Adaptive OFDM Index Modulation for Two-Hop Relay-Assisted Networks

Adaptive OFDM Index Modulation for Two-Hop Relay-Assisted Networks

Shuping Dang, , Justin P. Coon, , and Gaojie Chen, This work was supported by the SEN grant (EPSRC grant number EP/N002350/1) and the grant from China Scholarship Council (No. 201508060323). The authors are with the Department of Engineering Science, University of Oxford, Parks Road, Oxford, U.K., OX1 3PJ; tel: +44 (0)1865 283 393, (e-mail: {shuping.dang, justin.coon, gaojie.chen}@eng.ox.ac.uk).
Abstract

In this paper, we propose an adaptive orthogonal frequency-division multiplexing (OFDM) index modulation (IM) scheme for two-hop relay networks. In contrast to the traditional OFDM IM scheme with a deterministic and fixed mapping scheme, in this proposed adaptive OFDM IM scheme, the mapping schemes between a bit stream and indices of active subcarriers for the first and second hops are adaptively selected by a certain criterion. As a result, the active subcarriers for the same bit stream in the first and second hops can be varied in order to combat slow frequency-selective fading. In this way, the system reliability can be enhanced. Additionally, considering the fact that a relay device is normally a simple node, which may not always be able to perform mapping scheme selection due to limited processing capability, we also propose an alternative adaptive methodology in which the mapping scheme selection is only performed at the source and the relay will simply utilize the selected mapping scheme without changing it. The analyses of average outage probability, network capacity and symbol error rate (SER) are given in closed form for decode-and-forward (DF) relaying networks and are substantiated by numerical results generated by Monte Carlo simulations.

Index modulation, OFDM, two-hop relay system, adaptive modulation, decode-and-forward relay.

I Introduction

Classic modulation schemes, e.g. phase-shift keying (PSK) and quadrature amplitude modulation (QAM), map a sequence of information bits to a modulated data symbol with a certain amplitude and phase, and such a modulated data symbol can be represented by a point in the two-dimensional constellation diagram [1]. With rapid increasing demands of reliability and data transmission rate, however, these classic modulation schemes might not be able to satisfy the challenging communication requirements for the next generation networks in terms of spectrum and energy efficiency [2]. As a result, there has been a growing interest in design and optimization of modulation schemes which can achieve a high spectrum and/or energy efficiency [3]. Two of the most promising schemes are spatial modulation (SM) and orthogonal frequency division multiplexing (OFDM) index modulation (IM), which are capable of adding one extra modulation dimension in addition to the conventional amplitude and phase dimensions [4, 5, 6, 7]. SM is designed for multiple-input-multiple-output (MIMO) systems and exploits the spatial dimension, while OFDM IM is mainly applied to OFDM systems by conveying information via the frequency dimension. In practice, OFDM IM is easier to be implemented, since the deployment of MIMO under a limited correlation level among all antennas may not be affordable for most communication systems in terms of cost and physical size [8, 9], for example device-to-device (D2D) communication networks in which all nodes are nothing more than idle mobile devices [10]. Therefore, OFDM IM is now being regarded as an attractive modulation methodology for the 5G networks [11].

At the meantime, relay-assisted communications have been proved to be an effective approach to provide reliable communication service against propagation attenuation, multipath fading as well as shadowing, and are also regarded as one of the most promising technologies for the next generation cellular networks [12]. Moreover, with recent research achievements on multicarrier relay systems, the performance of relay-assisted communications can be further enhanced in terms of multiplexing and/or diversity gains with a moderate system complexity [13, 14, 15]. Due to the maturity and benefits of multicarrier relay systems, it is of an increasingly high interest to integrate it with other popular communication paradigms [16]. In recent years, some works related to SM also considered involving relays to further improve system performance and proposed the concepts of distributed SM (DSM) and dual-hop SM (Dh-SM) [17, 18].

