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Manytomany spacetime network coding for amplifyandforward cooperative networks: node selection and performance analysis
EURASIP Journal on Wireless Communications and Networking volume 2014, Article number: 48 (2014)
Abstract
In this paper, the multinode amplifyandforward cooperative communications for a network of N nodes is studied via the novel concept of manytomany spacetime network coding (M2MSTNC). Communication under the M2MSTNC scheme is performed over two phases: (1) the broadcasting phase and (2) the cooperation phase. In the former phase, each node broadcasts its data symbol to all the other nodes in the network in its allocated time slot, while in the latter phase, simultaneous transmissions from N1 nodes to a destination node take place in their time slot. In addition, the M2MSTNC scheme with optimal node selection (i.e., M2MSTNCONS) is proposed. In this scheme, the optimal relay node is selected based on the maximum harmonic mean value of the source, intermediate, and destination nodes’ scaled instantaneous channel gains. Theoretical symbolerrorrate analysis for Mary phase shift keying (MPSK) modulation is derived for both the M2MSTNC and M2MSTNCONS schemes. Also, the effect of timing synchronization errors and imperfect channel state information on the SER performance and achievable rates is analytically studied. It is shown that the proposed M2MSTNCONS scheme outperforms the M2MSTNC scheme and is less sensitive to timing offsets and channel estimation errors. It is envisioned that the M2MSTNCONS scheme will serve as a potential manytomany cooperative communication scheme with applications spanning sensor and mobile ad hoc networks.
1 Introduction
Network coding has recently emerged as an important design paradigm for wireless networks that allows multinode communications and also improves data distribution and network throughput [1]. Cooperative communications have also attracted much attention in the wireless literature as an effective means of jointly sharing transmissions of distributed single antenna nodes to exploit spatial diversity gains and mitigate channel fading and interference [2]. As most conventional multinode cooperative communication schemes are not directly applicable to information exchange across many geographically distributed nodes, wireless network coding has become increasingly attractive.
A few recent works have proposed the use of wireless network coding for multinode cooperative communications in wireless networks. For instance, in [3], the concept of wireless network cocast (WNC) that employs wireless network coding is proposed to achieve aggregate transmission power and delay reduction while achieving incremental diversity in locationaware networks. In [4], complex field network coding (CFNC) was employed to achieve a full diversity gain and a throughput as high as 1/2 symbol per user per channel use. However, research thus far had not fully exploited the joint potential of wireless network coding and cooperative diversity until the introduction of the novel concept of spacetime network coding (STNC) [5, 6]. In [5], the multipointtopoint (M2P) and pointtomultipoint (P2M) spacetime network codes were proposed to allow multiple source transmissions within a timedivision multiple access (TDMA) framework to a common node and the reverse common node transmission to multiple destinations, respectively. It was also shown that for a network of N nodes deploying M2PSTNC or P2MSTNC, only 2N time slots are required while achieving a diversity order of N per transmitted symbol. In [7], differential spacetime network coding (DSTNC) has been proposed for multisource cooperation to counteract the challenges of imperfect synchronization and channel estimation while achieving full diversity. Specifically, the authors analyze the pairwise error probability and derive the design criteria for DSTNC. Antieavesdropping spacetime network coding (AESTNC) has been proposed in [8] for secure cooperative communications against eavesdropping, while achieving full diversity. Manytoone STNC has been proposed in [9] for clusterbased cooperative communications to achieve spatial diversity and improve spectral efficiency. In [10], the symbol error (SER) of STNC is analyzed in independent but not necessarily identically distributed Nakami m fading channels. Specifically, exact and asymptotic SER expressions are derived for MPSK and MQAM modulations, and the impact of the fading parameter m, relay location, power allocation, and nonorthogonal codes on the SER are examined.
In [6], the manytomany spacetime network coding (M2MSTNC) for a network of N decodeandforward (DF) nodes is proposed to achieve a diversity order of N1 per node over a total of 2N time slots while maintaining a stable network throughput of 1/2 symbol per time slot per node. The operation of the M2MSTNC scheme is based on the assumption of N1 perfectly synchronized simultaneous transmissions in every time slot of the cooperation phase. However, the work in [6] did not analyze the impact of timing offsets on the network performance. In practice, simultaneous transmissions from multiple relay nodes are extremely challenging due to the imperfect timing synchronization. Most research in cooperative communications when focusing on simultaneous transmissions from distributed relay nodes assume perfect timing synchronization [4, 11, 12]. Overlooking the impact of timing synchronization errors could lead to detrimental effects on the network performance [13]. Also, channel state information errors at the receiving nodes are inevitable in practice [14]. Such errors could drastically diminish diversity gains and thus must be carefully characterized.
Based on the foregoing discussion, this work aims at better exploiting the potentials of the M2MSTNC communication scheme for amplifyandforward (AF) cooperative networks by (1) characterizing the symbol error rate performance for Mary phase shift keying (MPSK) modulation and (2) analyzing the impact of timing synchronization errors and channel estimation errors on the SER performance. To reduce the number of simultaneous transmissions while allowing N distributed AF nodes to exchange their data symbols, achieving a diversity order of N1 per node, the M2MSTNC scheme is augmented with optimal node selection (i.e., M2MSTNCONS). This work also analyzes the SER performance of the proposed M2MSTNCONS scheme and studies the impact of timing synchronization errors and imperfect channel state information.
Although selection in cooperative networks is not a new concept (e.g., see [15, 16]), the novelty of this work is manifested by augmenting it with a manytomany communications scheme to achieve full diversity and mitigate the adverse effects of timing offsets and channel estimation errors. The main contributions of this paper are summarized as follows:

Proposed the M2MSTNC scheme with optimal node selection (i.e., M2MSTNCONS) and analytically proved that it achieves full diversity order.

Analytically studied the effect of timing offsets and channel estimation errors on the performance of the M2MSTNC and M2MSTNCONS schemes.

