Beamforming in Ad Hoc Networks: MAC Design and Performance Modeling
© Khalil Fakih et al. 2009
Received: 1 February 2008
Accepted: 4 January 2009
Published: 22 February 2009
We examine in this paper the benefits of beamforming techniques in ad hoc networks. We first devise a novel MAC paradigm for ad hoc networks when using these techniques in multipath fading environment. In such networks, the use of conventional directional antennas does not necessarily improve the system performance. On the other hand, the exploitation of the potential benefits of smart antenna systems and especially beamforming techniques needs a prior knowledge of the physical channel. Our proposition performs jointly channel estimation and radio resource sharing. We validate the fruitfulness of the proposed MAC and we evaluate the effects of the channel estimation on the network performance. We then present an accurate analytical model for the performance of IEEE 802.11 MAC protocol. We extend the latter model, by introducing the fading probability, to derive the saturation throughput for our proposed MAC when the simplest beamforming strategy is used in real multipath fading ad hoc networks. Finally, numerical results validate our proposition.
Ad hoc networks seem to be a promising solution for wireless access networks in beyond 3G system. Traditionally, the research in these networks assumes the use of omnidirectional antennas. In this case, while two nodes are communicating using a given channel, MAC protocols such as IEEE 802.11 require all other nodes in the vicinity to stay silent. With smart antennas, when two nodes are communicating, their neighbors may communicate simultaneously, depending on the directions or channels of transmission.
Mainly, the smart antenna systems can be classified into two kinds: switched beam systems and adaptive array systems. The switched beam systems comprise only basic switching between separate predefined beams. In adaptive array systems, signal-processing methods are used to increase the capacity and the coverage, to ameliorate the link quality and to improve the spatial reuse. Moreover, avoidance or suppression of interferences can be added to these systems. Clearly, adaptive systems are more beneficial but more complex than switched beam systems.
In one-hop communication systems (i.e., cellular networks), the use of smart antenna enables the network operators to enhance the wireless network capacity. In multihop networks, which are expected to experience an enormous traffic increase, exploiting the potential of these antennas improves the spectrum efficiency, extends the coverage range, and alleviates the interferences by taking advantage of the interference suppression capabilities. In fact, because of the higher gain, the transmission range is longer, which can lead to longer battery life, better connectivity, fewer hops, and lower latency. Furthermore, due to the narrower beamwidth, the interference is reduced (or canceled) and therefore, the throughput is increased. Interestingly, beamforming techniques have been proven as a promising solution to improve the performance of ad hoc networks. Using these techniques, the signal can be directed in some privileged directions or channels. Therefore, an increasing in per-link capacity as well as number of communicating nodes can be obtained.
In ad hoc networks, the nodes share the same physical channel. Thus, an efficient MAC protocol should be designed to control the channel access and decrease the amount of collisions. Although various MAC schemes have been extensively studied using omnidirectional antennas, they cannot be applied directly to networks where smart antennas are used. In the literature, a tremendous number of MAC schemes has been proposed to support the directivity [1–6]. Nevertheless, in order to improve the network performance, many authors consider some unrealistic assumptions (because of their cost or their infeasibility) such as:
locating the nodes by an external hardware as GPS ,
splitting the main channel into two subchannels ,
assuming that the signal strength is carried only by the Line-of-Sight (LOS) component between two nodes ,
assuming a simplified antenna radiation pattern such as flat-top pattern or cone-sphere pattern .
As it can be seen, the proposed MAC protocols are so far from being realistic . In fact, using external hardware may not be cost effective and also it may not be the appropriate solution in multipath environments. Likewise, using two channels, two transceivers are needed and the front-end becomes complex and expensive in cost and in power. On the other hand, the most enhanced directional antennas in the market cannot radiate power only in tight direction. Rather, they have significant side lobes. Moreover, directional antennas are typical for environments characterized by strong LOS components. Such assumption is not always valid. For example, in indoor environments a significant angular spread is expected and the performance of directional antennas may be worse than omnidirectional ones [7, 8].
