A New Hierarchical M -ary DCSK Communication SystemDesign and AnalysisCAI, GUOFA; FANG, YI; HAN, GUOJUN; WANG, LIN; CHEN, GUANRONG

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Publication details:CAI, GUOFA., FANG, YI., HAN, GUOJUN., WANG, LIN., & CHEN, GUANRONG. (2017). A New HierarchicalM -ary DCSK Communication System: Design and Analysis. IEEE Access, 5, 17414-17424. [8013020].https://doi.org/10.1109/ACCESS.2017.2740973

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A New Hierarchical M-ary DCSK CommunicationSystem: Design and AnalysisGUOFA CAI1, (Member, IEEE), YI FANG1, (Member, IEEE),GUOJUN HAN1, (Senior Member, IEEE), LIN WANG2, (Senior Member, IEEE),AND GUANRONG CHEN3, (Fellow, IEEE)1School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China2Department of Communication Engineering, Xiamen University, Xiamen 361005, China3Department of Electronic Engineering, City University of Hong Kong, Hong Kong

Corresponding author: Yi Fang (fangyi@gdut.edu.cn)

This work was supported in part by NSF of China under Grant 61701121, Grant 61771149, Grant 61501126, Grant 61401099, andGrant 61671395, in part by the Hong Kong Research Grants Council through the GRF under Grant CityU 11208515, in part by NSF ofGuangdong Province under Grant 2016A030310337, in part by the Guangdong Innovative Research Team Program underGrant 2014ZT05G157, in part by the Science and Technology Program of Guangzhou under Grant 201607010155, in part by theGuangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2017), and in part by the China Post-DoctoralScience Foundation under Grant 2017M612612.

ABSTRACT Multiresolution M -ary differential chaos shift keying (MR-M -DCSK) modulation using non-uniformly spaced phase constellation is a promising technique that can satisfy different bit-error-rate (BER)requirements within one symbol over multipath fading channels. However, the BER performance alwaysdeteriorates as the modulation order M becomes larger. To overcome this shortcoming, a new hierarchicalsquare-constellation-based M -DCSK communication system using non-uniformly spaced distance constel-lation is proposed, which can be easily extended to other-constellation scenarios, e.g., rectangular, star, andasymmetrical square constellations. Furthermore, an adaptive transmission scheme is designed bymodifyingthe distance between the two constellation points in the last hierarchical level. In addition, theoreticalBER expressions of the proposed systems are derived over multipath Rayleigh fading channels, which areconsistent with the simulated results. Analytical and simulated results show that the proposed system notonly can provide lower energy consumption as compared with the conventional MR-M -DCSK system butalso can efficiently adjust BER performance based on the signal-to-noise ratio information. Therefore, theproposed system can serve as a desirable alternative for energy-efficient short-rangewireless-communicationapplications.

INDEX TERMS Hierarchical square-constellation-basedM -ary differential chaos shift keying (M -DCSK),bit error rate (BER), multipath Rayleigh fading channel, adaptive transmission.

I. INTRODUCTIONAs a promising chaotic modulation technique, differen-tial chaos shift keying (DCSK) has very desirable per-formance with relatively low complexity, and hence hasdrawn much attention in the past two decades. In contrastto the conventional differential phase shift keying (DPSK)system, the DCSK system not only inherits the advantageof requiring no channel state information (CSI) for detec-tion, but also is more robust to multipath fading [1]–[5].More importantly, DCSK modulation is particularly suit-able for ultra-wideband (UWB) applications [6]–[8]. To fur-ther expand its scope of applications, a large volume ofresearch work has been dedicated to designing network

coding schemes [9], [10], cooperative schemes [2], [11]–[13],multi-input multi-output schemes [3]–[5], [14]–[16], andsimultaneous wireless-information-and-power transferschemes [17], under the condition of DCSK modulation.In recent years, DCSK has also played an increasingly impor-tant role in short-range wireless-communication applications,such as wireless sensor networks (WSNs).

Unfortunately, conventional DCSK systems require aradio-frequency (RF) delay line at both the transmitter andthe receiver, which is rather difficult to be implemented.To address this weakness, a code-shifted DCSK (CS-DCSK)system based on Walsh codes has been conceived to removethe RF delay line at the receiver [18]. Following this work,

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G. Cai et al.: New Hierarchical M-ary DCSK Communication System: Design and Analysis

a multicarrier DCSK (MC-DCSK) system [19] and a phase-separated DCSK (PS-DCSK) system [20] have been furtherdeveloped to avoid using delay lines in both the transmitterand the receiver. In the MC-DCSK and PS-DCSK systems,the orthogonal sinusoidal signals (i.e., carriers) instead of thetime-delayed signals are transmitted.

In conventional DCSK-related systems, the reference-chaotic signal and information-bearing signal are transmittedsuccessively using the same energy in each transmissionperiod, thus suffering from relatively low energy efficiencyand spectral efficiency. To boost the spectral efficiency,some variants have been explored [19]–[26]. Specifically,the generalized CS-DCSK and high-data-rate CS-DCSK sys-tems have been developed in [21] and [22], respectively.These modified CS-DCSK systems can dramatically improvethe spectral efficiency, since more than one information-bearing signals are transmitted in each transmission period.Also, an orthogonal multilevel DCSK system has been intro-duced in [23], which achieves outstanding performance butdecreases some spectral efficiency.