Demonstrated that the M2MSTNCONS scheme is more resistant to timing offsets and channel estimation errors than its counterpart M2MSTNC scheme, in terms of the SER performance as well as achievable rate.
Due to the envisioned merits of the M2MSTNCONS scheme, its potential applications may include but are not limited to clusterbased communications for cooperative spectrum sensing and decision fusion in cognitive radio networks [17], and also for reliable and energyefficient inter and intracluster data gathering in wireless sensor networks [18]. Moreover, the M2MSTNCONS scheme can be used for improved network connectivity in clustered mobile ad hoc networks [19]. It is envisioned that the M2MSTNCONS scheme will serve as a potential candidate for manytomany cooperative communications in amplifyandforward cooperative networks.
In the rest of this paper, the system model of the M2MSTNC scheme is presented in Section 2. The signal model of the proposed M2MSTNCONS scheme is discussed in Section 3, while the theoretical symbol error rate of both the M2MSTNC and M2MSTNCONS schemes is analyzed in Section 4. The impact of timing offsets and channel estimation errors on the performance of both schemes is characterized in Sections 5, and 6, respectively. Simulation results are contrasted with the analytical results in Section 7. Finally, conclusions are drawn in Section 8.
2 System model
The M2MSTNC system model is based on a wireless network with N single antenna amplifyandforward nodes denoted S_{1}, S_{2}, …, S_{ N } for N ≥ 4. Each node S_{ j } for j ∈ {1, 2, …, N} is assumed to have its own data symbol x_{ j } to exchange with all the other N  1 nodes in the network. In this work, the channel between any two nodes is modeled as flat Rayleigh fading with additive white Gaussian noise (AWGN). Let h_{j,i} denote a generic channel coefficient representing the channel between any two nodes S_{ j } and S_{ i } for j ≠ i, and h_{j,i} is modeled as a zeromean complex Gaussian random variable with variance {\sigma}_{j,i}^{2} (i.e., {h}_{j,i}\sim \mathcal{C}\mathcal{N}\left(0,{\sigma}_{j,i}^{2}\right)). The squared channel gain h_{j,i}^{2} is an exponential random variable with mean {\sigma}_{j,i}^{2}. Also, the channel h_{j,i} between nodes S_{ j } and S_{ i } is assumed to be reciprocal (i.e., h_{i,j} = h_{j,i}) as in timedivision duplexing (TDD) systems, with perfect channel estimation at each node. Moreover, the channel coefficients are assumed to be quasistatic throughout the network operation. Finally, perfect timing synchronization between all the N nodes in the network is assumed.
The cooperative communication between all the nodes (depicted in Figure 1 for N = 4) is performed over a total of 2N time slots and is split into two phases (N time slots each): (a) the broadcasting phase (BP) and (b) the cooperation phase (CP). The communication under the two phases will be detailed in the following subsections and is expressed in matrix form as
2.1 Broadcasting phase
In the broadcasting phase, a source node S_{ j } is assigned a time slot T_{ j } in which it broadcasts its own data symbol x_{ j } to the N  1 other nodes S_{ i } in the network for i ∈ {1, 2, …, N} for i ≠ j. For source separation at each receiving node, each transmitted symbol x_{ j } is spread using a signature waveform c_{ j }(t) where it is assumed that each node knows the signature waveforms of all the other nodes. The crosscorrelation of c_{ j }(t) and c_{ i }(t) is {\rho}_{j,i}=\u3008{c}_{j}\left(t\right),{c}_{i}\left(t\right)\u3009\triangleq \left(1/{T}_{s}\right){\int}_{0}^{{T}_{s}}{c}_{j}\left(t\right){c}_{i}^{\ast}\left(t\right)\mathit{\text{dt}} for j ≠ i, with ρ_{j,j} = 1 and T_{ s } being the symbol duration. Thus, the signal received at node S_{ i } for i ≠ j in time slot T_{ j } is expressed as
where {P}_{j}^{\mathrm{B}} is the transmit power in the broadcasting phase at node S_{ j }, and h_{j,i} is the Rayleigh flat fading channel coefficient between nodes S_{ j } and S_{ i }. Also, n_{j,i}(t) is the additive noise process at node S_{ i } due to the signal transmitted by node S_{ j }, modeled as a zeromean complex Gaussian random variable with variance N_{0}. To extract data symbol x_{ j } at node S_{ i }, the received signal y_{j,i}(t) (given in (2)) is crosscorrelated with the signature waveform c_{ j }(t) to obtain
where {n}_{j,i}\sim \mathcal{C}\mathcal{N}\left(0,{N}_{0}\right). Upon completion of the broadcasting phase, each node S_{ i } will have exchanged its data symbol x_{ i } with the other nodes and received a set of N  1 signals {\left\{{y}_{j,i}\right\}}_{j=1,j\ne i}^{N} comprising symbols x_{1}, …, x_{i1},x_{i+1}, …, x_{ N } for j ≠ i from all the other nodes in the network. Node S_{ i } then performs a matched filtering operation on each of the received signals y_{j,i}, and the signaltonoise ratio (SNR) at the output of the matched filter is expressed as [2]
The received signals at each node at the end of the broadcasting phase are expressed as
where the i th row represents the signals received at node S_{ i }, while the j th column represents the signals received in time slot T_{ j } from node S_{ j }.
2.2 Cooperation phase
The cooperation phase involves two operations: (1) signal transmission and (2) multinode signal detection, which are discussed in the following subsections, respectively^{a}.
