IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 64, NO. 8, AUGUST 2017
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Carrier Index Differential Chaos Shift Keying Modulation Guixian Cheng, Lin Wang, Senior Member, IEEE, Weikai Xu, Member, IEEE, and Guanrong Chen, Fellow, IEEE
Abstract—A new carrier index differential chaos shift keying (DCSK) modulation method combining index modulation (IM) with multicarrier DCSK (MC-DCSK) is proposed. Two schemes employing two different index selectors and detectors to implement IM and demodulation are introduced. The new designed index selectors are one-to-one mappings between the index symbols and the carrier activation patterns, which can avoid the disaster caused by the previous designed selectors. Analytical bit error rate expressions of the two systems are derived over the additive white Gaussian noise as well as multipath Rayleigh fading channels. Simulation results verify the superiority of one scheme in energy efficiency and another scheme in spectral efficiency, respectively, compared with MC-DCSK. Index Terms—Detection, differential chaos shift keying (DCSK), energy efficiency, index modulation (IM), multicarriers, spectrum efficiency.
I. I NTRODUCTION ECENTLY, index modulation (IM) as a new technique has been developed for multicarrier systems, which uses the indices of the subcarriers as a kind of resource to carry information [1]. IM is developed from the subcarrier IM technique, which conveys the information in an on-off keying fashion, and needs a perfect feedforward from the transmitter to the receiver to implement the mapping method for the subcarrier index bits [2]. Since IM was proposed, it has inspired a new wave of research generating many schemes combined with orthogonal frequency-division multiplexing (OFDM) [1]–[3] and other communication techniques [4]. In the past two decades, chaotic modulations have drawn a great deal of attention in wireless communication applications due to their low-power, low-complexity, excellent anti-fading, and anti-intercept capabilities [5]–[11]. As the most important and promising chaotic modulation, differential chaos shift keying (DCSK) only needs a simple noncoherent demodulator without reproducing an exact replica of chaos sequences nor using any channel estimators, which makes DCSK more practical at the expense of some performance
R
Manuscript received July 19, 2016; revised August 29, 2016; accepted September 12, 2016. Date of publication October 13, 2016; date of current version July 31, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61271241, and in part by the Hong Kong GRF under Grant CityU 11208515. This brief was recommended by Associate Editor F. Lau. (Corresponding author: Lin Wang.) G. Cheng is with the Department of Communication Engineering, Xiamen University, Fujian 361005, China, and also with the School of Physics and Electronic Science, Guizhou Normal University, Guiyang 550001, China. L. Wang and W. Xu are with the Department of Communication Engineering, Xiamen University, Fujian 361005, China (e-mail:
[email protected]). G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSII.2016.2613093
degrading in comparison with chaos shift keying [8]. Besides, DCSK systems are more robust in multipath fading channels than differential phase shift keying schemes [6]. Some variants of DCSK have been developed from DCSK. Multicarrier DCSK (MC-DCSK) is one of the variants, with higher data rate and energy efficiency comparing with DCSK, which transmits the chaotic reference sequence over a predefined subcarrier frequency and transmits the multiple modulated data streams over the remaining subcarriers [8]. Since the combination of IM and DCSK transmission techniques remains open, a new communication scheme named carrier index DCSK (CI-DCSK) is proposed in this letter, which is based on IM and MC-DCSK. The main contributions of this letter are summarized as follows. First, two new CI-DCSK schemes are designed. Secondly, analytical bit error rate (BER) expressions of the two schemes over the additive white Gaussian noise (AWGN) and multipath Rayleigh fading channels are derived. Finally, simulation results demonstrate that the two CI-DCSK schemes outperform MC-DCSK in energy efficiency and spectral efficiency, respectively. This letter is organized as follows. Two schemes of CI-DCSK are presented in Section II. Performance analysis of CI-DCSK is carried out in Section III. In Section IV, simulation results are provided. Finally, Section V concludes the investigation. II. S YSTEM M ODEL OF