NOLTA, IEICE
Paper

M -ary modulation scheme based on
separation of deterministic chaotic
dynamics for noncoherent chaos-based
communications
Shintaro Arai 1 a) , Yoshifumi Nishio 2 b) , and Takaya Yamazato 3 c)
1

Dept. of Communication Network Engineering, Kagawa National College ofTechnology
551 Kohda, Takuma-cho, Mitoyo, Kagawa 769-1192, Japan

2

Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan

3

Institute of Liberal Arts and Science, Nagoya University
Furo-cho, Chikusa-ku, Nagoya 464-8063, Japana)arai@cn.kagawa-nct.ac.jp
nishio@ee.tokushima-u.ac.jpc)yamazato@m.ieice.orgb)Received July 25, 2013; Revised November 27, 2013; Published April 1, 2014
Abstract: The present paper proposes a novel M -ary modulation scheme based on separation
and reconstruction of deterministic chaotic dynamics for noncoherent chaos-based communications. The M -ary modulation scheme can transmit b-bit data at a time by using M = 2b
distinct sequences. In order to generate M distinct sequences and recover b-bit data, theproposed system separates the chaotic dynamics having the chaotic sequence by using M interleavers, and reconstructs the original sequence based on the chaotic dynamics from theseparated one by using M deinterleavers. In advance, each data symbol among M symbols is
allocated to each interleaver-deinterleaver pair. The transmitter selects an interleaver corresponding to a data symbol and separates the order of samples of the chaotic sequence using
the selected interleaver. The receiver feeds the received sequence into all M deinterleavers andoutputs M reconstructed sequences. The proposed system can reconstruct the original chaotic
sequence only when the correct deinterleaver, which becomes paired with the interleaver, isselected. Therefore, the receiver can recover b-bit data by analyzing the chaotic dynamics ofeach reconstructed sequence. We carry out computer simulations and evaluate performancesof the proposed M -ary modulation scheme.
Key Words: chaos, noncoherent chaos-based communications, M -ary modulation, deterministic chaotic dynamics

