Many-to-many space-time network coding for amplify-and-forward cooperative networks: node selection and performance analysis

Abstract

In this paper, the multinode amplify-and-forward cooperative communications for a network of N nodes is studied via the novel concept of many-to-many space-time network coding (M2M-STNC). Communication under the M2M-STNC 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 N-1 nodes to a destination node take place in their time slot. In addition, the M2M-STNC scheme with optimal node selection (i.e., M2M-STNC-ONS) 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 symbol-error-rate analysis for M-ary phase shift keying (M-PSK) modulation is derived for both the M2M-STNC and M2M-STNC-ONS 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 M2M-STNC-ONS scheme outperforms the M2M-STNC scheme and is less sensitive to timing offsets and channel estimation errors. It is envisioned that the M2M-STNC-ONS scheme will serve as a potential many-to-many 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 location-aware 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 space-time network coding (STNC) [5, 6]. In [5], the multipoint-to-point (M2P) and point-to-multipoint (P2M) space-time network codes were proposed to allow multiple source transmissions within a time-division 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 M2P-STNC or P2M-STNC, only 2N time slots are required while achieving a diversity order of N per transmitted symbol. In [7], differential space-time 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. Anti-eavesdropping space-time network coding (AE-STNC) has been proposed in [8] for secure cooperative communications against eavesdropping, while achieving full diversity. Many-to-one STNC has been proposed in [9] for cluster-based 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 M-PSK and M-QAM modulations, and the impact of the fading parameter m, relay location, power allocation, and non-orthogonal codes on the SER are examined.

In [6], the many-to-many space-time network coding (M2M-STNC) for a network of N decode-and-forward (DF) nodes is proposed to achieve a diversity order of N-1 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 M2M-STNC scheme is based on the assumption of N-1 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 M2M-STNC communication scheme for amplify-and-forward (AF) cooperative networks by (1) characterizing the symbol error rate performance for M-ary phase shift keying (M-PSK) 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 N-1 per node, the M2M-STNC scheme is augmented with optimal node selection (i.e., M2M-STNC-ONS). This work also analyzes the SER performance of the proposed M2M-STNC-ONS 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 many-to-many 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 M2M-STNC scheme with optimal node selection (i.e., M2M-STNC-ONS) 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 M2M-STNC and M2M-STNC-ONS schemes.

• Demonstrated that the M2M-STNC-ONS scheme is more resistant to timing offsets and channel estimation errors than its counterpart M2M-STNC scheme, in terms of the SER performance as well as achievable rate.

Due to the envisioned merits of the M2M-STNC-ONS scheme, its potential applications may include but are not limited to cluster-based communications for cooperative spectrum sensing and decision fusion in cognitive radio networks [17], and also for reliable and energy-efficient inter- and intra-cluster data gathering in wireless sensor networks [18]. Moreover, the M2M-STNC-ONS scheme can be used for improved network connectivity in clustered mobile ad hoc networks [19]. It is envisioned that the M2M-STNC-ONS scheme will serve as a potential candidate for many-to-many cooperative communications in amplify-and-forward cooperative networks.

In the rest of this paper, the system model of the M2M-STNC scheme is presented in Section 2. The signal model of the proposed M2M-STNC-ONS scheme is discussed in Section 3, while the theoretical symbol error rate of both the M2M-STNC and M2M-STNC-ONS 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 M2M-STNC system model is based on a wireless network with N single antenna amplify-and-forward nodes denoted S₁, S₂, …, 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 zero-mean 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 time-division duplexing (TDD) systems, with perfect channel estimation at each node. Moreover, the channel coefficients are assumed to be quasi-static 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

$$S_1\cdots S_j\cdots S_N{S}_1\cdots S_i\cdots S_N$$

$$\begin{array}{l}{T}_1\\ ⋮\\ T_j\\ ⋮\\ T_N\end{array}\overline{\underset{\text{Broadcasting phase}}{\underbrace{\left[\begin{array}{lllll}{x}_1& \cdots & 0& \cdots & 0\\ ⋮& \ddots & ⋮& \cdots & ⋮\\ 0& \cdots & x_j& \cdots & 0\\ ⋮& \cdots & ⋮& \ddots & ⋮\\ 0& \cdots & 0& \cdots & x_N\end{array}\right]}}}\begin{array}{l}{T}_{N+1}\\ ⋮\\ T_{N+i}\\ ⋮\\ T_{2N}\end{array}\overline{\underset{\text{Cooperation phase}}{\underbrace{\left[\begin{array}{lllll}0& \cdots & {\mathcal{X}}_i^1& \cdots & {\mathcal{X}}_N^1\\ ⋮& \ddots & ⋮& \cdots & ⋮\\ {\mathcal{X}}_1^i& \cdots & 0& \cdots & {\mathcal{X}}_N^i\\ ⋮& \cdots & ⋮& \ddots & ⋮\\ {\mathcal{X}}_1^N& \cdots & {\mathcal{X}}_i^N& \cdots & 0\end{array}\right]}}}.$$

(1)

Figure 1

M2M-STNC scheme with N = 4 nodes. (a) Broadcasting phase. (b) Cooperation phase.

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 cross-correlation of c _j (t) and c _i (t) is ${\rho }_{j,i}=〈c_j\left(t\right),c_i\left(t\right)〉\triangleq \left(1/T_s\right){\int }_0^{T_s}{c}_j\left(t\right)c_i^{\ast }\left(t\right)\mathit{\text{dt}}$ 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

$$y_{j,i}\left(t\right)=\sqrt{P_j^{\mathrm{B}}}{h}_{j,i}{x}_j{c}_j\left(t\right)+n_{j,i}\left(t\right),$$

(2)

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 zero-mean complex Gaussian random variable with variance N₀. To extract data symbol x _j at node S _i , the received signal y_j,i(t) (given in (2)) is cross-correlated with the signature waveform c _j (t) to obtain

$$y_{j,i}=〈y_{j,i}\left(t\right),c_j\left(t\right)〉=\sqrt{P_j^{\mathrm{B}}}{h}_{j,i}{x}_j+n_{j,i},$$

(3)

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 x₁, …, x_i-1,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 signal-to-noise ratio (SNR) at the output of the matched filter is expressed as [2]

$${\gamma }_{j,i}^{\text{BP}}=\frac{P_j^{\mathrm{B}}|h_{j,i}{|}^2}{N_0}.$$

(4)

The received signals at each node at the end of the broadcasting phase are expressed as

$$\mathbf{Y}=\left[\begin{array}{lllll}-& y_{2,1}& \cdots & y_{N-1,1}& y_{N,1}\\ y_{1,2}& -& \cdots & y_{N-1,2}& y_{N,2}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ y_{1,N-1}& y_{2,N-1}& \cdots & -& y_{N,N-1}\\ y_{1,N}& y_{2,N}& \cdots & y_{N-1,N}& -\end{array}\right],$$

(5)

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

$${\mathcal{X}}_k^i\left(t\right)=\sum _{\begin{array}{c}m=1\\ m\ne i,m\ne k\end{array}}^N{\beta }_{m,k,i}{y}_{m,k}{c}_m\left(t\right),$$

(6)