CI-DCSK In [1], index selectors map p1 index bits to a carrier activation pattern out of C(n, m) possible candiatas, where n is the number of all indexed carriers and m is the number of the active carriers. For each subblock of OFDM-IM, the maximum number of bits that can be carried by the selection of m active subcarriers is given by p1 = log2 C(n, m). Since the number of the carrier activation patterns is larger than the number of index symbols, when the receiver detects an invalid carrier activation pattern, a catastrophic result occurs. A. Transmitter In order to design a new communication system combining IM with DCSK, which can avoid the disaster mentioned in [1]. It is better to get an one-to-one mapping between carrier activation patterns and index symbols, so the former formula for p1 is revised as p1 = log2 C(n, m). And, there are only two situations satisfying the above formula. In both of the cases, n is 2p1 . In the first case, m is equal to 1, while in the other case, m is 2p1 − 1. To classify the first case, i.e., m = 1 refers to CI-DCSK1, the other case, i.e., m = 2p1 − 1 to CI-DCSK2. One can find that CI-DCSK1 is the same as CI-DCSK2 when p1 = 1. By setting the value of p1 , either system can offer different data rates and performances. The block diagram of the CI-DCSK schemes is shown in Fig. 1. At the transmitter, the total number of available
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Fig. 1.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 64, NO. 8, AUGUST 2017
Block diagram of the CI-DCSK schemes.
subcarriers is 2p1 + 1, one of which transmits the reference chaotic signal, and the remaining 2p1 subcarriers are indexed from 1 to 2p1 . In both schemes, the input information sequence is first divided into v groups, each has p bits, i.e., Sv = [av,1 , . . . , av,p1 , av,p1 +1 , . . . , av,p1 +p2 ] and v = 1, 2, . . . , V, where the first p1 bits and the last p2 bits are index bits and modulated bits, respectively. Each modulated bit av,j = [0, 1] with p1 + 1 ≤ j ≤ p1 + p2 is changed to bv,i = [1, −1] with 1 ≤ i ≤ p2 by polarity transformation. By the first p1 bits, the index selector will choose m active carriers out of 2p1 index carriers. The remaining p2 bits will be conveyed by DCSK through those active carriers with one subcarrier transmitting one bit. Hence, it is concluded that m equals to p2 . Based on the modulated bits, the index selectors will generate a sequence of dv,i for all the 2p1 available subcarriers, if the subcarrier is active, dv,i will be generated by the modulated bits, with the order of the modulated bits consistent with the order of the subcarrier indices; otherwise, dv,i will be zero. For each symbol duration, by p1 mapping bits, selector of CI-DCSK1 chooses the subcarrier of the corresponding index as the active carrier, while the selector of CI-DCSK2 chooses the subcarriers of the rest indices as the active carriers. Then, the remaining p2 bits are transmitted by the active carriers. For example, in CI-DCSK1, when the index bits av,1 av,2 = 01, among four available subcarriers indexed from 1 to 4, the active carrier is the one with index 2. Then, the only modulated bit will be transmitted by the carrier labeled 2 through DCSK modulation. All of the remaining three subcarriers will transmit zero. In CI-DCSK2, when the index bits av,1 av,2 = 01, among four indexed subcarriers, the indices of active ones are 1, 3, and 4. Three modulated bits will be transmitted by the subcarriers labeled 1, 3, and 4, in turn, by DCSK modulation. The remaining subcarrier labeled 2 will transmit zero. DCSK modulation is a transmitted-reference digital communication scheme. For each symbol duration, the DCSK signal contains two pieces of chaotic signal samples, the first piece as the reference, followed by its noninverted or inverted copy depending on the transmitted information bit (“0” or “1”). In this letter, the logistic map xk+1 = 1 − 2xk2 is selected the chaotic system. The reference chaotic signal as β cv (t) = k=1 cv,k h(t − kTc ) is generated by pulse shaping, where β is the spreading factor, h(t) is the square-root-raisedcosine filter and Tc is the chip time. To this end, the transmitted signal of the two CI-DCDK schemes can be expressed as 2p1 s(t) = cv (t) dv,i cos(2π fi t + φi ) + cos(2π f0 t + φ0 ) (1) i=1
where φi represents the phase angle of the carrier modulation, and fi is the frequency of the ith subcarrier. For these CI-DCSK systems, the subcarriers are chosen to be the same as the carriers of MC-DCSK, which are orthogonal over the chip duration [8].
