Manytomany spacetime network coding for amplifyandforward cooperative networks: node selection and performance analysis
 Mohammed W Baidas^{1}Email author and
 Allen B MacKenzie^{2}
https://doi.org/10.1186/16871499201448
© Baidas and MacKenzie; licensee Springer. 2014
Received: 8 November 2013
Accepted: 12 March 2014
Published: 26 March 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.
Keywords
Amplifyandforward Channel estimation errors Network coding Node selection Power allocation Timing synchronization1 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.
2.1 Broadcasting phase
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
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
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
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, ${\mathcal{I}}_{m,k,i}=0$.
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 $P={P}_{m}^{\mathrm{B}}+\sum _{i=1,i\ne m}^{N}{P}_{m,\text{opt},i}^{\mathrm{C}}$.
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
where γ j, i BP + γ j, i CP = γ_{j,i}, as defined in (17).
4.2 M2MSTNCONS
4.2.1 Asymptotic upper bound
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
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
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
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
Based on (70) and (77), it is clear that channel estimation errors increase the noise variance, which in turn reduces the resulting SNR at the output of the matched filter. Thus, with channel estimation errors, increasing the broadcasting and/or transmit power cannot indefinitely increase the SNR. Additionally, in the case of perfect CSI (i.e., ε_{j,i} = 0, ∀j, i ∈ {1, 2, …, N}, and j ≠ i), then the SNR expressions in (70) and (77) reduce to (4) and (18), respectively.