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VirtualMIMO feedback and transmission for wireless ad hoc networks
EURASIP Journal on Wireless Communications and Networking volume 2016, Article number: 261 (2016)
Abstract
Ina wireless ad hoc network, singleantenna nodes can cooperate in their transmissions to create a virtual multipleinput multipleoutput (MIMO) channel for maximizing the spatial multiplexing gain. However, implementing virtualMIMO transmission can result in overwhelming overhead, such as channel state information (CSI) feedback and local data exchange, as the number of cooperating nodes increases. To suppress this overhead, we apply the principle of virtualMIMO transmission to CSI feedback and design a novel framework for supporting parallel feedback streams, called virtualMIMO feedback. Building on this feedback framework, we optimize virtualMIMO transmission techniques and quantify their throughput gains in the presence of CSI and localexchange overhead. To be specific, the outage capacities of these virtualMIMO techniques are analyzed, and the corresponding bandwidth allocation for local data exchange and longrange transmission is optimized in a way to maximize overall spectral efficiency. Finally, the improvement of spectral efficiency in proposed virtualMIMO feedback and transmission is demonstrated by the simulation results.
Introduction
The multipleinput multipleoutput (MIMO) technology was a breakthrough in wireless communication as it improved the capacity of wireless system without increasing bandwidth and transmission power [1]. In singleuser MIMO system, the spatial degrees of freedom (DoF) have been utilized to attain diversity and array gains by either maximum ratio combining (MRC) or spacetime codes [2]. Moreover, MIMO techniques have been developed to achieve the full multiplexing gain in multiuser system by using space division multiple access (SDMA) schemes [3]. In practice, linear precoding schemes such as zeroforcing (ZF) and block diagonalization have been studied in imperfect channel state information (CSI) at transmitters and their matching scheduling algorithms achieving multiuser diversity gain have been proposed in [4–7]. Also, various existing studies in MIMO systems have been considered where a transmitter acquires CSI from receivers through the finite rate feedback by quantizing the estimated channel vectors; these studies show that the required number of feedback bits for each channel vector should be scaled with the number of antennas and transmit power to compensate for the rate loss due to limited feedback [4].
In modern cellular systems such as LTEadvanced, cooperative MIMO transmission schemes have been proposed to mitigate the othercell interference in cellular networks. By allowing that the neighboring base stations (BSs) share both user data and CSI via highcapacity backhaul links, cooperative MIMO creates a multiuser MIMO system with one BS equipped with a large number of antennas, called network MIMO or coordinated multipoint transmission and reception (CoMP) in Third Generation Partnership Project (3GPP) standardization. In network MIMO, a joint transmission between BSs is performed to send the desired data to the corresponding users, which provides a considerable performance gain in multicell interference channels [8, 9]. However, the network MIMO requires a huge amount of CSI feedback for all direct and interfering links as well as the desired data exchange through the backhaul links between BSs, which makes practical implementation difficult.
The cooperative MIMO system in a cellular network has been explored in the context of collaboration between singleantenna nodes in order to exploit spatial DoF in wireless ad hoc networks, called virtualMIMO (VMIMO). In [10, 11], a singlesource node broadcasts its desired signal to neighboring nodes in the same cluster with which they form the spacetime block coding (STBC) towards its corresponding destination node, which achieves the diversity gain in the distributed network. Similar studies in [12, 13] show that the cooperative beamforming provides the power gain at the desired receiver as well as interference avoidance at the unintended receivers. Recently, the efficient clustering algorithm with neighboring nodes for constructing VMIMO and the design of cooperative precoding that maximizes data rate are proposed in [14, 15]. However, these existing studies of VMIMO are limited to singlesource to singledestination link and employ the localexchange overhead that linearly increases with the number of cooperative nodes [10–15].
In an effort to improve the spectral efficiency in VMIMO systems, the optimal channel and power allocation between local and longrange transmission between transmitter and receiver clusters was proposed to minimize the outage capacity in [16]. In [17, 18], algorithms for efficient local data exchange were proposed, where multiple sources broadcast their desired signals over the same spectrum with simple signaling and perform the corresponding beamforming that maximizes the system performance. However, previous literature showed that CSI feedback of all channel links between receiver cluster and transmitter cluster are required to realize the cooperative beamforming, which significantly degrades the spectral efficiency as the number of nodes participating in the cooperation increases.