With respect to the single-carrier DCSK system, theMC-DCSK system enhances both energy and spectral effi-ciency [19], while the PS-DCSK system [20] only doublesthe data rate. Aiming to reduce the implementation com-plexity of the MC-DCSK system, an orthogonal-frequency-division-multiplexing DCSK system has been formulatedin [24]. As an extension of the quadrature chaos-shift keying(QCSK, i.e., 4-DCSK) modulation [25], a multiresolutionM -ary DCSK (MR-M -DCSK) system has been designedbased on the non-uniformly spaced phase constellationsin [26], which achieves both lower energy consumptionand higher spectral efficiency. This system can provide dif-ferent quality of services (QoSs) for the transmitted bitswithin a symbol subject to their specific bit-error-rate (BER)requirements. Recently, some research attempt has beendevoted to investigating a generalized constellation-basedM -DCSK communication system framework [27], in whichthe performance of both square and circle-constellation-based M -DCSK systems are extensively compared. Theresults have shown that the former can realize lower energyconsumption in comparison with the latter, and thus canbetter fit to power-constrained WSN applications. As com-pared with conventional modulation schemes, hierarchical-modulation schemes possess the unequal-error-protectionproperty, and hence are able to satisfy more requirements ofwireless-communication applications [28]–[30].

Motivated by the above discussions, a new hierarchicalsquare-constellation-basedM -DCSK communication systemis proposed in this paper, which can meet different BERrequirements for different transmitted bits within one symbol.Distinguished from the MR-M -DCSK system, the proposedsystem exploits the distance vector to realize a more flexibleBER service. Moreover, the new system can be extendedto the hierarchical other constellations-based M -DCSK sys-tems, e.g., rectangular, star and asymmetrical square constel-lations. Furthermore, an adaptive transmission scheme for the

proposed system is designed by utilizing the last elementsof the distance vectors. In addition, the theoretical errorperformance of the proposed system is carefully analyzedover multipath Rayleigh fading channels. Both analyticaland simulated results indicate that this hierarchicalM -DCSKsystem outperforms the conventional MR-M -DCSK system,and the designed adaptive scheme can satisfy a more flexibleBER requirement based on the signal-to-noise ratio (SNR)information. Therefore, the proposed system appears to bean excellent candidate for low-complexity and low-powerwireless-communication applications.

The major contributions of this work are asfollows.

1) A new hierarchical square-constellation-based M -DCSKcommunication system framework is proposed, whichcan be generalized to other types of constellations, e.g.,rectangular, star and asymmetrical square constellations.Moreover, an adaptive transmission scheme is designedfor the proposed system.

2) The BER expressions of the proposed system are derivedover multipath Rayleigh fading channels. Furthermore,the corresponding analysis methodology is general-ized to the hierarchical rectangular-constellation-basedM -DCSK system.

3) Both analytical and simulated results show that the newsystem accomplishes a noticeable performance gain thanthe conventional MR-M -DCSK system, and the designedadaptive scheme can efficiently modify the BER perfor-mance of the component bits within a symbol based onthe SNR information.

The rest of this paper is organized as follows.Section II introduces the hierarchical square-constellation-based M -DCSK communication system. Section III derivestheoretical BER expressions of the proposed system overmultipath Rayleigh fading channels. Section IV providesan adaptive transmission scheme for the proposed system.Section V performs simulations to demonstrate the supe-riority of the new design. Finally, Section VI gives someconcluding remarks.

II. HIERARCHICAL SQUARE-CONSTELLATION-BASEDM-DCSK SYSTEMIn this section, a new hierarchical square-constellation-basedM -DCSK communication system is proposed, with the blockdiagram illustrated in Fig. 1.

A. HIERARCHICAL SQUARE-CONSTELLATION-BASEDM-DCSK SYSTEMFig. 2 presents the constellation of a hierarchical square-based 64-DCSK signal. As seen from the figure, d ik and dqk(k = 1, 2) denote the distances between the decision bound-ary in the k-th hierarchical level and the decision boundaryin the (k + 1)-th level, respectively; and d i3 and d

q3 denote the

half distances between the two constellation points in the lasthierarchical level, respectively. Define two types of distance

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FIGURE 1. Block diagram of a hierarchical square-constellation-based M-DCSK communication system.

FIGURE 2. Constellation of a hierarchical square-based 64-DCSK signal.

vectors as follows:

di = [d i1, di2, . . . , d

im], (1)

and

dq = [dq1 , dq2 , . . . , d

qm], (2)

where d i1, di2, . . . , d

im−1 and dq1 , d

q2 , . . . , d

qm−1 are the deci-

sion boundaries for the two distance vectors, respectively,d ik > d ik+1, d

qk > dqk+1, and m = log2

√M .

