Distributed Energy Spectral Efficiency Optimization for Partial/Full Interference Alignment in MultiUser MultiRelay MultiCell MIMO Systems
Abstract
The energy spectral efficiency maximization (ESEM) problem of a multiuser, multirelay, multicell system is considered, where all the network nodes are equipped with multiple antenna aided transceivers. In order to deal with the potentially excessive interference originating from a plethora of geographically distributed transmission sources, a pair of transmission protocols based on interference alignment (IA) are conceived, which may be distributively implemented in the network. The first, termed the fullIA protocol, avoids all intracell interference (ICI) and othercell interference (OCI) by finding the perfect interferencenulling receive beamforming matrices (RxBFMs). The second protocol, termed as partialIA, only attempts to null the ICI. Employing the RxBFMs computed by either of these protocols mathematically decomposes the channel into a multiplicity of noninterfering multipleinput–singleoutput (MISO) channels, which we term as spatial multiplexing components (SMCs). The problem of finding the optimal SMCs as well as their power control variables for the ESEM problem considered is formally defined and converted into a convex optimization form with the aid of carefully selected variable relaxations and transformations. Thus, the optimal SMCs and power control variables can be distributively computed using both the classic dual decomposition and subgradient methods. The performance of both protocols is characterized, and the ESEM algorithm conceived is compared to a baseline equal power allocation (EPA) algorithm. The results indicate that indeed, the ESEM algorithm performs better than the EPA algorithm in terms of its ESE. Furthermore, surprisingly the partialIA protocol outperforms the fullIA protocol in all cases considered, which may be explained by the fact that the partialIA protocol is less restrictive in terms of the number of available transmit dimensions at the transmitters. Given the typical cell sizes considered in this paper, the pathloss sufficiently attenuates the majority of the interference, and thus the fullIA protocol overcompensates, when trying to avoid all possible sources of interference. We have observed that, given a sufficiently high maximum power, the partialIA protocol achieves an energy spectral efficiency (ESE) that is 2.42 times higher than that attained by the fullIA protocol.
I Introduction
Future wireless cellular networks are required to satisfy everincreasing area spectral efficiency (ASE) demands in the context of densely packed heterogeneous cells, where both relay nodes (RNs) and smallcells [1, 2] are employed. However, these changes will result in severe cochannel interference (CCI), since future networks will aim for fully exploiting the precious wireless spectrum by relying on a unity frequency reuse factor [3]. Furthermore, owing to the growing energy costs, a system’s energy efficiency is becoming a major concern [4]. Against this backdrop, in this paper we aim for maximizing the energy spectral efficiency (ESE) of the downlink (DL) of a decodeandforward (DF) [5] relayaided multipleinput–multipleoutput orthogonal frequency division multiple access (MIMOOFDMA) multicell network that employs the technique of interference alignment (IA).
IA was first introduced in [6, 7, 8], and it was further popularized in [9, 10]. In [10], Cadambe et al. described the main concept of IA and established the attainable degrees of freedom (DoF), when employing IA for completely avoiding interference in a network supporting userpairs. The principle of IA is that, instead of dividing the wireless resources amongst all users (often termed as orthogonalization), each user aligns his/her transmissions into a predetermined subspace, referred to as the interference subspace, at all the other receivers, so that the remaining subspace at all receivers becomes free of interference. Thus, the attainable DoFs in a system supporting userpairs is when employing IA, instead of obtained through orthogonalization [10]. This becomes highly favorable, as increases.
Hence, IA has been advocated as a viable technique of managing the uplink (UL) cochannel interference of multicell networks [11, 12]. Explicitly, IA is suitable for the UL, since the number of receive antennas (RAs) at the basestation (BS) is typically higher than the number of transmit antennas (TAs) at each user equipment (UE). Thus, the potentially higher number of signal dimensions available at the receiver can be exploited for aligning the CCI into a predetermined interference subspace, so that the BS can receive the transmissions of its own UEs without CCI. However, this is not feasible in the DL, since each UE has access to a low number of receive dimensions. This challenge was successfully tackled by the DL transmission scheme of [13], which relies on specifically designing transmit precoding (TP) matrices for reducing the number of transmit dimensions at the BSs, thus facilitating DL IA at the UEs. In contrast to other IA techniques, such as [14, 15, 16, 17, 18], the technique presented in [13] does not require cooperation among the BSs for exchanging channel state information (CSI), and IA is accomplished distributively. Furthermore, this technique facilitates IA in systems relying on arbitrary antenna configurations with the aid of frequency or timeextension, which is capable of substantially expanding the total number of transmit and receive dimensions in a multicarrier system such as OFDMA. In [19], the technique of [13] was generalized to an arbitrary number of BSs and UEs, where each of them is equipped with an arbitrary number of antennas. Furthermore, the authors of [19] employed the semiorthogonal user selection scheme of Yoo et al. [20] for maximizing the achievable SE. However, relaying was not considered in [19] and each UE was limited to receiving a single spatial stream.
