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Intrinsic interference mitigating coordinated beamforming for the FBMC/OQAM based downlink
EURASIP Journal on Advances in Signal Processing volume 2014, Article number: 86 (2014)
Abstract
In this work, we propose intrinsic interference mitigating coordinated beamforming (IIMCBF)based transmission strategies for the downlink of multiuser multipleinputmultipleout (MIMO) systems and coordinated multipoint (CoMP) systems where filter bank based multicarrier with offset quadrature amplitude modulation (FBMC/OQAM) is employed. Our goal is to alleviate the dimensionality constraint imposed on the stateoftheart solutions for FBMC/OQAMbased space division multiple access that the total number of receive antennas of the users must not exceed the number of transmit antennas at the base station. First, two IIMCBF algorithms are developed for a singlecell multiuser MIMO downlink system. The central idea is to jointly and iteratively calculate the precoding matrix and decoding matrix for each subcarrier to mitigate the multiuser interference as well as the intrinsic interference inherent in FBMC/OQAMbased systems. Second, for a CoMP downlink scenario where partial coordination among the base stations is considered, the application of coordinated beamformingbased transmission schemes is further investigated. An appropriate IIMCBF technique is proposed. Simulation results show that when the number of transmit antennas at the base station is equal to the total number of receive antennas of the users, the proposed IIMCBF algorithm outperforms the existing transmission strategies for FBMC/OQAMbased multiuser MIMO downlink systems. Moreover, we evaluate the performances of the IIMCBF schemes in the downlink of multiuser MIMO systems and CoMP systems where the total number of receive antennas of users exceeds the number of transmit antennas at the base station. It is observed that by employing the IIMCBF techniques, FBMC/OQAM systems achieve a similar bit error rate (BER) performance as its orthogonal frequency division multiplexing with the cyclic prefix insertion (CPOFDM)based counterpart while exhibiting superiority in terms of a higher spectral efficiency, a greater robustness against synchronization errors, and a lower outofband radiation. In the presence of residual carrier frequency offsets, FBMC/OQAM systems provide a much better performance compared to the CPOFDMbased system, which corroborates the theoretical analysis that FBMC/OQAM systems are more immune to the lack of perfect synchronization. In addition, numerical results with respect to the convergence behavior of the IIMCBF techniques are presented, and the computational complexity issue is also addressed.
1 Introduction
As a promising alternative to orthogonal frequency division multiplexing with the cyclic prefix insertion (CPOFDM), filter bank based multicarrier modulation (FBMC) has received great research attention in recent years. Using spectrally wellcontained synthesis and analysis of filter banks at the transmitter and at the receiver [1, 2], FBMC has an agile spectrum. Thereby, the outofband radiation is lower compared to CPOFDM, and it is consequently beneficial to choose FBMC over CPOFDM for asynchronous scenarios [3, 4] or to achieve an effective utilization of spectrum holes [5, 6]. Moreover, in systems where filter bank based multicarrier with offset quadrature amplitude modulation (FBMC/OQAM) is employed, the fact that the insertion of the CP is not required as in CPOFDMbased systems leads to a higher spectral efficiency.
In FBMC/OQAM systems, the real and imaginary parts of each complexvalued data symbol are staggered by half of the symbol period [2, 7] such that the desired signal and the intrinsic interference are separated in the real domain and in the pure imaginary domain, respectively. Different approaches of canceling the intrinsic interference have been proposed based on different assumptions on the frequency selectivity of the channel. In [8] and [9] where receive processing techniques have been developed for multipleinputmultipleout (MIMO) FBMC/OQAM systems, it is assumed that the channel frequency responses of adjacent subcarriers do not vary. Consequently, the intrinsic interference is canceled by taking the real part of the resulting signal after the equalization.
To alleviate the constraint on the frequency selectivity of the channel, a zero forcing (ZF)based approach has been proposed in [10] for multistream transmissions in a MIMO FBMC/OQAM system where the channel is not restricted to flat fading. More details of the performance analysis of this algorithm have been presented in [11]. However, the work in [10] and [11] is limited to the case where the number of receive antennas does not exceed the number of transmit antennas. In addition, the authors have shown numerically and have also pointed out that their proposed approach only provides a satisfactory performance in an asymmetric configuration, i.e., when the number of transmit antennas is larger than the number of receive antennas. Based on the concept of mitigating the intrinsic interference mentioned above for pointtopoint MIMO FBMC/OQAM systems, the authors in [12] have adapted the conventional spatial TomlinsonHarashima precoder (STHP) to an FBMC/OQAMbased multipleinputsingleoutput broadcast channel (MISOBC) which results in a new nonlinear precoder. It is known that nonlinear precoders have a higher computational complexity compared to linear precoders. Moreover, the nonlinear precoding technique in [12] is restricted to the case where each user is equipped with only a single receive antenna. On the other hand, a block diagonalization (BD)based linear precoder has been developed in [13] for the FBMC/OQAMbased multiuser MIMO downlink with space division multiple access (SDMA). It adopts the central idea of BD [14] to mitigate the multiuser interference and then uses the ZFbased approach [10] to deal with the intrinsic interference cancelation for the resulting equivalent singleuser transmissions. Consequently, this algorithm inherits the drawback of the ZFbased scheme such that it also fails to achieve a good performance in a symmetric multiuser MIMO downlink setting, where the number of transmit antennas at the base station is equal to the total number of receive antennas of the users. In addition, this linear precoder suffers from the dimensionality constraint that the total number of receive antennas of the users must not exceed the number of transmit antennas at the base station. Note that similarly as in these publications, we focus on scenarios where the channel on each subcarrier is flat fading in this work.