Beside these unrealistic assumptions, another critical point has to be considered. In fact, the validation of the proposed paradigms has been carried out through discrete event simulators. The common characteristics of all these simulators are the lack of supporting the physical layer behavior (including the physical channel model ) and the huge simulation time. Thus, in addition to an enhanced MAC protocol, analytical models would be needed to overcome these problems. Although a considerable work is achieved to explore analytically the distributed coordination function (DCF) behavior of IEEE 802.11 MAC protocol , little work has been done when using smart antennas in ad hoc networks. Moreover, in the latter case, the properties of the physical channel such as multipath fading are not considered, and the smartness is treated as point-to-point directivity as we stated before.
In this work, we propose a MAC protocol with channel tracker algorithm for ad hoc networks when using beamforming techniques. Our proposition consists of implementing a proactive channel tracker algorithm in parallel with an enhanced MAC protocol to exploit the beamforming techniques to their fullest. For the sake of completeness, we explore in this work the importance of using smart antenna systems in ad hoc networks by using an analytical study. This paper is a continuation of earlier works [10, 11].
Our contribution can be outlined as follows. (a) We overview the pertinent works on the design and analytical modeling of MAC protocols in ad hoc networks when using smart antenna systems. (b) We propose a new MAC protocol (BMAC) using beamforming techniques. Besides, we use a channel tracker algorithm in order to estimate channel coefficients between nodes. (c) By simulation, we validate our proposition and we evaluate the overheads introduced by the channel tracker algorithm on the network. (d) We propose an accurate analytical model for evaluating the IEEE 802.11 performances. (e) We extend our latter proposition to support beamforming techniques.
Mainly, this paper will be divided into two complementary parts: the first one focuses on the MAC design, while the second deals with the analytical modeling of the performance of that design.
2. BMAC: A Novel MAC Design
2.1. Related Works
In the literature, two works attempt to survey MAC protocols in ad hoc networks when using smart antennas [12, 13]. In , the four-way handshaking of the IEEE 802.11 medium access is considered as the main criterion to categorize the surveyed MAC protocols. In , the authors classify the MAC protocols based on the access scheme which defines two major MAC categories: random access protocols and scheduled protocols. The first category represents an adequate solution for ad hoc networks and most of the works have been done using this scheme. These works are further classified into three groups: pure-RTS/CTS protocols, tone-based protocols, and other protocols using additional control packets.
A novel carrier sensing (CS) mechanism called directional virtual CS (DVCS) and a scheme estimating the nodes direction called angle of arrival (AoA) caching are proposed in . The nodes update the AoA every time they receive a newer signal. In , the problem is alleviated by assuming that the gain in both omnidirectional mode and directional mode is the same. The control messages are sent in omnidirectional mode, while the data and the acknowledgment are exchanged using the beam receiving the highest power in the previous communication. In , a circular RTS is proposed to scan the medium. The authors in  propose a solution to overcome the hidden terminal problem. Moreover, they identify the transmitter and the receiver forbidden zones where the nodes are subject to interferences. In , the authors present another instances of hidden terminal; hidden terminal due to unheard RTS/CTS messages and hidden terminal due to the asymmetry in gain. They propose a multihop RTS MAC protocol to deal with these problems and to exploit the extended transmission range of directional antennas.
We note that these previous works have not fully exploited the benefits of adaptive arrays such as the ability to increase the spectrum efficiency, to extend the range of coverage and to form nulls in the directions of interferences. For these aims, little work has been done in literature. In , Yang proposed a MAC protocol called adaptive beamforming carrier sense multiple access/collision avoidance (ABF-CSMA/CA). In order to apply a directional RTS or CTS, a training sequence precedes these messages to estimate the channel. Another MAC protocol presented in  splits the main channel into two subchannels, with some predefined constraints.
2.2. BMAC Protocol
We propose a novel MAC protocol which performs channel gathering and medium sharing, jointly. Unlike other protocols, the Beamformed MAC (BMAC) does not require external devices to determine node locations. Our proposition is based on the channel and not on the position. The channel is estimated for further use when applying beamforming techniques, in order to couple the energy in the best way between the source and the destination and to restrain multiuser interferences. Thus, better connectivity and network capacity can be obtained. To prevent themselves from accessing pairs in communication sessions, the neighbors look up the updating frequency in their channel tables. If the tuple concerning a node is out of date then this node is considered busy.