2.2.1 Signal transmission
In the cooperation phase, each node S_{ i } acts as the destination node in time slot T_{N+i} for i ∈ {1, 2, …, N} and receives simultaneous transmissions from the other N  1 nodes. In particular, each node S_{ k } with k ≠ i forms a linearly coded signal {\mathcal{X}}_{k}^{i}\left(t\right) which is composed from the received N  2 signals of the k th row of matrix Y in (5), excluding the received signal from node S_{ i }. Node S_{ k } then transmits {\mathcal{X}}_{k}^{i}\left(t\right) which is given by
where c_{ m }(t) is the signature waveform associated with symbol x_{ m }, and β_{m,k,i} is the normalization factor, as defined by [2]
From (6), it should be noticed that node S_{ k } relays the received signals from the other N  2 nodes. Moreover, the received signal at node S_{ i } during time slot T_{N+i} is given by
where α_{m,i} is defined as
In (8), w_{ i }(t) is the zeromean N_{0} variance additive noise process at node S_{ i }, and {\stackrel{\u0304}{w}}_{i}\left(t\right) is the equivalent noise term which can be expressed as
The total power at source node S_{ m } associated with exchanging symbol x_{ m } with the other N  1 nodes in the network is given by {P}_{m}={P}_{m}^{\mathrm{B}}+{P}_{m}^{\mathrm{C}}, where P m B=δ m BP_{ m } is the broadcast power and {P}_{m}^{\mathrm{C}}=\sum _{i=1,i\ne m}^{N}{P}_{m,i}^{\mathrm{C}}={\delta}_{m}^{\mathrm{C}}{P}_{m} is the total cooperative power, with 0 < δ m B ≤ 1 and δ m C = 1  δ m B being the power allocation fractions to the broadcasting and cooperation phases, respectively. In addition, {P}_{m,i}^{\mathrm{C}} is the total cooperative power associated with relaying symbol x_{ m } to destination node S_{ i } for i ≠ m such that {P}_{m,i}^{\mathrm{C}}={\delta}_{m,i}^{\mathrm{C}}{P}_{m}^{\mathrm{C}} with 0\le {\delta}_{m,i}^{\mathrm{C}}\le 1. Thus, {P}_{m,i}^{\mathrm{C}} is given by {P}_{m,i}^{\mathrm{C}}=\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne m\end{array}}^{N}{P}_{m,k,i}^{\mathrm{C}} with each relaying node S_{ k } for k ≠ m and k ≠ i being allocated cooperative power {P}_{m,k,i}^{\mathrm{C}}={\delta}_{m,k,i}^{\mathrm{C}}{P}_{m,i}^{\mathrm{C}} with 0\le {\delta}_{m,k,i}^{\mathrm{C}}\le 1. Without any loss of generality, it is assumed that all the transmit power associated with transmitting symbol x_{ m } is the same for all the N nodes (i.e., {P}_{m}=P={P}_{m}^{\mathrm{B}}+{P}_{m}^{\mathrm{C}}, ∀m ∈ {1, 2, …, N}).
2.2.2 Multinode signal detection
Upon receiving signal {\mathcal{Y}}_{i}\left(t\right), a multinode signal detection operation is performed by node S_{ i } to extract each of the N  1 symbols x_{ j }, for j ∈ {1, 2, …, N}_{j≠i}. This is achieved by passing the received signal {\mathcal{Y}}_{i}\left(t\right) through a matched filter bank (MFB) of N  1 branches, matched to the corresponding nodes’ signature waveforms c_{ j }(t), yielding
where ρ_{m,j} is the correlation coefficient between c_{ m }(t) and c_{ j }(t). The output of the MFB can be put in a vector form of all the N  1{\mathcal{Y}}_{i,j}’s signals as {\mathcal{Y}}_{i}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{R}_{i}{A}_{i}{x}_{i}+{\stackrel{\u0304}{w}}_{i}, where {\mathcal{Y}}_{i}={\left[{\mathcal{Y}}_{1,i},\dots ,{\mathcal{Y}}_{i1,i},{\mathcal{Y}}_{i+1,i},\dots ,{\mathcal{Y}}_{N,i}\right]}^{T}, and x_{ i } = [x_{1}, …, x_{i1},x_{i+1}, …, x_{ N }]^{T}. In addition, {\stackrel{\u0304}{w}}_{i}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\left[{\stackrel{\u0304}{w}}_{1,i},\dots ,{\stackrel{\u0304}{w}}_{i1,i},{\stackrel{\u0304}{w}}_{i+1,i},\dots ,{\stackrel{\u0304}{w}}_{N,i}\right]}^{T}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\sim \phantom{\rule{0.3em}{0ex}}\mathcal{C}\mathcal{N}\left(\mathbf{0},{N}_{0}\left(\mathbf{I}+{G}_{i}\right){R}_{i}\right) and R_{ i }, A_{ i } and I, G_{ i } are (N  1) × (N  1) matrices with I being the identity matrix with R_{ i } being defined as
and the diagonal matrices A_{ i } and G_{ i } are, respectively, written as
and
with {g}_{j,i}^{2} being defined as {g}_{j,i}^{2}=\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^{N}{\beta}_{j,k,i}^{2}{h}_{k,i}{}^{2} for j ≠ i. The received signal vector {\mathcal{Y}}_{i} can then be decorrelated (assuming matrix R_{ i } is invertible) as {\stackrel{\u0304}{\mathcal{Y}}}_{i}={R}_{i}^{1}{\mathcal{Y}}_{i}={A}_{i}{x}_{i}+{\stackrel{\u0304}{\stackrel{\u0304}{w}}}_{i}, where {\stackrel{\u0304}{\stackrel{\u0304}{w}}}_{i}={R}_{i}^{1}{\stackrel{\u0304}{w}}_{i} and {\stackrel{\u0304}{\stackrel{\u0304}{w}}}_{i}\sim \mathcal{C}\mathcal{N}\left(\mathbf{0},{N}_{0}{R}_{i}^{1}\left(\mathbf{I}+{G}_{i}\right)\right). Thus, at node S_{ i }, the decorrelated received signal {\stackrel{\u0304}{\mathcal{Y}}}_{j,i} corresponding to symbol x_{ j } is obtained as
where {\stackrel{\u0304}{\stackrel{\u0304}{w}}}_{j,i}\sim \mathcal{C}\mathcal{N}\left(0,{N}_{0}{\varrho}_{j,i}\left(1+{g}_{j,i}^{2}\right)\right), and ϱ_{j,i} is the j th diagonal element of matrix {R}_{i}^{1}. Without loss of generality, it is assumed that ρ_{j,i} = ρ for all j ≠ i and thus [6]
It should be noted that upon the completion of the broadcasting and cooperation phases, each node S_{ i } for i = 1, 2, …, N has received N  1 signals containing symbol x_{ j } for j = 1, 2, …, N and j ≠ i; a direct signal from the source node S_{ j } in the broadcasting phase and N  2 signals from nodes S_{ m } for m ≠ i and m ≠ j, in the cooperation phase. The instantaneous SNR at the output of the matched filter at node S_{ i } corresponding to symbol x_{ j } is given by