Although the case of highly frequency selective channels is beyond the scope of this paper, we also refer to two techniques designed for such scenarios to complete the review of the stateoftheart transmission schemes for FBMC/OQAMbased multiuser downlink systems. The linear precoder in [15] has a structure of a filter applied on each subcarrier and its two adjacent subcarriers at twice the symbol rate. It also only focuses on the setting where the number of transmit antennas at the base station is not smaller than the total number of receive antennas of the users. Moreover, it only allows each user to have a single receive antenna, and consequently only one data stream can be transmitted to each user. In [16], transmission schemes for FBMC/OQAMbased multiuser MISO downlink systems have been developed also considering highly frequency selective channels. The authors have devised two different minimum mean square error (MMSE) approaches. A closeform solution is provided in the first one, while the second scheme involves a joint transmitter and receiver design via an iterative procedure. Note that in the aforementioned publications on the FBMC/OQAMbased multiuser downlink, the impact of the residual carrier frequency offsets (CFOs) has not been investigated.
To the best of the authors’ knowledge, FBMC/OQAMbased coordinated multipoint (CoMP) techniques have not been studied in the literature. In the context of CPOFDM, the corresponding research has been very fruitful, and CoMP is known as one of the advanced communication techniques that are able to provide benefits of reduced intercell interference and enhanced cell edge throughput [17–21]. In this work, we focus on downlink CoMP and the schemes that belong to the category of joint transmission [18]. When the full cooperation between the base stations of adjacent cells is assumed, the channel state information (CSI) and signals for all users are shared by the base stations. In this case, a virtual multiuser MIMO downlink setting is formed, where the transmit antennas are geographically separated. Thereby, the transmission strategies that have been developed for the singlecell multiuser MIMO downlink can be employed. Nevertheless, such a full cooperation scheme is not practical due to issues such as it requires excessive information exchange resulting in a large signaling overhead, and the CSI of all users is very hard to acquire [19]. As a more realistic solution, partial cooperation schemes have been proposed in [19, 21, 22], where the users are classified into two categories, cell (or in some papers [19, 21], cluster that consists of multiple cells) interior users and cell (cluster) edge users. The base stations of adjacent cells (clusters) transmit the same signals to the cell (cluster) edge users and coordinate beamforming techniques that rely on the limited cooperation between the cells (clusters) (e.g., the exchange of the beamforming matrices for cell (cluster) edge users) are employed to suppress the intracell (cluster) and intercell (cluster) interference. For these downlink CoMP scenarios, it is more likely that the total number of receive antennas of the users served by one base station is larger than the number of transmit antennas. Thus, transmission strategies that are able to tackle such a case are required. Note that in the aforementioned publications on CPOFDMbased downlink CoMP, perfect synchronization is assumed. However, the asynchronous nature of the interference in the downlink CoMP setting is emphasized in [23]. It has been shown in [23] that the lack of perfect synchronization causes a performance degradation. Such a fact greatly motivates the use of FBMC as a replacement of CPOFDM, as FBMC is more robust against synchronization errors compared to CPOFDM.
In this paper, we design intrinsic interference mitigating coordinated beamforming (IIMCBF) schemes^{a} for the FBMC/OQAMbased multiuser MIMO downlink systems and CoMP downlink systems without restricting the configuration with respect to the number of transmit antennas and the number of receive antennas. First, considering the symmetric singlecell multiuser MIMO downlink setting where the number of transmit antennas at the base station is equal to the total number of receive antennas of the users, we propose to compute the precoding matrix and the decoding matrix jointly in an iterative procedure for each subcarrier. Different choices of the decoding matrix in the initialization step are recommended for different scenarios. For CPOFDMbased multiuser MIMO downlink systems, there have been some publications on coordinated beamforming techniques [19, 24] proposed to cope with the dimensionality constraint imposed on BDbased precoding algorithms [14]. Inspired by these works, an IIMCBF scheme specifically for FBMC/OQAMbased systems is developed to alleviate the same dimensionality constraint that all stateoftheart transmission strategies for FBMC/OQAMbased multiuser downlink settings suffer from. It handles the mitigation of the multiuser interference as well as the intrinsic interference. Moreover, we investigate FBMC/OQAMbased CoMP techniques for the first time and provide an extension of the IIMCBF scheme designed for the FBMC/OQAMbased multiuser MIMO downlink system. To evaluate the performance of the proposed IIMCBF algorithms, we have performed extensive simulations. The bit error rate (BER) as well as sum rate performances are shown, and the convergence behavior of the developed coordinated beamforming techniques is also investigated via numerical simulations. It can be observed that the number of iterations required for the convergence in these IIMCBF schemes is acceptable. Thus, the additional computational complexity is not prohibitive compared to the closeform algorithms which fail in scenarios where the total number of receive antennas of the users exceeds the number of transmit antennas. In addition, we also investigate the effects of the residual CFOs and demonstrate the superiority of the FBMC/OQAMbased system over its CPOFDMbased counterpart in the tolerance of synchronization errors.
The remainder of the paper is organized as follows: Section 2 introduces the data model of a singlecell multiuser MIMO FBMC/OQAM system and reviews two stateoftheart transmission strategies for such a system. In Section 3, the two proposed algorithms IIMCBF 1 and IIMCBF 2 are described in detail for the case where the number of transmit antennas at the base station is equal to the total number of receive antennas of the users and the case where the former is smaller than the latter, respectively. Section 4 focuses on the CoMP downlink and presents another coordinated beamformingbased transmission scheme, namely ‘IIMCBF 3’. Numerical results are shown in Section 5, before conclusions are drawn in Section 6.