To overcome a huge amount of feedback overhead in the cooperative transmission scheme, this paper targets the implementation of a VMIMO system that achieves the full multiplexing gain for a given number of existing nodes and proposes a VMIMO feedback that supports multistream transmission to increase the accuracy of CSI at transmitters; this has not been addressed yet in the previous literature to the best of authors’ knowledge. Moreover, the optimal resource allocation for cooperative transmission is analyzed to maximize the spectral efficiency with a given localexchange overhead and the feedback channel resources. The contributions of this paper are summarized as follows.

VMIMO technique is proposed for CSI feedback, where the receivers simultaneously transmit their CSIs to the associated transmitters, which are cooperatively decoded at the transmitter cluster. The proposed feedback scheme provides a significant increase of total feedback bits by exploiting the spatial multiplexing gain over the orthogonal feedback channel access of traditional MIMO system which separately allocates the feedback channeluse to each receiver [4, 5]. Based on VMIMO feedback, we analyze the optimal resource partitioning between VMIMO feedback and data transmission channels that maximizes the spectral efficiency of cooperative ZF beamforming in a distributed ad hoc network. As the proposed VMIMO feedback method improves feedback channel capacity, it provides the increase of achievable MIMO transmission rate under optimal resource partitioning.

In the presence of CSI feedback and local data exchange overhead, we analyze the outage capacity of VMIMO transmission due to the discrepancy between localexchange and longrange data transmission channel quality. To meet the outage capacity, the optimal number of VMIMO nodes that maximizes the spectral efficiency is derived as a function of the distance between nodes, transmit power, and target data rate between transmitter and the corresponding receiver.
The remainder of this paper is organized as follows. The system model is introduced in Section 2, and the cooperative CSI feedback using VMIMO is described in Section 3. In Section 4, the cooperative ZF transmission is proposed and the corresponding performance gain and the resource optimization between local data exchange and longrange transmission are analyzed in a wireless ad hoc network. Section 5 provides simulation results, and the concluding remarks are followed in Section 6.
Notation: Bold upper case and lower case letters denote matrices and vectors, respectively. (·)^{T}, (·)^{†}, and ∥·∥ represent the transpose, Hermitian transpose, and the Frobenius norm of a matrix, respectively. \(\Gamma (a, x)=\int _{x}^{\infty } t^{\alpha 1}e^{t}dt\) is the complementary incomplete gamma function. \(\mathbb {E} \{\cdot \}\) and Q(x) denote the expectation over the random channel realizations and Qfunction of x, respectively.
System model
In this paper, we consider a wireless ad hoc network comprising M transmitters communicating with M corresponding receivers, where each of all transmitters and receivers is equipped with a single antenna and the distance of the receivers from the corresponding transmitters is much larger than the distance between transmitters as illustrated in Fig. 1. For this scenario, we adopt the VMIMO scheme that transmitters cooperate to perform ZF joint precoding, called cooperative ZF precoding for supporting M data streams, without requiring cooperation between receivers that causes a significant complexity on the implementation. Under the frequencydivision multiplexing (FDM) system, let h _{ i } denote the M×1 forward channel vector from the ith transmitter to all receivers and g _{ i } denote the M×1 reverse channel vector from ith receiver to all transmitters. All channel coefficients are modeled as independent and identically distributed (i.i.d.) random variables following \({\mathcal {CN}(0,1)}\) corresponding to rich scattering, and the relevant channels are perfectly estimated at both transmitters and receivers. Note that the distances between transmitters are relatively small to ensure that the direct lineofsight (LOS) path shows stronger power contribution than the other non lineofsight (NLOS) path due to the scattering factors [19], where the local channels among transmitters are considered to be a dominant LoS path in a short propagation distance, denoted as d _{ l }. It follows that the capacity between transmitters is given as
where \(P_{l}=P_{T}\, d_{l}^{\alpha }, P_{T}\) is the transmission power at each transmitter, N _{ o } is noise spectral density, and α is the pathloss exponent. Rician channel is represented as \(h_{R}= \sqrt {K \over K+1 }h_{\text {LOS}} + \sqrt {1 \over K+1 }h_{\text {NLOS}}\), where h _{LOS} is the fixed channel with unit norm and h _{NLOS} follows \({\mathcal {CN}(0,1)}\) under the assumption of K≫1. Over the highspeed local links, transmitters firstly exchange their data by frequencydivision multiple access (FDMA) and then transmit the corresponding shared data through the cooperative ZF precoding after CSI feedback is completed. The joint precoder V=[v _{1},v _{2},⋯,v _{ M }] is equal to the matrix H ^{†}(H H ^{†})^{−1} with columns normalized to have unit norm where H=[h _{1},⋯,h _{ M }] [20]. Assuming perfect CSI for all transmitters, the throughput for the ith transmitterreceiver link is represented as
where \(P_{o}=P_{T}\, d_{o}^{\alpha }\) and d _{ o } is the distance between transmitters and receivers, satisfying \({{\textbf {h}_{i}^{T}}{\textbf {v}_{j}}}=0, i \ne j\).