where {\gamma}_{j,i}^{\text{BP}} is an exponential random variable as in (4) with mean {\lambda}_{j,i}^{\text{BP}}=\frac{{N}_{0}}{{P}_{j}^{\mathrm{B}}{\sigma}_{j,i}^{2}}, and {\gamma}_{j,i}^{\text{CP}} is
It is clear from (18) that {\gamma}_{j,i}^{\text{CP}} is adversely affected by the noise amplification due to the simultaneous transmissions of the N2 nodes. The achievable rate between source node S_{ j } and destination node S_{ i } is given by
and the total achievable rate by node S_{ j } is expressed as {\mathcal{R}}_{j}^{\text{M2MSTNC}}=\sum _{i=1,i\ne j}^{N}{\mathcal{R}}_{j,i}^{\text{M2MSTNC}}\left({\gamma}_{j,i}\right). It should be noted that the M2MSTNC scheme requires stringent timing synchronization between the relaying nodes, and synchronizing all the distributed N nodes, as will be discussed later in this paper, is practically prohibitive.
3 Spacetime network coding with optimal node selection
When node S_{ i } acts as a destination node in its assigned time slot T_{N+i}, the intermediate node the transmitted signal of which results in the highest cumulative SNR value for symbol x_{ m } of source node S_{ m } for m ≠ i is selected. Specifically, for each source node, optimal relaying nodes are selected and then all the nodes selected for at least one source node transmit simultaneously. The node selection metric used by the destination node S_{ i } to determine the optimal node S_{ k } to ‘relay’ symbol x_{ m } received from source node S_{ m } for k ≠ i and k ≠ m is based on the scaled harmonic mean of the instantaneous source, intermediate and destination nodes’ scaled channel gains, as follows^{b}[15, 20, 21]
where X m, k B = P m Bh_{m,k}^{2} and X m, k, i C = P m, k, i Ch_{k,i}^{2} are exponential random variables corresponding to the broadcast transmission of symbol x_{ m } from source node S_{ m } to intermediate node S_{ k } with transmit power {P}_{m}^{\mathrm{B}} and the cooperative transmission of symbol x_{ m } from intermediate node S_{ k } to the destination node S_{ i } with cooperative transmit power P m, k, i C = P m,i C. Thus, the scaled harmonic mean values corresponding to symbol x_{ m }, for m ≠ i at node S_{ k } for k ≠ i and k ≠ m when node S_{ i } is the destination node is summarized in a matrix form as
For node S_{ i } to receive symbol x_{ m } for m ≠ i, the optimally selected node to forward symbol x_{ m } among the N  2 nodes that received independent copies of symbol x_{ m } during the broadcasting phase is defined by {k}_{m,i}^{\text{opt}}=arg\underset{k=1,2,\dots ,N}{\text{max}}{\left\{{\gamma}_{m,k,i}\right\}}_{k\ne i,k\ne m}. Hence, in time slot T_{N+i} for each symbol x_{ m } for m ≠ i, the system reduces to a source node S_{ m }, a destination node S_{ i }, and an optimally selected node for the transmission of x_{ m }. Thus, each symbol x_{ m } is associated with a set of indicator functions in the form of {\mathcal{I}}_{m,i}={\left\{{\mathcal{I}}_{m,k,i}\right\}}_{k=1,k\ne i,k\ne m}^{N}, where {\mathcal{I}}_{m,k,i} for k ≠ i, k ≠ m acts as a binary indicator function when node S_{ i } is the receiving node, while S_{ k } is the optimally selected node transmitting signal y_{m,k} corresponding to symbol x_{ m }. Hence, {\mathcal{I}}_{m,k,i} is defined by {\mathcal{I}}_{m,k,i}=1 if k={k}_{m,i}^{\text{opt}}; otherwise, {\mathcal{I}}_{m,k,i}=0.
As before, each node S_{ k } then possibly forms a linearly coded signal {\mathcal{Z}}_{k}^{i}\left(t\right) from its received signals in the broadcasting phase and transmits it to node S_{ i } during time slot T_{N+i}. Specifically, {\mathcal{Z}}_{k}^{i}\left(t\right) is composed from the received signals of the k th row of matrix Y in (5) in the form of
It should be noted that if node S_{ k } is not an optimal node to forward any of the x_{ m } for m ≠ i, m ≠ k data signals to node S_{ i }, then {\mathcal{Z}}_{k}^{i}\left(t\right)=0; otherwise, node S_{ k } is an optimal node to forward at least one symbol x_{ m } and {\mathcal{Z}}_{k}^{i}\left(t\right)\ne 0. Following the steps of the previous section, the received signal at node S_{ i } during time slot T_{N+i} is given by
where {\widehat{\alpha}}_{m,i} is defined as
with {\stackrel{\u0304}{h}}_{m,\text{opt},i} being the channel coefficient between the source node S_{ m } and the optimally selected node to forward symbol x_{ m } to node S_{ i } for m ≠ i, as implied by k={k}_{m,i}^{\text{opt}}, and β_{m,opt,i} is the scaling factor defined in (7). Also, {\u0125}_{m,\text{opt},i} is the channel coefficient between the optimally selected node and node S_{ i } for the transmission of symbol x_{ m } for m ≠ i. In (23), {\u0175}_{i}\left(t\right) is the equivalent noise term which can be expressed as
where n_{m,opt,i} is the noise sample at the optimally selected node by node S_{ i } for the transmission of symbol x_{ m }, for m ≠ i. It should be noted that under the M2MSTNCONS scheme, the total cooperative transmit power associated with relaying symbol x_{ m } to node S_{ i } is set to {P}_{m,\text{opt},i}^{\mathrm{C}}={P}_{m,i}^{\mathrm{C}}={\delta}_{m,i}^{\mathrm{C}}{P}_{m}^{\mathrm{C}}, where {P}_{m,\text{opt},i}^{\mathrm{C}} is the cooperative transmit power allocated to the optimally selected node. Thus, the total power associated with transmitting symbol x_{ m } is given by P={P}_{m}^{\mathrm{B}}+\sum _{i=1,i\ne m}^{N}{P}_{m,\text{opt},i}^{\mathrm{C}}.