210
Nonlinear Theory and Its Applications, IEICE, vol. 5, no. 2, pp. 210–221c
IEICE2014

DOI: 10.1588/nolta.5.210

1. Introduction
Communications using chaotic systems are one of interesting topics in the ﬁeld of engineering chaos [1–24]. Chaotic signals, which are generated in a deterministic rule, have following several properties:
non-periodic, random-like, bounded, and wide-band signals, sensitivity to initial conditions, diﬃculty
to predict their future behavior from past observations. During the past few decades, many researchers
have focused these properties and have addressed several topics about chaos-based communications:spread-spectrum communication systems and their related modulation schemes [1–9], multiple-access
schemes [10–13], chaos-based coding schemes [14–19], their applications [20–22], etc.
In general, chaos-based communication systems are classiﬁed into two main categories: coherent
and noncoherent systems. Coherent systems need identical chaotic sequence between the transmitterand receiver for demodulation. In other words, the receiver must reproduce replicas of the basis
signals (unmodulated carriers) to recover data. On the other hand, noncoherent systems need not
reproduce the basis signals for demodulation at the receiver. Thus, demodulation is performed basedon the received signal. Additionally, the noncoherent systems are categorized according to decoding
operation: correlator-type and non-correlator-type. In correlator-type systems, data is demodulated
by correlation operation without reproduction of replicas of the basis signals, in whereas in the noncorrelator-type systems, data is recovered without correlation operation. Instead of correlation operation, demodulation is performed based on characteristics of chaos having the received sequence. We
consider that the noncoherent system is the unique communication system using chaos. Diﬀerential
chaos shift keying (DCSK) receiver [1] and the optimal receiver [2] are well-known as typical noncoherent correlator-type and non-correlator-type systems, respectively. Moreover, it is also important
to develop a suboptimal receiver, which has a performance equivalent to or similar to the optimalreceiver, using more eﬃcient algorithms [4, 23]. In this study, we focus on the noncoherent noncorrelator-type system as the chaos-based communication. Note that the noncoherent system has
diﬀerent meanings between chaos-based communications and standard ones. The noncoherent system
in standard communications means that the receiver recovers data without carrier phase information.
Namely, the concept of noncoherent system diﬀers from each other.
Development of the noncoherent chaos-based communication system is essential to diﬀerentiate
from standard ones. However, the communication property of the noncoherent system is lower ascompared with the standard one since the noncoherent system recovers data without the basis signals,
as described above. Therefore, we consider that it is important to advance noncoherent chaos-based
communication systems for improving the communication property.
In our previous study, we focused on the deterministic chaotic dynamics, which is one of characteristics of chaos, and proposed the error-correcting scheme based on the chaotic dynamics [24]. Ourerror correction focuses on two successive chaotic sequences generated from the same chaotic map;the second sequence is generated with an initial value that is determined from the last value of theﬁrst sequence. Here, we connect the initial value of the second sequence and the last value of theﬁrst one, and create a long sequence. In this case, the created sequence is regarded as the successive
chaotic sequence having the same chaotic dynamics. This feature gives the receiver additional information to correctly recover data and thus improves the bit error performance. Based on this feature,
we designed the error-correcting scheme for noncoherent chaos-based communications. Computer
simulations conﬁrmed that the advantage gained in the bit error rate (BER) performance of our error
correction is about 1–1.5 dB compared to a conventional method (w/o coding). From the previous
results, we concluded that the chaotic dynamics had a great inﬂuence on the demodulation of chaos
communications.
The key point of our error correction is to apply successive chaotic sequences based on the chaotic
dynamics between identical symbols. Here, let us focus on the order of the chaotic sequence of eachsymbol. As a matter of fact, samples of the chaotic sequence are generated based on the identicalchaotic dynamics. Next, let us consider that the order of the samples is separated using a speciﬁcrule. In this case, the separated sequence loses the characteristic of the chaotic dynamics. In order torecover the original sequence based on the chaotic dynamics, we need to reconstruct the order of the211samples using the inverse of the rule. We consider that these operations are applied for improving thecommunication property of the noncoherent chaos-based communications as additional information.
The purpose of this study is to propose and examine a novel M -ary modulation scheme using theseparation and reconstruction of chaotic dynamics for noncoherent chaos-based communications. The
M -ary modulation scheme can transmit b-bit data at a time by using M = 2b distinct sequences,i.e., one sequence for each of the 2b expresses b-bit sequences [25]. This study generates M distinct
sequences by separating the chaotic dynamics having the chaotic sequence. In addition, the receiverof this study recovers b-bit data by reconstructing the original sequence based on the chaotic dynamics from the received one. In order to separate and reconstruct the chaotic dynamics, we use an−1
interleaver πm and deinterleaver πm, respectively (m : 0, 1, · · · , M − 1). In the proposed M -ary modulation, we prepare M interleavers and M deinterleavers for the proposed transmitter and receiver. In
advance, each data symbol among M symbols is allocated to each interleaver-deinterleaver pair. The
proposed transmitter selects an interleaver corresponding to a data symbol and separates the order ofsamples of the chaotic sequence using the selected interleaver. The proposed receiver feeds a received
sequence into all M deinterleavers and outputs M reconstructed sequences. In the proposed system,
we can reconstruct the original chaotic sequence only when the correct deinterleaver, which becomes
paired with the interleaver, is selected. Thus, the receiver can recover b-bit data by analyzing thechaotic dynamics of each reconstructed sequence. We carry out computer simulations and evaluate
performances of the proposed M -ary modulation scheme.
Note that the present paper does not compare the communication performance between the proposed M -ary modulation scheme and the traditional M -ary one. As one of traditional M -ary communication schemes, there is the M -ary spread-spectrum (M -ary/SS) communication system [26]. The
M -ary/SS system is known as the noncoherent communication system. However, the concept of thisnoncoherent system diﬀers from chaos-based one, as described above. In addition, this system performs correlation operation using shared spreading sequences between the transmitter and receiver.
Thus, it is diﬃcult to easily compare the performance of our scheme to the traditional one due todiﬀerent concepts and constraint conditions. The present paper puts emphasis on comparison withconventional noncoherent chaos-based communication and evaluation of the communication propertyof our scheme.
The present paper is organized as follows. In Section 2, we introduce a concept of our proposed
M -ary modulation scheme. The proposed method is presented in Section 3, and its performance isevaluated in Section 4. Finally, we conclude the present paper in Section 5.

