Abstract
ABSTRACT In this paper, we present an efficient amplitude shift keying (ASK)-aided orthogonal chaotic vector PSK (ASK-OCVPSK) non-coherent modulation scheme with considerations of different quality of service (QoS) demands. In our ASK-OCVPSK design, the reference chaotic signal is generated recursively from the shifted chaotic vectors orthogonalized by the Gram-Schmidt transform buffered in the shift register. Then, we embed the information on both the position and the amplitude of the chaotic sequence. The binary information bitstream is divided into two segments, which are modulated, respectively, by the parallel concatenated position shift keying (PSK) scheme and the ASK scheme. Next, the ASK symbol is further modulated by the PSK modulated chaotic vector, and the resultant symbol is transmitted together with the reference chaotic vector. Notably, with the aid of the shift register, only one chaotic vector is transmitted and the reference chaotic sequences can be easily regenerated recursively at the receiver for information recovery. Furthermore, we derive the theoretical bit error rate (BER) expressions for the proposed system over additive white Gaussian noise (AWGN) channel. Then, the spectrum efficiency and the complexity of the proposed scheme are analyzed and compared with those of the counterpart schemes. Simulations are performed over the AWGN channel and multipath Rayleigh fading channel, and the results demonstrate that the theoretical BER matches the simulation results. Moreover, the results verify that the presented scheme are more efficient and more reliable than the counterpart schemes and can provide different BER performances without changing the modulator structure to meet the different QoS demands of the users.
INDEX TERMS Non-coherent chaotic modulation, QoS demand, orthogonal chaotic vector position shift keying, amplitude shift keying, BER.
I. INTRODUCTION
Chaotic sequences have been widely used to spread the spectrum and improve the security of information transmission [1] - [4] , which can be used to modulate the information coherently or non-coherently. Compared with the coherent modulation scheme, the non-coherent chaotic receiver is more practical since the reference chaotic signal is transmitted and the complex chaotic synchronization circuit is removed [4] .
Among non-coherent chaotic communication systems, the non-coherent differential chaos shift keying (DCSK) [5] scheme has been considered the most outstanding alternative for spread-spectrum communications [6] because of its superior performance such as excellent bit error rate (BER), powerful near-far resilience and low implementation complexity. However, the transmission efficiency is lower than the coherent system due to the transmission of the reference chaotic signal.
Recently, various schemes have been proposed to improve the efficiency and the reliability of the DCSK scheme. In order to reduce the reference chaotic sequence number used by each user data bit and thereby improve the efficiency, the high-efficiency DCSK (HE-DCSK) scheme presented in [7] recycles each reference sample in DCSK and can transmit 2 data bits in one sample sequence, while the improved DCSK (I-DCSK) [8] uses a time-reversal operation to generate a reference signal orthogonal to the data carrier signal and then sum up these two signals into one time slot to double the data rate, and similarly, the short reference DCSK (SR-DCSK) scheme [9] shortens the reference signal such that it occupies less than half of the bit duration, thus the data rate and energy efficiency are enhanced. In order to enhance the bit error rate (BER) performance, noise reduction DCSK (NR-DCSK) [10] generates duplicate chaotic samples instead of different chaotic samples to be used as a reference sequence and then averages them at the receiver to reduce the noise variance. Moreover, with the aid of the multiple carriers, [11] presents a multi-carrier differential chaos shift keying (MC-DCSK) scheme, which utilizes multi-carrier transmission to enhance the data rate. Then [12] extends the MC-DCSK scheme and proposes the orthogonal frequency division multiplex (OFDM) based DCSK scheme to serve the multiple users. Furthermore, high dimension modulation is also exploited to improve the efficiency as well as the reliability. The quadrature chaos shift keying (QCSK) QCSK combines the DCSK and quadrature phase shift keying (QPSK) [13] and employs a Hilbert transform to generate the orthogonal signal in two dimensions, achieving twice the data rate as that of DCSK. β-ary DCSK [14] is an enhanced version of QCSK, which can further increase the data rate and provide different quality of service (QoS) for users. The orthogonal chaotic vector shift keying (OCVSK) [15] scheme utilizes the Gram-Schmidt method to extend the transform to higher dimensions to further improve the transmission efficiency. However, in the OCVSK receiver which performs the linearly concatenated Gram-Schmidt process, the noise power will increase due to the imperfect propagation and the errors induced by the non orthogonal reference signal, which degrades the BER performance.
