A new scheme to enhance the performance of permutation index–differential chaos shift keying communications system

In this paper, a new scheme based on permutation index–differential chaos shift keying is proposed, modeled, and evaluated in AWGN channel environment. Data is sent by frames, and each frame is headed by a single reference signal and followed by some information-bearing signals. Modulation is performed through permutations of a reference signal according to the mapped data. At the receiver, each incoming information-bearing signal undergoes all inverse permutation possibilities to perform a correlation with the delayed and stored version of the received reference signal. To decode the information bits, the detector selects the highest correlator outputs. The proposed scheme named single reference–permutation index–differential chaos shift keying is an enhanced version of PI-DCSK, and uses a single reference signal for multiple information-bearing ones. Hence, the energy requirement is saved by almost a half. The bit error performance is studied using the baseband system model and analytically tested using Gaussian approximation method. Results show the BER performance outperforms other standard and recently developed differentially coherent chaos systems, including Permutation Index–DCSK by an average of 2.25 dB. Moreover, the analytical form which is developed to predict the bit error rate (BER) is validated by simulation. Results demonstrate the performance in AWGN is closely matching with the simulation results, particularly at high SNR.

methodologies to improve the quality of the chaotic signal and to guarantee a high level of security and randomness. Synchronization between two matching chaotic systems to provide confident communication is considered in [25]. The work in [26] presents the generation of random number and a chaos-key that is suitable for data encryption applications. Using segmented logistic map to improve the chaotic map in the design of chaotic encryption algorithm is proposed in [27], while the work in [28] uses a neural network chaos algorithm for the development of dynamic key encryption and decryption algorithms for wireless communication security. The position of nodes in wireless sensor networks is estimated using discrete chaotic mapping in [29]. A software-defined radio system is used in [30] to study the behavior of chaotic sequences in a spreadspectrum system and show that these sequences can be used as spreading sequences in spread spectrum systems. Using the field programmable gate arrays (FPGAs) to implement the synchronization between two chaotic systems with heterogeneous dynamics was the focus in [31]. There are several other wireless applications that use non-coherent chaos-based communication such as multiple antennas [32], cooperative networks, network coding and power line communications [33].
The main contribution of this paper is as follows: we develop an SR-PI-DCSK which uses a single reference signal for multiple information-bearing ones. We compare BER performance with the other differentially and permutation-based DCSK that uses a dedicated reference signal for each information symbol. The developed model is mathematically analyzed and verified by simulation under various factors and different channel conditions.
The remainder of this paper is organized as follows. Sections 2 and 3 present the transmitted signal frame and permutation process, respectively. Section 4 describes the scheme for the proposed SR-PI-DCSK system. The details of system analysis are discussed in Sects. 5 and 6. The proposed system complexity analysis is evaluated in Sect. 7. The details of experimental results with discussion are provided in Sect. 8. Finally, Sect. 9 concludes the paper.

System signal format
The transmitted signal frame of SR-PI-DCSK is shown in Fig. 1. Each frame consists of a reference signal segment with a length of β samples followed by L information-bearing signals. Information-bearing signal is nothing but only a permutated version of the reference signal. Permutation order is determined by the information bits with the length of n, for example, to send 2 information bits, we need to set up 2 2 permutations from the reference signal while 3 bits require 9 different permutations.

Permutation process
Changing of samples order in each chaotic segment is achieved using permutation process. The reordering process is determined by the permutation matrix. The permutation matrix P j1 is square matrix with dimension equal to the chaotic segment length. A chaotic signal with the spreading factor of β can be permutated in β! − 1 permutation. Hence for information bits vector a = (a 1 a 2 . . . a n ) with length of n, we will select one distinguished permutation out of β! − 1 and to have permutation mapping such away that a j → P j and P j ∈ {P 1 P 2 . . . P 2 n } . For example, if X 0 = x 1 , . . . , x 4 , and the permutation matrix is Then the permutated output can be written as X 0 P j1 = x 4 , x 2 , x 3 , x 1 . Furthermore, P j is the permutation performed in the chaotic sequence to present a j data set.