In this paper, we aim for maximizing the system’s attainable ESE, defined as a counterpart of ASE [21], where the latter has the units of , while the former is measured in . This ESE metric has also been utilized in [22, 23, 24, 25, 26, 27]. The authors of [22] considered ESE maximization (ESEM) of both the UL and the DL of a cellular network, while providing both the optimal solution method and a lowercomplexity heuristic method. However, the effects of interference were not quantified in the system model of [22], since only a single cell was considered. Additionally, no relaying was employed. In [23], ESEM was performed in a multicell setting, where the CCI was eliminated with the aid of BS cooperation [28] and zeroforcing beamforming (ZFBF). However, the authors of [23] have not considered the benefits of multiple antenna aided nodes or relaying. As a further advance, the energyefficiency of a relay aided system was considered in [24], where the objective function (OF) of the optimization problem considered was formulated by incorporating both the spectral efficiency (SE) and the energy dissipated. Nevertheless, these two metrics must be appropriately weighted, which is still an open challenge. Thus, the ESE metric was not formally optimized.
In fact, the maximization of the ESE metric is typically formulated as a fractional (in this case, quasiconcave) programming problem [29, 30, 31], which relies on the classic solution methods of the bisection search [31], and on Dinkelbach’s method [29], as employed in [23, 25, 26]. However, the bisection search requires the solution of multiple convex feasibility problems, while Dinkelbach’s method requires the solution of multiple concave subtractive optimization problems. The total number of algorithmic iterations may become prohibitive in both cases. Hence, we opt for employing a beneficial method based on the CharnesCooper variable transformation [30, 32], allowing us to solve the ESEM problem by solving a single concave optimization problem and to demonstrate its benefits to the wireless communications community.
Let us now elaborate further by classifying the cochannel interference as intracell interference (ICI) and othercell interference (OCI). In the DL considered, the former describes the interference that a RN or UE may receive from the BS within its own cell, where multiple concurrent transmissions are also intended for other RNs or UEs, while the latter describes the interference originating from sources located in other cells.
We now provide a concise list of the contributions presented in this paper.

We evaluate the ESEM of IA employed in a realistic MIMOOFDMA system involving multiple cells, multiple relays and multiple users. Although ESEM has been studied intensely in recent years [22, 23, 24], these contributions typically consider single cells providing coverage without the assistance of relaying, or do not exploit the benefits of multiple antenna aided transceivers. Additionally, although IA was employed recently in [14, 33, 34, 35, 36], these contributions focus on userpair networks, rather than on multiuser cellular networks and the associated challenges of implementing IA require further research in the latter scenario. More importantly, previous contributions typically aim for investigating its SE benefits, while the achievable ESE of using IAbased protocols has not been explored at all. Green communications has become increasingly important, but the quantitative benefits of IA have not been documented in the context of energyefficient communications. Therefore, in this contribution we seek to deepen the research community’s understanding of IA from an ESE perspective. Furthermore, a more realistic multicell MIMOOFDMA relayaided network is considered in this treatise, where multiple users are supported by each BS and multiple relays. Therefore, the system model considered inevitably becomes challenging. As a beneficial result, the protocols and solutions provided in this paper can be more readily applied to real network scenarios, when compared to the existing IA literature, which focuses only on the user interference network. In contrast to our previous contributions [25, 26, 27], this treatise investigates a multiple antenna aided multicell system. Although a multiple antenna assisted system was also studied in our previous contribution [27], only a single macrocell was considered and no IA was employed for avoiding the ICI imposed by both the simultaneously transmitting BS and RNs.

