Chaos-Based Communication Systems

2.1. Coherent systems

In coherent systems, a local synchronized copy of each sample function has to be produced at the receiver. When the transmitted signal is corrupted by the random noise, it will be challenging to synchronize the local generated carrier with that in the transmitters as in coherent shift keying (CSK) [1]. The idea of CSK is to map each information bit to chaos bases signal say f1 and f2. If “+1” is to be sent, then chaos signal from generator f1 is to be transmitted with one bit duration, and if “−1” is to be sent, chaos signal from generator f2 is transmitted with same bit duration. The receiver should generate exact copy from f1 and f2 to recover the information. This is done by using dedicated synchronization circuits [3].

2.2. Non-coherent systems

This type of chaos receivers offers simple solution for synchronization problem by estimating one unique parameter such as power. This is achieved by multiplying and integrating the product of received sample signal with itself to estimate the signal energy. Chaos ON OFF keying (COOK) and non-coherent chaos shift keying are a practical implementation of this idea [4]. In COOK, the signal is transmitted within bit duration only if information “+1” is to be transmitted. Otherwise, no transmission is taken place at “−1” bit duration. A bit control switch is used to control the emission of signal at the transmitter output. However, obtaining an optimum threshold to distinguish between signal sets does not depend only on signal power at the correlator output but also on the noise power estimation that is the major drawback of such systems [1].

2.3. Differentially coherent systems

Another scheme is developed where a reference signal is followed by another reference signal modulated by the information bit and called information carrying signal. This arrangement is known as differentially coherent systems. Here, every bit is presented by two sample functions. In the case of bit 1 transmission, the information is sent by transmitting two identical sample functions. For bit 0, the reference signal is transmitted and followed by an inverted copy of the sample function. General structure of the receiver is based on how to correlate the reference signal with information bearing signal.

Differentially coherent systems show better bit error rate (BER) performance among other existing chaos-based systems and in different channel conditions [5]. In spite of some structure complexity, hardware design is studied and tested.

In this chapter, standard differentially coherent schemes are described and tested by computer simulations, analytical expression to estimate BER in additive white Gaussian channel is obtained using Gaussian approximation method [6, 7]. A brief description of recently developed system is discussed.

3.1. Differential chaos shift keying

Differential chaos shift keying (DCSK) transmitter structure is shown in Figure 1. Each information bit is represented by twin of successive chaotic signal slots with length of samples, where 2M represents the spreading factor. First time slot contains a reference signal and second slot contains the information bearing signal. That is simply a delayed version of the reference signal multiplied by the information bit. Thus, the instantaneous value of the transmitted signal at any instant can be written as

Average bit energy for a single bit can be given by:

where V(.) is the variance operator.

There are many chaotic maps which can be used as a signal source [1, 8]. However, symmetric tent map is selected to produce the chaotic signal due to its simplicity. Its discrete form is given by the equation x_n + 1 = 1 − 2|xn| where x is uniformly distributed between 1 and −1. It can be easily shown that E(x) = 0, Vx=13and Vx2=445[1] where E(.) represents the average operator [6, 9, 10].

Received signal sample ri = si + ζi is received via noisy channel characterized by Gaussian distribution where noise sample ζi is stationary random process withE(ζ) = 0 and its power spectral density given by Vζ=No2. The received sample is multiplied by its delayed version r_i − M and the multiplication output is integrated over half bit duration M. Assuming that synchronization is achieved perfectly at the DCSK receiver shown in Figure 3. Then, the correlator output Z_DCSK at the end of bit duration can be described.

Average value of the correlator output can be determined by the value of information bit in the first term, while other terms will have mean value of zero due to their statistical independence [1, 6, 9, 11]. The correlator output is passed to the detector with zero threshold value to minimize BER as described in (4).

As the chaotic signal x is stationary and xi is statistically independent from ζj at any(i, j), correlator output ZDCSK tends to have Gaussian distribution at sufficient value of M. Therefore, BER analytical evaluation of DCSK is obtained by calculating the means and variances of conditional probability of P(ZDCSK| b = 1) and P(ZDCSK| b =  − 1), respectively; then theoretical estimate of BER can be calculated as (5)

where erfcy=2π∫y∞e−t22dt.

Two major drawbacks of DCSK systems are as follows: (1) data rate is reduced by half because of the need of separate reference signal and (2) a technical issue can be generated from continuous change of switch position in Figure 2.

3.2. CDSK

Sending reference signal and information bearing signals in separate time slot will result in data rate reduction by half. Instead, orthogonality between each chaotic signal and its delayed version can be utilized efficiently by adding the generated chaotic signal with the modulated version of the previous signal. This scheme is known as correlation delay shift keying (CDSK) [6]. Information bit is sent by transmitting a signal as the sum of a chaotic sequence xi and of the delayed chaotic sequence multiplied by the information signal blx_i − L, where l is the bit counter and L is the amount of sequence to be delayed. Hence, the transmitted signal of CDSK at any instant i is given by

where L ≥ M and Eb = 2MV(x).