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]

$${\beta }_{m,k,i}=\sqrt{\frac{P_{m,k,i}^{\mathrm{C}}}{P_m^{\mathrm{B}}|h_{m,k}{|}^2+N_0}}.$$

(7)

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

$$\begin{array}{l}{\mathcal{Y}}_i\left(t\right)=\sum _{\begin{array}{c}k=1,k\ne i\end{array}}^N{h}_{k,i}{\mathcal{X}}_k^i\left(t\right)+w_i\left(t\right)=\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\alpha }_{m,i}{x}_m{c}_m\left(t\right)+{\bar{w}}_i\left(t\right),\end{array}$$

(8)

where α_m,i is defined as

$${\alpha }_{m,i}=\sqrt{P_m^{\mathrm{B}}}\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne m\end{array}}^N{\beta }_{m,k,i}{h}_{m,k}{h}_{k,i}.$$

(9)

In (8), w _i (t) is the zero-mean N₀ variance additive noise process at node S _i , and ${\bar{w}}_i\left(t\right)$ is the equivalent noise term which can be expressed as

$${\bar{w}}_i\left(t\right)=w_i\left(t\right)+\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne m\end{array}}^N{\beta }_{m,k,i}{h}_{k,i}{n}_{m,k}\left(t\right)c_m\left(t\right).$$

(10)

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

$${\mathcal{Y}}_{j,i}=〈{\mathcal{Y}}_i\left(t\right),c_j\left(t\right)〉=\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\alpha }_{m,i}{x}_m{\rho }_{m,j}+{\bar{w}}_{j,i},$$

(11)

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 ${\mathcal{Y}}_i=R_i{A}_i{x}_i+{\bar{w}}_i$, where ${\mathcal{Y}}_i={\left[{\mathcal{Y}}_{1,i},\dots ,{\mathcal{Y}}_{i-1,i},{\mathcal{Y}}_{i+1,i},\dots ,{\mathcal{Y}}_{N,i}\right]}^T$, and x _i = [x₁, …, x_i-1,x_i+1, …, x _N ]^T. In addition, ${\bar{w}}_i={\left[{\bar{w}}_{1,i},\dots ,{\bar{w}}_{i-1,i},{\bar{w}}_{i+1,i},\dots ,{\bar{w}}_{N,i}\right]}^T\sim \mathcal{C}\mathcal{N}\left(\mathbf{0},N_0\left(\mathbf{I}+G_i\right)R_i\right)$ 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

$$R_i=\left[\begin{array}{llllll}1& \cdots & {\rho }_{1,(i-1)}& {\rho }_{1,(i+1)}& \cdots & {\rho }_{1,N}\\ ⋮& \ddots & ⋮& ⋮& \cdots & ⋮\\ {\rho }_{(i-1),1}& \cdots & 1& {\rho }_{(i-1),(i+1)}& \cdots & {\rho }_{(i-1),N}\\ {\rho }_{(i+1),1}& \cdots & {\rho }_{(i+1),(i-1)}& 1& \cdots & {\rho }_{(i+1),N}\\ ⋮& \cdots & ⋮& ⋮& \ddots & ⋮\\ {\rho }_{N,1}& \cdots & {\rho }_{N,(i-1)}& {\rho }_{N,(i+1)}& \cdots & 1\end{array}\right],$$

(12)

and the diagonal matrices A _i and G _i are, respectively, written as

$$A_i=\text{diag}\left[{\alpha }_{1,i},\dots ,{\alpha }_{(i-1),i},{\alpha }_{(i+1),i},\dots ,{\alpha }_{N,i}\right],$$

(13)

and

$$G_i=\text{diag}\left[g_{1,i}^2,\dots ,g_{i-1,i}^2,g_{i+1,i}^2,\dots ,g_{N,i}^2\right],$$

(14)

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 ${\bar{\mathcal{Y}}}_i=R_i^{-1}{\mathcal{Y}}_i=A_i{x}_i+{\overline{\stackrel{̄}{w}}}_i$, where ${\overline{\stackrel{̄}{w}}}_i=R_i^{-1}{\bar{w}}_i$ and ${\overline{\stackrel{̄}{w}}}_i\sim \mathcal{C}\mathcal{N}\left(\mathbf{0},N_0{R}_i^{-1}\left(\mathbf{I}+G_i\right)\right)$. Thus, at node S _i , the decorrelated received signal ${\bar{\mathcal{Y}}}_{j,i}$ corresponding to symbol x _j is obtained as

$${\bar{\mathcal{Y}}}_{j,i}=\sqrt{P_j^{\mathrm{B}}}\left(\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N{\beta }_{j,k,i}{h}_{j,k}{h}_{k,i}\right)x_j+{\overline{\stackrel{̄}{w}}}_{j,i},$$

(15)

where ${\overline{\stackrel{̄}{w}}}_{j,i}\sim \mathcal{C}\mathcal{N}\left(0,N_0{\varrho }_{j,i}\left(1+g_{j,i}^2\right)\right)$, 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]

$${\varrho }_{j,i}=\frac{1+(N-3)\rho }{1+(N-3)\rho -(N-2){\rho }^2}\triangleq {\varrho }_{N-1}.$$

(16)

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

$${\gamma }_{j,i}={\gamma }_{j,i}^{\text{BP}}+{\gamma }_{j,i}^{\text{CP}},$$

(17)

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

$$\begin{array}{ll}{\gamma }_{j,i}^{\text{CP}}& =\frac{P_j^{\mathrm{B}}}{N_0{\varrho }_{N-1}\left(1+\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N\frac{P_{j,k,i}^{\mathrm{C}}|h_{k,i}{|}^2}{P_j^{\mathrm{B}}|h_{j,k}{|}^2+N_0}\right)}\\ \times \left(\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N\frac{P_{j,k,i}^{\mathrm{C}}\left|h_{j,k}{|}^2\right|h_{k,i}{|}^2}{P_j^{\mathrm{B}}|h_{j,k}{|}^2+N_0}\right).\end{array}$$

(18)

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 N-2 nodes. The achievable rate between source node S _j and destination node S _i is given by

$$\begin{array}{l}\begin{array}{l}{\mathcal{R}}_{j,i}^{\text{M2M-STNC}}\left({\gamma }_{j,i}\right)\\ =\frac{1}{2N}\underset{2}{log}\left(1+\frac{P_j^{\mathrm{B}}|h_{j,i}{|}^2}{N_0}+\frac{P_j^{\mathrm{B}}\left(\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N\frac{P_{j,k,i}^{\mathrm{C}}\left|h_{j,k}{|}^2\right|h_{k,i}{|}^2}{P_j^{\mathrm{B}}|h_{j,k}{|}^2+N_0}\right)}{N_0{\varrho }_{N-1}\left(1+\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N\frac{P_{j,k,i}^{\mathrm{C}}|h_{k,i}{|}^2}{P_j^{\mathrm{B}}|h_{j,k}{|}^2+N_0}\right)}\right),\end{array}\end{array}$$