However, there exists residual crosslink interference in limited CSI feedback channel and the corresponding throughput is denoted by
where \(\hat {\textbf {v}}_{i}\) is the normalized ith column vector of \(\hat {\textbf {H}}^{\dag }(\hat {\textbf {H}}\hat {\textbf {H}}^{\dag })^{1}\) given \(\hat {\textbf {H}}=[\hat {\textbf {h}}_{1},...,\hat {\textbf {h}}_ M]\) and \(\hat {\textbf {h}}_{i}\) is quantized by a Bbit random codebook that minimizes the chordal distance, for analytical simplicity^{1}. The residual interference is bounded as \({\mathbb E}\left [ \textbf {h}_{i} \hat {\textbf {v}}_{j} ^{2}\right ] \le 2^{{B\over M1}}\) [4, 5].
VirtualMIMO feedback
Note that total available bandwidth W is partitioned into W _{ f } for CSI feedback and W _{ d } for data transmission to implement VMIMO transmission, where W _{ f }+W _{ d }=W. In this section, we propose a novel method of realizing collaborative spatial CSI feedback channels, called VMIMO feedback. VMIMO feedback allows the system bandwidth to be more efficiently partitioned and utilized for data transmission and feedback as well.
VirtualMIMO feedback: algorithm and performance
To achieve the VMIMO feedback scheme as illustrated in Fig. 1, each of M receivers simultaneously transmits their estimated channel vectors {h _{1},...,h _{ M }} to the transmitters, where the M feedback streams are jointly detected by the cooperative ZF filter over whole feedback channel. To implement the cooperative ZF filter at transmitterside, the transmitters are required to exchange the received feedback signal and the CSI of the reverse channels G=[g _{1},...,g _{ M }], where g _{ i } is the channel from the ith receiver to all transmitters, which incurs relatively small overhead of channel exchange due to the highspeed intracluster links. Moreover, the reverse channel G can be estimated at the corresponding transmitters using uplink sounding signals in FDD. For VMIMO feedback, the average feedback bits of h _{ i } per unit time can be written as [21]
where \(\bar {P}_{o}=P_{R} \, {d_{o}}^{\alpha }, P_{R}\) is the transmission power at receiver cluster and \(({\bar{\mathbf{r}}_{\text{ZF}}^{i}})^{\dag }\) is the ith row vector of \({\bar {\textbf {R}}_{\text {ZF}}}= (\textbf {G}^{\dag }\textbf {G})^{1}\textbf {G}\). For simplicity, we denote that the expected achievable rate of feedback channel between the ith receiver and transmitters as \(R_{m}={\log _{2}(e)e^{1/{\bar {P}_{o}\over M}} { \Gamma \left (0,{M \over { {\bar P}_{o}} }\right)}}\).
As an alternative VMIMO feedback scheme, we consider MRC based orthogonal feedback scheme over each of M orthogonal feedback channel resource. By allowing local exchange of the received feedback signals and performing MRC at transmitters, the MRC filter to decode CSI of h _{ i } is determined as \(\left (\bar{\mathbf{r}}_{\text{MRC}}^{i}\right)^{\dag }=\textbf {g}_{i}^{\dag }/\textbf {g}_{i} \) and then the average feedback bits for h _{ i } are given by [21]
The MRCbased feedback obtains the array gain, but it shows low spectral efficiency due to the orthogonal channel division as M becomes large. The performance of VMIMO feedback compared with MRC feedback is characterized in the different value of SNR as follows.