Without loss of generality, assume that d im ≥ dqm. Then,define two priority vectors as

pi = [pi1, pi2, . . . , p

im−1, p

im]

=

[ d i1d im,d i2d im, . . . ,

d im−1d im

, 1], (3)

and

pq = [pq1, pq2, . . . , p

qm−1, p

qm]

=

[ dq1d im,dq2d im, . . . ,

dqm−1d im

,dqmd im

]. (4)

Thus, different BER values can be obtained by modifying thedistance vectors. The bit in the k-th position is protected withhigher priority than that in the (k+1)-th position, for a largerratio pik/p

ik+1 or p

qk/p

qk+1.

B. HIERARCHICAL SQUARE-CONSTELLATION-BASEDM-DCSK SYSTEMSuppose that a normalized chaotic reference signal cx =[cx,1, cx,2, . . . , cx,β ] is generated by a chaotic generator, e.g.,the logistic map

xk+1 = 1− 2x2k , k = 1, 2, . . . , β, (5)

where β is the spreading factor.When the signal cx is passed through a Hilbert transform,

a quadrature signal cy will be obtained. The two indepen-dent orthogonal signals are used to constitute the encodedinformation. Referring to Fig. 1, the encoded informationis expressed as ms = ascx + bscy, where the index s(s = 1, . . . ,M ) presents the symbol, and as and bs denotethe x-axis coordinate value and the y-axis coordinate valuecorresponding to a constellation point, respectively, with a2s+b2s 6= 1. In particular, as and bs can be computed by thedistance vectors di and dq. Moreover, the reference and infor-mation signals are expressed as xr (t) =

∑β

k=1 cx,kp(t − kTc)and xi(t) =

∑β

k=1 ms,kp(t − kTc), respectively, where Tcis the chip time, p(t) is the square-root-raised-cosine filter,∫ (k+1)TckTc

p(t − kTc)2dt = Ep,∑β

k=1 cx,kcy,k = 0, and∑β

k=1 c2x,k =

∑β

k=1 c2y,k = 1. Consequently, the trans-

mitted signal for the hierarchical square-constellation-basedM -DCSK system is formulated as

s(t) =√2xi(t) cos(2π f0t)−

√2xr (t) sin(2π f0t), (6)

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where f0 is the frequency of the sinusoidal carriers, withf0 � 1/Tc.

The impulse response function of a multipath fading chan-nel is given by h(t) =

∑Ll=1 αlδ(t−τl), where L is the number

of paths, and τl and αl are the path delay of the lth path andthe channel coefficient, respectively. Thus, the received signalcan be written as

r(t) = h(t)⊗ s(t)+ n(t), (7)

where ⊗ is the convolution operator and n(t) is widebandadditional white Gaussian noise (AWGN)with zeromean andthe variance N0/2.At the receiver end, the in-phase and quadrature carriers

are utilized to separate the received signal r(t). The sep-arated signals are first processed by matched filters, andsubsequently the filtered signals are sampled every kTc timeunits. The sampled reference signals will be correlated withthe sampled information signal or their Hilbert-transformedversion. The decision vector z = (za, zb) can be obtained afterthe correlation operation. It should be noted that the CSI, i.e.,h =

∑Ll=1 α

2l , can be estimated by a method in [27], [31].

Here, we simply assume that the receiver knows a perfectCSI. Prior to the decision process, the original decision vectorshould be first converted to z′ = z/(hEp) = (z′a, z

′b).

Generally, a search algorithm should be used to obtain thetransmitted bits. To be specific, the distances between thedecision vector and all constellation points, i.e., (z′a − as)

2+

(z′b − bs)2, are compared to find the bits for the positionof the minimum distance corresponding to a constellationpoint. However, as the value of M increases, the algorithmcomplexity will become higher. In fact, the receiver needs toperfectly know all elements of the distance vectors to make adecision.

Because square-based M -DCSK constellations can bereferred to as the superposition of several QCSK constel-lations based on the distance vectors di and dq, QCSKde-mapper and mapper in [25] are used to make the deci-sion in this paper, where the decision mechanism is shownin Fig. 3. As an example, the decision process for ahierarchical square-based 16-DCSK constellation is shownin Fig. 4. It is assumed that the bits ‘‘0000′′ are transmittedand the decision vector is presented by (z1, z2), as shown inFig. 4 (a). Based on the QCSK de-mapping rule, one can eas-ily obtain the two high-priority (HP) bits ‘‘00′′ because z1 > 0and z2 > 0. After that, the two HP bits are re-mapped by theQCSK constellation with the distance vector (d i1, d

q1 ). Based

on (z1, z2) and (d i1, dq1 ), one can obtain the decision vector

for the low-priority (LP) bits, denoted by (z1 − d i1, z2 − dq1 ).

According to the resultant vector (z1−d i1, z2−dq1 ) as shown in

Fig. 4(b), one can finally get the two LP bits, as ‘‘01′′. Hence,the decision process has lower complexity than the searchalgorithm when a large value of M is adopted. Note that, insuch a decision process, the receiver only needs to know thedistances d i1, d

i2, . . . , d

im−1 and d

q1 , d

q2 , . . . , d

qm−1 for making

the decision.