B. Receiver In the two schemes, the received signal is given by r(t) =
L
αl δ(t − τl ) ⊗ s(t) + n(t)
(2)
l=1
where L is the number of paths, αl and τl are the channel coefficient and the path delay of the lth path, ⊗ is the convolution operator, and n(t) is a wideband AWGN with zero mean and power density of N0 /2. αl are assumed to be independent Rayleigh distribution random variables. If the number of paths is one, with the channel coefficient being one, it is the AWGN case; otherwise, it is the Rayleigh fading case. This multipath Rayleigh fading channel mode is commonly used in spread-spectrum wireless communication systems [6], [8], [10]. In these schemes, the receivers not only need to detect the modulation bits on the active carriers, but also need to detect the index of the active carrier or the inactive carrier by two different detectors. The received signal is first separated by 2p1 + 1 corresponding orthogonal modulated carrier frequencies. After filtered by the matched filters, the signals are sampled at every kTc instant. Then, the demodulated discrete signals are recorded into two matrices, namely, the reference chaotic signals are stored in matrix A1×β , while the data signals are stored in matrix B2p1 ×β . Finally, z1×2p1 = A × BT is put into the detector. In CI-DCSK1, a detector based on maximum energy is applied, which computes the absolute value of the components in z and finds the maximum. Based on the index of the maximum energy carrier, p1 index bits can be recovered. The only modulated bit can be recovered by the sign of the component in z which owns the maximum energy. While in CI-DCSK2, a minimum energy based detector is applied, which computes the absolute value of the components in z and finds the minimum. Based on the index of the minimum energy carrier, p1 index bits can be recovered. To recover the modulated bits, 2p1 − 1 values which own the largest 2p1 − 1 absolute values, are selected out of z. Based on the sign of the chosen values, 2p1 − 1 modulated bits can be recovered, the order of which is consistent with the order of the carrier indices. Combining the index bits before the modulated bits, the transmitted bits are recovered.
III. P ERFORMANCE A NALYSIS ON CI-DCSK In the following analysis, assume that the largest multipath delay is much shorter than the symbol duration, i.e., 0 < τlmax β. Thus, the intersymbol interference is negligible [6], [8]. Besides, assume that the channel is slowly fading and the channel coefficients are constant during the transmission time of a symbol.
CHENG et al.: CI-DCSK MODULATION
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A. Output of the Correlator The decision variable for the ith active carrier of the vth data stream at the output of the correlator is Dv,i with 1 ≤ i ≤ p2 . And, the decision variable for the jth inactive carrier of the vth data stream at the output of the correlator is Dv,j with 1 ≤ j ≤ 2p1 −p2 . The inactive carrier transmits no information, thus the receiver only receives noise from this carrier. Thus L L β i Dv,i ≈ αv,l cv,k−τl bv,i + nv,k αv,l cv,k−τl + nv,k k=1
Dv,j ≈
β k=1
l=1
j nv,k
l=1
L
(3)
αv,l cv,k−τl + nv,k
(4)
l=1
where αv,l and τl are the channel coefficient and the path delay of the lth path affecting the vth sequence, respectively; nv,k , j niv,k , and nv,k are three independent zero-mean Gaussian noises with the power spectral density of N0 /2, which are caused by the reference carrier, the ith active carrier and the jth inactive carrier, respectively. For a large spreading factor, the following approximated expression is used [7], [8], [10]: β
cv,k−τl cv,k−τq ≈ 0, l = q.