However, to the best of the authors’ knowledge, the link between two-hop relay networks and OFDM IM is still lacking, which motivates us to construct an analytical framework of OFDM IM in two-hop relay-assisted networks in order to integrate the merits of both. Also, motivated by a variety of adaptive modulation and transmit antenna selection schemes proposed for SM and proved to be capable of significantly enhancing system performance [19, 20, 21], we would also like to apply the adaptive modulation mechanism in OFDM IM for two-hop relay networks. Based on these considerations, we propose the adaptive OFDM IM for two-hop relay networks in this paper. In particular, a two-hop relay network with one source, one relay and one destination is considered and the mapping scheme for bit sequence to the index of active subcarrier is dynamically selected according to the channel state information (CSI) at source and relay. By such an adaptive mapping scheme selection process, the system reliability can be improved.

In spite of mapping scheme adaptation, our proposed scheme is different from those traditional OFDM IM schemes, in which only a fixed number of subcarriers are active in each instant and thus will result in a low spectral efficiency [22, 23, 6]. Instead, an efficient activation mechanism with multiple active subcarriers is used in our proposed scheme to map the incoming bit stream to a corresponding subcarrier activation pattern. On the other hand, despite the higher spectral efficiency provided by adopting such an efficient activation mechanism with an indeterministic number of active subcarriers, there is a small possibility that all subcarriers are inactive. In this special case, the amplitude and phase modulation (APM) data symbol cannot be transmitted. We term this zero-active subcarrier dilemma, which hinders the utilization of a variable number of active subcarriers. In this paper, an additional benefit brought by the subcarrier adaptation is that we can employ a dual-mode transmission protocol to solve the zero-active subcarrier dilemma.

Following by these core ideas, the contributions of this paper are listed infra:

1. We propose an adaptive OFDM IM scheme for two-hop relay networks, by which the mapping schemes are dynamically selected based on instantaneous CSI and a mechanism allowing reducing throughput for gaining reliability can be constructed. Meanwhile, for implementation in practice, we propose two feasible mapping scheme selection methodologies, which can be applied depending on the design specification of the relay node, i.e. whether the relay is capable of performing mapping scheme selection or not.

2. We propose a dual-mode transmission protocol which can solve the zero-active subcarrier dilemma and transmit the APM symbol when all selected subcarriers are inactive.

3. We analyze the outage performance of the proposed modulation schemes and derive the exact expressions and asymptotic expressions in the high signal-to-noise ratio (SNR) region for the average outage probability in closed form; we analyze the average network capacity and derive its closed-form expression; we also investigate the error performance and give a closed-form approximation of the average symbol error rate (SER).

The rest of this paper is organized as follows. In Section II, the system model of adaptive OFDM IM in two-hop networks is introduced. After that, we specify the mapping scheme selection and performance evaluation metrics in Section III. The analyses of average outage performance, network capacity and error performance of the proposed systems are carried out in Section IV, Section V and Section VI, respectively. Then, all analyses are numerically verified and the simulation results are discussed in Section VII. Finally, Section VIII concludes the paper.

Ii System Model

Ii-a System framework

Consider a two-hop OFDM IM system with one source, one relay, one destination and subcarriers, which is operating in slow frequency-selective Rayleigh fading channels. Also, out of the subcarriers will be selected by a certain criterion to form a mapping scheme (a.k.a. a codebook) for OFDM IM, where . In each instant, there is a variable-length equiprobable stream required to be transmitted at the source, where denoting the index of the chosen APM constellation symbol for the th active subcarrier and is the order of APM; denoting the index of the subcarrier activation pattern and is the corresponding number of active subcarriers. Supposing that the lengths of and are and , the length of the entire variable-length stream is . The -bit stream is modulated by the subcarrier activation pattern, and out of subcarriers will be activated to convey the APM symbols generated according to these -bit streams , ,, by the conventional -ary APM scheme (e.g. -PSK or -QAM). Note that, normally the mapping scheme is preset in classic OFDM IM systems and remains fixed, but in this paper, we assume it can be dynamically selected according to the CSI at both source and relay. More details of the mapping scheme and its adaptation mechanisms will be given latter in Section III and Fig. 2.