We provide a sophisticated generalization of the IA protocol considered in [13]. Explicitly, in contrast to [13], the proposed IA protocol accounts for three cells, for an arbitrary number of users in each cell, for an arbitrary antenna configuration and for simultaneous direct as well as relayaided transmissions. This is accomplished through the careful design of precoding, transmit and receive beamforming matrices in order to ensure that IA is achieved. In particular, the number of guaranteed spatial dimensions available at the BSs, RNs and UEs must be judiciously chosen. Furthermore, we conceive of two transmission protocols in this work, which may be implemented distributively at each BS. The first protocol is termed as fullIA, which invokes IA for avoiding the interference arriving from all transmitters. This is the intuitive choice, as advocated by the existing literature [11, 13, 19] highlighting its benefits in terms of achieving the optimal DoF. For example, it was also employed in [13], but for a simpler system model having no relays. The second protocol proposed is unlike that of [13] and it is termed as partialIA, which only aims for avoiding the ICI using IA, while ignoring the effect of OCI when making scheduling decisions. The partialIA protocol therefore reduces the computational burden of having to estimate the DL CSI of the othercell channel matrices at the receivers, albeit this might be expected to reduce the system’s performance due to neglecting the OCI. We compared the performance of these two protocols and found that, as a surprise, the reducedcomplexity partialIA protocol is potentially capable of achieving a higher ESE than the fullIA protocol. Explicitly, the partialIA protocol achieves a higher ESE, since more simultaneous transmissions may be scheduled due to its relaxed constraint on the number of transmit dimensions available. Furthermore, in contrast to the protocol proposed in [19], ours is a twophase protocol, which is specifically designed for relayaided networks and does not limit the number of spatial streams available to each UE.

Employing the beamforming matrices calculated from either the fullIA or partialIA protocols results in a list of spatial multiplexing components (SMCs)^{1}^{1}1These SMCs are detailed further in Section III, which correspond to the specific data streams that the BSs can choose to support. Finding the optimal SMCs as well as the optimal power control variables associated with these optimal SMCs is formally defined as a networkwide optimization problem. Unlike in our previous work [25, 26, 27], we decompose the networkwide multicell optimization problem in order to formulate a subproblem for each BS using the technique of primal decomposition [37], thus eliminating the need for the highoverhead backhaulaided message passing amongst the BSs. Each of these subproblems is then converted into a convex form with the aid of various variable relaxations and transformations, which can then be optimally and distributively solved using the dual decomposition and subgradient methods of [37].
The organization of this paper is as follows. We introduce our system model in Section II and describe the proposed transmission protocols in Section III. Subsequently, the ESEM optimization problem considered is formulated in Section IV, where the solution method is developed as well. Our numerical results along with our further discussions are presented in Section V. Finally, our conclusions are given in Section VI along with our future research ideas.
Ii System model
In this work, a multicell DL MIMOOFDMA network, relying on a radical unity frequency reuse factor is considered. The ubiquitous OFDMA technique is employed for avoiding the severe frequencyselective fading encountered in wideband communication systems. Additionally, OFDMA allows for transmission symbol extensions in the frequencydomain [13], which are required by the proposed IAbased transmission protocol described in Section III.
As depicted in Fig. 1, each macrocell is divided into three sectors, and it is assumed that the employment of directional antennas and the nonlineofsight (NLOS) pathloss attenuates the interference power, with the exception of the OCI received from the first tier of interfering cells and the ICI from the serving BS and RNs of each macrocell. Therefore, we may focus our attention on the central region seen at the left of Fig. 1, which we term as an OCI region. Thus, each DL transmission within an OCI region is subjected to OCI from two macrocells. Furthermore, each sector of Fig. 1 is supported by RNs, which are located at a fixed distance from its associated BS and evenly spaced within the sector, as seen at the right of Fig. 1. The ratio of the BSRN distance to the cell radius is denoted by . Additionally, UEs are uniformly distributed within each sector. The system has access to OFDMA subcarriers, each characterized by a wireless bandwidth of Hertz. The BSs, DF RNs, and UEs are respectively equipped with , and antennas. It is assumed that all BSs and RNs are synchronized, and that the transceivers employ complexvalued symbol constellations to convey their data.