Notation Matrices and vectors are denoted by boldfaced uppercase and lowercase letters, respectively. We use the superscripts ^{T}, ^{H}, and ^{−1} for transpose, Hermitian transpose, and matrix inversion, respectively. An M×M identity matrix is denoted by I_{ M }. The Frobenius norm of a matrix is denoted by ∥·∥_{F}. Moreover, Re{·} symbolizes the real part of the input argument, while Im{·} represents the imaginary part. For a matrix A, A(m,n) denotes its (m,n)th entry.
2 System model
In a multiuser MIMO downlink system where SDMA is employed, one base station equipped with {M}_{\mathrm{T}}^{\left(\mathit{\text{BS}}\right)} transmit antennas transmits to Q users at the same time and on the same frequency. The number of receive antennas of the q th user is denoted by {M}_{{\mathrm{R}}_{q}}, and the total number of receive antennas of all users severed simultaneously is then {M}_{\mathrm{R}}^{\left(\text{tot}\right)}=\sum _{q=1}^{Q}{M}_{{\mathrm{R}}_{q}}. Assuming that the channel on each subcarrier can be treated as flat fading [10, 11, 13], the combined receive vector on the k th subcarrier and at the n th time instant is denoted by {\mathit{y}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right]={\left[\begin{array}{cccc}{\mathit{y}}_{1,k}^{\mathrm{T}}\left[\phantom{\rule{0.3em}{0ex}}n\right]& {\mathit{y}}_{2,k}^{\mathrm{T}}\left[\phantom{\rule{0.3em}{0ex}}n\right]& \cdots & {\mathit{y}}_{Q,k}^{\mathrm{T}}\left[\phantom{\rule{0.3em}{0ex}}n\right]\end{array}\right]}^{\mathrm{T}}\in {\in}^{{M}_{\mathrm{R}}^{\left(\text{tot}\right)}}where the received signals of all Q users are stacked and can be represented by
Here {\mathit{H}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{\mathrm{R}}^{\left(\text{tot}\right)}\times {M}_{\mathrm{T}}^{\left(\text{BS}\right)}} denotes the combined channel matrix of all Q users^{b} and is written as
where {\mathit{H}}_{q,k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{{\mathrm{R}}_{q}}\times {M}_{\mathrm{T}}^{\left(\text{BS}\right)}} represents the channel frequency response between the base station and the q th user, q=1,2,…,Q. The data vector {\mathit{d}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{R}}^{d} with the total number of spatial streams d=\sum _{q=1}^{Q}{d}_{q} is expressed as
where {\mathit{d}}_{q,k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{R}}^{{d}_{q}} denotes the desired signal for the q th user on the k th subcarrier and at the n th time instant when (k+n) is even^{c}, and d_{ q } denotes the number of spatial streams sent to the q th user. The terms c_{ i ℓ }d_{ ℓ }[ i] in (1) contribute to the intrinsic interference and are pure imaginary if the prototype pulse satisfies the perfect reconstruction property [7, 12], where ℓ=k−1, k, k+1, i=n−3,…,n+3, and (ℓ,i)≠(k,n). The coefficients c_{ i ℓ } (cf. Table 1) represent the system impulse response determined by the synthesis and analysis filters. The PHYDYAS prototype filter [25] is used, and the overlapping factor is chosen to be K=4. For more details about FBMC/OQAM systems, the reader is referred to [7]. Moreover, {\mathit{n}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{\mathrm{R}}^{\left(\text{tot}\right)}} denotes the combined additive white Gaussian noise vector with variance {\sigma}_{n}^{2}, and the noise is assumed to be spatially uncorrelated. For the q th user, the noise autocorrelation matrix is written as {\sigma}_{n}^{2}{\mathit{I}}_{{M}_{{\mathrm{R}}_{q}}}.
Furthermore, {\mathit{F}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{\mathrm{T}}^{\left(\text{BS}\right)}\times d} contains the precoding matrices for all users
where {\mathit{F}}_{q,k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{\mathrm{T}}^{\left(\text{BS}\right)}\times {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}}, q=1,2,…,Q are calculated to mitigate the multiuser interference by employing, e.g., BD [14] such that a multiuser MIMO downlink system is decoupled into parallel equivalent singleuser transmissions. Here, M T_{ q }(eq) symbolizes the resulting equivalent number of transmit antennas for the q th user. It is determined by the precoding scheme employed to suppress the multiuser interference, which will be explained in detail in Section 3. In addition, {\mathit{G}}_{q,k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}\times {d}_{q}}, q=1,2,…,Q are the transmit beamforming matrices for the equivalent singleuser systems. Note that throughout this work, equal power allocation on the spatial streams and subcarriers is assumed.