FIGURE 3. Decision mechanism for the hierarchical square-constellation-based M-DCSK system.

FIGURE 4. A graphic example for the decision process.

C. EXTENSION TO OTHER TYPES OF CONSTELLATIONSThe proposed system can be generalized to other types of con-stellations, such as rectangular, star and asymmetric squareconstellations. In the following, two examples are used toillustrate the flexibility of the distance vectors.

In the first example, consider the asymmetric hierarchicalsquare-based 16-DCSK constellation, as shown in Fig. 5(a).In this case, the HP bit b1 is determined by the first cluster,while the HP bit b2 is determined by the second cluster, whichis involved in the first cluster. Furthermore, the LP bit b3is determined by the third cluster, while the LP bit b4 isdetermined by the specific signal point involved in the thirdcluster. The asymmetric hierarchical square-based 16-DCSKhas three embedded sub-constellations and thus it can providefour levels of BERs.

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G. Cai et al.: New Hierarchical M-ary DCSK Communication System: Design and Analysis

FIGURE 5. Constellations of (a) a asymmetric hierarchical square-based16-DCSK signal, and (b) a hierarchical star-based 8-DCSK signal.

In the second example, consider the hierarchical star-based8-DCSK constellation, as shown in Fig. 5(b). This type ofconstellation efficiently combines the distance vector withthe phase vector, and hence it inherits both advantages ofthe hierarchical square-based and phase-based constellations.The HP bit b1 is determined by the distance vector, while thetwo LP bits b2 and b3 are determined by the phase vector.

III. PERFORMANCE ANALYSISIn this section, the theoretical BER performance for thehierarchical square-constellation-based M -DCSK system iscarefully analyzed over multipath fading channels. Besides,the corresponding analytical methodology is generalizedto the hierarchical rectangular-constellation-based M -DCSKsystem.

A. ENERGY AND CHANNELSThe average symbol energy for the hierarchical square-basedM -DCSK constellation can be computed as

Es =(di(di)T + dq(dq)T + 1

)Ep. (8)

FIGURE 6. Constellations of (a) the row/column for a hierarchicalsquare-based 16-DCSK signal, and (b) the row/column for a hierarchicalsquare-based 64-DCSK signal.

According to [31], when the inter-symbol interference(ISI) is negligible, the means of the two elements in thedecision vector z′ can be expressed as

E(z′a) = as,

E(z′b) = bs, (9)

where E(•) denotes the expectation operator. Moreover, thevariances of the two elements in z′ can be measured by

var(z′a) = var(z′b) =EsN0

2hE2p+

βN 20

4(hEp)2, (10)

where var(•) denotes the variance operator.For convenience, assume that the L paths of a multipath

fading channel are mutually independent and have the iden-tical channel gain. Accordingly, the probability density func-tion (PDF) of the instantaneous symbol-SNR γs = (hEs)/N0is given by

f (γs) =γ L−1s

(L − 1)!γ̄ Lcexp

(−γs

γ̄c

), (11)

where γ̄c = (Es/N0)E(α2l ) is the average symbol-SNR perchannel, l = 1, 2, . . . ,L, and

∑Ll=1 E(α

2l ) = 1.

B. BER PERFORMANCETo derive the BER expression of the hierarchical square-constellation-based M -DCSK system, we first consider theconstellations shown in Fig. 6. To simplify the exposition, weuse d = [d1, d2, . . . , dm] to represent di or dq here.For the hierarchical square-based 16-DCSK constellation

shown in Fig. 6(a), the average BER for bit b1, conditioned onγs, is given by (12), as shown at the top the next page, whereρ =

2γs√4γs+2β

and erfc(x) =∫∞

x exp(−t2)dt .1 Additionally,the average BER for bit b2, conditioned on γs, is expressedby (13), as shown at the top the next page.

On the other hand, considering the hierarchical square-based 64-DCSK constellation shown in Fig. 6(b), the averageBER for the bit b1, conditioned on γs, can be calculatedas (14), as shown at the top the next page, where d+ =[d1, d2 + d3] and d− = [d1, d2 − d3]. Likewise, the averageBERs for the bits b2 and b3, conditioned on γs, can beformulated as (15) and (16), as shown at the top the next page,respectively.

1The expression of Es is dependent on the constellation. Precisely speak-ing, considering the square-based constellation, Ep should be calculatedby (8). Especially, if one only considers the constellations shown in Fig. 5,the average energy reduces to Es = (ddT + 1)Ep.