Meanwhile, we assume that a single bit in represents the activation state of a given subcarrier in an on-off keying (OOK) manner. In other words, for a given -bit stream , the selected subcarriers having the same sequence numbers with bits ‘1’ will be activated, on which different APM symbols are transmitted. That is to say, and the number of active subcarriers is equal to the hamming weight of the -bit stream . More specifically, the activation state matrix can be designed as

 S(k)=diag{bS(k,1),bS(k,2),…,bS(k,NS)}, (1)

where is either ‘0’ or ‘1’ denoting the th entry in , . As a result, by OFDM IM, the information contained in the entire -bit stream is now conveyed by both of -ary constellation symbols and the activation state matrix .

Here we further assume that there is no direct transmission link between source and destination due to deep fading or severe propagation attenuation and a half-duplex forwarding protocol is adopted at the relay, so that two orthogonal temporal phases are required for a complete symbol transmission from source to destination. In the first phase, the generated OFDM IM symbol at the source is transmitted to the relay in slow frequency-selective Rayleigh fading channels. Assuming decode-and-forward (DF) relaying protocol is adopted at the relay, once the distorted symbol is received by the relay, it will be first decoded and regenerated according to the relay’s own mapping scheme for the second hop. Then the relay transmits the regenerated symbol to the destination. Finally, the distorted version of the new symbol is received by the destination and then decoded to retrieve the complete -bit stream. For clarity, the system block diagram of the source is illustrated in Fig. 1, which is also highly similar to the transmission part of the relay.

Ii-B Channel model

We assume the wireless channels in the first and second hops to be slow frequency-selective faded and their gains to be exponentially distributed but with different means and due to different propagation attenuations. Note that, the slow or quasi-static property referred to here indicates that the channel gains are random but would remain unchanged for a sufficiently large period of time [24]. Based on such an assumption, the overheads caused by transmitting the selected mapping scheme via feedforward links to relay and destination for decoding purposes are negligible [19]. This justifies the feasibility of our proposed adaptive OFDM IM scheme in practice. Denoting the set of all subcarriers as , the probability density function (PDF) and the cumulative distribution function (CDF) of the channel gain , , can be written as

 fi(s)=exp(−s/μi)/μi  ⇔  Fi(s)=1−exp(−s/μi) (2)

where denotes the index of the first or second hop. Meanwhile, because of the fundamental assumptions of OFDM, it is supposed that these channel gains in each hop are statistically independent111We assume a block-fading model in frequency akin to systems that employ a resource block frame/packet structure (e.g., Long-Term Evolution (LTE)), and hence the independent and identically distributed (i.i.d.) assumption in frequency for each hop can be justified [15]..

Ii-C APM scheme

In this paper, we adopt -PSK as the APM scheme222As has been proved for SM systems and similarly for OFDM IM systems, a constant-envelope APM is preferable over non-constant-envelope APM when the spatial or frequency indices are employed for information transmission, since it can offer a higher power efficiency and a reduced-complexity decoding process [25, 26]. In other words, -PSK outperforms -QAM. This is the reason of using -PSK in this paper.. Meanwhile, the analytical results in the following sections can also be used for -QAM cases if some precoding schemes (e.g. dynamic power allocation on constellation points) can be applied on each data symbol so that a constant envelope is achieved [27, 28]. Without loss of generality, we normalize the data symbol by , .

Iii Mapping Scheme Selections and Performance Evaluation Metrics

Now let us focus on the key property of the proposed system, i.e. dynamical mapping scheme selection according to CSI. First, here we assume CSI can be perfectly known at source and relay without delay and overhead. Then, the set of all mapping schemes is denoted by and the cardinality of , i.e. the total number of possible mapping schemes is given by , where denotes the binomial coefficient. It should be noted that the OFDM blocks are normally generated by inverse fast Fourier transform (IFFT), which will require , .

Iii-a Signal model and dual-mode transmission protocol

In order to perform mapping scheme selection, we have to construct the signal model and transmission protocol as well as some important performance metrics. Now, let us first focus on the construction of the transmit OFDM block (a.k.a. transmit signal vector) before adopting a certain mapping scheme. Assuming a cyclic prefix (CP) with sufficient length is applied to the time-domain transmit OFDM block produced by a -point IFFT, the OFDM signaling model can be considered at a subcarrier level [29]. A general form of the transmit OFDM block in frequency domain without interblock interference can be written as

 x=[x(1),x(2),…,x(NT)]T∈CNT×1, (3)

where denotes the matrix transpose operation.