(5)
k=1
Thus, the means and variances of Dv,i and Dv,j are approximated as follows [8]:
E Dv,i ≈ Var Dv,i ≈
L l=1 L l=1
⎜ F|Dv,j | (y) = erf ⎜ ⎝ L
y
2 v l=1 αv,l Ec N0
+
N02 β 2
⎟ ⎟ ⎠
(12)
L
− 21 2β N 2 v f|Dv,i | (y) = 2π αv,l Ec N0 + 0 4 ⎡ l=1 ⎛
2 ⎞ L 2 v y − l=1 αv,l Ec ⎟ ⎢ ⎜ ⎟ ⎜− ×⎢ exp ⎣ ⎝ L N02 β ⎠ 2 v 2 l=1 αv,l Ec N0 + 4 ⎛
2 ⎞⎤ L 2 v y + l=1 αv,l Ec ⎟⎥ ⎜ ⎥. ⎟ + exp⎜ − ⎝ 2 L N0 β ⎠⎦ 2 v 2 l=1 αv,l Ec N0 + 4 (13)
2 v αv,l Ec bv,i
2 v αv,l Ec N0 +
(6) N02 β 4
(7)
E Dv,j ≈ 0 (8) L 2 v 2 N β l=1 αv,l Ec N0 Var Dv,j ≈ + 0 (9) 2 4 β where Ecv = k=1 c2v,k . By the previous analysis, BER of the system Pe is composed of two parts, the BER of the index bits Pi and modulated bits Pm . Since there are p1 index bits and p2 modulated bits for each transmitting p bits, Pe is given by p1 p2 Pe = Pi + Pm (p = p1 + p2 ). (10) p p B. Derivation of BER Expressions for CI-DCSK1 In this scheme, the number of active subcarrier is 1, while the number of inactive subcarriers is 2p1 − 1. For each symbol duration, the total number of the transmitted bits is p1 + 1, which is consist of p1 index bits and one modulated bit. 1) BER for Index Bits: When the largest absolute value of the decision variable for the inactive carriers |Dv,j | is larger than the absolute value of the decision variable for the active carrier |Dv,i |, an error will occur. Based on the cumulative distribution function (CDF) of the quotient of two random variables, the symbol error rate (SER) of index bits Psi1 is derived as Psi1 = Pr Dv,i < max Dv,j +∞
2p1 −1 = 1− F|Dv,j | (y) f|Dv,i | (y)dy (11) 0
where F|Dv,j | (y) is the CDF of |Dv,j | shown in (12) with 1 ≤ j ≤ 2p1 − 1, and f|Dv,i | (y) is the probability density function (PDF) of |Dv,i | shown in (13) with i = 1. The CDF and PDF are deduced from the fact that the absolute values of the decision variables follow a folded normal distribution. The absolute values of the decision variables for the inactive 2p1 −1 carriers are independent and identically distributed random variables, thus max(|Dv,j |) is the maximum order statistic,
2p1 −1 whose CDF is F|Dv,j | (y) . Here ⎞ ⎛
Based on (12) and (13), (11) can be simplified as 2p1 −1 +∞ √ 2 2y Psi1 = 1 − erf √ π (2pγb + β) 0 pγb + β 2 (2y − pγb ) (2y + pγb )2 · exp − + exp − dy 4pγb + 2β 4pγb + 2β (14)
L
2 Ev /(pN ), p = p + p , and p = 1. where γb = 2 l=1 αv,l 0 1 2 2 c Based on Psi1 , BER of index bits for CI-DCSK1, Pi1 is given by [4]
Pi1 = Psi1 /p1 .
(15)
2) BER for Modulated Bits: In the CI-DCSK1 scheme, two different cases may lead to an error occurring in the modulated bits. One is when there is no error in the index symbol detection, but an error is detected in the modulated bits. In this case, the BER of the modulated bits on the active carrier PDCSK1 is derived as ⎡ − 1 ⎤ 2 4 1 2β ⎦ + (16) PDCSK1 = erfc⎣ 2 pγb p2 · γb2 where γb , p and β are the same as those in (14). In the second case, an error is detected in the index symbol, where the modulated bits are detected by the sign of a wrong correlator, thus the probability of correct detection is 0.5. Based on the above two cases, the BER for the modulated bits Pm1 is given by Pm1 = PDCSK1 (1 − Psi1 ) + 0.5Psi1 .
(17)
Putting (15) and (17) into (10), BER of CI-DCSK1 Pe1 is derived. When L = 1, α = 1, it is the AWGN case.