For each subcarrier , the complexvalued channel matrix associated with the wireless link spanning from the BS of macrocell to RN belonging to macrocell is denoted by^{2}^{2}2Superscript indices refer to the transmitter, while subscript indices refer to the receiver. Additionally, a prime symbol refers to a potentially interfering transmission source. . The channel matrix associated with the link spanning from the BS of macrocell to UE and belonging to macrocell on subcarrier is denoted by . Furthermore, the channel matrix associated with the link between RN belonging to macrocell and UE belonging to macrocell on subcarrier is denoted by . All channel matrices are assumed to have a full rank, as is often the case for wireless DL channels. For simplicity, the channel matrices associated with the same transceivers are combined across subcarriers to give the blockdiagonal channel matrices , and , respectively. For example, we have
(1) 
The channel matrices account for both the smallscale frequencyflat Rayleigh fading, as well as the largescale pathloss between the corresponding transceivers. In this system model, the transceivers are either stationary or moving sufficiently slowly for ensuring that the channel matrices can be considered timeinvariant for the duration of a scheduled transmission period. However, the channel matrices may evolve between each transmission period. Furthermore, it is assumed that the transceivers’ antennas are spaced sufficiently far apart for ensuring that the associated transmissions experience i.i.d. smallscale fading, which are drawn from complex i.i.d. normal distributions having a zero mean and a unit variance. The system uses timedivision duplexing (TDD) and hence the associated channel reciprocity may be exploited for predicting the CSI of the slowly varying DL channels from the received UL signal. Furthermore, by assuming the availability of lowrate errorfree wireless backhaul channels, the CSI associated with the wireless intracell RNUE links may be fed back to the particular BS in control, so that it may make the necessary scheduling decisions.
Additionally, each receiver suffers from complexvalued additive white Gaussian noise (AWGN) having a power spectral density of . Due to both regulatory and safety concerns, the maximum instantaneous transmission power of each BS and each RN is limited, which are denoted by and , respectively. We stipulate the idealized simplifying assumption that OFDMA modulation and demodulation is performed perfectly for all the information symbols.
Iii Transmission protocol design
Each BS may convey information to the UEs by either using a direct BSUE link, or by utilizing a RN to create a twohop BSRN and RNUE link, which requires two transmission phases. Thus, each transmission period is split into two halves. Due to the simultaneous transmissions from multiple sources, both the level of ICI and OCI in the network is likely to be detrimental to the achievable ESE. In order to avoid both types of interference, the technique of IA is employed, which requires the careful design of both the transmit beamforming matrix (TxBFMs) of the BSs and of the RNs, as well as the receive beamforming matrix (RxBFMs) of the RNs and of the UEs. As relaying links may be utilized in this system, the design of these matrices is different for the two transmission phases. Hence they are described separately in the following. Additionally, both the fullIA and partialIA protocols will be described sidebyside. To elaborate a little further, the fullIA protocol aims for completely avoiding both the ICI and OCI in both the first and second transmission phases by employing IA, while the partialIA protocol only aims for avoiding the ICI in both transmission phases, thus dispensing with estimating the OCI channel matrices at each receiver.
Furthermore, the proposed schemes crucially rely on the singular value decomposition (SVD), where the columns of the left and right singular matrices are composed of the left and right singular vectors of the associated matrix. These left and right singular vectors may be further partitioned into the leftmost and rightmost parts, which correspond to the nonzero and zero singular values, respectively. This structure is illustrated in detail in Fig. 2.
Iiia Beamforming design for the first phase
In the first phase, only the BSs are transmitting to both the RNs and the UEs. Therefore, the only source of interference is constituted by the neighboring BSs associated with the same OCI region, which may be avoided by carefully designing the TxBFMs at the BSs, as well as the RxBFMs at the RNs and the UEs in a distributive manner. Initially, a TP, denoted by , is randomlygenerated for each BS , where is the number of symbols transmitted by each BS during the first phase, which is accurately defined in Section IIIA3. The matrix has a full column rank and its entries are complexvalued. These TPs are invoked for reducing the number of transmit dimensions for each BS from to , thus facilitating IA at the receivers. Furthermore, the columns of these TP matrices are normalized so that the power assigned to each transmission remains unaffected. By employing these TPs, the precoded channel matrices of the first phase are given by
(2) 
and
(3) 
respectively for the BSRN and BSUE links.