In this paper, we present an amplitude shift keyingaided orthogonal chaotic vector position shift keying (ASK-OCVPSK) non-coherent modulation scheme with consideration of different QoS demands of the users, to further improve the efficiency and reliability of non-coherent chaotic modulation systems. Specifically, by employing the shift register at both the transmitter and the receiver, the reference chaotic signal is generated recursively and only one chaotic vector is required to be transmitted for one information-bearing symbol. At the transmitter, a chaotic vector with length M is combined with the buffered M − 1 chaotic vectors in the shift register. Next, the resultant M × M chaotic matrix is orthogonalized and normalized by the Gram-Schmidt process. Subsequently, the last M th chaotic vector is fed back to the shift register to be combined with the M − 2 chaotic vectors stored in the shift register, which constitutes the new M − 1 chaotic vectors for next round of orthogonalization and normalization, and acts as the reference sequence for the information symbol. At the receiver, the reference chaotic sequence is re-generated in a similar way by using the shift register.
Furthermore, the information is embedded in both the amplitude and the position of the chaotic vectors respectively. In other words, it is modulated by the combined parallel chaotic position shift keying (PSK) and the amplitude shift keying (ASK). The binary information bitstream with length log 2 M + log 2 N is divided into two parallel segments, the first log 2 M bits are used to select the orthogonal and normalized chaotic vectors and PSK is performed, the other segment including the bits from log 2 M + 1 to log 2 M + log 2 N is modulated by ASK. Subsequently, the informationbearing PSK modulated chaotic vector is used to modulate the ASK symbol. Then the resultant information-bearing signal is combined with the reference chaotic vector and transmitted over the wireless channel. Notably, by changing the value of M and N , the presented ASK-OCVPSK scheme can provide different level of modulation schemes to adapt to channel conditions, hence different levels of QoS can be achieved, thus allowing the system to transmit the information adaptively and efficiently.
More explicitly, the contributions of this paper include: (1) both the position and the amplitude of chaotic vectors are exploited to transmit the information; (2) the correlation among the chaotic vectors is reduced by applying the GramSchmidt process and higher reliability is achieved; (3) with the aid of the shift register, only one reference chaotic vector is required to be transmitted and the reference chaotic signal can be recovered recursively at the receiver; (4) different modulation levels can be adopted to meet different QoS demands of users adaptively. This paper is organized as follows. In Section II, the implementation of the ASK-OCVPSK transmitter and receiver is presented. In Section III, we derive the BER expressions for the proposed scheme. Subsequently, in Section IV, both the spectrum efficiency and the complexity are analyzed, and simulations are performed over the additive white Gaussian noise (AWGN) channel and multipath Rayleigh channel to compare with the analytical results and the counterpart schemes. Conclusions are given in Section V.
II. ASK-OCVPSK SCHEME
In this section, we present details of the ASK-OCVPSK scheme, including the reference chaotic signal generation method and the chaotic modulation scheme at the transmitter, as well as the retrieval of information at the receiver.
A. ASK-OCVPSK TRANSMITTER Fig. 1 illustrates the proposed ASK-OCVPSK transmitter. As shown in the figure, similar to the structure of DCSK modulator, the reference chaotic signal and the informationbearing modulated signal constitute the transmitted signal s k (t). Details about the generation of the reference chaotic signal and the information bearing signal are given as below. 