When selecting an arbitrary th mapping scheme, , which selects out of subcarriers to form a subset for OFDM IM, the reduced OFDM block determined by the -bit stream is given by

 x(k)=[x(m1,1),x(m2,2),…,x(mNS,NS)]T∈CNS×1, (4)

where

 x(mn,n)={χmn,    n∈NA(k)0,        otherwise (5)

corresponds to the data symbol transmitted on the th subcarrier, and is the subset of active subcarriers out of selected subcarriers when is transmitted.

Now, in order to avoid the zero-active subcarrier dilemma, which is defined as the case when all selected subcarriers are required to be inactive (i.e. and thus ), we propose a dual-mode transmission protocol, which consists of the regular transmission mode and the complementary transmission mode.

If , i.e. is not an all-zero bit stream, consequently, the regular transmission mode is activated and the received signal for the th hop can be expressed by[30]

 yi(k)=[yi(m1,1),yi(m2,2),…,yi(mNS,NS)]T=√PtNA(k)Hi(c)x(k)+wi∈CNS×1, (6)

where is the vector of independent complex additive white Gaussian noise (AWGN) samples on each subcarrier at the th hop, whose entries obey , and is the noise power; is a diagonal channel state matrix for the th hop when the th mapping scheme is selected; is a uniform transmit power adopted at both source and relay. Due to the normalization of APM symbols, we obtain the received SNR in the th hop for the th subcarrier as

 γi(k,n)=PtNA(k)N0bS(k,n)|hi(c,n)|2,     ∀ n∈NS(c). (7)

However, for (we denote for this case), i.e. is an all-zero bit stream, one of those unselected subcarriers will all be activated to undertake the transmission of data symbol in the th hop. In this case, the received signal transmitted on the complementary subcarrier in the th hop becomes

 yi(1,~ni)=hi(~ni)χm~n+wi(~ni), (8)

where is a complex AWGN on the complementary subcarrier, and the SNRs of the complementary subcarrier as well as other selected subcarriers become

 γi(1,~ni)=PtN0|hi(~ni)|2  and  γi(1,n)=0, ∀ n∈NS(c). (9)

We term this special case complementary transmission mode, so that at least one data symbol can still be transmitted on the standby subcarrier when all selected subcarriers are inactive333The information conveyed by can be crucial for system coordination, e.g. synchronization.. Based on the specification above for the dual-mode transmission protocol, we can have the average transmission rate in bit per channel use (bpcu) as follows

 ¯B=log2(2NS)+Ek∈K{max{1,NA(k)}log2(M)}=NS+log2(M)2NS(1+2NS−1NS)  [bpcu], (10)

where denotes the expected value of the random variable enclosed.

Iii-B Decentralized mapping scheme selection methodology

Here, we propose two feasible mapping scheme selection methodologies, which can be applied depending on the design specification of the relay node, i.e. whether the relay is capable of performing mapping scheme selection or not. We first introduce the decentralized mapping scheme as follows. When both source and relay can get access to perfect CSI and carry out the subcarrier adaptation process, in order to optimize the outage performance and error performance, the mapping scheme selections are independently performed at the source and relay according to the criterion444We denote the index of all-one -bit stream as , i.e. . In this case, all selected subcarriers will be active, so that a comprehensive mapping scheme selection considering the qualities of all channels can be performed.

 ^ci=argmaxc∈C⎧⎨⎩∑n∈NS(c)γi(2NS,n)⎫⎬⎭, (11)

where denotes the index of the optimal mapping scheme selected for the th hop. Because the mapping scheme selections are independently performed at both source and relay, we term this selection methodology the decentralized mapping scheme selection.