2. Concept of the proposed M -ary modulation scheme
This section introduces a concept of the proposed M -ary modulation scheme using Fig. 1. Thisﬁgure shows the separation and reconstruction of the chaotic sequence. In this ﬁgure, xi denotes asample of the chaotic sequence. As described in Section 1, the proposed scheme applies the chaotic−1
dynamics and its separation and reconstruction using the interleaver πm and deinterleaver πm. Notethat a sequence separated by m-th interleaver can be reconstructed to the original sequence by m-th
deinterleaver.
Here, let us consider two diﬀerent situations: correct or incorrect pair of the interleaver and deinterleaver. In the situation of the correct pair (see Fig. 1(a)), the order of the deinterleaved sequencecorresponds to the original chaotic sequence. When we draw a return map using the deinterleaved
sequence, all (xk , xk+1 ) points of the sequence overlap to lines of the original chaotic map. On theother hand, in the situation of the incorrect pair, almost points of the sequence do not overlap to theoriginal chaotic map since the incorrect deinterleaver cannot reconstruct the original chaotic sequence,as shown in Fig. 1(b). Thus, we can distinguish whether the sequence based on the chaotic dynamics
or not by analyzing the chaotic property of the reconstructed sequence.
We use this characteristic (i.e., the loss of the chaotic dynamics) as M -ary modulation for noncoherent chaos-based communications. This characteristic gives the receiver additional information tocorrectly recover data.

212

Fig. 1.

Concept of the proposed M -ary modulation scheme.

Fig. 2.

Block diagram of the proposed M -ary modulation.

3. System model of the proposed M -ary modulation scheme
Figure 2 shows a block diagram of the proposed M -ary modulation scheme. In this scheme, thetransmitter and receiver have M interleavers and M deinterleavers, respectively, for separating andreconstructing the chaotic dynamics.
The communication performance of the proposed method depends on the design of interleaverdeinterleaver pairs. Thus, it is important that how to design separation patterns of interleavers. In
order to increase the decoding property, we design separation patterns of interleavers with following
constraint conditions. We randomly allocate numbers for a separation pattern and set a uniqueseparation pattern with each interleaver-deinterleaver pair. Note that each separation pattern never213

Fig. 3.

M -ary modulation using separation of the chaotic dynamics.

matches other patterns except oneself. In addition, it is preferable that even a part of each patterndoes not overlap other ones. As an example, we consider following separated sequences when N = 8.
Original sequence: x0 , x1 , x2 , x3 , x4 , x5 , x6 , x7
Separated sequence A: x2 , x1 , x4 , x3 ,x7 , x6 , x0 , x5
Separated sequence B: x2 , x1 , x4 , x3 ,x6 , x5 , x7 , x0
As one can see, four samples (x2 , x1 , x4 , x3 ) fully overlap between two separated sequences. This caseinﬂuences the decoding process because a part of the sequence is reconstructed as the original one by
the incorrect deinterleaver. Thus, we have to avoid overlaps among separation patterns of interleavers
wherever possible. In advance, each data symbol among M symbols is allocated to each interleaver
and deinterleaver pair.