The first algorithm in our proposition, called channel acquisition (CA), is proactive. In previous works, some authors assumed the availability of the destination location, others used AoA methods or external hardware as GPS to determine the node location. In indoor applications where a large angular spread is expected, the AoA methods may not be suitable to determine the positions of the nodes. Moreover, the potential of beamforming techniques will not be fully exploited if only the node location is known. For these reasons, we can see the importance to implement a proactive channel tracker algorithm in parallel with an enhanced MAC protocol to exploit the beamforming techniques to their fullest. This algorithm consists in transmitting a training sequence (pilot symbols) periodically each (acquisition period). When receiving this training sequence, the channel to the corresponding node is estimated by applying the LMS algorithm . Then, the channel coefficients and the node identifier are saved in a specific table called channel table.
The acquisition period is calculated with respect to the coherence time of the channel ( ). The coherence time is related to the maximum Doppler frequency: where is equal to , is the carrier frequency, and is the speed of light. Thus, low mobility (quasistatic) environments are the most suitable environments for our proposition. However, if the nodes are involved in a high-mobility scenario, the load of this algorithm may be unsupportable. As will be shown in the simulation and analytical results, wise choice of maintains an acceptable channel estimation for immediate use and alleviates the resulting overheads.
We note that if we apply the "on-demand" channel estimation procedure (which involves less overheads on the network), only the channel toward the destination will be available. In this case, we can improve the quality of service of the communication link between the source and the corresponding destination but we cannot alleviate the interferences since we do not have the estimation of the channels toward these interferences.
Our MAC is based on IEEE 802.11 in order to ensure interoperability with current deployed WLAN modem. Under the assumption of using a half-duplex transceiver at each node, a packet exchange occurs as indicated in the state diagram. Some points have to be considered.
If the offered traffic load is sufficiently high, the network will be congested almost all the time. Consequently, the data packet will not have any priority over the training sequence (TS) packets and the data transmission will be interrupted each to transmit these sequences (if not, the estimated channel versions will be expired and the beamforming will not work properly). In this case, the amount of data lost by the omnidirectional transmission for the TS packets depends on the acquisition frequency.
If this is not the case, the CA algorithm can be stopped and the data transmission can proceed. Thus, the channel table for the nodes in the vicinity will be expired and the corresponding pair of nodes is considered busy.
(ii) Equipped with an antenna array of elements, the source node calculates the transmit Bd weights in order to make nulls toward the high noisy neighbors ( is the degree of freedom) and to couple the energy toward the intended destination. These high noisy neighbors can be seen as the channels having the maximum energy (i.e., the potential interferences with respect to the current node). Then, a BRTS can be transmitted using the calculated Bd (see the index on Figure 1).
Providing that an estimated channel version of all neighbor nodes is available, the zero forcing transmit beamforming algorithm is used. However, the traditional beamforming can be used. In the latter case, only the channel between the source and the destination will be used and the nulling capabilities cannot be exploited .
(iii) When receiving the BRTS control message, the nodes in the vicinity update their SNAV to prevent themselves from accessing this pair of nodes (source and destination). In fact, when using a Bd toward such destination, other nodes having near channels can receive the messages as well as this destination (see the index on Figure 1).
(iv) When receiving the BRTS, the destination node calculates the exceeded power for further transmitted power correction and then it sends omnidirectional CTS (OCTS) message containing this correction factor. Using this parameter, the source can adjust the transmission power to a certain level in order to maintain prespecified link quality. By that, a simple power control mechanism is implemented and the energy is saved.
We note that, BCTS cannot be used in this scheme because the version of the estimated channel (estimated with omnidirectional antenna) which is available at the current destination, does not take into account the transmit Bd. To use BRTS and BCTS in the same scheme, we have to implement a joint adaptive beamforming between the source and the destination. This strategy will be time consuming and it is not appropriate for ad hoc networks. From a cross-layer point of view, any joint transmit receive beamforming (iterative optimization) will inundate the network by the overheads and will produce network instability  (see the index on Figure 1).
(v) For receiving the OCTS message, the source can use the conjugate of the transmit Bd vector, namely, Bd*. It was shown in  that a strong network duality holds for TDD networks, in which the optimum receive Bds are the conjugates of the optimum transmit vectors (see the index on Figure 1).
(vi) After the exchange of the control messages, the source uses the Bd vector toward the destination (Bd) to send the data packets. As we will see in the next section, the link capacity will be improved and a higher global capacity will be obtained due to the spatial reuse improvement.
(vii) Once the data transmission/reception is completed, an ACK is transmitted and a CA session is enabled, to inform the neighbors about the availability of this pair (see the index on Figure 1).