2.1 Straightforward extension of the transmission strategy as in case of CPOFDM
In some publications on MIMO FBMC/OQAM systems, such as [8] and [9], it is assumed that the channel stays constant across adjacent subcarriers and during consecutive symbol periods, i.e.,
Since the precoding is performed on a persubcarrier basis, i.e., the calculation of the precoding matrices for a certain subcarrier is solely determined by the channel on the same subcarrier, the precoding matrices are also the same on adjacent subcarriers. Therefore, the combined received signal on the k th subcarrier and at the n th time instant can be expressed as
where {\stackrel{~}{\mathit{d}}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right] contains the realvalued desired signal and the pure imaginary intrinsic interference
Considering {\stackrel{~}{\mathit{d}}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right] as an equivalent transmitted signal, (5) resembles the data model of a CPOFDMbased multiuser MIMO downlink system. Consequently, transmission strategies that have been developed for multiuser MIMO CPOFDM downlink systems can be straightforwardly extended to their FBMC/OQAMbased counterparts where only one additional step is required, i.e., taking the real part of the resulting signal after the multiplication by the decoding matrix
where {\mathit{D}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{\mathrm{R}}^{\left(\text{tot}\right)}\times d} is the combined blockdiagonal decoding matrix on the k th subcarrier and at the n th time instant that contains the decoding matrices {\mathit{D}}_{q,k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{{\mathrm{R}}_{q}}\times {d}_{q}}, q=1,2,…,Q, for the Q users, respectively. It is worth mentioning that there is no cooperation among the users, and the decoding matrix for each user is computed separately.
The concept of this transmission scheme is simple, does not induce much additional processing compared to CPOFDMbased systems, and directly applies the stateoftheart transmit as well as receive processing techniques developed for the CPOFDMbased multiuser MIMO downlink. Nevertheless, it relies on the impractical assumption that the channel is flat fading and time invariant. In case of frequency selective channels, this transmission strategy fails to completely eliminate the intrinsic interference inherent in FBMC/OQAM systems and thus suffers from a performance degradation as also shown in the simulations section.
2.2 Block diagonalizationbased approach
In [13] a BDbased precoding algorithm has been proposed for FBMC/OQAMbased multiuser MIMO downlink systems, where {M}_{\mathrm{T}}^{\left(\text{BS}\right)}\ge {M}_{\mathrm{R}}^{\left(\text{tot}\right)}. First, the BD algorithm [14] is used to calculate the first part of the precoding matrix F_{q,k}[ n], q=1,2,…,Q, for the Q users to mitigate the multiuser interference. By rendering F_{q,k}[ n] for the q th user to lie in the null space of all the other users’ combined channel matrix, it is ensured that
Consequently, the received signal of the q th user is expressed as
where {\mathit{H}}_{q,k}\left[\phantom{\rule{0.3em}{0ex}}n\right]{\mathit{F}}_{q,k}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{{\mathrm{R}}_{q}}\times {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}}, q=1,2,…,Q, can be treated as the equivalent channels for parallel singleuser transmissions. Let us expand the real part of the received signal of the q th user
To cancel the intrinsic interference, the second term on the righthand side of (10) should be rendered as zero. This can be achieved by computing the second part of the precoding matrix such that [13]
where F_{ q }G_{ q } represents the precoding matrix for the q th user on a certain subcarrier and at a certain time instant, and H_{ q } denotes the channel matrix for the q th user on the same subcarrier and at the same time instant. From now on, the time and frequency indices are ignored as the precoding is performed on a persubcarrier basis. It should be noted that the optimization of the power allocation is originally incorporated in the BDbased technique [13]. Nevertheless, this part of the algorithm is not reviewed in detail here, since optimizing the power allocation is beyond the scope of this paper and equal power allocation is assumed. In the simulations section, when using the BDbased technique as a benchmark scheme, equal power allocation is also adopted to ensure a fair comparison.
This approach outperforms the straightforward extension of the CPOFDM case in the sense that it is able to tolerate a certain level of the frequency selectivity of the channel. However, it suffers from the dimensionality constraint that the number of transmit antennas at the base station has to be larger than or equal to the total number of receive antennas of the users, i.e., {M}_{\mathrm{T}}^{\left(\text{BS}\right)}\ge {M}_{\mathrm{R}}^{\left(\text{tot}\right)}. For the case where {M}_{\mathrm{T}}^{\left(\text{BS}\right)}={M}_{\mathrm{R}}^{\left(\text{tot}\right)}, this scheme is not able to provide a satisfactory performance as shown later in the simulations section.
3 Coordinated beamforming for the singlecell multiuser MIMO downlink
Based on the philosophy on the precoding design for multiuser MIMO downlink settings first proposed in [26], the precoding matrix for each user is decomposed into two parts as already shown in (4). For the q th user, the first part F_{ q } eliminates the multiuser interference^{d}. The second part G_{ q } plays the role of the transmit beamforming matrix for each equivalent singleuser transmission after the multiuser interference cancelation. Due to the fact that the intrinsic interference resides in FBMC/OQAMbased systems, we propose to further decouple G_{ q } into two parts, i.e.,
where {\mathit{G}}_{q,1}\in {\mathbb{C}}^{{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}\times {M}_{{x}_{q}}} is computed to suppress the intrinsic interference, and {\mathit{G}}_{q,2}\in {\mathbb{R}}^{{M}_{{x}_{q}}\times {d}_{q}} is used for the spatial mapping.