Subsequently, for the th hop, the selection criterion of the complementary subcarrier is given by

 ~ni=argmaxn∈N∖NS(^ci)|hi(n)|2. (12)

Iii-C Centralized mapping scheme selection methodology

However, the decentralized mapping scheme selection methodology will result in a high system complexity at the relay node, which might not be practical as the relay is normally an idle user mobile device with relatively limited processing capability [31]. We thereby propose an alternative suboptimal mapping scheme selection methodology which is performed only at the source and thus relaxes the design requirements of the relay. Because the mapping scheme selection is only performed at the source, we term this the centralized mapping scheme selection. Specifically, the mapping scheme is only selected at the source and will be simply followed by the relay. The mapping scheme selection criterion is given by

 ^c=argmaxc∈C⎧⎨⎩∑n∈NS(c)min{γ1(2NS,n),γ2(2NS,n)}⎫⎬⎭. (13)

In a similar manner to the decentralized case, we can select the contemporary subcarrier for a whole link by the criterion

 ~n=argmaxn∈N∖NS(^c)min{|h1(n)|2,|h2(n)|2}. (14)

By either the decentralized or centralized mapping scheme selection methodology, as long as the th mapping scheme is selected, the -bit stream will be mapped to activate a subset of from subcarriers. To achieve this, we first need to relabel the subcarriers in by the new indices in ascending order. To be clear, the indices in the original full set are termed absolute indices, whereas the indices in the selected subset are termed relative indices555Because the values of indices per se do not affect the mathematical analysis, with a slight abuse of notation, we do not specifically distinguish them and denote a generic subcarrier index as in this paper.. An example of the mapping scheme selection process from to can be observed in Fig. 2, given and . Furthermore, for clarity, an example of the OFDM IM mapping table when and is given in Table I (Because , BPSK is adopted for APM and here we stipulate that data symbols and correspond to bit ‘1’ and ‘0’, respectively).

Iii-D Performance evaluation metrics

Iii-D1 Average outage probability

We define the outage event of the proposed systems as follows [32].

Definition 1

An outage occurs when the SNR of any of the active subcarriers in either the first or the second hop falls below a preset outage threshold .

By fundamental probability theory, we can determine the conditional outage probability when is transmitted by

 (15)

where denotes the probability of the event enclosed. Therefore, the average outage probability over can be written as

 ¯Po(s)=Ek∈K{Po(s|k)}, (16)

which is an important metric used in this paper to characterize the system reliability.

Iii-D2 Average network capacity

To jointly consider throughput and reliability, we should also take network capacity into consideration to further investigate the performance of the proposed systems. According to the max-flow min-cut theorem [33], the average network capacity in two-hop networks is given by

 ¯C=EHi(^ci),hi(~ni)i∈{1,2},k∈K{C(k)}  [bit/s/Hz], (17)

where is the network capacity when is transmitted and can be written as

 (18)

Iii-D3 Average symbol error rate

To analyze the error performance, we have to first specify the detection method used in our proposed systems. Although the complexity of the ML detection will increase exponentially with the APM order and the number of selected subcarriers , it can provide the optimal error performance and serves as a good reference to other suboptimal detection methods [34]. Therefore, we adopt the ML detection method at the relay and destination to demodulate the received signal and retrieve the transmitted bits. However, due to the dual-mode transmission protocol adopted in this paper, the ML detection method has to be tailored accordingly. In order to effectively perform detection, we first need to construct a concatenated symbol block by and as , where represents the vector/diagonal matrix concatenation operation666Here we define: for arbitrary vectors and , ; for arbitrary diagonal matrices and , .. The set of all possible is denoted as and . Subsequently, the detection criterion of the ML detector for the received concatenated signal distorted from the authentic transmitted concatenated OFDM IM symbol in the th hop is given by

 ^X(^k)=argminX(k)∥∥∥˙Yi(˙k)−√PtNA(k)⟨Hi(^ci),hi(~ni)⟩X(k)∥∥∥F,, (19)

where denotes the Frobenius norm of the enclosed matrix/vector.