(19)

and the total achievable rate by node S _j is expressed as ${\mathcal{R}}_j^{\text{M2M-STNC}}=\sum _{i=1,i\ne j}^N{\mathcal{R}}_{j,i}^{\text{M2M-STNC}}\left({\gamma }_{j,i}\right)$. It should be noted that the M2M-STNC 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 Space-time 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]

$${\gamma }_{m,k,i}\triangleq {\mu }_H\left(X_{m,k}^{\mathrm{B}};X_{m,k,i}^{\mathrm{C}}\right)=\frac{X_{m,k}^{\mathrm{B}}{X}_{m,k,i}^{\mathrm{C}}}{X_{m,k}^{\mathrm{B}}+X_{m,k,i}^{\mathrm{C}}},$$

(20)

where X m, k B = P m B|h_m,k|² and X m, k, i C = P m, k, i C|h_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

$$x_1\cdots x_{i-1}{x}_{i+1}\cdots x_N$$

$${\mathbf{\Gamma }}_i=\overline{\left[\begin{array}{llllll}-& \cdots & {\gamma }_{i-1,1,i}& {\gamma }_{i+1,1,i}& \cdots & {\gamma }_{N,1,i}\\ ⋮& \cdots & ⋮& ⋮& \cdots & ⋮\\ {\gamma }_{1,i-1,i}& \cdots & -& {\gamma }_{i+1,i-1,i}& \cdots & {\gamma }_{N,i-1,i}\\ {\gamma }_{1,i+1,i}& \cdots & {\gamma }_{i-1,i+1,i}& -& \cdots & {\gamma }_{N,i+1,i}\\ ⋮& \cdots & ⋮& ⋮& \cdots & ⋮\\ {\gamma }_{1,N,i}& \cdots & {\gamma }_{i-1,N,i}& {\gamma }_{i+1,N,i}& \cdots & -\end{array}\right]}\begin{array}{l}{S}_1\\ ⋮\\ S_{i-1}\\ S_{i+1}\\ ⋮\\ S_N\end{array}.$$

(21)

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

$${\mathcal{Z}}_k^i\left(t\right)=\sum _{\begin{array}{c}m=1\\ m\ne i,m\ne k\end{array}}^N{\beta }_{m,k,i}{y}_{m,k}{\mathcal{I}}_{m,k,i}{c}_m\left(t\right).$$

(22)

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

$$\begin{array}{l}{\widehat{\mathcal{Y}}}_i\left(t\right)=\sum _{\begin{array}{c}k=1,k\ne i\end{array}}^N{h}_{k,i}{\mathcal{Z}}_k^i\left(t\right)+w_i\left(t\right)=\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\widehat{\alpha }}_{m,i}{x}_m{c}_m\left(t\right)1+{ŵ}_i\left(t\right),\end{array}$$

(23)

where ${\widehat{\alpha }}_{m,i}$ is defined as

$${\widehat{\alpha }}_{m,i}={\beta }_{m,\text{opt},i}{\bar{h}}_{m,\text{opt},i}\sqrt{P_m^{\mathrm{B}}}{ĥ}_{m,\text{opt},i},$$

(24)

with ${\bar{h}}_{m,\text{opt},i}$ 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, ${ĥ}_{m,\text{opt},i}$ 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), ${ŵ}_i\left(t\right)$ is the equivalent noise term which can be expressed as

$${ŵ}_i\left(t\right)=w_i\left(t\right)+\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\beta }_{m,\text{opt},i}{ĥ}_{m,\text{opt},i}{n}_{m,\text{opt},i}\left(t\right)c_m\left(t\right),$$

(25)

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 M2M-STNC-ONS 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 ${\widehat{\mathcal{Y}}}_{j,i}=\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\widehat{\alpha }}_{m,i}{x}_m{\rho }_{m,j}+{ŵ}_{j,i}$which in vector form is expressed as ${\widehat{\mathcal{Y}}}_i=R_i{Â}_i{x}_i+{ŵ}_i$. In particular, ${ŵ}_i={\left[{ŵ}_{1,i},\dots ,{ŵ}_{i-1,i},{ŵ}_{i+1,i},\dots ,{ŵ}_{N,i}\right]}^T\sim \mathcal{C}\mathcal{N}(\mathbf{0},N_0(\mathbf{I}+{Ĝ}_i\left)R_i\right)$, with R _i being defined in (12), while ${Â}_i$ and ${Ĝ}_i$ are defined as

$${Â}_i=\text{diag}\left[{\widehat{\alpha }}_{1,i},\dots ,{\widehat{\alpha }}_{(i-1),i},{\widehat{\alpha }}_{(i+1),i},\dots ,{\widehat{\alpha }}_{N,i}\right],$$

(26)

and

$${Ĝ}_i=\text{diag}\left[{ĝ}_{1,i}^2,\dots ,{ĝ}_{i-1,i}^2,{ĝ}_{i+1,i}^2,\dots ,{ĝ}_{N,i}^2\right],$$

(27)

with ${ĝ}_{j,i}^2$ being defined as ${ĝ}_{j,i}^2={\beta }_{j,\text{opt},i}^2|{ĥ}_{j,\text{opt},i}{|}^2$ for j ≠ i. The decorrelated signal ${Ŷ}_{j,i}$ is given by

$${Ŷ}_{j,i}={\beta }_{j,\text{opt},i}{\bar{h}}_{j,\text{opt},i}\sqrt{P_j^{\mathrm{B}}}{ĥ}_{j,\text{opt},i}{x}_j+{\widehat{ŵ}}_{j,i},$$

(28)

where ${\widehat{ŵ}}_{j,i}\sim \mathcal{C}\mathcal{N}\left(0,N_0{\varrho }_{N-1}\left(1+{\beta }_{j,\text{opt},i}^2|{ĥ}_{j,\text{opt},i}{|}^2\right)\right)$. 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

$${\widehat{\gamma }}_{j,i}={\gamma }_{j,i}^{\text{BP}}+{\widehat{\gamma }}_{j,i}^{\text{CP}},$$

(29)

where ${\widehat{\gamma }}_{j,i}^{\text{CP}}$ is given by

$${\widehat{\gamma }}_{j,i}^{\text{CP}}=\frac{P_{j,i}^{\mathrm{C}}{P}_j^{\mathrm{B}}\left|{ĥ}_{j,\text{opt},i}{|}^2\right|{\bar{h}}_{j,\text{opt},i}{|}^2}{N_0{\varrho }_{N-1}\left(P_j^{\mathrm{B}}|{\bar{h}}_{j,\text{opt},i}{|}^2+P_{j,i}^{\mathrm{C}}|{ĥ}_{j,\text{opt},i}{|}^2+N_0\right)},$$

(30)

which at high SNR can be tightly approximated as [2]

$$\begin{array}{ll}{\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}& \simeq \frac{P_{j,i}^{\mathrm{C}}{P}_j^{\mathrm{B}}\left|{ĥ}_{j,\text{opt},i}{|}^2\right|{\bar{h}}_{j,\text{opt},i}{|}^2}{N_0{\varrho }_{N-1}\left(P_j^{\mathrm{B}}|{\bar{h}}_{j,\text{opt},i}{|}^2+P_{j,i}^{\mathrm{C}}|{ĥ}_{j,\text{opt},i}{|}^2\right)}\\ ={\mu }_H\left(X_{j,\text{opt}}^{\mathrm{B}};X_{j,\text{opt},i}^{\mathrm{C}}\right),\end{array}$$