3.1 The IIMCBF 1 algorithm
We first propose a coordinated beamformingbased transmission scheme IIMCBF 1 for symmetric multiuser MIMO downlink settings where {M}_{\mathrm{T}}^{\left(\text{BS}\right)}={M}_{\mathrm{R}}^{\left(\text{tot}\right)}. BD [14] is employed to calculate the first part of the precoding matrices {\mathit{F}}_{q}\in {\mathbb{C}}^{{M}_{\mathrm{T}}^{\left(\text{BS}\right)}\times {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}}, q=1,2,…,Q. For the q th user, define a matrix {\stackrel{~}{\mathit{H}}}_{\mathrm{e}}\in {\mathbb{C}}^{({M}_{\mathrm{R}}^{\left(\text{tot}\right)}{M}_{{\mathrm{R}}_{q}})\times {M}_{\mathrm{T}}^{\left(\text{BS}\right)}} as
which contains the channel matrices of all the other users. The precoding matrix F_{ q } for the q th user is obtained as {\mathit{F}}_{q}={\stackrel{~}{\mathit{V}}}_{{\mathrm{e}}_{(q,0)}}\in {\mathbb{C}}^{{M}_{\mathrm{T}}^{\left(\text{BS}\right)}\times {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}}, where {\stackrel{~}{\mathit{V}}}_{{\mathrm{e}}_{(q,0)}} contains the last {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)} right singular vectors that form an orthonormal basis for the null space of {\stackrel{~}{\mathit{H}}}_{\mathrm{e}}[14]. The resulting equivalent number of transmit antennas {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}={M}_{\mathrm{T}}^{\left(\text{BS}\right)}\sum _{g=1,g\ne q}^{Q}{M}_{{\mathrm{R}}_{g}} is equal to {M}_{{\mathrm{R}}_{q}}. The reader is referred [14] for more details of the BD algorithm.
For the q th user, we propose to jointly and iteratively update the second part of its precoding matrix G_{ q } and its decoding matrix. An equivalent channel matrix {\mathit{H}}_{{\mathrm{e}}_{q}} is defined as
where {\mathit{D}}_{q}\in {\mathbb{R}}^{{M}_{{\mathrm{R}}_{q}}\times {d}_{q}} is the realvalued decoding matrix.
The proposed coordinated beamforming algorithm is summarized as follows:

Step 1: Initialize the decoding matrix {\mathit{D}}_{q}^{\left(0\right)}\in {\mathbb{R}}^{{M}_{{\mathrm{R}}_{q}}\times {d}_{q}}, set the iteration index p to zero, and set a threshold ε for the stopping criterion. The decoding matrix is generated randomly if the current subcarrier is the first one; otherwise, set the decoding matrix as the one calculated for the previous subcarrier [24].

Step 2: Set p←p+1 and calculate the equivalent channel matrix {\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)} in the p th iteration as
{\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}={\mathit{D}}_{q}^{{(p1)}^{\mathrm{T}}}{\mathit{H}}_{q}{\mathit{F}}_{q}\in {\mathbb{C}}^{{d}_{q}\times {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}}.(15) 
Define a matrix
{\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}=\left[\begin{array}{ll}\text{Im}\left\{{\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}\right\}& \text{Re}\left\{{\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}\right\}\end{array}\right]\in {\mathbb{R}}^{{d}_{q}\times 2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}}.(16) 
Step 3: Calculate the precoding matrix {\mathit{G}}_{q}^{\left(p\right)}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\mathit{G}}_{q,1}^{\left(p\right)}{\mathit{G}}_{q,2}^{\left(p\right)} for the p th iteration. First, we perform the singular value decomposition (SVD) of {\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)} as
{\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}={\mathit{U}}_{q,1}^{\left(p\right)}{\mathit{\Sigma}}_{q,1}^{\left(p\right)}{\mathit{V}}_{q,1}^{{\left(p\right)}^{\mathrm{T}}}.(17) 
Denoting the rank of {\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)} as {r}_{q}^{\left(p\right)}, we define {\mathit{V}}_{q,1,0}^{\left(p\right)}\in {\mathbb{R}}^{2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}\times {M}_{{x}_{q}}} as containing the last {M}_{{x}_{q}}=2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}{r}_{q}^{\left(p\right)} right singular vectors that form an orthonormal basis for the null space of {\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}. Hence, {\mathit{G}}_{q,1}^{\left(p\right)} for the p th iteration can be obtained via
{\mathit{V}}_{1,0}^{\left(p\right)}=\left[\begin{array}{l}\text{Re}\left\{{\mathit{G}}_{q,1}^{\left(p\right)}\right\}\\ \text{Im}\left\{{\mathit{G}}_{q,1}^{\left(p\right)}\right\}\end{array}\right]\in {\mathbb{R}}^{2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}\times {M}_{{x}_{q}}}.(18) 
To further calculate {\mathit{G}}_{q,2}^{\left(p\right)}, the following equivalent channel matrix after the cancelation of the intrinsic interference for the p th iteration is defined as
{\stackrel{~}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}=\text{Re}\left\{{\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}{\mathit{G}}_{q,1}^{\left(p\right)}\right\}\in {\mathbb{R}}^{{d}_{q}\times {M}_{{x}_{q}}}.(19) 
Further calculate the SVD of {\stackrel{~}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)} and define {\mathit{V}}_{q,2,1}^{\left(p\right)}\in {\mathbb{R}}^{{M}_{{x}_{q}}\times {d}_{q}} as containing the first d_{ q } right singular vectors. Thereby, {\mathit{G}}_{q,2}^{\left(p\right)} is obtained as {\mathit{G}}_{q,2}^{\left(p\right)}={\mathit{V}}_{q,2,1}^{\left(p\right)}.

