special structure and the information can be recovered from the correlation between the two parts of the basis functions.

If the propagation conditions are so good that the basis function(s) can be regenerated at the receiver, then digital modulation schemes using conventional orthonormal (typically periodic3 ) basis functions, and orthonormal chaotic basis functions can achieve similar levels of noise performance. it is fundamentally easier to regenerate a periodic basis function than a chaotic one.We conjecture, therefore, that the noise performance of digital chaotic modulation with coherent correlation receivers will always lag behind that of equivalent modulation schemes using periodic basis functions.

DCSK is a popular method for transmitting binary information using a chaotic signal as a carrier. It is noncoherent in nature and does not require synchronization between the transmitter and the receiver. In DCSK, a reference chaotic waveform x(t), is transmitted during the first half of each bit period of the input data stream. If the bit to be transmitted is a "1," the same waveform is transmitted again during the second half

of the bit period, while if the bit is a "0," its additive inverse -x(t) is transmitted. In the normal DCSK receiver, the signal is delayed by half a bit period and correlated with the undelayed signal to get the decision variable for producing the output data stream.

A conventional modulation scheme is referred to as the conventional counterpart of a chaotic one if the chaotic and conventional modulation schemes are characterized by the same signal-space diagram and if the demodulators used to recover the digital information have the same configuration. We will show that the noise performance of a chaotic modulation scheme is equal to that of its conventional counterpart provided that the energy per bit is kept constant and that the basis functions, if more than one is used, are orthogonal.

The noise performance of a digital communications scheme is determined by the probability distribution of the observation variable. If corresponding terms in the expressions for the observation variables of a chaotic and a conventional modulation

scheme have the same properties, then the noise performance of the two schemes is identical.

A. Signal-Space Diagrams

The signal-space and basis function characterizations of conventional digital modulation schemes have been extended to chaotic ones. Figure 1 shows that the signal-space diagrams for the coherent binary antipodal CSK and coherent binary phase-shift keying (BPSK) modulation schemes are the same. Two basis functions are required in coherent DCSK and differentially coherent DCSK. Comparing the signal-space diagram of DCSK (see Fig. 2) with that of coherent frequency-shift keying (FSK), we conclude that these modulation schemes can be characterized by identical signal-space diagrams.

Fig. 1. Signal-space diagram for binary antipodal CSK and BPSK

Fig. 2. Signal-space diagram of coherent DCSK, differentially coherent

DCSK, and coherent FSK.

B. Generalized Block Diagram of Correlator-Based Receivers

It was first shown for the simplest cases, and then in for every chaotic modulation scheme, that modulated chaotic signals can be demodulated using correlation in one of three ways:

1) a coherent correlation receiver, where the basis functions are recovered by chaotic synchronization;

2) a noncoherent correlator-based receiver, where the reference signal is equal to the received signal;

3) a differentially coherent receiver, where the information is carried by the correlation between the reference and information-bearing chips.

As in conventional telecommunications systems, coherent receivers offer the best noise performance, while the differentially coherent technique is much less sensitive to channel distortion and imperfections. The difference between the coherent and differentially coherent schemes is primarily in the manner in which the reference signals are generated for the correlators. Therefore, we will analyze coherent and differentially coherent receiver configurations using the common block diagram shown in Fig. 3.

Fig. 3. General block diagram of a digital chaotic communications receiver.

C. Estimation Problem

In chaotic communications, the basic functions are inherently nonperiodic chaotic signals which vary from bit to bit. By virtue of the chaotic basis functions, two estimation problems may appear in chaotic communications, degrading the noise performance and limiting the attainable data rate.

If the propagation conditions are such that coherent detection is impossible, then chaotic switching with DCSK basis functions and a differentially coherent receiver (DCSK, for short), offers the best possible performance for a chaotic digital modulation

scheme. In the limit, the noise performance of DCSK lags only 3 dB behind that of DPSK with autocorrelation demodulation. In this case, the choice of periodic or chaotic basis functions is determined by the propagation conditions. In particular, the multipath performance of a DCSK system can be improved by increasing the transmission bandwidth. the performance bounds for chaotic modulation schemes to the limits for conventional narrow-band modulation techniques, the comparison is not fair in the sense that chaotic modulation is intended for use as an inherently wide-band communications system. The advantage of DCSK is that the fall-off in its performance in a wideband multipath channel is more gradual than that of an equivalent narrow-band modulation scheme.

Fig. 4. Simulated noise performance curves for DPSK and wideband DCSK

in a single-ray channel (solid and dashed, respectively) and a multipath channel

where coherent detection is impossible (dash-dot and dotted, respectively).

While DCSK disimproves by about 4 dB, DPSK fails completely.

Fig. 4 shows the performance degradation in narrow-band DPSK (BT = 1 , classical DPSK with optimum receiver configuration and wide-band DCSK (BT = 17 ) systems operating in single-ray and multipath channels. The bit duration was set to 2µs in both cases. Although the single-ray performance of DCSK is worse than that of DPSK, its multipath performance is significantly better. Therefore, DCSK offers a performance advantage over DPSK in multipath environments when the propagation conditions are so poor that coherent detection is not possible.