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G. Cai et al.: New Hierarchical M-ary DCSK Communication System: Design and Analysis

P(4,d, b1|γs) =14

erfc

2(d1 + d2)hEp√4EshN0 + 2βN 2

0

+ erfc

2(d1 − d2)hEp√4EshN0 + 2βN 2

0

=

14

erfc

2(d1+d2)hEsdidiT+dqdqT+1√4EshN0 + 2βN 2

0

+ erfc

2(d1−d2)hEsdidiT+dqdqT+1√4EshN0 + 2βN 2

0

=

14

(erfc

((d1 + d2)ρ

didiT + dqdqT + 1

)+ erfc

((d1 − d2)ρ

didiT + dqdqT + 1

))(12)

P(4,d, b2|γs) =14

(2erfc

(d2ρ

didiT + dqdqT + 1

)− erfc

((2d1 + d2)ρ

didiT + dqdqT + 1

)+ erfc

((2d1 − d2)ρ

didiT + dqdqT + 1

))(13)

P(8,d, b1|γs) =18

(erfc

((d1 − d2 − d3)ρ

didiT + dqdqT + 1

)+ erfc

((d1 − d2 + d3)ρ

didiT + dqdqT + 1

)+ erfc

((d1 + d2 − d3)ρ

didiT + dqdqT + 1

)+ erfc

((d1 + d2 + d3)ρ

didiT + dqdqT + 1

))=

12

(P(4,d+, b1|γs)+ P(4,d−, b1|γs)

)(14)

P(8,d, b2|γs) =18

(2erfc

((d2 + d3)ρ

didiT + dqdqT + 1

)− erfc

((2d1 + d2 + d3)ρ

didiT + dqdqT + 1

)+ erfc

((2d1 − d2 − d3)ρ

didiT + dqdqT + 1

)+2erfc

((d2 − d3)ρ

didiT + dqdqT + 1

)− erfc

((2d1 + d2 − d3)ρ

didiT + dqdqT + 1

)+ erfc

((2d1 − d2 + d3)ρ

didiT + dqdqT + 1

))=

12

(P(4,d+, b2|γs)+ P(4,d−, b2|γs)

)(15)

P(8,d, b3|γs) =18

(4erfc

(d3ρ

didiT + dqdqT + 1

)+ 2erfc

((2d2 − d3)ρ

didiT + dqdqT + 1

)+ erfc

((2d1 − 2d2 − d3)ρ

2ddT + 1

)− 2erfc

((2d2 + d3)ρ

didiT + dqdqT + 1

)− erfc

((2d1 − 2d2 + d3)ρ

didiT + dqdqT + 1

)− 2erfc

((2d1 − d3)ρ

didiT + dqdqT + 1

)+ erfc

((2d1 + d3)ρ

didiT + dqdqT + 1

)+ 2erfc

((2d1 + 2d2 − d3)ρ

didiT + dqdqT + 1

)− erfc

((2d1 + 2d2 + d3)ρ

didiT + dqdqT + 1

))(16)

Based on the recursive algorithm described in [32] and theabove expressions, the following expressions can be obtained.When i < m, the average BER bi is given by

P(√M ,d, bi|γs)

=12

(P

(√M2,d+, bi|γs

)+ P

(√M2,d−, bi|γs

)),

(17)

where d+ = [d1, d2, . . . , dm−1, dm−1 + dm] andd− = [d1, d2, . . . , dm−1, dm−1 − dm].When i = m, the average BER bm is expressed as

P(√M ,d, bm|γs)

=12m

2m−1∑i=1

2m−1∑j=1

12

((−1)j+1erfc

((d0(i)− B(j))ρ

didiT + dqdqT + 1

))

+

2m−1∑i=1

1+2m−1∑j=1

12

((−1)jerfc

((d1(i)−B(j))ρ

didiT+dqdqT+1

)),(18)

where d0 and d1 are the position vectors of the constella-tion points for LP bits 0 and 1, respectively, B(j) = 0.5(x2j−1 + x2j), j = 1, 2, . . . ,m, and xj is the coordinate forthe jth symbol. The method for generating these parametersis available in [32].

Finally, the BER expressions for the hierarchical square-constellation-based M -DCSK system can be derived.Specifically, the BER of the k-th bit, i.e., bk (k =

1, 3, 5, . . . , 2m− 1), in the hierarchical square-constellation-based M -DCSK system, is given by

Pb(M ,di,dq, bk ) =∫∞

0P(√

M ,di, bk |γs)f (γs)dγs.

(19)

Furthermore, the BER of the n-th bit, i.e., bn (n =

2, 4, 6, . . . , 2m), is given by

Pb(M ,di,dq, bn) =∫∞

0P(√

M ,dq, bn|γs)f (γs)dγs.

(20)

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G. Cai et al.: New Hierarchical M-ary DCSK Communication System: Design and Analysis

FIGURE 7. Constellation of hierarchical rectangular-based 8-DCSK signal.

C. EXTENSION TO HIERARCHICALRECTANGULAR-CONSTELLATION-BASED M-DCSK SYSTEMThe above BER results can be immediately extended to therectangular-constellation scenario. Fig. 7 shows the constel-lation of a hierarchical rectangular-based 8-DCSK signal.To guarantee the consistency with the definition for the hier-archical square-basedM -DCSK constellation, one should setm = log2

√M/2. The two distance vectors can be written

as di = [d i1, di2, . . . , d

im+1] and dq = [dq1 , d

q2 , . . . , d

qm].