Hence, the conditional SER for the th hop can be written as

 Pe−i(˙X(˙k)|Hi(^ci),hi(~ni))=∑^X(^k)≠˙X(˙k)Pe−i(˙X(˙k)→^X(^k)|Hi(^ci),hi(~ni)). (20)

Subsequently, the unconditional SER can be expressed as

 Pe−i(˙X(˙k))=EHi(^ci),hi(~ni){Pe−i(˙X(˙k)|Hi(^ci),hi(~ni))}. (21)

Due to the bottleneck effect of a two-hop network, the end-to-end SER can be approximated by777For simplicity, we neglect the circumstance that an erroneously estimated and retransmitted signal at the relay can be again ‘erroneously’ estimated to the correct signal at the destination, as the probability is small.

 Pe(˙X(˙k))≈Pe−1(˙X(˙k))+Pe−2(˙X(˙k))−Pe−1(˙X(˙k))Pe−2(˙X(˙k)). (22)

Now, we can define the average SER as

 ¯Pe=E˙X(˙k)∈X{Pe(˙X(˙k))}, (23)

which will be used in this paper to evaluate the error performance of the proposed systems.

Iii-E Comparison benchmarks

Iii-E1 OFDM IM without mapping scheme adaptation

Without mapping scheme adaptation, the OFDM IM scheme will activate out of subcarriers in each instant to convey information [6]. The transmission rate of the OFDM IM without mapping scheme adaptation is given by

 Bclassic=NT2log2(M)+⌊log2(NTNT/2)⌋      [bpcu], (24)

where represents the floor function.

It can be seen by comparing (10) with (24) that the transmission rate of the OFDM IM scheme without mapping scheme adaptation is higher than that with mapping scheme adaptation. Another merit of the OFDM IM without mapping scheme adaptation is that it always activates subcarriers, so that the zero-active subcarrier dilemma does not exist. However, because the OFDM IM scheme without mapping scheme adaptation utilizes the index of subcarrier set to convey information, it is expected that some possible subcarrier sets will never be used in this scheme and therefore the spectral efficiency is degraded without any benefit [6]. Also, without subcarrier selection, the reliability of the systems will be much lower than that with our proposed adaptive scheme.

Iii-E2 Fpsk

As a well known modulation scheme [22], we also take FPSK as a comparison benchmark in this paper. In FPSK, only one subcarrier will be activated and the index of the active subcarrier is used for conveying information. The transmission rate is given by

 BFPSK=log2(M)+⌊log2(NT)⌋        [bpcu]. (25)

The FPSK scheme would lead to a low-complexity system, since the detection process is much simpler than our proposed adaptive OFDM IM scheme and the zero-active subcarrier dilemma does not exist either. On the other hand, the transmission rate of the systems using FPSK might be significantly lower than that of systems using our proposed schemes, because the spectral efficiency of FPSK in frequency dimension is too low by activating only a single subcarrier in each instant. Detailed numerical comparisons in terms of outage performance, network capacity and error performance for the adaptive OFDM IM scheme and these two benchmarks will be given in Section VII. The transmission rates of these three schemes are plotted in Fig. 3 for illustration purposes.

Iv Outage Performance Analysis

Iv-a Decentralized mapping scheme selection

Iv-A1 Complementary transmission mode

When is required to be transmitted, the complementary transmission mode will be activated and the standby subcarrier will be employed to convey the APM symbol . By (12), the standby subcarrier is the th worst among all subcarriers and the only one active subcarrier in the complementary transmission mode. Therefore, the outage probability regarding the subcarrier in the th hop is given by

 Po−i(s|k=1)=Fi(NT−NS)(sN0Pt), (26)

where

 Fi(ξ)(s)=NT∑n=ξ(NTn)(Fi(s))n(1−Fi(s))NT−n (27)

denotes the outage probability of the th order statistic of the channel gain among subcarriers in the th hop [35]. Hence, the outage probability of the complementary transmission mode is

 Po(s|k=1)=Po−1(s|k=1)+Po−2(s|k=1)−Po−1(s|k=1)Po−2(s|k=1). (28)

Iv-A2 Analysis of average outage probability

In order to analyze the outage performance when , we propose and prove a lemma as follows.

Lemma 1

For , the generic conditional outage probability for the th hop can be determined by

 Po−i(s|k)=NT−NA(k)+1∑ξ=NT−NS+1Υ(k,ξ)Fi(ξ)(sN0NA(k)Pt), (29)

where .