1) REFERENCE CHAOTIC SIGNAL
First, we use the logistic map [3] to generate the chaotic sequences recursively, which can be expressed as
where c n denotes the n th element of the chaotic sequence, c 1 denotes the initial value of the chaotic sequences. Next, the chaotic sequence x k for the k th information symbol with length M is fed into a buffer, and is combined with the orthogonal and normalized chaotic vectors e 
where · denotes the inner product operation, and e 
It can be observed from the Table. 1 that the orthonormalized vectors e k i (1 ≤ i ≤ M ) are still chaotic since the evaluated largest Lyapunov exponents [16] are positive for different M . After the pulse shaping, the reference chaotic signal is expressed as below
where e k M ,m is the m th sample of the vector e k M , T c is the chip duration of one chaotic sample, and h T (t) is defined as the impulse response of the square-root-raised-cosine filter, which is normalized to have unit energy as
As mentioned above, the information is embedded on both the amplitude and the position of chaotic matrix. Specifically, the information bitstream is divided into two segments by a serial to parallel converter. As shown in Fig. 1 , the first segment, denoted as b 1 . . . b log 2 M , is fed into the upper 'Bits to symbol converter' to generate the decimal symbol j, while the second part, denoted as b log 2 M +1 . . . b log 2 M +log 2 N , is fed into the bottom 'Bits to symbol converter' to generate the decimal symbol p. Next, the decimal symbol j is modulated by a PSK scheme. Namely, j is used to select a specific vector e k j from the M ×M reference signal matrix E k presented above.
The other decimal symbol p is modulated by an N -ary ASK scheme, and the normalized information-bearing amplitude value A p is obtained as
Subsequently, A p is modulated by the chaotic vector e k j , and the resultant modulated signal is denoted by A p e k j .
3) The TRANSMITTED SIGNAL The transmitted signal s k (t) consists of the reference chaotic signal e k M (t) and the information-bearing signal A p e k j for the k th information symbol, which can be expressed as
where f is the frequency of the carrier which is assumed to satisfy f >>
B. ASK-OCVPSK RECEIVER Fig. 2 illustrates the receiver of the proposed ASK-OCVPSK system. The received signal corresponding to the k th information symbol, which is denoted as r k (t), is given by
where ζ (t) is a zero-mean stationary Gaussian noise with power spectral density of N 0 /2. Subsequently, r k (t) is multiplied with the synchronized quadrature and in-phase carriers, respectively. Then the resultant signals, f k (t) and g k (t), are fed into two identical parallel filters matched with the impulse signal h T (t) given by Eq. (6) to maximize the output signal-to-noise ratio at the n th sampling time t = nT c where n ∈ 1, 2, · · ·. The resultant samples denoted by f n and g n can be used to retrieve the reference chaotic sequences and the information.
1) REFERENCE CHAOTIC SIGNAL RECOVERY
The samples f n and g n are stored in two parallel buffers respectively. For the k th information symbol, the samples
k with a length of M . Similarly, the samples g (k−1)M +1 , . . . , g (k−1)M +M constitute the vector g k which also has a length of M . Specifically, f k and g k can be expressed as
Block diagram of the ASK-OCVPSK receiver. VOLUME 5, 2017 Assuming that both the sinusoidal carrier and the chip are perfectly synchronized, the m th elements in f k and g k , which are denoted by f k m and g k m , can be obtained as
where 1 ≤ m ≤ M , ψ k m denotes the Gaussian noise added to the m th sample of the reference vector e k , ξ k m denotes the Gaussian noise added to the m th sample of the informationbearing vector A p e k . Both of them are independent Gaussian random variables with mean values and variances given as [17] 
where 
, which is the corresponding contaminated version of E k . Then the vectors stored in the shift register are shifted and updated with the one reference chaotic vector for each incoming symbol.
2) INFORMATION RETRIEVAL
Next, using the autocorrelation property of the chaotic sequences, we can demodulate the received signal by calculating the inner product of f 
where f k i,m denotes the m th sample of the reference vector f k i , and g k j,m denotes the m th sample of the information bearing vector g k j . Thus, the symbolj can be easily obtained by selecting the index of the maximum value of ρ(i, j) ( 
Then the symbolÃ p is calculated bỹ
Finally, using the 'symbol to bits converters', we can retrieve the two parallel bitstreamsb 1 . . .b log 2 M and b log 2 M +1 . . .b log 2 M +log 2 N respectively fromj andÃ p .