(31)

where it can be verified that ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$ is the scaled harmonic mean of two exponential random variables

$$X_{j,\text{opt}}^{\mathrm{B}}=\frac{P_j^{\mathrm{B}}|{\bar{h}}_{j,\text{opt},i}{|}^2}{N_0{\varrho }_{N-1}}\text{and}{X}_{j,\text{opt},i}^{\mathrm{C}}=\frac{P_{j,i}^{\mathrm{C}}|{ĥ}_{j,\text{opt},i}{|}^2}{N_0{\varrho }_{N-1}},$$

(32)

with means ${\lambda }_{j,\text{opt},i}^{\mathrm{B}}=\frac{N_0{\varrho }_{N-1}}{P_j^{\mathrm{B}}{\bar{\sigma }}_{j,\text{opt},i}^2}$ and ${\lambda }_{j,\text{opt},i}^{\mathrm{C}}=\frac{N_0{\varrho }_{N-1}}{P_{j,i}^{\mathrm{C}}{\widehat{\sigma }}_{j,\text{opt},i}^2}$, respectively. Note that ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$ 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 ${\lambda }_{j,i}^{\text{BP}}=\frac{N_0}{{\delta }_j^{\mathrm{B}}P{\sigma }_{j,i}^2}$, ${\lambda }_{j,\text{opt},i}^{\mathrm{B}}=\frac{N_0{\varrho }_{N-1}}{{\delta }_j^{\mathrm{B}}P{\bar{\sigma }}_{j,\text{opt},i}^2}$ and ${\lambda }_{j,\text{opt},i}^{\mathrm{C}}=\frac{N_0{\varrho }_{N-1}}{{\delta }_{j,i}^{\mathrm{C}}(1-{\delta }_j^{\mathrm{B}})P{\widehat{\sigma }}_{j,\text{opt},i}^2},$ ∀i, j ∈ {1, 2, …, N}_j≠i. Thus, ${\widehat{\gamma }}_{j,i}$ is redefined as ${\bar{\widehat{\gamma }}}_{j,i}={\gamma }_{j,i}^{\text{BC}}+{\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$. The achievable rate between source node S _j and destination node S _i under the M2M-STNC-ONS scheme is given by

$$\begin{array}{l}\begin{array}{l}{\mathcal{R}}_{j,i}^{\text{M2M-STNC-ONS}}\left({\widehat{\gamma }}_{j,i}\right)\\ =\frac{1}{2N}\underset{2}{log}\left(1+\frac{P_j^{\mathrm{B}}|h_{j,i}{|}^2}{N_0}+\frac{P_{j,i}^{\mathrm{C}}{P}_j^{\mathrm{B}}\left|{ĥ}_{j,\text{opt},i}{|}^2\right|{\bar{h}}_{j,\text{opt},i}{|}^2}{N_0{\varrho }_{N-1}\left(P_j^{\mathrm{B}}|{\bar{h}}_{j,\text{opt},i}{|}^2+P_{j,i}^{\mathrm{C}}|{ĥ}_{j,\text{opt},i}{|}^2+N_0\right)}\right).\end{array}\end{array}$$

(33)

Thus ${\mathcal{R}}_j^{\text{M2M-STNC-ONS}}=\sum _{i=1,i\ne j}^N{\mathcal{R}}_{j,i}^{\text{M2M-STNC-ONS}}\left({\widehat{\gamma }}_{j,i}\right)$ is the total rate achievable by node S _j .

4 Symbol error rate performance analysis

4.1 M2M-STNC

In general, the SER for M-PSK modulation conditional on the channel state information (CSI) for SNR γ is given by [22]

$${\mathrm{\Psi }}_{\left|\right\{h\}}\left(\gamma \right)=\frac{1}{\pi }{\int }_0^{(M-1)\pi /M}exp\left(-\frac{b_{\text{psk}}\gamma }{\stackrel{2}{sin}\theta }\right)d\theta ,$$

(34)

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

$$P_{\text{SER}}^{\text{M2M-STNC}}={\mathrm{\Psi }}_{|{\left\{h_{j,i}\right\}}_{j,i=1,j\ne i}^N}\left({\gamma }_{j,i}^{\text{BP}}+{\gamma }_{j,i}^{\text{CP}}\right),$$

(35)

where γ j, i BP + γ j, i CP = γ_j,i, as defined in (17).

4.2 M2M-STNC-ONS

Denoting the moment generating function (MGF) of a random variable Z with probability density function (PDF) p _Z (z) as

$${\mathcal{M}}_Z\left(s\right)={\int }_{-\infty }^{\infty }{e}^{-\mathit{\text{sz}}}{p}_Z\left(z\right)\mathit{\text{dz}},$$

(36)

and averaging the conditional SER over the Rayleigh fading channel statistics, the approximate SER expression is given by

$$\begin{array}{ll}{\tilde{P}}_{\text{SER}}^{\text{M2M-STNC-ONS}}& =\mathrm{\Psi }\left({\gamma }_{j,i}^{\text{BP}}+{\bar{\widehat{\gamma }}}_{j,i}^{\mathrm{C}}\right)=\frac{1}{\pi }{\int }_0^{(M-1)\pi /M}\\ \times {\mathcal{M}}_{{\gamma }_{j,i}^{\text{BP}}}\left(\frac{b_{\text{psk}}}{{sin}^2\theta }\right){\mathcal{M}}_{{\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}}\left(\frac{b_{\text{psk}}}{{sin}^2\theta }\right)d\theta ,\end{array}$$

(37)

where the approximation is due to the use of ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$ 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]

$${\mathcal{M}}_{{\gamma }_{j,i}^{\text{BP}}}\left(s\right)=\frac{1}{1+s{\delta }_j^{\mathrm{B}}P{\sigma }_{j,i}^2/N_0}.$$

(38)

To determine the MGF of ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$, the cumulative distribution function (CDF) of ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$ is derived as

$$P_{{\widehat{\gamma }}_{j,i}^{\text{CP}}}\left(\gamma \right)=\text{Pr}\left[\underset{\begin{array}{c}k=1,2,\dots ,N\\ k\ne i,k\ne j\end{array}}{\text{max}}{\widehat{\gamma }}_{j,k,i}\le \gamma \right]=\prod _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N{P}_{{\widehat{\gamma }}_{j,k,i}}\left(\gamma \right),$$

(39)

where

$${\widehat{\gamma }}_{j,k,i}=\frac{P_{j,i}^{\mathrm{C}}{P}_j^{\mathrm{B}}\left|h_{j,k}{|}^2\right|h_{k,i}{|}^2}{N_0{\varrho }_{N-1}\left(P_j^{\mathrm{B}}|h_{j,k}{|}^2+P_{j,i}^{\mathrm{C}}|h_{k,i}{|}^2\right)},$$