In this scenario, the average transmitted energy is also givenby Es = (didiT + dqdqT + 1)Ep.Using Eqs. (12)-(18) as the root for this algorithm, the

BER of the k-th bit, i.e., bk (k = 1, 3, 5, . . . , 2m − 1),in the hierarchical rectangular-constellation-based M -DCSKsystem, is given as

Prb(M ,di,dq, bk ) =

∫∞

0P(√

2M ,di, bk |γs)f (γs)dγs,

(21)

while the BER of the nth bit, i.e., bn (n = 2, 4, 6, . . . , 2m), isexpressed by

Prb(M ,di,dq, bn) =

∫∞

0P

(√M2,dq, bn|γs

)f (γs)dγs.

(22)

Note that, based on Eqs. (12)-(18) and Eqs. (21)-(22), thedistance vectors di and dq for the hierarchical rectangu-lar constellations should be utilized to calculate the BERsof the hierarchical rectangular-constellation-basedM -DCSKsystem.

IV. AN ADAPTIVE TRANSMISSION SCHEMEAs discussed in Section II, the receiver does not need thevalues of d im and dqm when demodulating the received signal.Based on this property, an adaptive transmission scheme isdesigned in this section.

To begin with, in order to achieve different BERs fordifferent transmitted bits within a symbol and to ensure thetransmitted energy be no more than that of the MR-M -DCSKsystem [26], the distance vectors di and dq should be subjectto the following constraints:

d ik ≥ 2d ik+1, k = 1, 2, . . . ,m− 1,

dqk ≥ 2dqk+1, k = 1, 2, . . . ,m− 1,

didiT+ dqdqT ≤

12. (23)

FIGURE 8. A graphic example for an adaptive hierarchical square-based64-DCSK constellation.

Then, to effectively realize adaptive transmission, an exam-ple for the hierarchical square-based 64-DCSK constellationis presented in Fig. 8, showing the design of the distancevectors. In this example, it is assumed that di = dq = d.The principle of the design is adjusting the distance betweenthe two constellation points in the last hierarchical levelwhile keeping the remaining distances unchanged. By thisoperation, the decision boundaries remain the same but theconstellation points move farther from or closer to the deci-sion boundaries. As shown in Fig. 8, the decision boundariesremain unvaried as the distance d3 increases or decreases.At the receiver, one can obtain the transmitted bits correctlywithout using d3.Finally, the effect of the distance d im and dqm on the BER

is further illustrated. According to Eqs. (14)-(16), the BERperformance is changedwhenmodifying d3. Therefore, basedon the SNR information, one can adjust the distances d imand dqm to achieve the target BERs for HP bits, while allow-ing the LP bits to be transmitted as reliably as possible.In actual implementation, to satisfy different target BERrequirements for transmitted bits, the initial elements of di

and dq can be obtained by numerically solving the BERexpressions (12)-(20).

V. NUMERICAL RESULTS AND DISCUSSIONSIn this section, the BER performance of the hierarchicalsquare-constellation-based M -DCSK system is presentedover AWGN and multipath Rayleigh fading channels, thusverifying the superiority of the design and the accuracy ofthe analysis. Moreover, the feasibility of the proposed systemfor UWB transmission environments is discussed via simu-lations. Unless otherwise stated, the parameters used for themultipath channel and proposed systems in simulations areset as follows: number of paths is L = 3, channel gains are

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FIGURE 9. BER curves of the hierarchical square-constellation-based(a) 16-DCSK system, and (b) 64-DCSK system over an AWGN channel. Thespreading factors used are β = 5 and 80.

E(α21) = E(α22) = E(α23) = 1/3, time delays are τ1 = 0,τ2 = 2, τ3 = 5, and the distance vectors are di = dq = d.Note also that, for the hierarchical 16-DCSK system, b1 andb2 denote the HP bits while b3 and b4 represent the LP bits.For the hierarchical 64-DCSK system, b1 and b2 denote theHP bits while b5 and b6 represent the LP bits.

A. BER COMPARISON BETWEEN THEORETICAL ANALYSISAND SIMULATED RESULTSFig. 9 compares the theoretical and simulated BER curves ofthe hierarchical square-constellation-based 16-DCSK systemand 64-DCSK system over an AWGN channel, where thespreading factors are set to β = 5 and 80, and the dis-tance vectors for M = 16 and M = 64 are set to d =[√10/5,

√10/10] and d = [2

√42/21,

√42/21,

√42/42],

respectively. Referring to Fig. 9(a), the theoretical BERcurves match well with the simulated BER curves for bothspreading factors, which validates the performance analysis.Similar observations can be obtained from the results forM = 64 (see Fig. 9(b)).In Fig. 10, the theoretical and simulated BER curves

of the hierarchical square-constellation-based 16-DCSKsystem and 64-DCSK system are compared over a multipath

FIGURE 10. BER curves of the hierarchical square-constellation-based(a) 16-DCSK system, and (b) 64-DCSK system over a multipath Rayleighfading channel. The spreading factors used are β = 80 and 160.

Rayleigh fading channel, where the spreading factors are setto β = 80 and 160, and distance vectors are the same asthat in Fig. 9. As can be seen, the theoretical result is alsohighly consistent with the simulated result, irrespective of themodulation order and spreading factor.