Step 4: Update the decoding matrix based on the equivalent channel matrix after the cancelation of the intrinsic interference where only the processing at the transmitter is considered
{\stackrel{~}{\mathit{H}}}_{{\text{etx}}_{q}}^{\left(p\right)}=\text{Re}\left\{{\mathit{H}}_{q}{\mathit{F}}_{q}{\mathit{G}}_{q}^{\left(p\right)}\right\}\in {\mathbb{R}}^{{M}_{{\mathrm{R}}_{q}}\times {d}_{q}}.(20) 
When the MMSE receiver^{e} is used, the decoding matrix has the following form
{\mathit{D}}_{q}^{\left(p\right)}={\stackrel{~}{\mathit{H}}}_{{\text{etx}}_{q}}^{\left(p\right)}{\left({\stackrel{~}{\mathit{H}}}_{{\text{etx}}_{q}}^{{\left(p\right)}^{\mathrm{T}}}{\stackrel{~}{\mathit{H}}}_{{\text{etx}}_{q}}^{\left(p\right)}+{\sigma}_{n}^{2}{\mathit{I}}_{{d}_{q}}\right)}^{1}.(21) 
Step 5: Calculate the term Δ(G_{ q }) defined as
\Delta \left({\mathit{G}}_{q}\right)={\u2225{\mathit{G}}_{q}^{\left(p\right)}{\mathit{G}}_{q}^{(p1)}\u2225}_{\mathrm{F}}^{2},(22) 
which measures the change of the precoding matrix G_{ q }. If Δ(G_{ q })<ε, the convergence is achieved, and the iterative procedure terminates. Otherwise go back to Step 2.
It is important to note that in the special case where {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}={M}_{{\mathrm{R}}_{q}}={d}_{q}+1, we propose to compute the decoding matrix in the initialization step as follows. Consequently, it is observed that the coordinated beamforming technique only needs two iterations to converge. First, calculate the SVD of {\stackrel{\u030c}{\mathit{H}}}_{q}\in {\mathbb{R}}^{{M}_{{\mathrm{R}}_{q}}\times 2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}}
and let {\mathit{V}}_{q,1,0}^{\left(0\right)} contain the last (2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}{M}_{{\mathrm{R}}_{q}}) right singular vectors. Defining {\mathit{G}}_{q,1}^{\left(0\right)} via
we compute {\mathit{U}}_{q,2,1}^{\left(0\right)}\in {\mathbb{R}}^{{M}_{{\mathrm{R}}_{q}}\times {d}_{q}} from the SVD of \text{Re}\left\{{\mathit{H}}_{q}{\mathit{F}}_{q}{\mathit{G}}_{q,1}^{\left(0\right)}\right\} such that it contains the first d_{ q } left singular vectors. Then the decoding matrix for the initialization step is chosen as
3.2 The IIMCBF 2 algorithm
In multiuser MIMO FBMC/OQAM downlink systems where the total number of receive antennas of the users exceeds the number of transmit antennas at the base station, the BD algorithm [14] or the BDbased technique [13] cannot be employed to achieve the multiuser interference or the intrinsic interference suppression.
Therefore, we propose another iterative procedure to jointly compute the precoding matrix and the decoding matrix. Let us first define an equivalent combined channel matrix after the decoding at the user terminals as
Unlike the coordinated beamforming schemes in [19] or [24], the decoding matrices {\mathit{D}}_{q}\in {\mathbb{R}}^{{M}_{{\mathrm{R}}_{q}}\times {d}_{q}}, q=1,2,…,Q are forced to be realvalued. Although the BDbased concept cannot be employed on the physical channel due to the dimensionality constraint, it can be used on this equivalent channel.
The proposed IIMCBF 2 algorithm for the FBMC/OQAMbased multiuser MIMO downlink system is described in detail as follows:

Step 1: Initialize the decoding matrices {\mathit{D}}_{q}^{\left(0\right)}\in {\mathbb{R}}^{{M}_{{\mathrm{R}}_{q}}\times {d}_{q}} (q=1,…,Q), set the iteration index p to zero, and set a threshold ε for the stopping criterion. If the current subcarrier is the first one, the decoding matrices are generated randomly; otherwise, set the decoding matrices as those calculated for the previous subcarrier [24].

Step 2: Set p←p+1 and calculate the equivalent channel matrix {\mathit{H}}_{\mathrm{e}}^{\left(p\right)} in the p th iteration as
{\mathit{H}}_{\mathrm{e}}^{\left(p\right)}={\left[\begin{array}{llll}{\mathit{H}}_{{\mathrm{e}}_{1}}^{{\left(p\right)}^{\mathrm{T}}}& {\mathit{H}}_{{\mathrm{e}}_{2}}^{{\left(p\right)}^{\mathrm{T}}}& \cdots & {\mathit{H}}_{{\mathrm{e}}_{Q}}^{{\left(p\right)}^{\mathrm{T}}}\end{array}\right]}^{\mathrm{T}},(27) 
where {\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}={\mathit{D}}_{q}^{{(p1)}^{\mathrm{T}}}{\mathit{H}}_{q} is the equivalent channel matrix for the q th user in the p th iteration.