To extract symbol x_{ j }, the received signal {\widehat{\mathcal{Y}}}_{i}\left(t\right) is passed through a MFB, and the output of the j th branch is expressed as {\widehat{\mathcal{Y}}}_{j,i}=\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^{N}{\widehat{\alpha}}_{m,i}{x}_{m}{\rho}_{m,j}+{\u0175}_{j,i}which in vector form is expressed as {\widehat{\mathcal{Y}}}_{i}={R}_{i}{\xc2}_{i}{x}_{i}+{\u0175}_{i}. In particular, {\u0175}_{i}={\left[{\u0175}_{1,i},\dots ,{\u0175}_{i1,i},{\u0175}_{i+1,i},\dots ,{\u0175}_{N,i}\right]}^{T}\sim \mathcal{C}\mathcal{N}(\mathbf{0},{N}_{0}(\mathbf{I}+{\u011c}_{i}\left){R}_{i}\right), with R_{ i } being defined in (12), while {\xc2}_{i} and {\u011c}_{i} are defined as
and
with {\u011d}_{j,i}^{2} being defined as {\u011d}_{j,i}^{2}={\beta}_{j,\text{opt},i}^{2}{\u0125}_{j,\text{opt},i}{}^{2} for j ≠ i. The decorrelated signal {\u0176}_{j,i} is given by
where {\widehat{\u0175}}_{j,i}\sim \mathcal{C}\mathcal{N}\left(0,{N}_{0}{\varrho}_{N1}\left(1+{\beta}_{j,\text{opt},i}^{2}{\u0125}_{j,\text{opt},i}{}^{2}\right)\right). At the end of the broadcasting and cooperation phases, two signals comprising symbol x_{ j } are received at node S_{ i }; the first comes from the direct transmission in the broadcasting phase, while the other is from the optimally selected node in the cooperation phase. At the output of the matched filter, the instantaneous SNR is given by
where {\widehat{\gamma}}_{j,i}^{\text{CP}} is given by
which at high SNR can be tightly approximated as [2]
where it can be verified that {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}} is the scaled harmonic mean of two exponential random variables
with means {\lambda}_{j,\text{opt},i}^{\mathrm{B}}=\frac{{N}_{0}{\varrho}_{N1}}{{P}_{j}^{\mathrm{B}}{\stackrel{\u0304}{\sigma}}_{j,\text{opt},i}^{2}} and {\lambda}_{j,\text{opt},i}^{\mathrm{C}}=\frac{{N}_{0}{\varrho}_{N1}}{{P}_{j,i}^{\mathrm{C}}{\widehat{\sigma}}_{j,\text{opt},i}^{2}}, respectively. Note that {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}} corresponds to the optimally selected node with the maximum harmonic mean. The means of {\gamma}_{j,i}^{\text{BP}}, {X}_{j,\text{opt}}^{\mathrm{B}} and {X}_{j,\text{opt},i}^{\mathrm{C}} are redefined, respectively, as {\lambda}_{j,i}^{\text{BP}}=\frac{{N}_{0}}{{\delta}_{j}^{\mathrm{B}}P{\sigma}_{j,i}^{2}}, {\lambda}_{j,\text{opt},i}^{\mathrm{B}}=\frac{{N}_{0}{\varrho}_{N1}}{{\delta}_{j}^{\mathrm{B}}P{\stackrel{\u0304}{\sigma}}_{j,\text{opt},i}^{2}} and {\lambda}_{j,\text{opt},i}^{\mathrm{C}}=\frac{{N}_{0}{\varrho}_{N1}}{{\delta}_{j,i}^{\mathrm{C}}(1{\delta}_{j}^{\mathrm{B}})P{\widehat{\sigma}}_{j,\text{opt},i}^{2}}, ∀i, j ∈ {1, 2, …, N}_{j≠i}. Thus, {\widehat{\gamma}}_{j,i} is redefined as {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}={\gamma}_{j,i}^{\text{BC}}+{\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}}. The achievable rate between source node S_{ j } and destination node S_{ i } under the M2MSTNCONS scheme is given by
Thus {\mathcal{R}}_{j}^{\text{M2MSTNCONS}}=\sum _{i=1,i\ne j}^{N}{\mathcal{R}}_{j,i}^{\text{M2MSTNCONS}}\left({\widehat{\gamma}}_{j,i}\right) is the total rate achievable by node S_{ j }.
4 Symbol error rate performance analysis
4.1 M2MSTNC
In general, the SER for MPSK modulation conditional on the channel state information (CSI) for SNR γ is given by [22]
where b_{psk}= sin2(π / M). The derived instantaneous SNR due to the cooperative transmission {\gamma}_{j,i}^{\text{CP}} in (18) is extremely difficult to manipulate [5]. Thus, only the conditional SER of symbol x_{ j } detected at node S_{ i } (for i ≠ j) is provided, which can be evaluated numerically as
where γ j, i BP + γ j, i CP = γ_{j,i}, as defined in (17).