Compared with structure of DCSK, structure of CDSK transmitter is characterized by replacing the switch by an adder as illustrated in Figure 4. Data rate is doubled when compared with DCSK because of reference time slot utilization [6]. Putting delay L = M, then the receiver of CDSK is similar to that DCSK and each received sample ri segment is correlated with the previous one r_i − M. Hence, correlator output ZCDSK can be computed as

It can be clearly observed that the correlator output ZCDSK contains more intra-signal and noise terms compared to DCSK. Hence, BER performance is expected to be lower. The cross terms in (7) is statistically independent and ZCDSK tends to have Gaussian distribution at sufficient value of M. Theoretical value of BER can be found by calculating the mean and variance of ZCDSK when the transmitted bit is +1 and −1, respectively. Decoding is performed according to the same rule in (4) and BER is given by [6].

3.3. High efficiency-differential chaos shift keying (HE-DCSK)

To enhance the bandwidth efficiency of DCSK and CDSK, a pair of information bits can be modulated using same reference signal by reusing each reference signal twice [9]. First, reference signal is modulated with information bit after delay of M sequence as a standard DCSK. Second, information bit is modulated after the delay of 3 M. Both modulated segments are added together in the second time slot. The scheme is illustrated in Figure 5. Thus, transmitted signal which is emitted from HE-DCSK transmitter can be written as:

where k is the pair sequence number. Signal is received through AWGN where each received signal is delayed and correlated twice, first, after M samples delay and second, after 3M as shown in Figure 6. The scheme represents an extended version of DCSK receiver, therefore the output of first modulator Zl can be given by:

Similarly, Z₂ can be calculated by Z2=∑i=1Mriri−3M. Average value of the first term in (10) contains the useful signal energy while the remaining terms are having zero mean. Information recovery is performed by comparing Z₁ and Z₂with zero-based threshold defined by the following equation.

Both correlator output Z₁ and Z₂ exhibit Gaussian distribution. BER of any correlator can be given as [9]

3.4. Time reversal-differential chaos shift keying (TR-DCSK)

The system is initially proposed [12] and developed by Albassam [13]. In this scheme, reference signal is generated and added to its time-reversed version. Hence, no separate time slot for reference signal is needed. This will generate a symmetric signal around the middle of bit duration. First half is transmitted directly and the second half is modulated with information bit. Provided that M is spreading factor, transmitted signal can be given by

Transmitter block diagram is illustrated in Figure 7.

Channel under investigation is AWGN and the received signal can be written as

At the receiver, each incoming noisy segment undergoes time reversal process. Hence, the output after the time reversal unit ri ′can be given as

Perfect bit synchronization is assumed where each incoming signal ri is correlated with time-reversed version ri ′. Due to signal symmetry, correlator output is integrated over the duration of M2, which is twice as in DCSK and CDSK. This is to avoid the effect of redundant signal components in the second half (i.e., >M2). The correlator output Z at the end of first bit duration can be given as

Similarly, BER rate can be readily shown to have

3.5. Energy efficient-differential chaos shift keying (EF-DCSK)

In all previous systems, each transmitted signal is composed of two separate segments such as reference signal and information bearing signal. A simplified system with minimum energy requirement is proposed in Ref. [14]. Simply, a chaos source generates a signal for one bit to be sent as a reference. Then, the transmitter will decide to send either same reference signal or newly generated one using a bit controlled switch as shown in Figure 9. For example, if information bit 1 is transmitted, delayed version of the reference signal is transmitted. Otherwise, the transmitter will generate a new signal. Therefore, each segment will play a dual role; one as information bearing signal at the time of bit generation and as a reference signal for the next bit duration. This eliminates the need for sending reference separately. Without loss of generality, we will consider the analysis for the first bit b where b ∈ {1, 0} and the transmitted signal Si at the ith instant can be represented as

The source emits M samples for each information bit in addition to the initial reference signal. Thus, the average bit energy transmitted can be found as.

Information decoding is performed by correlating each incoming signal ri with its delayed version and the correlation product is averaged over M. Information bit b~can be extracted by comparing correlator output with the predefined threshold as shown in Figure 10.

The received signal ri can be described as ri = si + ζi and the correlator output Zef can be formulated as

Signal energy estimation can be obtained only by taking the mean value of first term in (7). Ideally, this will be either zero or V_ar(x), all other terms are the zero mean. Obviously, it can be observed that the number of cross-terms of EF-CDSK correlator is less to that in CDSK. However, the distance between signal elements (average value of the correlator for each transmitted bit) is half compared to that in DCSK. Despite all that, the information can be decoded according to the following rule

where αth is the decoding threshold and it is given by Eb/2.

BER expression can be found by calculating mean and variance of the Gaussian distribution function of P(Z| b = 1) and P(Z| b =  − 1) and can be formulated as