(40)

and $P_{{\widehat{\gamma }}_{j,k,i}}\left(\gamma \right)=1-2\gamma \sqrt{{\widehat{\lambda }}_{j,k}^{\mathrm{B}}{\widehat{\lambda }}_{j,k,i}^{\mathrm{C}}}{e}^{-\gamma \left({\widehat{\lambda }}_{j,k}^{\mathrm{B}}+{\widehat{\lambda }}_{j,k,i}^{\mathrm{C}}\right)}{K}_1\left(2\gamma \sqrt{{\widehat{\lambda }}_{j,k}^{\mathrm{B}}{\widehat{\lambda }}_{j,k,i}^{\mathrm{C}}}\right)$ with ${\widehat{\lambda }}_{j,k}^{\mathrm{B}}$ and ${\widehat{\lambda }}_{j,k,i}^{\mathrm{C}}$ being defined as ${\widehat{\lambda }}_{j,k}^{\mathrm{B}}=\frac{N_0{\varrho }_{N-1}}{{\delta }_j^{\mathrm{B}}P{\sigma }_{j,k}^2}$ and ${\widehat{\lambda }}_{j,k,i}^{\mathrm{C}}=\frac{N_0{\varrho }_{N-1}}{{\delta }_{j,i}^{\mathrm{C}}\left(1-{\delta }_j^{\mathrm{B}}\right)P{\sigma }_{k,i}^2}$, respectively. Also, K₁(·) is the first-order modified Bessel function of the second kind [23]. At high SNR, K₁(·) can be approximated for small x as K₁(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 ${\widehat{\lambda }}_{j,k,i}^{\text{BC}}\triangleq {\widehat{\lambda }}_{j,k}^{\mathrm{B}}+{\widehat{\lambda }}_{j,k,i}^{\mathrm{C}}=\frac{N_0{\varrho }_{N-1}}{P}{\mathrm{\Phi }}_{j,k,i}$, where

$$\begin{array}{l}{\mathrm{\Phi }}_{j,k,i}=\frac{{\delta }_j^{\mathrm{B}}{\sigma }_{j,k}^2+{\delta }_{j,i}^{\mathrm{C}}\left(1-{\delta }_j^{\mathrm{B}}\right){\sigma }_{k,i}^2}{{\delta }_{j,i}^{\mathrm{C}}\left(1-{\delta }_j^{\mathrm{B}}\right){\delta }_j^{\mathrm{B}}{\sigma }_{j,k}^2{\sigma }_{k,i}^2}.\end{array}$$

(41)

Therefore, the PDF of ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$ can be obtained as

$$\begin{array}{l}{p}_{{\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}}\left(\gamma \right)=\sum _{\begin{array}{c}m=1\\ m\ne i,m\ne j\end{array}}^N{p}_{{\widehat{\gamma }}_{j,m,i}}\left(\gamma \right)\prod _{\begin{array}{c}k=1,k\ne m\\ k\ne i,k\ne j\end{array}}^N\left(1-e^{-\gamma {\widehat{\lambda }}_{j,k,i}^{\text{BC}}}\right),\end{array}$$

(42)

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 ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$ is quite difficult [20]; however, a useful relationship between the CDF of a random variable X and its MGF exists and is given by ${\mathcal{M}}_X\left(s\right)=s\mathcal{ℒ}\left\{P_X\right(x\left)\right\}$, with $\mathcal{ℒ}\{·\}$ 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 ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$ can be shown to be

$$\begin{array}{ll}{\mathcal{M}}_{{\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}}\left(s\right)& =1-\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N\frac{s}{s+{\widehat{\lambda }}_{j,k,i}^{\text{BC}}}+\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N\sum _{\begin{array}{c}m=1,m\ne k\\ m\ne i,m\ne j\end{array}}^N\\ \times \frac{s}{s+{\widehat{\lambda }}_{j,k,i}^{\text{BC}}+{\widehat{\lambda }}_{j,m,i}^{\text{BC}}}-\cdots +\frac{{(-1)}^{N}s}{s+\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N{\widehat{\lambda }}_{j,k,i}^{\text{BC}}}.\end{array}$$

(43)

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

$$\begin{array}{ll}{\tilde{P}}_{\text{SER}}^{\text{M2M-STNC-ONS}}& \approx \frac{1}{\pi }{\int }_0^{(M-1)\pi /M}\frac{1}{\left(1+\frac{b_{\text{psk}}{\delta }_j^{\mathrm{B}}P{\sigma }_{j,i}^2}{\stackrel{2}{sin}\left(\theta \right)N_0}\right)}\\ \times \left(1-\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N\frac{\frac{{\mathit{\text{Pb}}}_{\text{PSK}}}{\stackrel{2}{sin}\left(\theta \right)N_0{\varrho }_{N-1}}}{\frac{{\mathit{\text{Pb}}}_{\text{PSK}}}{\stackrel{2}{sin}\left(\theta \right)N_0{\varrho }_{N-1}}+{\mathrm{\Phi }}_{j,k,i}}+\cdots \right.\\ +\left.\frac{{(-1)}^N\frac{{\mathit{\text{Pb}}}_{\text{PSK}}}{\stackrel{2}{sin}\left(\theta \right)N_0{\varrho }_{N-1}}}{\frac{{\mathit{\text{Pb}}}_{\text{PSK}}}{\stackrel{2}{sin}\left(\theta \right)N_0{\varrho }_{N-1}}+\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N{\mathrm{\Phi }}_{j,k,i}}\right)d\theta .\end{array}$$

(44)

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]

$$\begin{array}{l}{\mathcal{M}}_{{\gamma }_{j,i}^{\text{BP}}}\left(s\right)\lesssim \frac{N_0}{s{\delta }_j^{\mathrm{B}}P{\sigma }_{j,i}^2}.\end{array}$$

(45)

Second, an asymptotic upper bound for ${\mathcal{M}}_{{\bar{\gamma }}_{j,i}^{\text{CP}}}\left(s\right)$ at high SNR can be determined by approximating e^x ≃ (1 + x) when x → 0 in the PDF of ${\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}$ defined in (31), which is now given by

$$p_{{\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}}\left(\gamma \right)\approx \sum _{\begin{array}{c}m=1\\ m\ne i,m\ne j\end{array}}^N{\lambda }_{j,m,i}^{\text{BC}}\left(1-\gamma {\lambda }_{j,m,i}^{\text{BC}}\right){\gamma }^{N-3}\prod _{\begin{array}{c}k=1,k\ne m\\ k\ne i,k\ne j\end{array}}^N{\lambda }_{j,k,i}^{\text{BC}}.$$

(46)

Since ${\lambda }_{j,k,i}^{\text{BC}}=\frac{N_0{\varrho }_{N-1}}{P}{\mathrm{\Phi }}_{j,k,i}$, so by substituting (46) into (36), it can be shown that

$${\mathcal{M}}_{{\bar{\widehat{\gamma }}}_{j,i}^{\text{CP}}}\left(s\right)\lesssim (N-2)!{\left(\frac{N_0{r}_{N-1}}{\mathit{\text{sP}}}\right)}^{N-2}\prod _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N{\mathrm{\Phi }}_{j,k,i}.$$

(47)

Finally, by substituting (45) and (47) into (37), the asymptotic upper-bound SER expression is obtained as

$$\begin{array}{l}\begin{array}{ll}{\bar{P}}_{\text{UB-SER}}^{\text{M2M-STNC-ONS}}& \lesssim {\left(\frac{N_0}{P}\right)}^{N-1}\frac{(N-2)!\Theta (N-1){\left({\varrho }_{N-1}\right)}^{N-2}}{{\delta }_j^{\mathrm{B}}{\sigma }_{j,i}^2{b}_{\text{PSK}}^{N-1}}\\ \times \prod _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N{\mathrm{\Phi }}_{j,k,i},\end{array}\end{array}$$

(48)

with Θ(N - 1) being defined as $\Theta (N-1)=\frac{1}{\pi }{\int }_0^{(M-1)\pi /M}$ (sin2(θ))^N-1dθ.