We now define and as the minimum number of receive dimensions at each RN and each UE, respectively, which are chosen by the network operator. Furthermore, only the specific values of and along with the number of antennas at each network node and the number of available subcarrier blocks affect the feasibility of IA, while and have no effect.
IiiA1 FullIA receiver design
In order to completely avoid the interference arriving from the neighboring BSs during the first phase, it is necessary for the precoded OCI channel matrices given by (2) and (3), to have intersecting left nullspaces. Firstly, the precoded OCI channel matrices may be concatenated for forming the interference matrices, for example
(4) 
for RN in macrocell , and
(5) 
for UE in macrocell . These matrices are associated with a left nullspace of at least and dimensions if
(6) 
and
(7) 
respectively. Therefore, to guarantee and receive dimensions at the RNs and UEs, respectively, is derived as
(8) 
The intersecting left nullspace may be found using the SVD on and , for RN and UE in macrocell , respectively. For example, the SVD of may be written as , where is the left singular matrix containing, as its columns, the left singular vectors of , while is a rectangular diagonal matrix whose diagonal entries are the singular values of ordered in descending value, and is the right singular matrix containing, as its columns, the right singular vectors of . The intersecting left nullspace may then be obtained as the rightmost columns of (corresponding to the zero singular values), and this is used as the RxBFM, , for RN in macrocell . A similar procedure is performed for obtaining the RxBFM, , for UE in macrocell in the first phase, where the rightmost columns of the corresponding left singular matrix are selected.
To summarize, the cost of implementing the fullIA protocol in the first transmission phase is the reduction of the number of available spatial transmission streams at each BS from to . Thus, if the RNs and UEs require a large number of spatial streams, the BSs have to substantially reduce the number of transmitted streams in order to accommodate IA. However, it is clear that should be higher than to ensure that the BSs become capable of transmitting. Following this procedure, the and total interference signal dimensions received at each RN and at each UE respectively have each been aligned to dimensions, leaving and receive signal dimensions free from interference at the RNs and UEs, respectively. Thus, IA has been successfully employed for reducing the number of spatial dimensions that the interference signals occupy.
IiiA2 PartialIA receiver design
Using this design philosophy, the OCI encountered during the first phase is ignored when designing the RxBFMs. However, since there is no ICI in the first phase since only the BSs are transmitting, there is no need to reduce the number of transmit dimensions at the BSs. Therefore,
(9) 
is chosen. Furthermore, the matched filter receiver design is adopted for maximizing the achievable SE [38]. In this case, the SVD is performed on the intracell precoded channel matrices, yielding for example
(10) 
and
(11) 
respectively, and the (resp. ) leftmost left (thus corresponding to the highest singular values) singular vectors are selected as the RxBFM for the RNs (resp. UEs) in the first phase.
In summary, IA is not required during the first transmission phase of the partialIA protocol, since the only transmitter within the same cell is the associated BS. Therefore, it is not necessary for the BSs to reduce the number of transmit dimensions available to them for the sake of avoiding interference.
IiiA3 Scheduling and transmitter design
Having designed the RxBFMs, the effective DL channel matrices can be written as
(12) 
or
(13) 
for RN and UE in macrocell , respectively. We term the rows of these matrices as the SMCs of the associated transceivers, since each SMC corresponds to a distinct virtual multipleinput–singleoutput (MISO) channel between the associated transmitter as well as receiver, and then multiple MISOs can be multiplexed for composing a MIMO channel. A set of SMCs is generated for each of the two transmission phases, and each BS then distributively groups these SMCs according to the semiorthogonal user selection algorithm^{3}^{3}3This selection method aims for reducing the power loss imposed by the channel inversion operation of the ZFBF matrix [20, 27]., as described in [20, 39], given a semiorthogonality parameter . For the first transmission phase, up to SMCs may be served simultaneously by each BS, while avoiding ICI. The set of groupings available for BS is denoted by^{4}^{4}4N.B. Each group additionally contains the SMCs selected for the second phase, as it will be discussed in Section IIIB3. . The SMCs belonging to group , which are denoted by , are then the rows of the effective scheduled DL matrix, denoted by for macrocell . In order to avoid ICI between these selected SMCs of group , macrocell applies the ZFBF matrix, given in (14), by as the right channel inverse before using its TP, , where is a realvalued diagonal matrix, which normalizes the columns of for ensuring that the power assigned to each transmission remains unaffected.