(viii) If a tuple (i.e., for node B) in the channel table of a node A is not updated each , where is a tradeoff factor, the node A assumes that node B is in a function mode and prevents itself from attempting to access this node, eliminating by that the deafness problem .
Finally, to summarize the main differences between our proposition and other propositions in the same context (i.e., DMAC ) we present in the following a brief comparison between BMAC and DMAC.
BMAC is channel-based however DMAC is position or location-based.
The BMAC works even in rich multipath scattering environment however DMAC shuts down if the angular spread is considerable. Moreover, if the sender and the receiver are not in LOS view, the performance of directional antenna may be worse than omnidirectional one.
The BMAC uses the adaptive beamforming techniques and not conventional directional antenna.
The radiation pattern of DMAC is very simplified and it is illustrated by a main lobe and by a small sphere representing the side lobes.
This simplified antenna radiation pattern is static. We mean that the node requires the position of the destination in order to steer the main lobe in the right direction. Moreover, this antenna radiation pattern imposes an aggressive simplification and the technological limits do not allow such "ideal" beam.
The BMAC is based on the Channel Acquisition subalgorithm to maintain an available channel estimation version for future use. This subalgorithm is exploited also as a virtual carrier sensing to prevent deafness and thus to avoid collision.
In DMAC the nodes location is determined by an external system.
DMAC does not perform power control (fixed beamwidth). In contrast, BMAC saves the energy by a simple optional power control mechanism.
DMAC uses DNAV (as DVCS) while BMAC uses SNAV as explained above.
The novelty of our proposition comes from both MAC and physical layer(application of beamforming techniques in ad hoc networks).
These points make the BMAC a realistic protocol. However DMAC (even if we assume that the determination of the nodes position is possible and the radiation pattern is feasible) will shut down in indoor application where the angle spread is expected to be very large.
2.3. Performances Evaluation of the BMAC Protocol
To evaluate the impact of the beamforming techniques and the channel-based protocol (BMAC) on ad hoc networks, we simulate through different random scenarios the three following MAC protocols: IEEE 802.11b (with omnidirectional antenna pattern), the basic DMAC  protocol (modified version of IEEE 802.11 MAC protocol to support pure directivity), and finally the BMAC. More attention will be focused on the BMAC to examine the effects of the tradeoff parameters and as well as channel evolution effects on the network performance. Note that relates the acquisition period to the coherence time and represents the tradeoff factor between the LOS and the non-LOS components of the channel.
2.3.1. Simulation Model
Firstly, to show the effectiveness of the BMAC, we used a high-traffic load model in order to put our network in a realistic congested condition. Using this traffic model, all the transmitters have always packets to send during the simulation. If the medium is available, they immediately perform a transmission. Otherwise, they push their packets in their stacks, and they wait until the medium becomes idle. Secondly, for channel-load evaluation purpose, we alleviate the network load and we simulate the BMAC in different environments: directive and nondirective environments, low-change and fast-change environments.
Many MAC protocols based on antenna directivity were proposed and performance improvements to the IEEE 802.11 MAC were shown. The propagation models used in these MAC are simplified and suitably do not take into account a certain number of physical phenomenon which can have an important impact on the network performance. The multipath propagation is one of these phenomenons. As we have seen, all the suggested protocols assume that the signal is carried out by the LOS path between two nodes.
where is the path loss, is the distance between the transmitter and the receiver, is a tradeoff factor between the LOS component and the random component of the channel (this parameter is equal to in our general simulation), is the antenna array response for the main AoA, is a Gaussian random vector with zero mean and, the index stands for the multipath effect. Note that we use a circular antenna array with half wavelength spaced elements and we consider eight antenna elements in our simulations.
where is the signal intended for node , is the channel vector between a predefined antenna element at node and the antenna array at node , is the transpose conjugate of the weight vector described in the following section, and contains both background noise and interferences coming from another nodes in the vicinity.
The simplest strategy to exploit the smartness of antenna arrays in ad hoc networks is to use standard beamforming, that is, to point the main lobe of the antenna array of the source in the direction of the destination. However, if the global CSI is available at the transmitter, it is possible to actively suppress the interferences as depicted in Figure 2. Beamforming algorithms can be formulated as centralized or decentralized game. In ad hoc networks and especially in civilian applications, where the available calculation power is moderate, a decentralized beamforming algorithm is preferred. In addition, because of the availability of all the channels toward the neighbors node, we will exploit only the zero-forcing algorithm in our work. In fact, the traditional beamforming does not perform interference rejection and therefore it is not so beneficial for ad hoc networks.