4.2.2 Diversity order analysis

The diversity order is given by $\mathbf{\Gamma }=-{lim}_{\text{SNR}\to \infty }$$log\left({\bar{P}}_{\text{UB-SER}}^{\text{M2M-STNC-ONS}}\right)/log\left(\text{SNR}\right)$, where SNR = P / N₀[2]. Clearly, the M2M-STNC-ONS scheme achieves a full diversity order of Γ = N - 1 per node.

It is noteworthy that the concept of many-to-many space-time 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 TDMA-based cooperative communications (i.e., without network coding and multiple-access transmissions), a total of N² 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 ‘in-phase’ 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 M2M-STNC and M2M-STNC-ONS schemes due to the timing offsets between the nodes in the cooperation phase.

5.1 Signal model under M2M-STNC 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 higher-order 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]

$${\mathcal{Y}}_i\left(t\right)=\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\alpha }_{m,i}\sum _{l=-1}^1{x}_m\left(n\right)c_m\left(t-l{T}_s-{\tau }_{i,m}\right)+{\bar{w}}_i\left(t\right),$$

(49)

where α_m,i is defined in (9), and ${\bar{w}}_i\left(t\right)$ is written as

$$\begin{array}{ll}{\bar{w}}_i\left(t\right)& =w_i\left(t\right)+\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne m\end{array}}^N{\beta }_{m,k,i}{h}_{k,i}\\ \times \sum _{l=-1}^1{n}_{m,k}\left(t-l{T}_s\right)c_m\left(t-l{T}_s-{\tau }_{i,m}\right).\end{array}$$

(50)

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,i-1 ≤ 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]

$${\mathcal{Y}}_{j,i}\left(n\right)=〈{\mathcal{Y}}_i\left(t\right),c_j\left(t\right)〉={\int }_{l{T}_s+{\Delta }_i}^{l{T}_s+T_s+{\Delta }_i}{\mathcal{Y}}_i\left(t\right)c_j^{\ast }(t-l{T}_s-{\Delta }_i)\mathit{\text{dt}},$$

(51)

with c _j (t) being zero outside the duration of T _s time units. Define the (N - 1) × (N - 1) cross-correlation matrix R _i (l) whose entries are modeled for l = 1, l = 0, and l = -1 as [28]

$$\begin{array}{l}\begin{array}{l}{\tilde{\rho }}_{m,j}^{\left(1\right)}\\ =\left\{\begin{array}{ll}\frac{1}{T_s}{\int }_{{\Delta }_i}^{{\tau }_{i,m}}{c}_m(t-{\tau }_{i,m})c_j^{\ast }(t-{\Delta }_i)\mathit{\text{dt}}={\rho }_{m,j}\left(\frac{{\tau }_{i,m}-{\Delta }_i}{T_s}\right),& {\tau }_{i,m}>{\Delta }_i\\ 0,& \text{otherwise}\end{array}\right.,\end{array}\end{array}$$

(52)

$$\begin{array}{ll}{\bar{\rho }}_{m,j}^{\left(0\right)}& =\frac{1}{T_s}{\int }_{\text{max}({\tau }_{i,m},{\Delta }_i)}^{min({\tau }_{i,m},{\Delta }_i)+T_s}{c}_m(t-{\tau }_{i,m})c_j^{\ast }(t-{\Delta }_i)\mathit{\text{dt}}\\ =\left\{\begin{array}{ll}{\rho }_{m,j}\left(1-\frac{{\Delta }_i-{\tau }_{i,m}}{T_s}\right),& {\Delta }_i\ge {\tau }_{i,m}\\ {\rho }_{m,j}\left(1-\frac{{\tau }_{i,m}-{\Delta }_i}{T_s}\right),& {\Delta }_i<{\tau }_{i,m}\end{array}\right.,\end{array}$$

(53)

and

$$\begin{array}{l}{\tilde{\rho }}_{m,j}^{(-1)}\\ =\left\{\begin{array}{ll}\frac{1}{T_s}{\int }_{{\tau }_{i,m}}^{{\Delta }_i}{c}_m(t-{\tau }_{i,m})c_j^{\ast }(t-{\Delta }_i)\mathit{\text{dt}}={\rho }_{m,j}\left(\frac{{\Delta }_i-{\tau }_{i,m}}{T_s}\right),& {\tau }_{i,m}<{\Delta }_i\\ 0,& \text{otherwise}\end{array}\right.,\end{array}$$

(54)

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 time-shift value. Intuitively, the smaller are the time shifts, the less severe are the timing synchronization errors. Now, let $R_i\left(0\right)={\bar{R}}_i$ be defined as

$${\bar{R}}_i=\left[\begin{array}{llllll}1& \cdots & {\bar{\rho }}_{1,(i-1)}^{\left(0\right)}& {\bar{\rho }}_{1,(i+1)}^{\left(0\right)}& \cdots & {\bar{\rho }}_{1,N}^{\left(0\right)}\\ ⋮& \ddots & ⋮& ⋮& \cdots & ⋮\\ {\bar{\rho }}_{(i-1),1}^{\left(0\right)}& \cdots & 1& {\bar{\rho }}_{(i-1),(i+1)}^{\left(0\right)}& \cdots & {\bar{\rho }}_{(i-1),N}^{\left(0\right)}\\ {\bar{\rho }}_{(i+1),1}^{\left(0\right)}& \cdots & {\bar{\rho }}_{(i+1),(i-1)}^{\left(0\right)}& 1& \cdots & {\bar{\rho }}_{(i+1),N}^{\left(0\right)}\\ ⋮& \cdots & ⋮& ⋮& \ddots & ⋮\\ {\bar{\rho }}_{N,1}^{\left(0\right)}& \cdots & {\bar{\rho }}_{N,(i-1)}^{\left(0\right)}& {\bar{\rho }}_{N,(i+1)}^{\left(0\right)}& \cdots & 1\end{array}\right],$$

(55)

and $R_i\left(1\right)=R_i^T(-1)={\tilde{R}}_i$, where ${\tilde{R}}_i$ is defined as follows [29]

$$\begin{array}{l}{\tilde{R}}_i=\left[\begin{array}{llllll}0& \cdots & {\tilde{\rho }}_{1,(i-1)}^{\left(1\right)}& {\tilde{\rho }}_{1,(i+1)}^{\left(1\right)}& \cdots & {\tilde{\rho }}_{1,N}^{\left(1\right)}\\ ⋮& \ddots & ⋮& ⋮& \cdots & ⋮\\ 0& \cdots & 0& {\tilde{\rho }}_{(i-1),(i+1)}^{\left(1\right)}& \cdots & {\tilde{\rho }}_{(i-1),N}^{\left(1\right)}\\ 0& \cdots & 0& 0& \cdots & {\tilde{\rho }}_{(i+1),N}^{\left(1\right)}\\ ⋮& \cdots & ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & 0& 0& \cdots & 0\end{array}\right].\end{array}$$