(14) 
The effective endtoend channel power gains are then given by the squares of the diagonal entries in . For SMC in group of macrocell corresponding to a direct first phase BSUE link, the effective channel power gain is denoted by , while the effective channel power gain of the OCI link, originating from macrocell serving SMC group to UE in macrocell , is obtained from the element of
(15) 
corresponding to SMC at UE of macrocell , and is denoted by . In the case of the fullIA protocol, all OCI is avoided, thus , . The effective channel power gains for the BStoRN links, corresponding to SMCpair^{5}^{5}5Relaying links contain both a SMC for the BSRN link and a SMC for the RNUE link. , may be similarly obtained and are denoted by , whereas an OCI link is denoted by .
IiiB Beamforming design for the second phase
During the second phase, both the BSs and the RNs may transmit. Therefore, in a similar fashion to the first phase, the BS in cell adopts the precoding matrix , while RN in cell adopts the precoding matrix , which are again complexvalued matrices having a full columnrank. Additionally, the columns of these TP matrices are normalized. Due to the additional interference imposed by the transmissions of the RNs, it is necessary to reduce the number of transmit dimensions at the BSs even further in order to facilitate IA at the DL receivers. Additionally, note that each TP matrix used at the RNs consist of columns, since the information received by each RN during the first phase must be conveyed to the intended UE. The precoded channel matrices used during the second phase are thus given by (note that the transmitter indices are and , since these may be intercell channel matrices)
(16) 
and
(17) 
IiiB1 FullIA receiver design
The receiver design used during the second phase depends on whether the BS or a RN is selected to serve each UE within the same macrocell. Each of the possible transmitters may be examined for the sake of finding the most beneficial choice. For example, assuming that BS transmits to UE during the second phase, the OCI and ICI channel matrices may be concatenated to form (18). However, when assuming for example, that RN of macrocell transmits to UE , the combined interference matrix is defined by (19).
(18)  
(19)  
Therefore, in order to guarantee having receive dimensions at each UE, we have
(20)  
In both cases described above, the SVD may again be employed for finding the intersecting left nullspace of the precoded interference matrix. The RxBFM, , at UE in macrocell used during the second phase is then given by the rightmost (thus corresponding to its zero singular values) number of columns in the left singular matrix of , when the BS is the activated transmitter. By contrast, when assuming that RN is the activated transmitter, the rightmost number of columns in the ordered left singular matrix of specify the RxBFM matrix.
In conclusion, the BSs once again have to reduce the number of spatial transmission streams available to them in order to facilitate IA. In this case, their number is reduced from to . Additionally, each RN reduces the number of streams available for them to transmit from to . On one hand, when the BS is selected as the active transmitter for a particular UE using the fullIA protocol, a total of interference signal dimensions are aligned to signal dimensions, leaving signal dimensions free from interference. Thus, IA has been successfully employed. On the other hand, when a RN is selected as the activated transmitter for a particular UE, there is a total of interference signal dimensions, which are aligned to signal dimensions. Therefore, IA is only feasible at the UEs if we have . The constraint given by
(21) 
is additionally enforced in the fullIA protocol, so that the CCI can still be nulled when and a RN is selected as the active transmitter. However, IA is not employed in this case.
IiiB2 PartialIA receiver design
Although the effects of OCI are ignored when using this protocol, the ICI must be avoided. Thus, the interference matrix, assuming for example that the BS is the selected transmitter for UE in macrocell , is then given by
(22) 
By contrast, if RN of macrocell is selected as the transmitter for UE , then the interference matrix is given by
(23)  
which implies that
(24) 
is satisfied for ensuring that the UEs are capable of finding approximate RxBFMs, which completely null the ICI.
Thus, UE may employ the number of rightmost left singular columns in as its RxBFM, when the BS is the activated transmitter. By contrast, assuming that RN is the activated transmitter, the number of rightmost left singular columns in specify the RxBFM.