III. THEORETICAL PERFORMANCE ANALYSIS
In this section, we first derive the BER expressions for the PSK scheme and the ASK scheme over the AWGN channel, then the BER expression for the ASK-OCVPSK modulated system can therefore be obtained.
A. BER FOR PSK SCHEME
When the information bitstreamb 1 . . .b log 2 M is modulated by the PSK scheme, the BER, which is denoted by P PSK b , can be deduced by the symbol error rate (SER) for the symbol j denoted as P PSK S (j).
1) SER OF THE SYMBOL j
Considering thatj corresponds to the maximum ρ(i, j), the probability of making a wrong decision for the transmitted symbol j, which is denoted by P PSK S (j), can be calculated by
where prob(•) denotes the probability operator, namely prob max{ρ(i, j),
, and max(•) is the maximization operator. The decision variable ρ(i, j) can be deduced with aid of the Eq. (11) and the orthogonality of e k i e k j when i = j as below
where E s is the symbol energy and normalized as E s = 1. Based on the central limit theorem, when M is large, ρ(i, j) can be approximately regarded as a Gaussian random variable [17] with the mean value and variance as below
Since the variable A p is independent of ρ(i, j), let prob(A p ) denote the probability that A p is transmitted, the SER of the symbol j is derived as
where P PSK SC (j|A p ) is the SER for the symbol j conditional on the transmission of the ASK symbol A p . For i = j, the random variables in Eq. (17) are independent and share the same distribution, thus P PSK SC (j|A p ) can calculated by
where f (r, A p ) is the probability density function of ρ(j, j) when A p is transmitted, which is given by
2) BER Note that, if the symbol j is wrongly demodulated, the number of wrong bits in the wrong bitstream would be from 1 to log 2 M . For 1 ≤ i ≤ M and i = j, ρ(i, j) with different i are independent of each other and share the same distribution, thus the probabilities for misjudging j to be the remaining symbols are assumed to be the same, namely equal to 1 M −1 . In this case, the number of wrong bits in the wrong bitstream has equal probability from 1 to log 2 M , thus the BER for the PSK-modulated bitstream can be derived from the SER of symbol j as
where P PSK S (j) is given by Eq. (20) , and θ (M ) is a factor that converts SER to BER for an M -ary symbol that has the same probability for misjudging itself to be the remaining symbols, and can be calculated by
where C v log 2 M is a combination operator which outputs the overall number of selecting v elements from log 2 M elements.
B. BER FOR ASK SCHEME
The BER expression the ASK scheme, namely P ASK b , can be derived based on the derived result of P PSK SC (j|A p ) in two cases that the symbol j is incorrectly or correctly demodulated.
In the case thatj is incorrect, if the symbolj is wrongly demodulated, namelyj = j,Ã p would be a random guess, thus the probability for the symbol A p to be wrongly demodulated would be equal to N −1 N . When the symbol A p is incorrectly demodulated, the number of wrong bits in the bitstream converted from the wrong symbolÃ p would be equally distributed from 1 to log 2 N , namely the BER for the ASK-modulated bitstream in this scenario is equal to
In another case thatj is correct, namelyj = j, using Eq. (15),Ã p can be expressed as
where
Let the middle of every two neighboring ASK symbols is set as the decision thresholds, we here only consider the situation that the symbol is misjudged to be its neighboring symbol for approximate derivation, thus the number of wrong bits in a wrong symbol is equal to 1 when using Gray coding scheme. Note that the BER for the first symbol and the last symbol in one frame is different from the other symbols due to the boundary.
To be more specific, let A 1 be the first symbol placed on the left, A N be the last symbol placed on the right, and d be the distance between two neighboring symbols, if the symbol A 1 is transmitted, the probability for it to be misjudged as A 2 would be equal to prob(
thus the BER in this scenario would be equal to [ 
Similarly, if symbol A N is transmitted, the BER in this scenario would be equal to [ 
, the BER in this scenario would be equal to
Accordingly, the BER for the bitstream modulated by the ASK scheme can be approximated as
where prob(
are calculated by the Monte Carlo method.