Step 3: Calculate the precoding matrices {\mathit{F}}_{q}^{\left(p\right)} (q=1,…,Q) in the p th iteration to cancel the multiuser interference based on the BD algorithm [14]. For the q th user, define a matrix {\stackrel{~}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}\in {\mathbb{C}}^{(d{d}_{q})\times {M}_{\mathrm{T}}^{\left(\text{BS}\right)}} as
{\stackrel{~}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}={\left[\begin{array}{llllll}{\mathit{H}}_{{\mathrm{e}}_{1}}^{{\left(p\right)}^{\mathrm{T}}}& \phantom{\rule{0.3em}{0ex}}\cdots \phantom{\rule{0.3em}{0ex}}& {\mathit{H}}_{{\mathrm{e}}_{q1}}^{{\left(p\right)}^{\mathrm{T}}}& {\mathit{H}}_{{\mathrm{e}}_{q+1}}^{{\left(p\right)}^{\mathrm{T}}}& \phantom{\rule{0.3em}{0ex}}\cdots \phantom{\rule{0.3em}{0ex}}& {\mathit{H}}_{{\mathrm{e}}_{Q}}^{{\left(p\right)}^{\mathrm{T}}}\end{array}\right]}^{\mathrm{T}},(28) 
which contains the equivalent channel matrices of all the other users that are calculated in Step 2. The precoding matrix {\mathit{F}}_{q}^{\left(p\right)} for the q th user in the p th iteration is obtained as {\mathit{F}}_{q}^{\left(p\right)}={\stackrel{~}{\mathit{V}}}_{{\mathrm{e}}_{(q,0)}}^{\left(p\right)}\in {\mathbb{C}}^{{M}_{\mathrm{T}}^{\left(\text{BS}\right)}\times {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}}, where {\stackrel{~}{\mathit{V}}}_{{\mathrm{e}}_{(q,0)}}^{\left(p\right)} contains the last {M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)} right singular vectors that form an orthonormal basis for the null space of {\stackrel{~}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}[14]. To this end, the multiuser MIMO downlink transmission is decoupled into parallel equivalent singleuser MIMO transmissions that will be considered in the following steps.

Step 4: Define a matrix {\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}\in {\mathbb{R}}^{{d}_{q}\times 2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}} for the q th user based on its equivalent channel matrix {\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}{\mathit{F}}_{q}^{\left(p\right)} after the cancelation of the multiuser interference
{\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}=\left[\begin{array}{ll}\text{Im}\left\{{\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}{\mathit{F}}_{q}^{\left(p\right)}\right\}& \text{Re}\left\{{\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}{\mathit{F}}_{q}^{\left(p\right)}\right\}\end{array}\right].(29) 
Step 5: Calculate the precoding matrix {\mathit{G}}_{q}^{\left(p\right)}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\mathit{G}}_{q,1}^{\left(p\right)}{\mathit{G}}_{q,2}^{\left(p\right)} for the q th user in the p th iteration. First, we perform the SVD of {\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)} and obtain {\mathit{V}}_{q,1,0}^{\left(p\right)}\in {\mathbb{R}}^{2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}\times {M}_{{x}_{q}}} as containing the last {M}_{{x}_{q}}=2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}{r}_{q}^{\left(p\right)} right singular vectors that form an orthonormal basis for the null space of {\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}, where {r}_{q}^{\left(p\right)} denotes the rank of {\stackrel{\u030c}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}. Hence, {\mathit{G}}_{q,1}^{\left(p\right)}\in {\in}^{{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}\times {M}_{{x}_{q}}} can be obtained via
{\mathit{V}}_{q,1,0}^{\left(p\right)}=\left[\begin{array}{l}\text{Re}\left\{{\mathit{G}}_{q,1}^{\left(p\right)}\right\}\\ \text{Im}\left\{{\mathit{G}}_{q,1}^{\left(p\right)}\right\}\end{array}\right]\in {\mathbb{R}}^{2{M}_{{\mathrm{T}}_{q}}^{\left(\text{eq}\right)}\times {M}_{{x}_{q}}}(30) 
such that (11) is fulfilled to achieve the mitigation of the intrinsic interference.

Now we define the following equivalent channel matrix after canceling the intrinsic interference for the q th user in the p th iteration
{\stackrel{\u0304}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)}=\text{Re}\left\{{\mathit{H}}_{{\mathrm{e}}_{q}}^{\left(p\right)}{\mathit{F}}_{q}^{\left(p\right)}{\mathit{G}}_{q,1}^{\left(p\right)}\right\}\in {\mathbb{R}}^{{d}_{q}\times {M}_{{x}_{q}}}.(31) 
Further calculate the SVD of {\stackrel{\u0304}{\mathit{H}}}_{{\mathrm{e}}_{q}}^{\left(p\right)} and define {\mathit{V}}_{q,2,1}^{\left(p\right)}\in {\mathbb{R}}^{{M}_{{x}_{q}}\times {d}_{q}} as containing the first d_{ q } right singular vectors. Then {\mathit{G}}_{q,2}^{\left(p\right)} is obtained as {\mathit{G}}_{q,2}^{\left(p\right)}={\mathit{V}}_{q,2,1}^{\left(p\right)}.

Step 6: Update the decoding matrix for each user based on the realvalued equivalent channel matrix where the processing at the transmitter and the procedure of taking the real part of the receive signal are taken into account
{\mathit{H}}_{{\text{etx}}_{q}}^{\left(p\right)}=\text{Re}\left\{{\mathit{H}}_{q}{\mathit{F}}_{q}^{\left(p\right)}{\mathit{G}}_{q}^{\left(p\right)}\right\}\in {\mathbb{R}}^{{M}_{{\mathrm{R}}_{q}}\times {d}_{q}},\phantom{\rule{1em}{0ex}}q=1,\dots ,\mathrm{Q.}(32) 
When the MMSE receiver is used, the update of the decoding matrix in the p th iteration for the q th user has the following form:
{\mathit{D}}_{q}^{\left(p\right)}={\mathit{H}}_{{\text{etx}}_{q}}^{\left(p\right)}{\left({\mathit{H}}_{{\text{etx}}_{q}}^{{\left(p\right)}^{\mathrm{T}}}{\mathit{H}}_{{\text{etx}}_{q}}^{\left(p\right)}+{\sigma}_{n}^{2}{\mathit{I}}_{{d}_{q}}\right)}^{1}.(33) 
Step 7: Calculate the term ξ^{(p)} that measures the residual multiuser and the interstream interference for the p th iteration defined as
{\xi}^{\left(p\right)}={\u2225\text{off}\left({\mathit{D}}^{{\left(p\right)}^{\mathrm{T}}}\text{Re}\left\{\mathit{H}{\mathit{F}}^{\left(p\right)}\right\}\right)\u2225}_{\mathrm{F}}^{2},(34) 
where off (·) indicates an operation of keeping all offdiagonal elements of its input matrix while setting its diagonal elements to zero. If ξ^{(p)}<ε, the convergence is achieved, and the iterative procedure terminates. Otherwise go back to Step 2.