Proof:

See Appendix A

Due to the DF forwarding protocol, the conditional outage probability considering two hops is derived by

 Po(s|k)=Po−1(s|k)+Po−2(s|k)−Po−1(s|k)Po−2(s|k). (30)

Because is equiprobable, the average outage probability is given by

 ¯Po(s)=12NS∑k∈KPo(s|k). (31)

By employing a power series expansion at , we can obtain the asymptotic expression of as

 ¯Po(s)∼(NTNT−NS)2NS[(1μ1)NT−NS+(1μ2)NT−NS](sN0Pt)NT−NS, (32)

by which we can find the diversity order to be .

Iv-B Centralized mapping scheme selection

Iv-B1 Complementary transmission mode

Because the mapping schemes used at the source and relay are the same by the centralized mapping scheme selection, we have to integrate the channels in two hops into a whole, which is termed a link. When DF forwarding protocol is adopted, due to the bottleneck effect, the PDF and CDF of the gain regarding a certain unsorted link can be determined from (2) to be

 ϕ(s)=exp(−s/μΣ)/μΣ  ⇔  Φ(s)=1−exp(−s/μΣ), (33)

where .

In a similar manner to the decentralized case, we can have the outage probability of the complementary transmission mode as

 Po(s|k=1)=Φ(NT−NS)(sN0Pt), (34)

where

 Φ(ξ)(s)=NT∑n=ξ(NTn)(Φ(s))n(1−Φ(s))NT−n (35)

denotes the outage probability of the th order statistic of the link gain among links.

Iv-B2 Analysis of average outage probability

Now, let us generalize the analysis to an arbitrary case when , and similarly obtain the generic conditional average outage probability by

 Po(s|k)=NT−NA(k)+1∑ξ=NT−NS+1Υ(k,ξ)Φ(ξ)(sN0NA(k)Pt). (36)

Then, by (16) and the assumption of equiprobable , the average outage probability can be calculated by (31) as well. By employing a power series expansion at , we can obtain the asymptotic expression of as

 ¯Po(s)∼12NS(NTNT−NS)(sN0μΣPt)NT−NS, (37)

by which we can find the diversity order to be .

V Network Capacity Analysis

V-a Decentralized mapping scheme selection

V-A1 Complementary transmission mode

By (18), we can reduce the expression of the network capacity of the complementary transmission mode to

 C(1)=12log2(1+min{γ1(1,~n1),γ2(1,~n2)}), (38)

in which the PDF of after performing mapping scheme selections can be expressed as and

 ψ(ξ,η)(xs)=xf1(ξ)(xs)(1−F2(η)(xs))+xf2(η)(xs)(1−F1(ξ)(xs)), (39)

where

 fi(ξ)(s)=dFi(ξ)(s)ds=NT!(Fi(s))ξ−1(1−Fi(s))NT−ξfi(s)(ξ−1)!(NT−ξ)!. (40)

Therefore, the average capacity over and , for the complementary transmission mode can be obtained as

 ¯C(1)=Eh1(~n1),h2(~n2){C(1)}=∫∞012log2(1+s)ψ(NT−NS,NT−NS)(sN0Pt)ds=ΛG(NT−NS,NT−NS,N0Pt). (41)

Here, is a defined special function, which can be explicitly written in (42). The expressions and can be derived in closed form in (43) and (44), where ; is the exponential integral function and represents an intermediate coefficient.

V-A2 Analysis of average network capacity

For , there will be active subcarriers at each hop, whose orders are denoted by and , where and for and , . The set of permutations of and is denoted as and . Therefore, in order to determine the conditional average capacity over and , we propose and prove a lemma as follows.

Lemma 2

For , the conditional average capacity over and is given by

 ¯C(k)=EH1(^c1),H2(^c2){C(k)}=∑Ξ1(k)∈D(k)∑Ξ2(k)∈D(k)((NS−NA(k))!NS!)2∑n∈NA(k)ΛG(ξn,ηn,N0NA(k)Pt). (45)
Proof:

See Appendix B.

As a result of the equiprobable assumption of , the average network capacity can be obtained by

 ¯C=Ek∈K{¯