3.1 Transmitter (M -ary modulation)
The transmitter consists of a chaotic signal generator and M interleavers, as shown in Fig. 3. A
chaotic sequence is ﬁrst generated by the chaotic signal generator. In this study, we use a skew tent
map, which is one of simple chaotic maps, for the generator, and it is described by Eq. (1).

xi+1 =⎧
2xi + 1 − a
⎪
⎪
⎪
⎨
1+a

(−1 ≤ xi ≤ a),⎪⎪−2xi + 1 + a⎪⎩
1−a(a < xi ≤ 1),(1)

where i = 0, 1, · · · , N − 1 (N is a chaotic sequence length) and a denotes a position of the top of the
skew tent map. Next, an arbitrary interleaver πm is selected according to an input data m. Finally,
the chaotic sequence is fed into the selected πm , and the order of samples of the sequence is separated
according to the separation pattern of πm for separating of the chaotic dynamics of the sequence. The
ﬁrst-generated-chaotic sequence S and the interleaved sequence Sm are given as follows.
S = x0 , x1 , · · · , xi , · · · , xN −1 ,Sm

= x0,m , x1,m , · · · , xi,m , · · · , xN −1,m ,

(2)
(3)

where x0 denotes an initial value that is selected at random, xi,m denotes a signal sample interleavedby πm . Thus, Sm is outputted as the transmitting sequence.

214

Fig. 4.

M -ary demodulation using reconstruction of the chaotic dynamics.

3.2 Channel and noise
The channel distorts the signal and corrupts it by noise. In this study, noise of the channel isassumed to be the additive white Gaussian noise (AWGN). Thus, the received sequence is given by
R = (r0 , r1 , · · · , rN −1 ) = Sm + AWGN.

3.3 Receiver (M -ary demodulation)

The receiver consists of M deinterleavers and an analysis unit of the chaotic dynamics of the sequence(second unit), as shown in Fig. 4. Since we consider the noncoherent system, the receiver memorizes
the chaotic map used at the transmitter. However, the receiver never knows the initial value of thechaotic sequence generated at the transmitter. In other words, the receiver cannot reproduce replicasof the basis signals.
The received sequence R is ﬁrst fed into all M deinterleavers, and the order of the sequence is−1
separated by each deinterleaver πm. The deinterleaved sequence Rm is given as follows.
Rm= (r0,m , r1,m , · · · , ri,m , · · · , rN −1,m ),

(4)

−1
.
where ri,m denotes a sample of the sequence deinterleaved by πm
Each deinterleaved sequence Rm is fed into the second unit, and the chaotic dynamics of eachsequence is analyzed. In this study, we apply the suboptimal detection algorithm [23, 24] for analyzing
the chaotic dynamics. This algorithm calculates the shortest distance between Nd -successive signal
samples of the received sequence and Nd -dimensional space made from Nd successive chaotic signal
samples generated by the skew tent map (Nd : 2, 3, · · ·). Moreover, this algorithm outputs the sum of
the distance, which achieves the minimum shortest distance within the set of l, from i = 0 to N − Nd .
Here, we deﬁne following two parameters: R i,m as the Nd -successive signal samples beginning with i−1of the sequence deinterleaved by πm, and l as the number of the straight lines in the Nd -dimensional
space. These parameters are given as follows.

R i,m= (ri,m , ri+1,m , ..., ri+Nd −1,m ),
Nd −1

l = 2

.

(5)
(6)

For calculating the shortest distance, we ﬁnd the closest point Pl between R i,m and the lth line using
the scalar product of the vector. When both edges of the lth line are deﬁned as P l and P l shownin Fig. 5, the closest point Pl is calculated by the following equation.

215

Fig. 5.

Pl

Calculation of the shortest distance.