The proposed scheme
To generate the signal in Fig. 1, the design of modulator and demodulator of SR-PI-DCSK system is shown in Figs. 2 and 3, respectively. A reference signal X is generated with the chaotic samples having a length of β . This reference signal is sent initially prior to the frame through switch S, to modulate the first information symbol, the same version of the reference signal is delayed and undergoes permutation operation according to the transmitted bits. The reference signal and its permutated version are related in a way that ensures very low correlation such that XP T j (X) = 0 [17], where P j (.) is the permutation operator. Thus, the first transmitted frame can be given by the baseband representation as: is the reference chaotic segment and P j1 X T β represents the first information signal which is nothing but reference signal delayed by β and permutated by P j1 mapped from information vector a j1 . Moreover, L is the frame length. In this paper, the analysis is simplified using the analysis of the first symbol only, similar methods can be applied on other following information symbols since all the symbols are statistically independent.
Under the assumption that the transmitted signal is received via AWGN channel and the transmitter and receiver are perfectly synchronized. The following operations are performed in the receiver: (1) The first incoming reference signal, is loaded into delay block and stored for correlation with other L symbol as shown in Fig. 2. (2) The following information-bearing signal is directly loaded to the permutation block to perform all possible M inverse permutations, where M = 2 n .
To detect the first information symbol, the received reference signal is delayed by β , (i.e., X β + W β ) is correlated with output of m th inverse permutation block output where m ∈ (1, 2, . . . , M) , hence, the output can be written as P −1 m (P j1 X β + W 0 ) where W 0 = (w 1 w 2 . . . w β ) is the noise vector and w k is AWGN noise sample having power spectral density of N 0 /2 . Therefore, the m th correlator output can be found as where SS represents desired signal component at m = j 1 , while SI is the inter signal interference components at m = j 1 . Inter signal interference results from the correlation between the chaotic signal and its permutated version. The NN is the noise to noise correlation that has a significant impact on the performance particularly at a large spreading factor of β.
The first estimated permutation index ĵ 1 is determined by correlator which is having the maximum output compared with the others, hence By estimating the correct ĵ 1 , the transmitter can predict which permutation which has been used to send the information. Table 1 compares the proposed system with other chaos-based systems, clearly, the proposed scheme offers several features compared with DCSK, CDSK, HE-DCSK, and TR-DCSK. With respect to energy per bits SR-PI-DCSK, the requirement is significantly reduced. This is due to the use of a single reference signal in each frame. Moreover, the proposed scheme offers superior spectral efficiency as each information-bearing signal is mapped with B bits and this reference is used multiple times in each frame. This arrangement guarantees spectral efficiency with respect to DCSK, CDSK, and HE-DCSK. On the other hand, reasonable system complexity is needed to achieve these features.

System analysis
The BER performance analysis is performed analytically using GA methods that are suitable for a system having large spreading factors. However, for low spreading factors, an accurate integration method is suggested [10].
For each L symbols sent, one reference signal is dedicated; therefore, LE symbol = (L + 1)E s , where E symbol is the average symbol energy while E s is average segment energy given by βV (X) and V(.) is the variance operator. For standard DCSK and CDSK and standard scheme [18-20, 22, 23], each information-bearing signal is accompanied by reference signal. Therefore, to send L symbols, we need to have L additional Since each symbol is carrying n bits, hence average bit energy is given by

Theoretical performance analysis
In this section, the performance analysis of the proposed scheme is analyzed in AWGN channel environment to predict the BER performance. The chaotic sequence is generated from the symmetric Tent map, given by the equation The chaotic sequence x is having uniform distribution between −1 and 1 with zero mean and computed variance V (x) = 1/3 and V (x 2 ) = 4/45 = 4/5V 2 (x) [11]. For the rest of the analysis, the following assumptions are considered (1) The correlation between the chaotic sequence and its permutated version is decaying quickly for a sufficient correlation window. (2) The correlation between the chaotic sequence and the noise sample is statistically independent. The Bit Error Probability of the proposed scheme is determined by calculating (1) erroneous permutation estimation Pr map , (2) erroneous probability of bits detection which largely depends on the number of bits n which calculated by the following Correct estimation of the transmitted sequence is done by selecting the maximum absolute value of one of the correlator outputs. This output results from the correlation between the delayed reference signal and inverse permutated information-bearing signal. Each correlator output can be modeled as a Gaussian random variable D m . For equiprobable transmitted sequence, the error probability of permutation index estimation conditioned by P j is given by where the D j1 and D m are decision variable at mth and j th 1 correlator output. The error will occur only if any value of D m can have a magnitude larger than D j1 . To detect the first information-bearing signal, the output of mth correlator can have two values and can be rewritten as The correlation components can be calculated as where w ′ i is the permutated noise sample. Based on the assumption that the correlation between noise sample w i and chaotic samples x j for all i and j is negligible, for sufficient value of β , then While E(D m ) ≈ 0 , similarly The output of each 2 n − 1 correlators D m for m = j 1 are statistically independent random values characterized by a normal distribution with zero mean and can be given as While the correlator output conditioned by correct permutation can be given as It is easier to calculate the probability of correct permutation which is occurred only if magnitude of Dj 1 > D 1 , Dj 1 > D 2 and . . . Dj 1 > D M , therefore the probability of correct map detection can be given as [22] where F Dm (y) is the commutative distribution function given by The overall BER of the system can be given by substitution on When L is very large, the overall BER of the system can be given by (21)