Low SNR: As \(\bar P_{o} \rightarrow 0 \), the number of average feedback bits in MRC based feedback is approximated as \(B_{\text {MRC}}^{i}\approx {{W_{f} \bar {P}_{o}} \over M} {\mathbb {E}}\left [{({\bar {\textbf {r}}_{\text {MRC}}^{i}})^{\dag } \textbf {g}_{i} ^{2}}\right ] \log _{2}e ={W_{f} \bar P_{o}} \log _{2}e\), while that of ZF based VMIMO is determined by \(B_{\text{ZF}}^{i}\approx {{W_{f} \bar P_{o}} \over M} {\mathbb {E}}\left [{({\bar {\mathbf{r}}_{\text{ZF}}^{i}})^{\dag } \mathbf{g}_{i} ^{2}}\right ] \log _{2}e ={{W_{f} \bar{P}_{o}}\over M} \log_{2}e\). Therefore, the cooperative decoding for CSI feedback under the bandwidth partitioning provides a higher performance gain than the VMIMO feedback scheme in the low SNR regime, where \(B_{MRC}^{i} = MB_{ZF}^{i}\).

High SNR: As \(\bar P_{o} \) increases, the average feedback bit in MRC based feedback method in (4) is approximated by
$$\begin{array}{*{20}l} B_{\rm{MRC}}^{i} &\approx {{W_{f}}\over {M}} \cdot { {\mathbb E}\left [\log_{2}\left({\bar P_{o}} \left ({\bar{\textbf{r}}_{{\rm{MRC}}}^{i}})^{\dag} \textbf{g}_{i}\right^{2}\right) \right ]} \\ &= {{W_{f}} \over {M}}\cdot \left(\log_{2} \left(\bar P_{o} \right) + {\mathbb E}\left[ \log_{2} \chi_{2M}^{2} \right]\right). \\ \end{array} $$(5)Moreover, the average feedback bit in ZF based feedback method in (3) is approximated by
$$\begin{array}{*{20}l} B_{\text{ZF}}^{i} &\approx W_{f} \cdot { {\mathbb{E}}\left[\log_{2}\left({{\bar{P}_{o}}\over M} \left ({\bar{\textbf{r}}_{{\text{ZF}}}^{i}})^{\dag} \textbf{g}_{i}\right^{2}\right) \right]} \\ &={{W_{f}}}\cdot \left(\log_{2} \left({\bar{P}_{o}}\over {M} \right) + {\mathbb{E}}\left[ \log_{2} {\chi_{2}^{2}} \right]\right)\\ \end{array} $$(6)where the \(\left  ({\bar {\textbf {r}}_{{\text {MRC}}}^{i}})^{\dag } \textbf {g}_{i}\right ^{2}\) and \(\left  ({\bar {\textbf {r}}_{{\text {ZF}}}^{i}})^{\dag }\textbf {g}_{i}\right ^{2}\) follow the distribution of \(\chi _{2M}^{2}\) and \({\chi _{2}^{2}}\), respectively [22]. Therefore, we conclude that \(B_{\text {ZF}}^{i}= MB_{\text {MRC}}^{i}\) in the high SNR regime.
Throughout this paper, we consider ZFbased feedback scheme as the suggested VMIMO feedback scheme, since it provides better performance gain than that of MRC based feedback scheme over the operational threshold (i.e., relatively high SNR).
In a realistic channel model, the number of feedback bits of h _{ i } over instantaneous channel G is represented as
where \({\bar B}_{\text {ZF}}^{i}\) is continuously changed in given channel realization. However, adopting instantaneous \({\bar B}_{\text {ZF}}^{i}\)bit quantization at each receiver causes several practical implementation issues due to the additional control signals required for reporting the dynamic quantization level to receivers at every channel realization and multiple channel codebooks for each allocated quantization level. Therefore, the average feedback bits \(B_{\text {ZF}}^{i}\) is allocated for each channel \(\hat {\textbf {h}}_{i}\) quantization for considering the feasibility of implementation, which approaches almost the same result as instantaneous \({\bar {B}}_{\text {ZF}}^{i}\)bit allocation in VMIMO feedback in cases of large \(\bar {P}_{o}\). Note that \({\bar {B}}_{\text {ZF}}^{i}\) and \({B}_{\text {ZF}}^{i}\) are approximated as
As shown in (8), the second term is negligible as SNR becomes large so that \(B^{i}_{\text {ZF}}={\bar {B}}^{i}_{\text {ZF}}\). This approximation is also verified by numerical results in Section 5. In practice, \(\lfloor B_{\text {ZF}}^{i}\rfloor \) bits are applied for the number of quantized VMIMO feedbacks since the feedbacks bits should be an integer to design a practical channel codebook, where ⌊·⌋ denotes the floor function.