(l) (l)(l)=p0 , p1 , · · · , pNd −1 = (ul · vl ) ul + P l ,

(7)

where
P l − P l,||P l − P l ||= R i,m − P l .

unit vector ul=vlThen, each outputted distanceis expressed as



(8)
(9)

−1
Dm , which is calculated using the sequence deinterleaved by πm
,

N −Nd

Dm

=


min ||Pl − R i,m ||,

i=0

l

(10)


Next, the receiver selects the smallest
Dm from all outputted
Dm values and outputs the
decoded data. As described above, a separating pattern of each interleaver-deinterleaver pair does
not correspond with other pairs. Namely, the single (correct) deinterleaver within the second unit canreconstruct the original chaotic sequence generated at the transmitter. When the successive signal
samples based on the chaotic dynamics is used for the calculation ofDm , the calculation value
becomes the smallest value from other calculations. Finally, the receiver outputs a decoded symbol−1corresponding to πm, which reconstructs Rm for the smallest
Dm .
This suboptimal detection algorithm have been proposed based on the optimal receiver [2], which
is based on an optimal classiﬁer which optimizes the bit error rate by selecting the symbol thatminimizes the a posteriori probability. Our suboptimal algorithm approximates the calculation of theprobability by calculating shortest distances. As simulation results, we have already conﬁrmed thatthe performance of our suboptimal algorithm is equivalent to the optimal one [23]. Therefore, it isconsidered that the calculation of the sum of the shortest distance is suitable for analyzing the chaotic
dynamics in our M -ary demodulation scheme although we do not try to use other decision criteria.

4. Evaluation of the proposed M -ary modulation
This section explains the computer simulation of the proposed M -ary modulation and evaluates its
performances. Experimental conditions are as follows. The parameter of the skew tent map is ﬁxed
as a = 0.05. The initial value of the chaotic sequence is set at random every transmitting data.
We assume following parameters: b = 2, 3 and 4, M = 4, 8 and 16, and N = 4–256. For calculationof the shortest distance, we use 4-dimensional space (Nd = 4). Based on these conditions, we iterate
the simulation 10,000 times and calculate BER performance.
Figures 6(a) and (b) show the BERs versus Eb /N0 (SNR per bit) for N = 32 and N = 64, respectively. We plot the performance of the proposed scheme with M = 4, 8 and 16 and the performanceof the suboptimal receiver as the binary noncoherent chaos-based communication (i.e., conventional
method). From both ﬁgures, we can observe that the BER performance of the proposed scheme
improves with increasing the number of M . Let us focus on BER = 1 × 10−4 of both ﬁgures. To
achieve this BER performance, the required Eb /N0 of the binary noncoherent system is about 18 dB.

216

Fig. 6.

BER vs. Eb /N0 (M = 4, 8 and 16).

On the other hand, in the case of the proposed scheme, the required Eb /N0 is 16 dB or less when
M = 16 (b = 4). Namely, to achieve BER= 1 × 10−4 , the proposed scheme with M = 16 realizes asaving of energy about 2 dB as compared with the binary system.
Figures 7(a) and (b) show the BER performances of the proposed and conventional one with thesame Eb /N0 for N = 32 and N = 64, respectively. To achieve the same Eb /N0 , we set each diﬀerent
N for each method. When we assume N = 32 and 64 of the proposed scheme with M = 16 (b = 4), Nof the conventional one becomes 8 (= 32/b) and 16 (= 64/b), respectively, for performing simulations
with the same Eb /N0 . As shown in Figs. 7(a) and (b), BERs of the proposed scheme show gains
over the conventional one. The maximum 1.4dB gain in error performance is obtained by using the
propose scheme. As the reason for this, we consider that the separation and reconstruction of the217

Fig. 7.