Adhering to the Pecora-Carroll drive-response concept, several chaos synchronization schemes have been successfully established. Chaos dynamics are deterministic but extremely sensitive to initial conditions. Even infinitesimal changes in the initial condition will lead to an exponential divergence of orbits. Moreover, a broad-spectrum

and noise-like chaotic signal is particularly appropriate for secure communications. Chaotic signal masking and chaotic modulation, have been developed for analog communication systems. The idea of chaotic masking is that the information signal is

masked by directly adding a chaotic signal at the transmitter. Later, the information-bearing signal is received at the receiving end of the communication and recovered after some signal processing operations. The idea behind chaotic modulation is that the information signal is injected into a chaotic system, or is modulated by means

of an invertible transformation so that spread-spectrum transmission is achieved.

In addition to the above approaches, several methods, such as signal reconstruction methods, nonlinear forecasting systems, and an adaptive filter approach, have also been developed. The shared feature of nonlinear forecasting systems and signal reconstruction methods is that a differential operator is deemed necessary inside the system design. However, using the differential operator is limited, in that system noises are also amplified in the system loop. Therefore, in practice, a serious problem may arise in the hardware implementation of secure communications. In an adaptive filter for the demodulating method, the adaptive filter reduces the effects of channel noise and also estimates the bifurcation parameter (signal of transmission). However, this method can only be applied to a particular logistic system. On the other hand, recent investigations have linked observer-based concepts to chaos synchronization, which constructs all of the state information from only the transmitted signal. A systematic

method, employing a nonlinear state observer, is proposed to resolve the chaotic synchronization of a class of hyperchaotic systems via a scalar transmitted signal. Although chaotic synchronization can be ensured due to the effect of transmitting the nonlinear terms, the implementation of nonlinear functions is more complicated in

practice.

M-DCSK is a generalization of the DCSK scheme described above. The reference waveform of DCSK (which is typically transmitted during the first half of the bit period) is broken up into k time slots distributed all over the bit duration. Each time

slot is 2i-chips long and contains chips forming the reference waveform R, followed by the same waveform multiplied by the data value being transmitted (we call this the data waveform T). k and i are assumed to be integers. A single bit is composed of k time slots with different reference and data waveforms. A single chip is represented as xng(t-nTc) , where {xn} is a chaotic sequence and g(t) is a known waveform of duration Tc . {xn} can be obtained by iterating a discrete time system at a rate fc =(1/Tc) , or by sampling the output of a continuous time system at the same rate. The case of a single time slot per bit (k = 1) corresponds to DCSK.

In coherent communications, the mean values of the received message points in the observation space coincide with the message points of the signal-space diagram; hence, coherent receivers offer the best noise performance for an additive white Gaussian

noise (AWGN) channel.

An inherently wideband signal with constant energy per bit can be generated by combining the DCSK technique with Frequency Modulation; this is the basis of FM-DCSK.

In FM-DCSK we exploit the fact that the instantaneous power of an FM signal does not depend on the modulation . Let the chaotic signal, which is slowly-varying compared

to the carrier, be the input of an FM modulator. In this case, the output power of the FM modulator is determined exclusively by the carrier, and the energy per bit becomes

constant. This wideband output from the FM modulator is modulated using the DCSK technique by the binary information to be transmitted.

Note that the FM-DCSK modulator contains two modulator circuits-a conventional FM and a DCSK modulator-where the input of the FM modulator is a lowpass chaotic signal. As with the conventional DCSK techniquc, every information bit is mapped to two sample functions: the first is the reference, while the second is the information-bearing part of the transmitted signal. T denotes the bit duration. The output s ( t ) of the transmitter passes through the channel in which it is corrupted by additive noise n(t). The channel selection filter is characterized by its impulse response h(t). The noisy received signal r(t) is applied to the input of the FM-DCSK demodulator.

The trellis-coded modulation (TCM) achieves a secure and efficient transmission, the chaotic modulation was combined with the trellis code and the channel coding gain is obtained from these two convolutions of trellis and chaos and also well in terms of rate efficiency and bit error rate (BER) performance. The trellis-coded chaotic modulation (or chaotic-coded modulation: CCM) is exploited using the TCM criterion, and the performance of it is investigated through the computer simulations.

In the receiver, we designed the adaptive decoding algorithm based on a sequential search and threshold. Simulation results showed the CCM improved the performance

according to the increasing number of the decoding complexity and constraint length, and clarified that the scheme had the tradeoff between decoding complexity and the performance.

A new Chaotic Modulation scheme with Trellis code to achieve a secure and high-performance transmission. The trellis-coded modulation (TCM) performs well in terms of rate efficiency and bit error rate (BER) performance. The trellis-coded chaotic modulation (or chaotic-coded modulation: CCM) is exploited using the TCM criterion, and the performance of it is investigated through the computer simulations.