B. PERFORMANCE COMPARISON BETWEEN THEPROPOSED SYSTEM AND THE MR-M-DCSK SYSTEMFig. 11 shows the BER curves of the hierarchical square-constellation-based 16-DCSK system and the MR-16-DCSKsystem [26] over a multipath Rayleigh fading channel,where the parameters are set to β = 160, d =

[√10/5,

√10/10] and θ = [π/4, π/8, π/16].2 For a fair

comparison, assume that the two systems have the sametransmitted energy. It can be observed that the proposedsystem provides better BER performance than the MR-16-DCSK system. For example, under a fixed channel condition(i.e., a fixed fading profile and a fixed noise variance σ 2

n ),the proposed system requires a relatively lower transmittedenergy to achieve the same BER level (e.g., BER=10−4) with

2The distance vector d = [√10/5,

√10/10] and the phase vector θ =

[π/4, π/8, π/16] are used for the proposed system and the MR-16-DCSKsystem, respectively.

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FIGURE 11. BER curves of the proposed system and MR-16-DCSK systemover a multipath Rayleigh fading channel. The spreading factor used isβ = 160.

FIGURE 12. BER curves of the the proposed system and MR-16-DCSKsystem over a UWB CM1 channel.

respect to the MR 16-DCSK system. Moreover, at Eb/N0 =

30 dB, the HP bits (b1, b2) in the proposed system achievesa BER of 2 × 10−4, while that in the MR-16-DCSK systemonly accomplishes a BER of 2 × 10−3. Thanks to the aboveadvantage, the proposed system should be more suitable forenergy-efficient wireless-communication applications, e.g.,WSNs.

To validate the feasibility of the design in UWB transmis-sion scenarios, the hierarchical square-constellation-based16-DCSK system and the MR-16-DCSK system are furtherexamined over an IEEE 802.15.4a CM1 channel, with cor-responding results shown in Fig. 12. In the simulation, theparameters used for the UWB channel model are set as fol-lows: the pulse duration time is Tc = 10 ns, the guard intervalis Tg = 90 ns, and the sample frequency is fs = 8 GHz.Furthermore, the distance vector and the phase vector usedhere are the same as that in Fig. 11. As can be seen, theproposed system also achieves better BER performance thanthe MR-16-DCSK system over UWB channels.

C. BER PERFORMANCE OF THE PROPOSED ADAPTIVETRANSMISSION SCHEMEFig. 13 presents the BER results of the adaptive transmis-sion schemes in the hierarchical square-constellation-based

FIGURE 13. BER curves of the adaptive transmission schemes in thehierarchical square-constellation-based (a) 16-DCSK system, and(b) 64-DCSK system, over a multipath Rayleigh fading channel. Thespreading factors used are β = 160. To implement the adaptivetransmission schemes, the distances d2 and d3 are adjusted for the casesof M = 16 and M = 64, respectively.

16-DCSK system and 64-DCSK system over a multi-path Rayleigh fading channel, where the spreading fac-tor is set to 160. The distance vectors in the case ofM = 16 are assumed as d = [

√10/5,

√10/10]

and d = [√10/5,

√10/20], while those in the case of

M = 64 are assumed as d = [2√42/21,

√42/21,

√42/42]

and d = [2√42/21,

√42/21,

√42/84]. For a given M , it

can be observed that different BER values can be obtained byadjusting dm. Accordingly, by using such an adaptive trans-mission scheme, the HP bits can be transmitted as reliablyas possible when the channel condition is bad, while bothHP and LP bits can be reliably transmitted when the channelcondition is good.

Next, the effect of the distance d2 on the BER performanceof the hierarchical square-constellation-based 16-DCSK sys-tem is discussed. The result is shown in Fig. 14, whered = [d1, d2] = [

√10/5, d2], β = 80, and SNR =

25 dB, 30 dB, 35 dB. Referring to Fig. 14(a), given a fixedSNR, there is intersection between the BER curves of the HPbits and the LP bits. Let d2, I be the value of d2 that make theBER of HP bits be equal to that of LP bits. one can observethat d2, I > d1/2 as SNR increases from 30 dB to 35 dB.

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FIGURE 14. Effect of the distance d2 on the BER of the hierarchicalsquare-constellation-based 16-DCSK system over a multipath Rayleighfading channel: (a) BER curves of the HP and LP bits vs. the distance d2,and (b) BER curves of the proposed system vs. the distance d2. Theparameters used areβ = 160,d1 =

√10/5 and SNR = 25 dB,30 dB,35 dB.

Hence, when d2 ≤ d1/2, one can find that there is no intersec-tion between the BER curves of the HP bits and the LP bits,thus ensuring that different bits have different BERs. Thisresult verifies the accuracy of the constraints (23). In addition,according to Fig. 14(b), there exists an optimal value of d2that can accomplish the minimum average BER for a givenSNR. For a generalized adaptive system, one can estimate theoptimal value of dm, i.e., dm,min = argmin

dmPavgb (d), based on

a simple computer search.