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 64, NO. 8, AUGUST 2017
C. Derivation of the BER Expression for CI-DCSK2 In this scheme, the number of active subcarriers is 2p1 − 1, and the number of inactive subcarrier is 1. For each symbol duration, the total number of the transmitted bits is p1 +2p1 −1, which is consist of p1 index bits and 2p1 − 1 modulated bit. 1) BER for Index Bits: In CI-DCSK2, when the absolute value of the decision variable for the only one inactive carrier is larger than the minimum absolute value of the decision variable for the active carriers, there will be an error occurring. Based on the CDF of the quotient of two random variables, the SER of index bits Psi2 is derived as Psi2 = Pr Dv,j > min Dv,i +∞
2p1 −1 = 1 − 1 − F|Dv,i | (y) f|Dv,j | (y)dy (18) 0
where f|Dv,j | (y) is the PDF of |Dv,j | and F|Dv,i | (y) is the CDF of |Dv,i |, as shown in (19) and (20), respectively, with 1 ≤ i ≤ 2p1 − 1 and j = 1. Similarly, the above CDF and PDF are deduced from |Dv,i | and |Dv,j | which follow a folded normal distribution. The absolute value of 2p1 − 1 correlator outputs from the active carriers are independent identically distributed random variables, hence min(|Dv,i |) is the minimum p order statistic, whose CDF is (1 − (1 − F|Dv,i | (y))2 1 −1 ). Here
⎞⎤
⎢ ⎜ y2 ⎜ ×⎢ ⎣exp⎝− L 2 v 2 l=1 αv,l Ec N0 /2 +
N02 β 4
⎛
⎟⎥ ⎥ ⎟ ⎠⎦
⎞
⎢ ⎜ 2 Ev y − Ll=1 αv,l ⎜ 1⎢ c ⎜ ⎢ F|Dv,i | (y) = ⎢erf ⎜ 2⎣ ⎝ L 2 v 2 l=1 αv,l Ec N0 +
Pi2 = Psi2 /p1 .
(22)
2) BER for Modulated Bits: Similarly, two different cases may lead to an error in the modulated bits. In the first case, there is no error detected in the index symbol, but an error is detected in modulated bits. In this case, the BER of the modulated bits on each of the 2p1 − 1 carriers is given by ⎡ − 1 ⎤ 2 p1 +1 2p1 2 1 β · 2 ⎦ PDCSK2 = erfc⎣ + (23) 2 pγb 2p2 · γb2 where γb , p, and β are the same as those in (21). In the second case, there is an error detected in the index symbol. Since the probability of wrong detection for all the p modulated bits are Psi2 · (1 − (0.5)2 1 −1 ), the BER for each modulated bit can be calculated by averaging the above error rates by 2p1 − 1. Based on the above two cases, the BER for the modulated bits is given by !