Complexity analysis
In this section, system complexity is evaluated against a number of essential components in each system. The evaluation is calculated based on the hardware resources required to transmit and receive single bits. Table 2 compares the transmitter of the proposed system with DCSK, CDSK, HE-DCSK, and TR-DCSK. The SR-PI-DCSK transmitter is less complex than CDSK, HE-DCK and TR-DCSK systems in terms of adder, multiplier and modulator. However, the proposed system used delay and permutations units and required to transmit L bits in each frame. The other systems required one bit delay and didn't have data permutation unit, except TR-DCSK that did not include delay unit and required one bit for data permutation.
The receiver of the proposed scheme requires L/n delay element which helps to reuse the structure of receiver L times as shown in Fig. 3. Detailed receiver complexity of the SR-PI-DCSK compared with other systems is illustrated in Table 3. The receiver of the proposed scheme is more complexity than other systems as it required more delay and inverse permutation functions.

Experimental results (theoretical and simulation)
In this section, different standard chaos schemes including the recently proposed scheme have been simulated and compared with the suggested system in AWGN channel environment and with various E b /N 0 . Furthermore, the proposed system has been compared with the standard PI-DCSK [19]. Analytic performance based on Eq. 24 has been compared with simulation results for different frame lengths. Finally, the effect of the spreading factors on the proposed scheme has been examined.
In Fig. 4, the SR-PI-DCSK is compared with the DCSK, CDSK, HE-DCSK, and TR-DCSK. The system outperforms the other systems by an average of 2 dB except for DCSK which exceeds the suggested system by 1 dB, although the proposed system offers a double bit rate. This is due to non-complete orthogonality at spreading factor β = 100 . When the spreading factor is increased to β = 150 , it can be easily noted that the SR-PI-DCSK system performance is enhanced by an average of 3 dB for all other systems as shown in Fig. 5. Obviously, the superiority in the performance continued while spreading factor increased to β = 200 and β = 300 as shown in Figs. 6 and 7, respectively. This can be explained by the fact that, compared to other system chaos signal, the average transmitted signal energy per bit for the proposed system has been saved by 2L/(L + 1) ≈ 50% compared to the standard systems to which corresponds to a gain or horizontal displacement of BER. In addition to the very low cross-correlation value at the correlator output between the interference component.  Table 4 shows summary of the performance of the SR-PI-DCSK, the proposed system outperform all others systems except DCSK at β = 100.
Comparison of the SR-PI-DCSK system performance at various frame lengths is demonstrated in Figs. 8, 9, 10 and 11. The comparison is performed at the increment trend of the spreading factor. It is obvious that the performance of the proposed system with frame length L = 3 is better than systems with the length of L = 2 and L = 1 or PI-DCSK [19]. It is evident that the performance is degraded at a high value of β due to noise to noise (NN on Eq. 12) which at the correlator output.
A theoretical estimation of BER for the proposed system is developed using GA and validated using computer simulation. Excellent agreement between theoretical expression in Eq. 24 and simulation results that shown in Figs. 12, 13, 14 and 15 for β = 100 , β = 150 , β = 300 and β = 500 , respectively. The results obtained clearly validate the expression derived.
Effect of increasing of spread factor β has been studied and is illustrated in Fig. 16 at fixed value of E b /N 0 (at 13 dB and 17 dB). It is clearly shown that the performance is enhanced as β increases and reaches its ideal performance at β = 100 , and this matches with the results shown in Figs. 4, 5, 6, and 7. After β reaching 100, the performance starts to decrease with β due to dominance of noise to noise expression (NN) at high spreading factor as illustrated by Eq. (24).

Conclusion
A new permutation-based DCSK system named single reference-permutation indexdifferential chaos shift keying (SR-PI-DCSK) is proposed, designed, and validated through simulation. A theoretical formula to predict the bit error rate is developed and verified. The proposed system sends information symbols in a frame; each frame consists of a reference signal followed by a sequence of L information-bearing signal. This reduces the energy requirement and enhances the BER. The BER performance is simulated and compared with DCSK, CDSK, HE-DCSK, and PI-DCSK schemes. Results illustrate the system outperform other chaos-based systems, particularly at large spreading factor. Comparison results illustrate the advantage of the proposed scheme specifically at moderate E b /N 0 over the other schemes. Based on the GA method, theoretical prediction for BER is developed and compared with the simulation results. Excellent matching between the derived form and simulation results is noticed at typical values of spread factor.