C. BER FOR ASK-OCVPSK SCHEME
Finally, the overall BER for the ASK-OCVPSK system can be obtained as
IV. PERFORMANCE ANALYSIS AND SIMULATION RESULTS
In this section, the spectrum efficiency and the complexity are analyzed, and simulations are performed over AWGN channel and Rayleigh fading channel to compare the ASK-OCVPSK scheme with the counterpart schemes [18] , [15] .
A. SPECTRUM EFFICIENCY ANALYSIS
According to the definition of spectrum efficiency given in [19] , which is η = bit rate total bandwidth , we can obtain the spectrum efficiency as belowgiven as below
where log 2 M + log 2 N is the symbol number, MT c is the duration for each symbol wherein M is the bit number in one symbol and T c is the bit duration,
represents the bandwidth occupied by the ASK-OCVPSK modulated signal and α(0 ≤ α ≤ 1) is the rolloff factor.
Similarly, the spectrum efficiency of the Phase-separated DCSK (PS-DCSK) system in [17] and the MC-DCSK system in [11] can be obtained as below
where Q is the number of subcarriers used in the MC-DCSK system. Fig. 3 illustrates and compares the spectrum efficiency of the ASK-OCVPSK system with that of the FIGURE 3. Comparisons of spectrum efficiency among the PS-DCSK [17] system, the MC-DCSK [11] system and the ASK-OCVPSK system with M = 32, N = 2 and α = 0.2.
PS-DCSK system and the MC-DCSK system, where single carrier systems are extended to multi-carrier systems with the same method given in [11] . It can be observed from Fig. 3 that the ASK-OCVPSK system has higher spectrum efficiency than that of the PS-DCSK system and the MC-DCSK system, thus the proposed ASK-OCVPSK scheme can increase the transmission rate as compared to its counterparts, which is desirable for non-coherent chaotic communications.
B. COMPLEXITY ANALYSIS
We here evaluate the system complexity using the number of spreading/despreading and correlation operations required to transmit one symbol [20] , and assume that the shifting operation complexity is equivalent to that of the correlation operation due to the same operand with the same length. For the ASK-OCVPSK scheme, from Eq. (4), we can see that only 2(M − 1) correlation operations are required by the proposed system to generate the last orthogonal vector e k M . Table. 2 lists the operation details for the ASK-OCVPSK scheme. Table 3 summarizes the complexity of the ASK-OCVPSK scheme, OCVSK scheme and β-ary DCSK scheme. Since the OCVSK transceiver [15] requires Gram-Schmidt process, thus the total spreading/despreading and correlation operation number is determined by 2M 2 . Additionally, in the β-ary DCSK system [18] , one Hilbert transform with size M requires 2M − 1 multiplications and is approximate to two correlation operations, thus the operation number is around 8. It can be observed from the table that for all M ≥ 3, the complexity of OCVSK scheme [15] is higher than ASK-OCVSK scheme, and the the β-ary DCSK [18] has the lowest complexity since the modulation is performed in 2-dimension space.
C. ANALYSIS OF BER
We first compare the simulated BER of ASK-OCVPSK with the theoretical BER given by Eq. (23) Subsequently, we investigate the BER performances of the presented scheme with different modulation order over the AWGN channel. As illustrated in Fig. 5 , different BER can be provided by changing the modulation order of the ASK scheme and the PSK scheme with no need of changing the presented modulator structure. It can also be observed that when the total transmitted bit number is the same, for example, when 5 + 3 bits and 6 + 2 bits are transmitted, the BER performances of the ASK-OCVPSK with higher level PSK modulation are better than the one with lower level PSK modulation in case that E b /N 0 is greater. The reason is that the PSK scheme provides the additional spreading gain brought by the chaotic sequences, which is more effective when the signal to noise ratio (SNR) is higher.