VirtualMIMO feedback: bandwidth partitioning
VMIMO is performed by the following sequential procedures: (1) local data exchange between neighboring transmitters, (2) CSI feedback from the receiver to transmitter, and (3) cooperative transmission from the transmitter cluster to the corresponding receivers. For optimal bandwidth allocation for VMIMO, it is required for considering both bandwidth for local exchange and data transmission/CSI feedback. However, the corresponding optimization requires the super centralunit having all channel information over the network to control bandwidth allocation, which are impractical in the wireless ad hoc environment. Therefore, we investigate the optimal partition of the bandwidth W for both VMIMO transmission and feedback assuming that the local data exchange has been ideally performed prior to the data transmission. This assumption is followed by the network model in Section 2, where a local data exchange links between transmitters have much shorter propagation distances than the data transmission link between transmitters and receivers. As a result, the bandwidth required for local data (intracluster) exchanges would be much smaller than the bandwidth for longrange (intercluster) since the highspeed channel links are provided between transmitters.
Note that the desired symbol is transmitted over (W−W _{ f }) and W _{ f } is used for CSI feedback from receivers to the corresponding transmitters. Using VMIMO feedback, the spectral efficiency is represented as
where the signaltointerference and noise ratio (SINR _{ i }) is denoted as
In (10), \(\hat {\textbf {v}}_{i}\) is the transmit beamforming vector for the ith data stream transmission which is determined by using the CSI \(\{\hat {\mathbf {h}}_{1}\),..., \(\hat {\mathbf {h}}_{M} \}\) received in virtualMIMO feedback with feedback bits of \(B_{\text {ZF}}^{i}=W_{f} \cdot R_{m}\). Although it is hard to solve the closedform of optimal \(W_{f}^{*}\) that maximizes (9), we derive the scaling of \(W_{f}^{*}\) with system parameters using \(\bar {C}_{W}={{(WW_{f})} \over {W}}\cdot {\mathbb {E}}\left [\sum _{i=1}^{M}\log _{2} (\mathsf {SIR}_{i})\right ]\), which provides the accurate result when P _{ o } is large enough to assume the interferencelimited environment. Note that we rewrite SINR _{ i } to SIR _{ i } and upperbound of the spectral efficiency using Jensen’s inequality, for analytical simplicity.
where the distribution of SIR under limited feedback on (a) is derived in [22], (b) follows from the fact that \(X={{2(M1)}\over {2}}\cdot { {\chi _{2}^{2}} \over \chi _{2(M1)}^{2}}\) is Fdistribution with mean \({M1}\over {M2}\) [23], and signaltointerference ratio (SIR _{ i }) is denoted as
As shown in (11), the spectral efficiency is scaled with \(\log _{2} \left ({2^{{ W_{f} R_{m} \over (M1)}}} \right)\) due to the quantization error while it is also scaled with (W−W _{ f }) with respect to W _{ f }. Therefore, the choice of W _{ f } shows the tradeoff between the performance loss due to the quantization error and the increase of transmission spectral efficiency. From (11), the optimal \(W_{f}^{*}\) is found by solving \(\frac {{\partial \bar C_{W} }}{{\partial W_{f} }} =0\) where
and then we obtain \(W_{f}^{*}\) as follows.
Therefore, the amount of bandwidth for CSI feedback is increased with \(\mathcal {O} (M\log _{2}(M))\) and inversely proportional to the feedback channel rate R _{ m } achieved by VMIMO feedback.
VirtualMIMO transmission: cooperative zeroforcing precoding
In this section, we propose VMIMO transmission that enables the conventional ZF precoding for cooperative data transmission via the local data exchange and VMIMO feedback proposed in the preceding section. While the Section 3 focused on the optimality of bandwidth between data link and feedback link between transmitter and the receivers, we studied further effects on the local data exchange for constructing VMIMO transmitters and the analysis on the optimal node clustering to maximize the spectral efficiency throughout this section.