4.2 M2MSTNCONS
Denoting the moment generating function (MGF) of a random variable Z with probability density function (PDF) p_{ Z }(z) as
and averaging the conditional SER over the Rayleigh fading channel statistics, the approximate SER expression is given by
where the approximation is due to the use of {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}} in (31) instead of {\widehat{\gamma}}_{j,i}^{\text{CP}} in (30). Additionally, {\mathcal{M}}_{{\gamma}_{j,i}^{\text{BP}}}\left(s\right) is defined as [2]
To determine the MGF of {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}}, the cumulative distribution function (CDF) of {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}} is derived as
where
and {P}_{{\widehat{\gamma}}_{j,k,i}}\left(\gamma \right)=12\gamma \sqrt{{\widehat{\lambda}}_{j,k}^{\mathrm{B}}{\widehat{\lambda}}_{j,k,i}^{\mathrm{C}}}{e}^{\gamma \left({\widehat{\lambda}}_{j,k}^{\mathrm{B}}+{\widehat{\lambda}}_{j,k,i}^{\mathrm{C}}\right)}{K}_{1}\left(2\gamma \sqrt{{\widehat{\lambda}}_{j,k}^{\mathrm{B}}{\widehat{\lambda}}_{j,k,i}^{\mathrm{C}}}\right) with {\widehat{\lambda}}_{j,k}^{\mathrm{B}} and {\widehat{\lambda}}_{j,k,i}^{\mathrm{C}} being defined as {\widehat{\lambda}}_{j,k}^{\mathrm{B}}=\frac{{N}_{0}{\varrho}_{N1}}{{\delta}_{j}^{\mathrm{B}}P{\sigma}_{j,k}^{2}} and {\widehat{\lambda}}_{j,k,i}^{\mathrm{C}}=\frac{{N}_{0}{\varrho}_{N1}}{{\delta}_{j,i}^{\mathrm{C}}\left(1{\delta}_{j}^{\mathrm{B}}\right)P{\sigma}_{k,i}^{2}}, respectively. Also, K_{1}(·) is the firstorder modified Bessel function of the second kind [23]. At high SNR, K_{1}(·) can be approximated for small x as K_{1}(x) ≈ 1 / x[23], and thus the CDF of {\widehat{\gamma}}_{j,k,i} simplifies to {P}_{{\widehat{\gamma}}_{j,k,i}}\left(\gamma \right)\approx 1{e}^{\gamma \left({\widehat{\lambda}}_{j,k}^{\mathrm{B}}+{\widehat{\lambda}}_{j,k,i}^{\mathrm{C}}\right)}. For convenience, define {\widehat{\lambda}}_{j,k,i}^{\text{BC}}\triangleq {\widehat{\lambda}}_{j,k}^{\mathrm{B}}+{\widehat{\lambda}}_{j,k,i}^{\mathrm{C}}=\frac{{N}_{0}{\varrho}_{N1}}{P}{\mathrm{\Phi}}_{j,k,i}, where
Therefore, the PDF of {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}} can be obtained as
where {p}_{{\widehat{\gamma}}_{j,m,i}}\left(\gamma \right)={\widehat{\lambda}}_{j,m,i}^{\text{BC}}{e}^{\gamma {\widehat{\lambda}}_{j,m,i}^{\text{BC}}} is the PDF of {\widehat{\gamma}}_{j,m,i}. Using (42) to determine the MGF of {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}} is quite difficult [20]; however, a useful relationship between the CDF of a random variable X and its MGF exists and is given by {\mathcal{M}}_{X}\left(s\right)=s\mathcal{\mathcal{L}}\left\{{P}_{X}\right(x\left)\right\}, with \mathcal{\mathcal{L}}\{\xb7\} being the Laplace transform of the parameter CDF [23]. Hence, by substituting {P}_{{\widehat{\gamma}}_{j,k,i}}\left(\gamma \right)\approx 1{e}^{\gamma \left({\widehat{\lambda}}_{j,k}^{\mathrm{B}}+{\widehat{\lambda}}_{j,k,i}^{\mathrm{C}}\right)} into (39), expanding the resulting product and then taking the Laplace transform, the MGF of {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}} can be shown to be
Thus, by substituting (38) and (43) into (37), the approximate SER performance for symbol x_{ j } detected at node S_{ i } for i ≠ j can be determined using
4.2.1 Asymptotic upper bound
An asymptotic upper bound on the SER performance is derived by first noticing that at high SNR, the MGF of {\gamma}_{j,i}^{\text{BP}} given in (38) can be asymptotically upper bounded as [2]
Second, an asymptotic upper bound for {\mathcal{M}}_{{\stackrel{\u0304}{\gamma}}_{j,i}^{\text{CP}}}\left(s\right) at high SNR can be determined by approximating e^{x} ≃ (1 + x) when x → 0 in the PDF of {\stackrel{\u0304}{\widehat{\gamma}}}_{j,i}^{\text{CP}} defined in (31), which is now given by
Since {\lambda}_{j,k,i}^{\text{BC}}=\frac{{N}_{0}{\varrho}_{N1}}{P}{\mathrm{\Phi}}_{j,k,i}, so by substituting (46) into (36), it can be shown that
Finally, by substituting (45) and (47) into (37), the asymptotic upperbound SER expression is obtained as
with Θ(N  1) being defined as \Theta (N1)=\frac{1}{\pi}{\int}_{0}^{(M1)\pi /M} (sin2(θ))^{N1}dθ.
4.2.2 Diversity order analysis
The diversity order is given by \mathbf{\Gamma}={lim}_{\text{SNR}\to \infty}log\left({\stackrel{\u0304}{P}}_{\text{UBSER}}^{\text{M2MSTNCONS}}\right)/log\left(\text{SNR}\right), where SNR = P / N_{0}[2]. Clearly, the M2MSTNCONS scheme achieves a full diversity order of Γ = N  1 per node.