p1 Pm2 = PDCSK2 (1 − Psi2 ) + Psi2 1 − (0.5)2 −1 / 2p1 − 1 . (24)
L − 12 N02 β π 2 v N0 f|Dv,j | (y) = αv,l Ec + 2 2 4 l=1 ⎡ ⎛
⎡
2 Ev , p = p + p , n = 2p1 where γb = (2p1 /(pN0 )) Ll=1 αv,l 1 2 c p 1 and p2 = 2 − 1. Based on Psi2 , the BER of index bits for CI-DCSK2 Pi2 is given by
(19)
⎟ ⎟ ⎟ ⎟ 2 N0 β ⎠ 4
⎛ ⎜ 2 Ev y + Ll=1 αv,l ⎜ c ⎜ + erf ⎜ ⎝ L 2 v 2 l=1 αv,l Ec N0 +
⎞⎤ ⎟⎥ ⎟⎥ ⎟⎥ ⎟⎥. 2 N0 β ⎠⎦ 4
(20) Based on (19) and (20), (18) can be simplified as +∞ 8n 2ny2 Psi2 = exp − π (2pγb + βn) 0 2pγb + nβ ⎞ ⎡ ⎛ ⎛ ⎟ ⎢ ⎜ y + pn γb 1 ⎜ ⎟ ⎜erfc⎜ ×⎢ 1 −
⎠ ⎣ ⎝ ⎝ n−1 2 1 4p γ + β b 2 n ⎛
⎞⎞n−1 ⎤
⎟⎟ ⎜ y − pn γb ⎟⎟ + erfc⎜
⎠⎠ ⎝ 1 4p 2 n γb + β
⎥ ⎥dy ⎦ (21)
Putting (22) and (24) into (10), BER of CI-DCSK2 Pe2 is derived. When L = 1, α = 1, it is the AWGN case. In this letter, only L independent identically distributed Rayleigh fading channels are considered, thus, the PDF of γb can be written as [6], [12] γbL−1 γb (25) exp − f (γb ) = γc (L − 1)!γ Lc where γ c = (Eb /N0 )E[αj2 ] = (Eb /N0 )E[αl2 ] (j = l), is the average bit-SNR per channel, and γb = ( Ll=1 αl2 )(Eb /N0 ), L 2 with l=1 E[αl ] = 1. Finally, the BER expression of the CI-DCSK schemes under multipath Rayleigh fading channel is given by +∞ Pmulti = Pe · f (γb )dγb . (26) 0
IV. N UMERICAL R ESULTS AND D ISCUSSIONS Figs. 2 and 3 show the BER performances of the CI-DCSK schemes over the AWGN channel and compare them with the obtained performances of the MC-DCSK scheme. Obviously, the simulation results and analytical BER expressions match very well, and both schemes outperform the traditional DCSK. In the following article, the performances of MC-DCSK scheme are compared to those of CI-DCSK schemes with the same number of transmitted bits per symbol in several different cases. When the numbers of transmitted bits per symbol are 4, 5, and 7, the corresponding numbers of index bits of CI-DCSK1 are 3, 4, and 6, which include 9, 17, and 65 carriers in the system, and for MC-DCSK the number M of carriers is 5, 6, and 8, respectively. Thus, the numbers of the transmitted bits per subcarrier are (4/9), (5/17), and (7/65) for CI-DCSK1, and (4/5), (5/6), and (7/8) for MC-DCSK. When comparing to MC-DCSK with the same number of bits per symbol, the spectral efficiency of CI-DCSK1 is reduced but the BER performance shown in Fig. 2 is significantly improved.
CHENG et al.: CI-DCSK MODULATION
Fig. 2. Performance comparisons between MC-DCSK and CI-DCSK1 with a spreading factor of β = 160 over the AWGN channel, with the same number of transmitted bits per symbol.
Fig. 3. Performance comparisons between MC-DCSK and CI-DCSK2 with a spreading factor of β = 80 over the AWGN channel, with the same number of transmitted bits per symbol.
When the numbers of transmitted bits per symbol are 5, 19, and 69, the corresponding numbers of index bits of CI-DCSK2 are 2, 4, and 6, which include 5, 17, and 65 carriers in the system, and for MC-DCSK the number M of carriers is 6, 20, 70, respectively. Thus, the numbers of transmitted bits per subcarrier are (5/5), (19/17), and (69/65) for CI-DCSK2, and (5/6), (19/20), and (69/70) for MC-DCSK. When comparing to MC-DCSK with the same number of bits per symbol, the spectral efficiency of CI-DCSK2 is obviously improved. As shown in Fig. 3, when the number of the bits per symbol is 5, the BER of MC-DCSK and CI-DCSK2 is almost the same, especially at high SNR. When the numbers of the bits per symbol are 19 and 69, the BER of MC-DCSK is a little better than that of CI-DCSK2. The BER of CI-DCSK2 decreases significantly at high SNR as shown in Fig. 3. Compared with MC-DCSK, CI-DCSK2 can be considered as a scheme with a higher spectral efficiency at the expense of a little degradation in BER performance. In the CI-DCSK2 scheme, the number of modulated bits is more than that of index bits, so the performance is mainly determined by the BER of modulated bits. With the number of index bits increasing, the average energy per modulated bit reduces and then the BER of modulated bits degrades, which leads to the BER of the system decreasing, as shown in Fig. 3. In the case of multipath Rayleigh fading channels, three paths L = 3 are considered, having equal average power gain as E[α12 ] = E[α22 ] = E[α32 ] = 1/3, with different time delays τ1 = 0, τ2 = 2 and τ3 = 5. Fig. 4 shows the accuracy of the computed derivations over the multipath Rayleigh fading channel for CI-DCSK1 and CI-DCSK2.