2.3.2. Simulation Results
We compare the performance of the simulated MAC protocols in randomly distributed topologies. Herein, the potential of beamforming techniques with respect to the simple directional antenna pattern is examined.
In the sequel, we will access the performance of the BMAC through an analytical study. For this aim, we first bring out in the next section an accurate analytical model to evaluate the performances of the DCF scheme of IEEE 802.11 MAC protocol. Then, we extend the latter analytical model to access the performance of BMAC.
3. Analytical Modeling
As we stated in the introduction, most of the works in ad hoc networks with or without smart antenna systems have been validated by using discrete event simulators. In recent years, some analytical models have been proposed to analyze IEEE 802.11 MAC protocol behaviors. The work in  is a prominent work in this domain. Another attempts can be found in [26–28]. In , Bianchi evaluated the performance of the DCF scheme with the assumption of ideal channel conditions. This saturation throughput is defined as the limit reached by the system throughput as the offered load increase. Recall that in this basic work and under ideal channel condition, the packet is lost only in the case of collision. Furthermore, the authors assumed that the packets collide with constant and independent probability called conditional collision probability. From a practical point of view, the problem is alleviated by skipping the impact of the finite-retry limits and some physical characteristics as the channel conditions and the antenna radiation pattern. Building up Bianchi's work, Wu et al.  dealt with one of the major limitations of the Markov model by including the finite retry limits. Ziouva and Antonakopoulos  introduced the concept of busy channel. Chatzimisios et al.  proposed a new performance analysis to calculate the packet delay and the packet drop probability.
So far, some works have been done to model analytically the effect of smart antennas on ad hoc networks. In , the author used a pie-slice antenna radiation pattern model and he neglected or simplified other physical parameters. In , many issues related to the deployment of directive antenna in ad hoc networks are discussed and analyzed. In this work, the transmission probabilities are taken independent from the MAC protocol. In , the authors suggested that the pie-slice models for the directional antenna exaggerate the system throughput. In , a MAC protocol exploiting the spatial diversity called SD-MAC is proposed. In this work, the authors extended a new approach to characterize the saturation throughput for multihop ad hoc networks using spatial diversity. The key feature in this work is the consideration of fading channels. As in , the packet loss probability (LP) due to collision is constant. Since fading can also occur, the packets can be lost without collision. Thus, the authors define the LP by where stands for the packet loss due to fading. Although the authors mention that the MIMO techniques and especially spatial diversity are used, they did not give an explicit expression of the packet LP that may coordinate with channel type and with the antenna array size. Furthermore, they suppose the availability of the channel sate information (CSI) without evaluating the effect or the cost of the channel estimation overheads on the network.
3.1. Modeling the IEEE 802.11 MAC Protocol
In this section, we propose a novel model combined with busy channel, retry limitation and nonsaturated condition. With improvements on the precedent models, our model adopts these three issues and brings out the new analytical throughput. We assume that the nodes in the network share the same physical properties and the number of nodes is fixed and finite.
For retry limitation, and are set to represent the maximum retry limit in MAC layer and in Physical (PHY) layer, respectively. As specified in IEEE 802.11, contention window (CW) size of a stage is , when . If , the CW size is held as . is the minimum contention window.
Here, we introduce an add-in state representing the idle stage of a single node. The parameter represents the probability that a node has a consequent packet to transmit after a success or failed transmission. Correspondingly, is the probability that a node meets no new packet from upper layer and turns into the stage to wait for new packets. At the waiting stage, a node keeps waiting slot by slot until it gets a new packet and moves into the backoff states.
For the convenience in demonstration, two intermediate points are involved between the idle stage and the backoff stages. They can be treated as two "pseudo states" for two instances in the function of nodes. The point named after a transmission is the moment when a node is requiring new packets from upper layer. The other one named before a transmission is the moment when a node is ready to send a new packet.