It is noteworthy that the concept of manytomany spacetime network coding with optimal node selection allows us to achieve full diversity of N  1 per network node with only 2N time slots. In conventional TDMAbased cooperative communications (i.e., without network coding and multipleaccess transmissions), a total of N^{2} time slots is required to achieve full diversity. Clearly, our scheme is more bandwidth efficient than conventional cooperative communication systems.
5 Timing synchronization analysis
It is well known that due to the diagonal structure of the broadcasting phase, as shown in (1), the problem of perfect timing synchronization is alleviated since within the TDMA framework, only one source node is allowed to transmit at any one time [24]. Moreover, the analysis so far assumed perfect ‘inphase’ synchronization among the transmitting nodes in the cooperation phase. However, simultaneous transmissions of the different nodes during the cooperation phase impose a major practical challenge, especially for a large number of the transmitting nodes distributed over a large network area. Clock mismatches of the geographically distributed nodes result in different transmission times. Also, the lack of tracking at the receiving node for all the other cooperative nodes and the lack of compensation for propagation delays can have detrimental effects on the network performance. Thus, this section aims at analyzing the degradation in the SER performance of the M2MSTNC and M2MSTNCONS schemes due to the timing offsets between the nodes in the cooperation phase.
5.1 Signal model under M2MSTNC scheme
In the cooperation phase, consider the scenario where node S_{ i } is the receiving node while the remaining distributed nodes S_{ m } for m ∈ {1, 2, …, N}_{m ≠ i} transmit asynchronously. Let τ_{i,m} be the time offset for each transmitting node S_{ m } during the i th time slot. Also, assume that each distributed node initiates and terminates its transmissions within T_{ s } time units of each other within each TDMA time slot. Moreover, the effect of the different propagation delays is manifested in the form of superposition of pulses from each node S_{ m } for m ∈ {1, 2, …, N}_{m≠i} that are shifted by τ_{i,m}. This implies that neighboring symbols will introduce intersymbol interference (ISI) to the desired symbol. In this work, only the ISI contribution from the neighboring symbols to the desired symbol is considered, while higherorder terms are neglected due to their smaller effect [14]. From (8), the received signal at node S_{ i } during the i th time slot is expressed as^{c}[25, 26]
where α_{m,i} is defined in (9), and {\stackrel{\u0304}{w}}_{i}\left(t\right) is written as
Without loss of generality, the random time shifts between the N  1 nodes and the receiving node S_{ i } are ordered such that 0 ≤ τ_{i,1} ≤ … ≤ τ_{i,i1} ≤ τ_{i,i+1} ≤ … ≤ τ_{i,N}<T_{ s }. As before, the received signal is then fed into a bank of (N  1) filters, matched to the nodes’ signature waveforms, and sampled at t = l T_{ s } + Δ_{ i }, where Δ_{ i } is the timing shift chosen by the receiving node S_{ i } to compensate for the average delay of the transmitting nodes. Thus, the received signal is given by [27]
with c_{ j }(t) being zero outside the duration of T_{ s } time units. Define the (N  1) × (N  1) crosscorrelation matrix R_{ i }(l) whose entries are modeled for l = 1, l = 0, and l = 1 as [28]
and
respectively, where R_{ i }(l) = 0, ∀ l > 1, {R}_{i}\left(l\right)={R}_{i}^{T}(l), and as before, it is assumed that ρ_{m,j} = ρ for m ≠ i. Furthermore, the time shifts are assumed to be uniformly distributed as (τ_{i,m}  Δ_{ i }) ∼ U[  Δ T_{ s } / 2, Δ T_{ s }/2] around the reference clock Δ_{ i }, ∀m ∈ {1, 2, …, N}_{m≠i}, where Δ T_{ s } ∈ [0, T_{ s }) is the maximum timeshift value. Intuitively, the smaller are the time shifts, the less severe are the timing synchronization errors. Now, let {R}_{i}\left(0\right)={\stackrel{\u0304}{R}}_{i} be defined as
and {R}_{i}\left(1\right)={R}_{i}^{T}(1)={\stackrel{~}{R}}_{i}, where {\stackrel{~}{R}}_{i} is defined as follows [29]
Thus, the output of the matched filter bank can be expressed as [27, 29]
where matrix A_{ i } is defined in (13), whereas x_{ i }(l + ς) is defined in general as
for ς ∈ {1, 0, 1}. Also, {\stackrel{\u0304}{w}}_{i}\left(l\right) is the noise vector with variance given by
where E[·] is the expectation operator, and matrix G_{ i } is defined in (14). As before, the vector {\mathcal{Y}}_{i}\left(l\right) can be decorrelated as {\stackrel{\u0304}{\mathcal{Y}}}_{i}\left(l\right)={R}_{i}^{1}{\mathcal{Y}}_{i}\left(l\right)={\u0154}_{i}{A}_{i}{x}_{i}(l+1)+{\stackrel{\u0304}{\stackrel{\u0304}{R}}}_{i}{A}_{i}{x}_{i}\left(l\right)+{\stackrel{\u0300}{R}}_{i}{A}_{i}{x}_{i}(l1)+{\stackrel{\u0304}{\stackrel{\u0304}{w}}}_{i}\left(l\right), where R_{ i } is as defined in (12) with offdiagonal elements equal to ρ, {\stackrel{\u0304}{\stackrel{\u0304}{R}}}_{i}={R}_{i}^{1}{\stackrel{\u0304}{R}}_{i}, {\u0154}_{i}={R}_{i}^{1}{\stackrel{~}{R}}_{i}^{T}, {\stackrel{\u0300}{R}}_{i}={R}_{i}^{1}{\stackrel{~}{R}}_{i}, and {\stackrel{\u0304}{\stackrel{\u0304}{w}}}_{i}\left(l\right)={R}_{i}^{1}{\stackrel{\u0304}{w}}_{i}\left(l\right) with