(56)

Thus, the output of the matched filter bank can be expressed as [27, 29]

$${\mathcal{Y}}_i\left(l\right)={\tilde{R}}_i^T{A}_i{x}_i(l+1)+{\bar{R}}_i{A}_i{x}_i\left(l\right)+{\tilde{R}}_i{A}_i{x}_i(l-1)+{\bar{w}}_i\left(l\right),$$

(57)

where matrix A _i is defined in (13), whereas x _i (l + ς) is defined in general as

$$\begin{array}{ll}{x}_i(l+\varsigma )& =\left[x_1(l+\varsigma ),\dots ,x_{i-1}(l+\varsigma ),\right.\\ {\left.x_{i+1}(l+\varsigma ),\dots ,x_N(l+\varsigma )\right]}^T\end{array}$$

(58)

for ς ∈ {-1, 0, 1}. Also, ${\bar{w}}_i\left(l\right)$ is the noise vector with variance given by

$$\begin{array}{l}E\left[{\bar{w}}_i\left(l\right){\bar{w}}_i^H(l+\varsigma )\right]\\ =\left\{\begin{array}{ll}{N}_0\left((\mathbf{I}+G_i){\bar{R}}_i^T+G_i{\tilde{R}}_i\right),& \text{if}\varsigma =1\\ N_0\left((\mathbf{I}+G_i){\bar{R}}_i+G_i{\tilde{R}}_i^T+G_i{\tilde{R}}_i\right),& \text{if}\varsigma =0\\ N_0\left((\mathbf{I}+G_i){\bar{R}}_i+G_i{\tilde{R}}_i^T\right),& \text{if}\varsigma =-1\\ 0,& \text{otherwise}\end{array}\right.,\end{array}$$

(59)

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 ${\bar{\mathcal{Y}}}_i\left(l\right)=R_i^{-1}{\mathcal{Y}}_i\left(l\right)={Ŕ}_i{A}_i{x}_i(l+1)+{\overline{\stackrel{̄}{R}}}_i{A}_i{x}_i\left(l\right)+{\grave{R}}_i{A}_i{x}_i(l-1)+{\overline{\stackrel{̄}{w}}}_i\left(l\right)$, where R _i is as defined in (12) with off-diagonal elements equal to ρ, ${\overline{\stackrel{̄}{R}}}_i=R_i^{-1}{\bar{R}}_i$, ${Ŕ}_i=R_i^{-1}{\tilde{R}}_i^T$, ${\grave{R}}_i=R_i^{-1}{\tilde{R}}_i$, and ${\overline{\stackrel{̄}{w}}}_i\left(l\right)=R_i^{-1}{\bar{w}}_i\left(l\right)$ with

$$\begin{array}{l}\begin{array}{l}E\left[{\overline{\stackrel{̄}{w}}}_i\left(l\right){\overline{\stackrel{̄}{w}}}_i^H(l+\varsigma )\right]\\ =\left\{\begin{array}{ll}{N}_0{R}_i^{-1}\left((\mathbf{I}+G_i){\bar{R}}_i^T+G_i{\tilde{R}}_i\right)R_i^{-T},& \text{if}\varsigma =1\\ N_0{R}_i^{-1}\left((\mathbf{I}+G_i){\bar{R}}_i+G_i{\tilde{R}}_i^T+G_i{\tilde{R}}_i\right)R_i^{-T},& \text{if}\varsigma =0\\ N_0{R}_i^{-1}\left((\mathbf{I}+G_i){\bar{R}}_i+G_i{\tilde{R}}_i^T\right)R_i^{-T},& \text{if}\varsigma =-1\\ 0,& \text{otherwise}\end{array}\right..\end{array}\end{array}$$

(60)

$${\bar{\mathcal{Y}}}_{j,i}\left(l\right)=\underset{\text{Desired symbol}}{\underbrace{{\alpha }_{j,i}{\overline{\stackrel{̄}{\rho }}}_{j,j}{x}_j\left(l\right)}}+\underset{\text{ISI symbols}}{\underbrace{\sum _{m=1,m\ne i}^N{\alpha }_{m,i}\left({\acute{\rho }}_{j,m}{x}_m(l+1)+{\acute{\rho }}_{j,m}{x}_m(l-1)\right)+\sum _{m=1,m\ne i,j}^N{\alpha }_{m,i}{\overline{\stackrel{̄}{\rho }}}_{j,m}{x}_m\left(l\right)}}+{\overline{\stackrel{̄}{w}}}_{j,i}\left(l\right),$$

(61)

The decorrelated received signal ${\bar{\mathcal{Y}}}_{j,i}\left(l\right)$ at the output of the j th MFB branch is given by

where ${\overline{\stackrel{̄}{\rho }}}_{j,j}$, is the j th diagonal element of matrix ${\overline{\stackrel{̄}{R}}}_i$, while ${\acute{\rho }}_{j,m}$ and ${\grave{\rho }}_{j,m}$ are the (j,m)th element of matrices ${Ŕ}_i$ and ${\grave{R}}_i$, respectively. Additionally, ${\overline{\stackrel{̄}{w}}}_{j,i}\left(l\right)\sim \mathcal{C}\mathcal{N}(0,{\overline{\stackrel{̄}{\varrho }}}_{j,i}{N}_0)$, where ${\overline{\stackrel{̄}{\varrho }}}_{j,i}$ is the j th diagonal element of matrix $R_i^{-1}\left((\mathbf{I}+G_i){\bar{R}}_i+G_i{\tilde{R}}_i^T+G_i{\tilde{R}}_i\right)R_i^{-T}$. Based on the above analysis, the instantaneous conditional signal-to-interference-plus-noise 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

$$\begin{array}{ll}{\gamma }_{j,i}& ={\gamma }_{j,i}^{\text{BP}}+{\widehat{\gamma }}_{j,i}^{\text{CP}}=\frac{P_j^{\mathrm{B}}|h_{j,i}{|}^2}{N_0}\\ +\frac{P_j^{\mathrm{B}}\left(\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne j\end{array}}^N\frac{P_{j,k,i}^{\mathrm{C}}\left|h_{j,k}{|}^2\right|h_{k,i}{|}^2}{P_j^{\mathrm{B}}|h_{j,k}{|}^2+N_0}\right){\overline{\stackrel{̄}{\rho }}}_{j,j}^2}{{\mathcal{I}}_{j,i}+{\overline{\stackrel{̄}{\varrho }}}_{j,i}{N}_0},\end{array}$$

(62)

where ${\mathcal{I}}_{j,i}$ is the ISI variance as defined by

$$\begin{array}{ll}{\mathcal{I}}_{j,i}& =\sum _{m=1,m\ne i}^N{P}_m^{\mathrm{B}}\left(\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne m\end{array}}^N\frac{P_{m,k,i}^{\mathrm{C}}\left|h_{m,k}{|}^2\right|h_{k,i}{|}^2}{P_m^{\mathrm{B}}|h_{m,k}{|}^2+N_0}\right)\left({\acute{\rho }}_{j,m}^2+{\grave{\rho }}_{j,m}^2\right)\\ +\sum _{m=1,m\ne i,j}^N{P}_m^{\mathrm{B}}\left(\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne m\end{array}}^N\frac{P_{m,k,i}^{\mathrm{C}}\left|h_{m,k}{|}^2\right|h_{k,i}{|}^2}{P_m^{\mathrm{B}}|h_{m,k}{|}^2+N_0}\right){\overline{\stackrel{̄}{\rho }}}_{j,m}^2,\end{array}$$