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Fig. 4. BER performances of CI-DCSK1 and CI-DCSK2 over a multipath Rayleigh fading channel, with spreading factor β = 80, and the numbers of index bits are p1 = 1, 6 and p1 = 1, 2, respectively.
V. C ONCLUSION A novel CI-DCSK communication scheme based on IM and MC-DCSK has been designed and analyzed in the brief. Based on two different index and detect methods exploited in the transceivers, two types of CI-DCSK are introduced. The good match between simulation results and analytical BER expressions, over AWGN as well as multipath Rayleigh fading channels, demonstrates that the proposed CI-DCSK schemes are quite accurate. In comparison with the traditional DCSK, the new CI-DCSK schemes achieve a higher data rate with lower energy consumption. In comparison with MC-DCSK, CI-DCSK1 gains better energy efficiency, while CI-DCSK2 achieves better spectral efficiency. R EFERENCES [1] E. Ba¸sar, Ü. Aygölü, E. Panayırcı, and H. V. Poor, “Orthogonal frequency division multiplexing with index modulation,” IEEE Trans. Signal Process., vol. 61, no. 22, pp. 5536–5549, Nov. 2013. [2] R. Abu-Alhiga and H. Haas, “Subcarrier-index modulation OFDM,” in Proc. IEEE 20th Int. Symp. Pers. Indoor Mobile Radio Commun., Tokyo, Japan, Sep. 2009, pp. 177–181. [3] R. Fan, Y. J. Yu, and Y. L. Guan, “Generalization of orthogonal frequency division multiplexing with index modulation,” IEEE Trans. Wireless Commun., vol. 14, no. 10, pp. 5350–5359, Oct. 2015. [4] G. Kaddoum, Y. Nijsure, and H. Tran, “Generalized code index modulation technique for high-data-rate communication systems,” IEEE Trans. Veh. Technol., vol. 65, no. 9, pp. 7000–7009, Sep. 2016. [5] G. Kolumban, M. P. Kennedy, and L. O. Chua, “The role of synchronization in digital communications using chaos. I. Fundamentals of digital communications,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. vol. 44, no. 10, pp. 927–936, Oct. 1997. [6] Y. Xia, C. K. Tse, and F. C. M. Lau, “Performance of differential chaos-shift-keying digital communication systems over a multipath fading channel with delay spread,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 51, no. 12, pp. 680–684, Dec. 2004. [7] W. Xu, L. Wang, and G. Chen, “Performance of DCSK cooperative communication systems over multipath fading channels,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 1, pp. 196–204, Jan. 2011. [8] G. Kaddoum, F.-D. Richardson, and F. Gagnon, “Design and analysis of a multi-carrier differential chaos shift keying communication system,” IEEE Trans. Commun., vol. 61, no. 8, pp. 3281–3291, Aug. 2013. [9] H. Yang, G.-P. Jiang, and J. Duan, “Phase-separated DCSK: A simple delay-component-free solution for chaotic communications,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 61, no. 12, pp. 967–971, Dec. 2014. [10] L. Wang, G. Cai, and G. R. Chen, “Design and performance analysis of a new multiresolution M-ary differential chaos shift keying communication system,” IEEE Trans. Wireless Commun., vol. 14, no. 9, pp. 5197–5208, Sep. 2015. [11] G. Kaddoum and E. Soujeri, “NR-DCSK: A noise reduction differential chaos shift keying system,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 63, no. 7, pp. 648–652, Jul. 2016. [12] J. G. Proakis and M. Salehi, Digital Communications. New York, NY, USA: McGraw-Hill, 2007.