The decorrelated received signal {\stackrel{\u0304}{\mathcal{Y}}}_{j,i}\left(l\right) at the output of the j th MFB branch is given by
where {\stackrel{\u0304}{\stackrel{\u0304}{\rho}}}_{j,j}, is the j th diagonal element of matrix {\stackrel{\u0304}{\stackrel{\u0304}{R}}}_{i}, while {\stackrel{\u0301}{\rho}}_{j,m} and {\stackrel{\u0300}{\rho}}_{j,m} are the (j,m)th element of matrices {\u0154}_{i} and {\stackrel{\u0300}{R}}_{i}, respectively. Additionally, {\stackrel{\u0304}{\stackrel{\u0304}{w}}}_{j,i}\left(l\right)\sim \mathcal{C}\mathcal{N}(0,{\stackrel{\u0304}{\stackrel{\u0304}{\varrho}}}_{j,i}{N}_{0}), where {\stackrel{\u0304}{\stackrel{\u0304}{\varrho}}}_{j,i} is the j th diagonal element of matrix {R}_{i}^{1}\left((\mathbf{I}+{G}_{i}){\stackrel{\u0304}{R}}_{i}+{G}_{i}{\stackrel{~}{R}}_{i}^{T}+{G}_{i}{\stackrel{~}{R}}_{i}\right){R}_{i}^{T}. Based on the above analysis, the instantaneous conditional signaltointerferenceplusnoise ratio (SINR) at the output of the MRC of node S_{ i } of symbol x_{ j } for j ≠ i, after further manipulation, is obtained as
where {\mathcal{I}}_{j,i} is the ISI variance as defined by
and it is assumed that the data symbols are statistically independent. Based on (62), finding a closed form solution for the SER for MPSK modulation is extremely difficult; therefore, a conditional SER given the channel knowledge is obtained by substituting (62) into (34) and then numerically evaluating it.
It should be noted that γ_{j,i} in (62) is composed of the SNR due to the broadcasting phase and the SINR due to the cooperation phase. Thus, it can be verified that if τ_{i,m}  Δ_{ i } = 0, ∀i, m ∈ {1, 2, …, N} and i ≠ m (i.e., perfect timing synchronization), then {\stackrel{\u0304}{\rho}}_{j,i}^{\left(0\right)}={\rho}_{j,i} and also {\stackrel{~}{\rho}}_{j,i}^{(1)}={\stackrel{~}{\rho}}_{j,i}^{\left(1\right)}=0 and thus the SINR γ_{j,i} in (62) reduces to that of (17), as {\widehat{\gamma}}_{j,i}^{\text{CP}} in (62) reduces to the one in (18).
5.2 Signal model under M2MSTNCONS scheme
From (23), the received signal at node S_{ i } is given by
where {\widehat{\alpha}}_{m,i} is defined in (24) and {\u0175}_{i}\left(t\right) is written as
Following the analysis of the M2MSTNC scheme and replacing matrices A_{ i } and G_{ i } with {\xc2}_{i} and {\u011c}_{i}, respectively (see (26) and (27)), the instantaneous conditional SINR of symbol x_{ j } at node S_{ i } can be shown to be
where {\widehat{\widehat{\varrho}}}_{j,i} is the j th diagonal element of matrix {R}_{i}^{1}\left(\left(\mathbf{I}+{\u011c}_{i}\right){\stackrel{\u0304}{R}}_{i}+{\u011c}_{i}{\stackrel{~}{R}}_{i}^{T}+{\u011c}_{i}{\stackrel{~}{R}}_{i}\right){R}_{i}^{T}, and
It is noteworthy that under perfect timing synchronization, (66) reduces to (29), as the SINR term due to the cooperation phase reduces to the SNR term of (30).
6 Imperfect channel state information
So far, perfect CSI has been assumed and in practice, such assumption is not valid. Channel estimation errors are possibly caused by inaccurate channel estimation/equalization, noise or Doppler shift. Conventionally, channel estimation is based on transmitting a known pilot ‘training’ sequence with a particular power, prior to data transmission. Inaccurate channel estimation results in a channel estimation error with variance, denoted as ε. At the end of the training phase, the receiving node has imperfect CSI for channel equalization and data detection. In the following subsections, the impact of channel estimation errors on the performance of the M2MSTNC and M2MSTNCONS schemes, assuming perfect timing synchronization, is studied and characterized.
6.1 M2MSTNC
In the broadcasting phase, the received signal at node S_{ i } from node S_{ j } with channel estimation error is expressed as
where {h}_{j,i}^{\epsilon} denotes the channel estimation error. Consequently, \sqrt{{P}_{j}^{\mathrm{B}}}{h}_{j,i}^{\epsilon}{x}_{j} is the added noise term that scales with the broadcasting power. Furthermore, the channel estimation error {h}_{j,i}^{\epsilon} is modeled as a zeromean complex Gaussian random variable with variance ε_{j,i}. Thus, the additional selfnoise term \sqrt{{P}_{j}^{\mathrm{B}}}{h}_{j,i}^{\epsilon}{x}_{j} is a zeromean complex Gaussian random variable with variance ε_{j,i}P j B. Equation (68) is rewritten as
where {n}_{j,i}^{\epsilon}\left(t\right)=\sqrt{{P}_{j}^{\mathrm{B}}}{h}_{j,i}^{\epsilon}{x}_{j}{c}_{j}\left(t\right)+{n}_{j,i}\left(t\right) is a zeromean Gaussian random variable with variance ε_{j,i}P j B + N_{0}. Thus, the SNR after matched filtering is given by
In the cooperation phase, the received signal at node S_{ i } is given by
where {\alpha}_{m,i}^{\epsilon} is defined as
and
As before, w_{ i }(t) is the zeromean N_{0}variance AWGN sample at node S_{ i } and {\stackrel{\u0304}{w}}_{i}^{\epsilon}\left(t\right) is the equivalent noise term, expressed as
After multinode signal detection, the received signal corresponding to symbol x_{ j } is given by