For implementing VMIMO feedback, it is assumed that the transmitters jointly detect feedback information to improve CSI quality. As illustrated in Fig. 2, the procedure for the VMIMO transmission can be described by three steps which is summarized as follows and analyzed in the subsections.

Step I: (Local data exchange) The ith transmitter broadcasts its desired signal to the (M−1) neighboring transmitters over β bandwidth. Since M data streams are exchanged between nodes over orthogonal bandwidth in FDMA manner, a total of β M bandwidth is used for local data exchange, to be transmitted towards the corresponding receivers.

Step II: (VMIMO feedback) Prior to the data transmission, each transmitter sends pilot signals to the receiver. Then, each receiver estimates CSI of channel links between all transmitters and the corresponding receiver. Finally, CSI is fed back to the transmitter through the reverse channel. All corresponding CSIs are obtained at all transmitters from joint VMIMO decoding suggested in Section 3.

Step III: (VMIMO transmission) Transmitter clusters design the cooperative ZF precoding to minimize the interuser interference then communicate with their corresponding receivers.
Local data exchange
Since the achievable data rate between intracluster transmitters should be higher than the data rate between transmitterreceiver pairs [16], the following condition is required for performing VMIMO transmission
where β is the bandwidth for local data exchange at each transmitter and R represents the target data rate at the ith transmitterreceiver link. Therefore, the minimum required bandwidth for local data exchange at the ith transmitter is represented as
Longrange data transmission
Infinite feedback channel
We consider the case of perfect CSI, corresponding to highresolution feedback (B _{ZF}→∞). Assuming that the distances between transmitters and the corresponding receivers are approximately same (or homogeneous), the achievable rate at the ith transmitterreceiver link is represented as (2). However, Due to the discrepancy between the local and longrange channel gain and the fluctuation of smallscale fading between transmitters and receivers, the achievable rate in (2) does not always support the target data rate R at the ith transmitterreceiver link. Therefore, it causes the outage probability of achievable rate so that the outage capacity of ith link is represented as follows.
where \(P_{\text {out}}=P(R_{ZF}^{i}<R)\). Moreover, the M β channels are used for the local exchange among M transmitters and thus the spectral efficiency of ZFbased VMIMO is defined as
where R _{ sum }=M·R. Since β is proportional to d _{ l }, the spectral efficiency becomes higher when d _{ l } becomes smaller. Also, large P _{ o } increases the spectral efficiency as the outage probability approaches zero.
Limited feedback channel
Given \(B_{\text {ZF}}^{i}\) feedback bits at each receiver, the residual interference due to the quantization error in ZF precoding is approximated as the sum of (M−1) exponential random variables in [21] and the achievable rate at the ith link is represented as
where \(\gamma =P_{o}/M, \delta ={P_{o} \over M}\cdot 2^{ {B_{\text{ZF}}\over {M1}}}\), \(x \sim {\chi _{2}^{2}}\) and \(y \sim \chi _{2(M1)}^{2}\). Similar to the case of perfect CSI, the outage capacity of the ith transmitterreceiver link in the limited feedback is derived as follows.
Proposition 1
Given \(B_{\text {ZF}}^{i}\) and P _{ o }, the outage capacity of ZF based VMIMO transmission is derived as
where t=2^{R}−1 and \(I_{r}={\left (1 \over \delta \right)^{M1} \left ({t\over \gamma } +{1 \over \delta } \right)^{(M1)}}\)
Proof
See Appendix. □
From proposition 1, the large number of M degrades the system performance as it increases the outage probability of each transmitterreceiver link. Moreover, I _{ r } represents the effect of quantization error due to limited feedback, which follows I _{ r }≤1 in given \(B_{\text{ZF}}^{i}\) feedback bits. The degrading factor of the residual interference I _{ r } goes to one when \(B_{\text{ZF}}^{i}\) is large enough so that the result of (20) approaches the outage capacity in (17).
The optimal number of active transmitters
As the number of nodes participating in VMIMO transmission increases, the multiplexing gain can be linearly increased with M. However, the large number of clustering nodes M also increases both localexchange overhead and outage probability so that increasing M is not always an optimal solution from the perspective of spectral efficiency since it requires a huge amount of the channel bandwidth for local exchange (see (18)). In this section, we provide the optimal number of nodes that maximizes (18) as follows. To obtain M ^{∗}, we differentiate (18) with M as follows.