BER Performance

In this section, we analyze the bit error rate (BER) performance of M-DCSK in additive white Gaussian noise (AWGN) channels. We assume a binary information source and real valued or binary chaotic spreading sequences. For every bit of information, the M-DCSK transmitter generates k chaotic sequences, one for each time slot and each of length i . Each sequence xl,i is followed by the same sequence multiplied by the data signal bm = +/-1 . The transmitted signal for a single time slot (the lth) is given by

Sl,j = { xl,j , 0<j<i and bmxl,j-i , i<j<2i }

The main warnings are that bit-error calculations based on moment calculations and

Gaussian assumptions ignore the chaotic dynamics and lead to inexact results which at best are lower bounds on the error rates, and further that the traditional correlation decoders are not always optimal.

Multiple Access

While multiple access schemes using DCSK have been proposed in the literature, one of the main advantages provided by M-DCSK over conventional DCSK is the ease with which the system can be adapted for multiple access communications. In M-DCSK, by varying the number of chips transmitted during each time slot (2i) we can support multiple users on the same communication channel without modifying the basic transmitter and receiver architectures. The spreading factor S of all the users can be maintained constant by varying the number of time slots transmitted per bit k period , such that the product S = 2ki is constant. Since and have been assumed to be integers, this means that the maximum possible number of users for a given value of S is equal to the number of integer factors of S . To fit more users into the channel, we can use a group of nearby spreading factors (S,S+1,S+2,â€¦) , so that all of them have similar bandwidths. Our scheme can also be combined with the multiple access scheme for DCSK proposed in for this purpose.

We now analyze the multiuser performance of M-DCSK. Consider N users transmitting simultaneously over the same channel. For each user, we can write the transmitted signal for a single time slot (the lth) as

Sn,l,j = { xn,l,j , 0<j<in and bn,p,xn,l,j, in<j<2in }

where 2in is the time slot width and kn is the number of time slots per bit for user n, and bn,p is the information bit being transmitted by the nth user during the time slot. { xn,l,j } is a real or binary valued chaotic sequence. The value of l ranges from 1 to kn for the nth user. We assume that all users use the same spreading factor S . Each user is assigned a pair of values (kn ,in ), such that the product S = 2knin is kept constant. The details of this assignment shall be shown not to significantly affect performance.

The closed form analytical expressions for the noise performance of the coherent antipodal CSK, coherent DCSK, and differentially coherent DCSK modulation schemes in an AWGN channel. We extended the waveform receiver concept to chaotic modulation schemes by showing that the shapes of the chaotic basis functions, which are different for each transmitted symbol, have no effect on the noise performance, provided that the energy per bit is kept constant and the basic functions are orthogonal.

We have shown that the noise performance of coherent antipodal CSK and coherent DCSK depends neither on the bit duration nor on the bandwidth of the telecommunications channel. The noise performance of coherent antipodal CSK and coherent DCSK is as good as that of conventional coherent BPSK and coherent FSK, respectively.

The noise performance of differentially coherent DCSK has also been expressed in closed form. We have shown that FM-DCSK need not be considered as a new modulation scheme; it is simply a variant of DCSK where the power of the transmitted signal, and consequently the energy per bit, is kept constant by introducing a secondary FM modulation.

As in the case of conventional suboptimum DPSK, the noise performance of differentially coherent DCSK depends on both the bit duration and the channel bandwidth. Since half of the energy per bit in DCSK is used to provide the reference signal for the demodulator, the noise performance of differentially coherent DCSK is 3 dB worse than that of suboptimum DPSK. A technique to reduce this loss, without transmitting a periodic signal, has been proposed in and further developed.

The advantages of various modulations are

Chaotic systems have many important properties, such as aperiodicity, sensitive dependence on initial conditions and topological transitivity, ergodicity and random-like behaviors, etc. Most properties are related to the fundamental requirements of conventional cryptography. So, chaotic cryptosystems have many useful and practical applications.

FM-DCSK performs extremely well over a multipath radio channel. The performance degradation even in the worst-case is less than 4 dB.

For optimum performance, the power spectral density of the transmitted signal must be constant.

A new chaotic communication regime based on the direct modulation of the parameters of the logistic map is proposed. In this regime, the logistic map is used as a chaotic carrier, whereas the speech signal acts as the modulating waveform. It is demonstrated that coding the speech signal in this manner results in whitening its power spectral density (PSD), and hence, a significant modeling

improvement is achieved.

chaotic masking method with feedback has been proposed for improving the robust synchronization in chaotic communication systems in the presence of any information signal.

In chaotic spread spectrum communication systems, the spreading sequence is a chaotic waveform. Systems using chaos shift keying (CSK), differential CSK (DCSK), its variants, and several other schemes have been reported. The main advantages over conventional systems are expected to be simplicity and lower cost, especially for low-power applications.