The fourth equation models that a node will not decrease its CW when the backoff stage reaches . Once the retransmission reaches the limit, no matter the current trial succeeds or fails, a node drops the present packet. The fifth equation shows that after a transmission, a node turns to the upper layer to obtain a new packet. The sixth equation describes that a node is ready to transmit if it has got a new packet. As shown in the seventh equation, a node is set to idle if it gets no new packet after a transmission, moreover, an idle node keeps waiting until there comes a new packet. Let with , be the stationary distribution of the chain. A closed-form solution can be obtained from this Markov chain. First, note that
Owing to the property of omnidirectional antenna, a node transmits with probability while the others have to keep silence, we have . This latter equation and (7) represent a nonlinear system with two unknowns and , which can be solved by numerical methods.
Considering such a scenario, a node A is transmitting data to its destination with its backoff counter ; the other nodes in the system remain silent and freeze their backoff counters due to the busy channel. Any backoff counter of silent nodes is , otherwise a collision would have happened when more than one node reach the zero of backoff counter at the same time. According to the DCF specifications, after the transmission, all the nodes wait for a DIFS time and then continue the decrement in the backoff counters. Therefore, except node A, all the others can access the channel after a period . Only when node A generates a new random backoff equal to zero for the next transmission, it will access again the channel after the period of one DIFS with a probability .
3.2. Modeling the Performance of Ad Hoc Networks When Using the Simplest Beamforming Strategy
The network consists of nodes uniformly distributed in a square area, each of which has neighboring nodes. That is, there are nodes in the omnidirectional coverage zone of each node. We assume that all the nodes are equipped with half wavelength spaced antenna-elements. In this section, we derive the saturation throughput for ad hoc networks when using maximum ratio transmission technique. Our main contribution is concentrated on developing the packet LP due to fading when using a simple beamforming strategy. However, the model can be extended to other beamforming algorithms. Recall that, the channel state information is needed at the transmitter to properly generate the correspondent Bd. Based on the new accurate analytical model for IEEE 802.11 proposed in previous section, we derive also the saturation throughput of the BMAC using the developed packet LP. In summary, our work in this section is divided into two parts: the first one is about the determination of the packet LP due to fading by using both analytical and empirical studies. While in the second, we use this probability to calculate the saturation throughput of a simplified version of BMAC.
3.2.2. Loss Probability Due to Fading
3.2.3. Saturation Throughput of the Beamformed MAC Protocol
In (19), and denote, respectively, the average time for successful transmission and failed transmission due to collision. They are calculated according to (11) and based on where . stands for the overheads. It is important to mention here that we transmit the training sequence in omnidirectional mode in order to estimate the channel coefficients without beamforming.
In the following and before computing the saturation throughput for data packets using beamforming techniques, two points have to be considered.
(i) The per-link performance enhancement due to the use of beamforming techniques is embedded in the LPF function. Higher number of antennas leads to lower LPF value, and then higher per-link capacity can be obtained. However, the performance improvement due to the spatial reuse will be modeled by a function called effective spatial reuse ( ) which denotes the number of simultaneous links that can coexist. Therefore, the number of stations that can be active at the same time will be .
The sources of loosing packets are three in our case:
loss due to the mismatching between the estimated version of the channel and the real channel. In fact, under perfect estimation of the channel coefficients, the only error that may occur from the use of this estimated channel occurs in the adaptation between the coherence time and the acquisition period. In such case and based on the outdated channel coefficients, the calculated Bd that maximizes the diversity will be a scaled version of , where stands for estimated and the superscript stands for transpose conjugate. Subsequently, weighting the real channel by this Bd will generate an additive factor that depends on the mismatching between the real channel and the estimated channel.
We consider the five events experienced by a typical user U  and we calculate the probability of each event based on and :
3.3. Numerical Example
This work focused on exploring the benefits of smart antennas and especially beamforming techniques in multipath fading ad hoc networks. For this aim, we proposed a novel protocol, named BMAC, to adapt the MAC functionalities to the new antenna paradigm. The results show that, in quasistatic scenarios, the BMAC offers a high throughput and better quality of service than the conventional directional MAC. Moreover, we devised a new accurate model for analytical evaluation of the performance of ad hoc networks when simple beamforming technique is used. Finally, a numerical example shows that the numerical results cope with the simulation ones.
Beyond this work, we aim to design and analyze a more general MAC protocol to support MIMO links in ad hoc networks. For near scope, we aim to take into account the hidden terminal problem in the proposed analytical model.
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