Then, (21) is derived as
Solving the above equation with M>0, we obtain the following proposition.
Proposition 2
Given R and P _{ o }, the optimal number of node M ^{∗} that maximizes the spectral efficiency of (18) is determined as
To characterize the decision of M ^{∗} with system parameters, M ^{∗} can be approximated as
in large P _{ o }, where the high transmission power and small R reduce the outage probability and increase the number of cooperative nodes, respectively. From the result in proposition 2, we suggest the following clustering operation to maximize the spectral efficiency for VMIMO transmission.

M ^{∗} activate node selection: note that M nodes are distributed over transmitter cluster, where M ^{∗}<M. The node selection of M ^{∗} of M is performed by the roundrobin manner, where it provides the fairness of data transmission opportunity, implementable with low complexity. In a roundrobin approach, the selected nodes give a priority of data transmission to other idle nodes in a next transmission phase.

VMIMO transmission for M ^{∗}: the selected M ^{∗} transmitters only share the desired data streams \(\{s_{1}, s_{2},..., s_{M^{*}}\}\phantom {\dot {i}\!}\) over the M ^{∗} β bandwidth as each of the selected M ^{∗} nodes broadcasts its own data in β bandwidth. Then, (M−M ^{∗}) idle nodes, which are not allowed to transmit their own data, can help improving the achievable rate in (2) by providing array gain (see Fig. 3). In that case, h _{ i } v _{ i }^{2} follows a chisquare distribution with DoF of 2(M−M ^{∗}+1); then, the outage capacity is given by
$$\begin{array}{*{20}l} {\tilde{R}_{\mathsf{opt}}}^{i} &= (1P_{\mathsf{out}})\cdot R \\ &=\left(1P\left(\mathbf{h}_{i}\mathbf{v}_{i}^{2} \le t \right)\right)\cdot R \\ &=\left(1P\left({\chi}_{2(MM^{*}+1)}^{2} \le t \right)\right)\cdot R \\ &= 1\left(1  {e^{ t}}\sum\limits_{i = 1}^{M  {M^{*}} + 1} {\frac{{{t^{i  1}}}}{{(i  1)!}}} \right) \cdot R \\ &= {e^{ t}}\sum\limits_{i = 1}^{M  {M^{*}} + 1} {\frac{{{t^{i  1}}}}{{(i  1)!}}} \cdot R. \end{array} $$(25)where \(t={{M^{*}} \over {P_{o}}}\left (2^{R}1\right)\). Therefore, the large number of idle nodes can reduce the outage probability and \(\tilde R_{\mathsf {opt}}^{i}\) approaches R as M goes to infinity.
Simulation results
In this section, the analysis of proposed VMIMO feedback and transmission schemes and the corresponding performance gains are demonstrated by simulation results. Firstly, we show that the average feedback bit allocation over VMIMO feedback channel approaches the same result as instantaneous feedback bit allocation in certain SNR regimes. Figure 4 depicts the relative difference between \(B^{i}_{\text {ZF}}\) and \({\bar {B}}^{i}_{\text {ZF}}\) in (3) and (7), where \(\xi = \frac {\left {\bar {B}}^{i}_{\text {ZF}}  B^{i}_{\text {ZF}}\right }{B_{\text {ZF}}^{i}}\). As the transmission power of feedback channel, \(\bar {P}_{o}\), increases, ξ converges to 0 and then it becomes smaller than 0.1 when \(\bar {P}_{o} >12\) dB. Therefore, using the average VMIMO feedback bits in the medium range of \(\bar {P}_{o}\), such as 10–15 dB, guarantees a marginal performance loss compared with the use of instantaneous feedback bits so that constant VMIMO feedback bits computed in average sense does not provide the significant performance loss rather than the use of variable feedback bits from instantaneous channel.
Figure 5 compares the throughput of ZF precoding given the average feedback bits and instantaneous feedback bits in VMIMO feedback. In the ergodic sense, we show that the sum throughput using instantaneous VMIMO feedback bits is highly close to that of average (equally allocated) VMIMO feedback bits given \(\bar {P}_{o}=10\) and 15 dB. Moreover, the performance of proposed VMIMO feedback and that of conventional feedback approach is compared, where the conventional feedback method considers the equal feedback bits for each receiver under the orthogonal feedback bandwidth. On the saturation region in high SNR, the 25 and 15 % of performance gain are shown in the proposed feedback method compared with the conventional method at \(\bar {P}_{o}=10\) and 15 dB, respectively.