Next, we compare the BER performances of our ASK-OCVPSK scheme with that of the β-DCSK in [18] and the OCVSK [15] system over the AWGN channel in FIGURE 6. BER comparisons of the β-ary DCSK [18] , the OCVSK [15] and the ASK-OCVPSK systems over AWGN channel. Fig. 6 . It can be observed that both the ASK-OCVPSK and OCVSK schemes outperform the β-DCSK thanks to the higher dimension modulation. Moreover, the proposed system outperforms the OCVSK system with improvement of more than 3 dB when the BER is at 10 −4 . The reason why our scheme achieves higher reliability is that in the OCVSK system [15] , the noise power is amplified when the Gram-Schmidt process is performed to obtain the reference chaotic signal for information recovery at the receiver, while in the proposed ASK-OCVPSK system, the reference chaotic signal is obtained with the use of the shift register and the Gram-Schmidt process is removed from the receiver, thus the noise power contained in the reference chaotic signal remains unchanged.
FIGURE 7.
Differences of the noise components for the orthogonal reference chaotic vector of the OCVSK [15] system and the ASK-OCVPSK (5+0 bits) system over AWGN channel with M = 32. Fig. 7 demonstrates the differences of the noise components in the OCVSK system and the ASK-OCVPSK system which are evaluated by |e i (T ) − e i (R)| 2 (1 ≤ i ≤ M ), namely the squares of the absolute difference between the orthogonal vector e i (T ) at the transmitter and the corresponding vector e i (R) at the receiver, for the proposed ASK-OCVPSK system and the OCVSK [15] system with different E b /N 0 values. It can be observed from Fig. 7 that in the OCVSK system, the noise power increases with the sequential Gram-Schmidt orthogonalization order, while in the ASK-OCVPSK system the noise differences remain the same. Fig. 8 compares the BER performances of ASK-OCVPSK and the counterpart schemes [18] , [15] with the same VOLUME 5, 2017 parameter setting over the multipath Rayleigh fading channel with average power gains E(λ 2 1 ) = 4/7, E(λ 2 2 ) = 2/7, E(λ 2 3 ) = 1/7 and time delays τ 1 = 0, τ 2 = 2, τ 3 = 5. It can be observed from Fig. 8 that when the value of E b /N 0 is above 5dB, both the ASK-OCVPSK and the OCVSK outperform the β-ary DCSK thanks to the higher dimension modulation. However, when the value of E b /N 0 is smaller, the BER performances of the proposed ASK-OCVPSK come close to those of the β-ary DCSK and are worse than those of the OCVSK. The reason is that in the ASK-OCVPSK receiver, the correct symbol is obtained if and only if the correlation output with i = j are larger than the other M − 1 outputs with i = j in Eq. (17) , which leads to the large power of M − 1 in the SER expression of Eq. (21) and thus the curve of the ASK-OCVPSK is more steeper.
V. CONCLUSION
In this paper, we present an efficient ASK-OCVPSK scheme to transmit information with both the amplitude and the position of orthogonalized chaotic sequences. In our design, with the aid of the shift register, the orthonormalized reference chaotic signal is generated recursively from the Gram-Schmidt process and only one chaotic vector is required to be transmitted since the receiver can easily regenerate the reference signal by using the shift register. On the other hand, the information is modulated in a parallel way by the ASK-aided chaotic PSK scheme with one segment used for the position selection of the chaotic vector and another segment modulated by ASK, then the ASK symbol is further modulated by the PSK modulated chaotic vector. Thus by using different segment sizes, the presented scheme can provide different QoS services for users. Subsequently we derive the theoretical BER expression for the ASK-OCVPSK system over an AWGN channel. Numerical simulation results verify the effectiveness of our derivations, and the theoretical spectrum efficiency, system complexity as well as the BER performances are compared among the ASK-OCVPSK and its counterpart systems. The results demonstrate that the proposed ASK-OCVPSK system achieves a better trade-off between the reliability and complexity than the high order DCSK modulation system based on a Hilbert transform and the OCVSK system do, which demonstrate that the presented ASK-OCVPSK scheme is efficient and adaptive to meet different QoS demands of the users.
WEIWEI RAO received the B.S. degree from Sun Yat-sen University, Guangzhou, China, in 2016, where he is currently pursuing the master's degree in communication engineering. He has an accepted authored paper and a co-authored paper, and the authored paper will be presented at the IEEE International Conference on Communications, France, and the co-authored paper is presented at the IEEE/CIC International Conference on Communications, China. (M'03) 
LIN ZHANG


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