VI. CONCLUSIONSIn this paper, a new hierarchical square-constellation-basedM -DCSK communication system has been proposed, whichis also applicable to other constellations, such as rectangular,star, and asymmetrical square constellations. To realize amore flexibility BER requirement, an adaptive transmissionscheme has been designed by adjusting the distance vec-tor based on the SNR information. Furthermore, the BERexpressions of the proposed system have been derived overmultipath Rayleigh fading channels. Both theoretical andsimulated results have been given to demonstrate the supe-riority of the new design and the accuracy of the analysis.

Compared with the MR-M -DCSK system, the proposed sys-tem can achieve remarkably better error performance. As aresult, the proposed system stands out as a good candidatefor energy-efficient wireless-communication applications.

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GUOFA CAI (M’17) received the B.Sc. degreein communication engineering from Jimei Univer-sity, Xiamen, China, in 2007, the M.Sc. degreein circuits and systems from Fuzhou University,Fuzhou, China, in 2012, and the Ph.D. degree incommunication engineering from Xiamen Univer-sity, Xiamen, China, in 2015. He is currently aResearch Fellow with the School of Electrical andElectronic Engineering, Nanyang TechnologicalUniversity, Singapore, while on temporary leave

from the School of Information Engineering, GuangdongUniversity of Tech-nology, China. His primary research interests include information theory andcoding, chaotic communications, and UWB communications.

Yi Fang (M’15) received the B.Sc. degree inelectronic engineering from East China JiaotongUniversity, China, in 2008, and the Ph.D. degree incommunication engineering, Xiamen University,China, in 2013. In 2012, he was a Research Assis-tant in electronic and information engineering, TheHong Kong Polytechnic University, Hong Kong.From 2012 to 2013, he was a Visiting Scholar inelectronic and electrical engineering, UniversityCollege London, U.K. From 2014 to 2015, he was

a Research Fellow with the School of Electrical and Electronic Engineering,Nanyang Technological University, Singapore. He is currently an Asso-ciate Professor with the School of Information Engineering, GuangdongUniversity of Technology, China. His currently research interests includeinformation and coding theory, LDPC/protograph codes, spread-spectrummodulation, and cooperative communications. He is an Associate Editor ofthe IEEE ACCESS.

GUOJUN HAN (M’12–SM’14) received the M.E.degree in electronic engineering from South ChinaUniversity of Technology, Guangzhou, China,and the Ph.D. degree from Sun Yat-sen Univer-sity, Guangzhou. From 2011 to 2013, he was aResearch Fellow with the School of Electrical andElectronic Engineering, Nanyang TechnologicalUniversity, Singapore. From 2013 to 2014, he wasa Research Associate with the Department of Elec-trical and Electronic Engineering, The Hong Kong

University of Science and Technology. He is currently a Full Professor andVice Dean of the School of Information Engineering with Guangdong Uni-versity of Technology, Guangzhou. His research interests include wirelesscommunications, coding, and signal processing for data storage.

LIN WANG (S’99–M’03–SM’09) received thePh.D. degree in electronics engineering from theUniversity of Electronic Science and Technologyof China, China, in 2001. From 1984 to 1986, hewas a Teaching Assistant with the MathematicsDepartment, Chongqing Normal University. From1989 to 2002, he was a Teaching Assistant, a Lec-turer, and an Associate Professor in applied math-ematics and communication engineering with theChongqing University of Post and Telecommuni-

cation, China. From 1995 to 1996, he was with theMathematics Department,University of New England, Australia. In 2003, he spent three months as aVisiting Researcher with the Center for Chaos and Complexity Networkswith the Department of Electronic Engineering, City University of HongKong. From 2013 to 2013, he was a Senior Visiting Researcher with theDepartment of ECE, University of California at Davis, Davis. From 2003to 2012, he was a Full Professor and the Associate Dean of the School ofInformation Science and Technology, Xiamen University, China. He hasbeen a Distinguished Professor with Xiamen University since 2012. Heholds 14 patents in the field of physical layer in digital communications. Hehas authored over 100 journal and conference papers. His current researchinterests are in the area of channel coding, joint source and channel coding,chaos modulation, and their applications to wireless communication andstorage systems.

GUANRONG CHEN (M’89–SM’92–F’97)received the M.Sc. degree in computer sciencefrom Sun Yat-sen University, Guangzhou, China,in 1981, and the Ph.D. degree in appliedmathemat-ics from Texas A&M University, College Station,TX, USA, in 1987. Hewas a tenured Full Professorwith the University of Houston, TX. He has beenthe Chair Professor and the Founding Director ofthe Center for Chaos and Complex Networks withthe City University of Hong Kong since 2000.

He received the 2011 Euler Gold Medal, Russia, and conferred HonoraryDoctorate by the Saint Petersburg State University, Russia, in 2011 and bythe University of Le Havre, Romandie, France, in 2014. He is a member ofthe Academia of Europe, a fellow of The World Academy of Sciences, and aHighly Cited Researcher in engineering as well as in mathematics accordingto Thomson Reuters.

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