(63)

and it is assumed that the data symbols are statistically independent. Based on (62), finding a closed form solution for the SER for M-PSK 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 ${\bar{\rho }}_{j,i}^{\left(0\right)}={\rho }_{j,i}$ and also ${\tilde{\rho }}_{j,i}^{(-1)}={\tilde{\rho }}_{j,i}^{\left(1\right)}=0$ 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 M2M-STNC-ONS scheme

From (23), the received signal at node S _i is given by

$${\widehat{\mathcal{Y}}}_i\left(t\right)=\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\widehat{\alpha }}_{m,i}\sum _{l=-1}^1{x}_m\left(l\right)c_m\left(t-l{T}_s-{\tau }_{i,m}\right)+{\bar{w}}_i\left(t\right),$$

(64)

where ${\widehat{\alpha }}_{m,i}$ is defined in (24) and ${ŵ}_i\left(t\right)$ is written as

$$\begin{array}{ll}{ŵ}_i\left(t\right)& =w_i\left(t\right)+\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\beta }_{m,\text{opt},i}{ĥ}_{m,\text{opt},i}\\ \times \sum _{l=-1}^1{n}_{m,\text{opt},i}\left(t-l{T}_s\right)c_m\left(t-l{T}_s-{\tau }_{i,m}\right).\end{array}$$

(65)

Following the analysis of the M2M-STNC scheme and replacing matrices A _i and G _i with ${Â}_i$ and ${Ĝ}_i$, respectively (see (26) and (27)), the instantaneous conditional SINR of symbol x _j at node S _i can be shown to be

$${\widehat{\gamma }}_{j,i}=\frac{P_j^{\mathrm{B}}|h_{j,i}{|}^2}{N_0}+\frac{P_j^{\mathrm{B}}\left(\frac{P_{j,i}^{\mathrm{C}}\left|{ĥ}_{j,\text{opt},i}{|}^2\right|{\bar{h}}_{j,\text{opt},i}{|}^2}{P_j^{\mathrm{B}}|{\bar{h}}_{j,\text{opt},i}{|}^2+N_0}\right){\overline{\stackrel{̄}{\rho }}}_{j,j}^2}{{Î}_{j,i}+{\widehat{\widehat{\varrho }}}_{j,i}{N}_0},$$

(66)

where ${\widehat{\widehat{\varrho }}}_{j,i}$ is the j th diagonal element of matrix $R_i^{-1}\left(\left(\mathbf{I}+{Ĝ}_i\right){\bar{R}}_i+{Ĝ}_i{\tilde{R}}_i^T+{Ĝ}_i{\tilde{R}}_i\right)R_i^{-T}$, and

$$\begin{array}{ll}{Î}_{j,i}& =\sum _{m=1,m\ne i}^N{P}_m^{\mathrm{B}}\left(\frac{P_{m,i}^{\mathrm{C}}\left|{ĥ}_{m,\text{opt},i}{|}^2\right|{\bar{h}}_{m,\text{opt},i}{|}^2}{P_m^{\mathrm{B}}|{\bar{h}}_{m,\text{opt},i}{|}^2+N_0}\right)\left({\acute{\rho }}_{j,m}^2+{\grave{\rho }}_{j,m}^2\right)\\ +\sum _{m=1,m\ne i,j}^N{P}_m^{\mathrm{B}}\left(\frac{P_{m,i}^{\mathrm{C}}\left|{ĥ}_{m,\text{opt},i}{|}^2\right|{\bar{h}}_{m,\text{opt},i}{|}^2}{P_m^{\mathrm{B}}|{\bar{h}}_{m,\text{opt},i}{|}^2+N_0}\right){\overline{\stackrel{̄}{\rho }}}_{j,m}^2\mathrm{.s}\end{array}$$

(67)

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 M2M-STNC and M2M-STNC-ONS schemes, assuming perfect timing synchronization, is studied and characterized.

6.1 M2M-STNC

In the broadcasting phase, the received signal at node S _i from node S _j with channel estimation error is expressed as

$$y_{j,i}^{\epsilon }\left(t\right)=\sqrt{P_j^{\mathrm{B}}}(h_{j,i}+h_{j,i}^{\epsilon })x_j{c}_j\left(t\right)+n_{j,i}\left(t\right),$$

(68)

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 zero-mean complex Gaussian random variable with variance ε_j,i. Thus, the additional self-noise term $\sqrt{P_j^{\mathrm{B}}}{h}_{j,i}^{\epsilon }{x}_j$ is a zero-mean complex Gaussian random variable with variance ε_j,iP j B. Equation (68) is re-written as

$$y_{j,i}^{\epsilon }\left(t\right)=\sqrt{P_j^{\mathrm{B}}}{h}_{j,i}{x}_j{c}_j\left(t\right)+n_{j,i}^{\epsilon }\left(t\right)$$

(69)

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 zero-mean Gaussian random variable with variance ε_j,iP j B + N₀. Thus, the SNR after matched filtering is given by

$${\gamma }_{j,i}^{\text{BP}}=\frac{P_j^{\mathrm{B}}|h_{j,i}{|}^2}{{\epsilon }_{j,i}{P}_j^{\mathrm{B}}+N_0}.$$

(70)

In the cooperation phase, the received signal at node S _i is given by

$${\mathcal{Y}}_i^{\epsilon }\left(t\right)=\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N{\alpha }_{m,i}^{\epsilon }{x}_m{c}_m\left(t\right)+{\bar{w}}_i^{\epsilon }\left(t\right),$$

(71)

where ${\alpha }_{m,i}^{\epsilon }$ is defined as

$${\alpha }_{m,i}^{\epsilon }=\sqrt{P_m^{\mathrm{B}}}\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne m\end{array}}^N{\beta }_{m,k,i}^{\epsilon }{h}_{m,k}{h}_{k,i},$$

(72)

and

$${\beta }_{m,k,i}^{\epsilon }=\sqrt{\frac{P_{m,k,i}^{\mathrm{C}}}{P_m^{\mathrm{B}}\left(|h_{m,k}{|}^2+{\epsilon }_{m,k}\right)+N_0}}.$$

(73)

As before, w _i (t) is the zero-mean N₀-variance AWGN sample at node S _i and ${\bar{w}}_i^{\epsilon }\left(t\right)$ is the equivalent noise term, expressed as

$${\bar{w}}_i^{\epsilon }\left(t\right)=w_i\left(t\right)+\sum _{\begin{array}{c}m=1,m\ne i\end{array}}^N\sum _{\begin{array}{c}k=1\\ k\ne i,k\ne m\end{array}}^N{\beta }_{m,k,i}^{\epsilon }\left(\left(h_{k,i}+h_{k,i}^{\epsilon }\right)\right)$$