Given W=50,P _{ o }=30 dB, and (d _{ l }/d _{ o })=0.3, Figs. 6 and 7 compare the spectral efficiency of ZF precoding in (9) under the MRC and ZF based VMIMO feedback schemes. In Fig. 6, it is shown that the performance gain of ZF VMIMO feedback over MRC feedback increases as \(\bar {P}_{o}=7, 10,\) and 13 dB when M=5. However, the performance of MRC based CSI feedback is rather better than the ZF VMIMO feedback in the low feedback power such as \(\bar {P}_{o}=7\) dB. In Fig. 7, the optimal resource partitioning based on (14) is plotted given \(\bar {P}_{o}=10\) dB and the different value of M. As derived in (14), the optimal bandwidth allocation for CSI feedback increases with M and the analytical results match the simulations.
In Fig. 8, the spectral efficiency of ZF based VMIMO transmission is compared in the distributed network, where P _{ o } = 20 dB, R = 2, (d _{ l }/d _{ o }) = 0.3, and W _{ f } = 30 with the transmission power of feedback channel, respectively. As shown in (20), the performance gain of ZF transmission in large M generates the strong residual interference given the finite rate feedback channel. The efficiency of ZF transmission of VMIMO feedback approaches that of perfect CSI as \(\bar P_{o}\) goes to large. Furthermore, the gain of ZF VMIMO feedback provides a dramatic increase of system performance than the MRCbased CSI feedback in large \(\bar P_{o}\) since it achieves the full multiplexing gain for CSI acquisition. Fig. 9 shows the spectral efficiency of ZF based VMIMO transmission and its optimal M ^{∗}. As derived in (23), the optimal solution in high P _{ o } is that all nodes participate in VMIMO, while the corresponding solution becomes noncooperative when P _{ o } approaches low SNR region. However, in the medium value of transmission power, increasing the number of cooperation nodes does not always provide a higher spectral efficiency as it requires a large amount of bandwidth for local data exchange at the transmitter cluster. Setting the parameters P _{ o }=10 dB, R=1, and (d _{ l }/d _{ o })=0.1,0.3, and 0.6, the transmitters and receivers are located within the distance of d _{ l }, respectively. Then, we compare the curves of spectral efficiency and analytical results in (23). While the increase of cooperative nodes provides a higher multiplexing gain, it also requires additional costs for local data exchange, as shown in (18). Therefore, the proposed VMIMO is more beneficial for the case of i) short distance between neighboring transmitters which provides highspeed local data link and ii) the short distance between the corresponding receivers that experiences a strong interuser interference channel, respectively.
Conclusions
In this paper, an efficient feedback and data transmission scheme using VMIMO has been proposed in a wireless ad hoc network. We show that the cooperative feedback improves the accuracy of CSI compared with the orthogonal channel feedback and provides abundant channel resources for data transmission by reducing the use of channel resource for CSI feedback. Given the proposed feedback scheme, the outage capacity of ZF based VMIMO transmission under the transmitter cooperation is analyzed and the optimal number of active nodes that maximize the spectral efficiency is proposed for VMIMO based clustering. Future works will examine the design of VMIMO feedback and transmission in randomly distributed transmitter and receiver nodes and propose hierarchical clustering and cooperation with VMIMO feedback in the existence of multiple transmitter and receiver clusters. Moreover, we leave the analysis on stochastic geometric approach under VMIMO system, where it can represent the effectiveness of the proposed algorithms on more realistic environment.
Appendix
Proof of proposition 1
Note that the outage probability of ZF precoding at the ith transmitterreceiver pair is \(P(R_{{\mathsf ZF}}^{i}<R)=P({\mathsf {SINR}}_{i}\) <t), where t=2^{R}−1. Then, the outage probability is derived as
Therefore, the outage capacity at the ith transmitterreceiver pair is given as
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Acknowledgements
This research was supported by ICT R&D program of MSIP/IITP. (No. B0126151012, Multiple access technique with ultralow latency and high efficiency for tactile internet services in IoT environments)
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The authors declare that they have no competing interests.
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Keywords
 Virtual MIMO
 CSI feedback
 Clustering