BER vs. Eb /N0 (Comparison with the same Eb /N0 ).

chaotic dynamics work eﬀectively. As described above, the correct deinterleaver can only reconstruct
the original chaotic sequence generated at the transmitter. In this case, other incorrect deinterleavers
further separate the order of samples of the received sequence, which has already separated the chaotic
dynamics at the transmitter. Namely, the sequence separated by the incorrect deinterleaver mostly
loses the characteristics of chaos. Thus, we consider that the error performance of the proposed scheme
is better than the conventional one with the same Eb /N0 since the interleaver and deinterleaver giverise to great diﬀerence in the characteristic between the correct sequence and other incorrect sequences.
Figure 8 plots the BERs versus the chaotic sequence length N for diﬀerent Eb /N0 values. For faircomparison, we show the performance of the conventional system with the same Eb /N0 . For example,
we compare the performance of the proposed scheme with N = 16 to the conventional one with N = 4

218

Fig. 8.

BER vs. N (M = 16).

when M = 16. From Fig. 8, it can be conﬁrmed that the performance of the proposed one clearly
increases as compared with the conventional one when N = 32, 64 and 128. Especially, the BER
of N = 64 with Eb /N0 = 16 dB improves from 1 × 10−3 to 1 × 10−4 or more using the proposed
scheme. From this result, we ﬁnd that the eﬀectiveness of using the separation and reconstructionof the chaotic dynamics for M -ary modulation. However, we also ﬁnd the degradation of the error
performance of the proposed scheme when N = 4, 8 and 256. First let us focus on BERs in theregion of small N (N = 4 and 8). As the reason for degradation of the performance in this region,
we consider the decrease of the number of separation patterns using the interleaver and deinterleaver.
For example, when N = 4, the number of the patterns is N ! = 4! = 24. When we choose 16 patterns
from 24 patterns, some samples certainly overlap among chosen patterns since the number of thepatterns is very low. In other words, the interleaver cannot fully separate the chaotic dynamics ofthe sequence in the case of low number of the patterns. Thus, the analysis error increases due tothe inﬂuence of low number of the patterns. Next, we focus on the BER when N = 256 (i.e., theregion of large N ). As the reason for the performance degradation, we consider an accumulation of
calculation error of the shortest distance. In general, the transmitting sequence is distorted in thechannel and is arrived at the receiver. Due to the inﬂuence of the channel, the calculation error of
the shortest distance occurs even if the chaotic dynamics of the received sequence is reconstructed by
the correct deinterleaver. As described in Eq. (10), the outputted shortest distance is the sum of Dmfrom i = 0 to N − Nd . Namely, the accumulation of calculation error increases with increasing N .
Thus, the accumulation of calculation error aﬀects an accuracy of demodulation. This degradation
phenomenon is often found in other noncoherent chaos-based communication systems [3, 5–7]. Several
studies about noncoherent systems have reported that the BER increases because the noise component
dominates over the gain in the required signal component when N becomes large at constant Eb /N0 .
This conclusion is similar to our discussion. We consider that the degradation with increasing N is
speciﬁc phenomenon of noncoherent chaos-based communication systems.
Therefore, in order to achieve the better error performance, we need to ﬁnd and set the following
suitable parameters: number of separation patterns of the interleaver and deinterleaver, and thechaotic sequence length N .

5. Conclusions

The present paper has proposed the novel M -ary modulation scheme using the characteristics ofchaos for noncoherent chaos-based communications. We have focused on the successive samples of thechaotic sequence based on the chaotic dynamics and have addressed the separation and reconstruction219

of the chaotic dynamics of the sequence, which can be applied as additional information to correctly
recover b-bit data, by using the interleaver and deinterleaver for M -ary modulation.
As results of the computer simulation, we have observed that the proposed scheme with M = 16
realizes a saving of energy about 2 dB as compared with the conventional system for achieving
BER = 1 × 10−4 . Moreover, we have conﬁrmed that the advantage gained in BER performance of the
proposed scheme is about 1.4 dB compared to the conventional one with the same Eb /N0 . Finally, it
has been concluded that it is necessary to ﬁnd and set the suitable parameters of the proposed system
in order to achieve the better error performance. We currently consider that the design method ofinterleaver-deinterleaver pairs with the constraint conditions, which is described in Section 3, is a
suboptimum method in this study. As an important future work, we would like to discuss whether
our design method is optimum.

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