To summarize, the BSs reduce the number of spatial streams available to them from to , while the RNs reduce the number of their spatial streams from to . On one hand, when the BS is selected as the active transmitter for the partialIA protocol, a total of interference signal dimensions are aligned to signal dimensions, leaving signal dimensions free from interference. Thus, IA has been successfully employed. On the other hand, when a RN is selected as the activated transmitter, there are a total of interference signal dimensions, which are aligned to signal dimensions. Therefore, IA is only feasible for . However, the aforementioned RxBFMs are still capable of nulling the CCI, when a RN is selected as the active transmitter in the partialIA protocol and we have . But in this case the constraint given by (21) is not required, since it is already satisfied by (24).
IiiB3 Scheduling and transmitter design
In a similar fashion to the first phase, the effective DL channel matrices are given by
(25) 
and
(26) 
when the BS or RN is activated as the transmitter for UE belonging to macrocell , respectively. The rows of the DL TxBFMs corresponding to each transmitter form the SMCs for that transmitter, and they may be grouped at each BS according to the semiorthogonal user selection algorithm described above. Furthermore, in the second phase, each BS can select up to number of SMCs to serve simultaneously while avoiding ICI, whereas each RN may select number of SMCs. At BS (or RN of macrocell ), the selected SMCs of group form the rows of its effective scheduled DL matrix, denoted by (or ). The ZFBF matrix employed by BS or by RN of macrocell in the second phase is then given by the right inverse (27) or (28), respectively, where the realvalued diagonal matrices of and are required for normalizing the columns of and , respectively.
(27) 
(28) 
The effective channel power gains in the second phase are thus given by the squares of the diagonal entries in and . The effective channel power gain of a BSUE SMC of group associated with macrocell and UE is denoted by , while the RNUE effective channel power gain of SMCpair associated with RN of macrocell and UE may be denoted by . Similar to the first phase, the effective channel power gain of the OCI link originating from the BS of macrocell serving group to UE in macrocell , is obtained from the specific element of
(29) 
corresponding to SMC at UE of macrocell , which is denoted by . On the other hand, the effective channel power gain of the OCI link, originating from RN of macrocell serving group to UE of macrocell , is obtained from the element of
(30) 
corresponding to SMC at UE of macrocell , and is denoted by . In the case of the fullIA protocol, all OCI is avoided, thus we have , .
IiiC Achievable spectral efficiency and energy efficiency
Since we have mathematically decomposed the MIMO channels into effective SISO channels, we may directly employ the Shannon capacity bound for characterizing the achievable ESE performance, rather than relying on bounds derived for MIMO channels [40]. We begin by defining the signaltointerferenceplusnoiseratio (SINR) of the direct link SMCs belonging to group and intended for UE of macrocell during the first and the second phase as
(31) 
and
(32) 
respectively, where the total received OCI in the first and second phase has been denoted by (33) and (34), respectively, where is a function of , representing the RN index (similar to used before) associated with the SMCpair . For simplicity^{6}^{6}6If the level of interference is strong enough, then more sophisticated methods, such as multiuser detection, may be employed., the interference that was not avoided using IA is treated as noise.
(33) 
(34) 
The set contains the power control variables denoted by , , , and , . On the other hand, the set contains the group selection indicator variables, , , where , when the SMC group has been selected for macrocell , and otherwise. The total noise power across all subcarriers is given by , while is the signal to noise ratio (SNR) difference between the SNR at the discreteinput–continuousoutput memoryless channel (DCMC) capacity and the actual SNR required by the specific modulation and coding schemes of the practical physical layer transceivers employed [41].
The SINR of the BSRN SMC belonging to group of macrocell and intended for RN may be expressed as
(35) 
while the SINR of the corresponding RNUE link may be formulated as
(36) 
where the total received OCI of the BSRN and RNUE links are given by (37) and (38), respectively.
(37) 
(38) 
The achievable SE of the direct first and second phase transmissions can be respectively written as
(39) 
and
(40) 
where the prelog factor of accounts for the fact that the transmission period has been split into two phases. When using the DF protocol, the achievable SE of the relaying link is limited by the weaker of the BSRN and RNUE links [5], which is given by
Thus the total achievable SE of macrocell is given by (42).