Note that the stopping criterion similar as (22) that tracks the change of the precoding matrix can also be adopted^{f}. In addition, for both IIMCBF schemes proposed in this section and Section 3.1, it is not required that the users are informed of the decoding matrices that are obtained at the base station while computing the precoding matrices. After the users acquire the information of the effective channel via channel estimation, the receive processing can be performed. For example, the MMSE receiver of the effective channel for each user can be employed.
4 Coordinated beamforming for the CoMP downlink
In this section, we focus on a CoMP downlink setting based on [19]. Note that in [19] a clustered cellular scenario is considered where each cluster contains multiple cells. Since full cooperation is assumed in each cluster, the downlink transmissions for each cluster resemble those of the singlecell multiuser MIMO downlink. Therefore, we simplify the scenario description of the CoMP downlink and only consider joint transmission of adjacent cells. It is assumed that the cell interior users only receive signals from their own base station and suffer only from the intracell interference, i.e., the multiuser interference as in the singlecell multiuser MIMO downlink scenarios. On the other hand, both the intracell interference and the intercell interference have an impact on the cell edge users. To assist the cell edge users to combat the interference and also deal with the greater path loss compared to the cell interior users, the base stations of the adjacent cells transmit the same signals to each cell edge user. An example of a twocell FBMC/OQAMbased CoMP downlink scenario is illustrated in Figure 1.
Assume that Cell 1 and Cell 2 are equipped with {M}_{\mathrm{T}}^{\left(\mathrm{BS}1\right)} and {M}_{T}^{\left(\mathrm{BS}2\right)} transmit antennas, respectively. The number of users served by each cell is denoted as Q. The received signal on the k th subcarrier and at the n th time instant of the q th user of Cell 1 as a cell interior user is expressed as
where {\mathit{H}}_{q,k,\left(1\right)}\left[\phantom{\rule{0.3em}{0ex}}n\right]\in {\mathbb{C}}^{{M}_{{R}_{q}}\times {M}_{\mathrm{T}}^{\left(\mathrm{BS}1\right)}} represents the channel matrix between the base station of Cell 1 and the q th user with {M}_{{R}_{q}} receive antennas, F_{k,(1)}[ n] symbolizes the precoding matrix for Cell 1, and d_{k,(1)}[ n] contains the signals for all users served by Cell 1. It can be seen that for a cell interior user, assuming that the intercell interference is negligible, the transmission from the base station in its own cell is the same as in a singlecell multiuser MIMO downlink system. For the g th user that is a cell edge user and receives the same signals from both Cell 1 and Cell 2, its received signal on the k th subcarrier and at the n th time instant has the following form:
Here the channel matrix, precoding matrix, and the data vector with respect to Cell 2 are denoted similarly as in (35) only with ‘(2)’ in the subscripts. Note that d_{g,k,(1)}[ n]=d_{g,k,(2)}[ n], i.e., the signals from Cell 1 and Cell 2 transmitted to the g th user^{g} are the same. To enable such FBMC/OQAMbased CoMP downlink transmissions, the mitigation of the intracell, intercell, and intrinsic interference has to be achieved. Therefore, we propose the following IIMCBF scheme that is an extension of the approach described in Section 3.2 and is also the outcome of adapting the Extended FlexCoBF algorithm for CPOFDMbased systems in [19] to FBMC/OQAMbased systems.
Consider M cells, and the m th cell serves Q_{ m } users simultaneously, m=1,2,…,M. It is assumed that for the m th cell, the users 1,2,…,L_{ m } are cell interior users, while the remaining (Q_{ m }−L_{ m }) users are cell edge users. The proposed IIMCBF 3 scheme is summarized as follows:

Step 1: Initialize the realvalued decoding matrices {\mathit{D}}_{q}^{\left(0\right)} (q=1,…,Q_{ m }) for the m th cell, where m=1,…,M. Set the iteration index p to zero, and set a threshold ε for the stopping criterion. If the current subcarrier is the first one, the decoding matrices are generated randomly; otherwise, set the decoding matrices as those calculated for the previous subcarrier [24].

Step 2: Set p←p+1 and calculate the equivalent channel matrix H e_{ m }(p) in the p th iteration as
{\mathit{H}}_{{\mathrm{e}}_{m}}^{\left(p\right)}={\left[\begin{array}{llll}{\mathit{H}}_{{\mathrm{e}}_{(1,m)}}^{{\left(p\right)}^{\mathrm{T}}}& {\mathit{H}}_{{\mathrm{e}}_{(2,m)}}^{{\left(p\right)}^{\mathrm{T}}}& \cdots & {\mathit{H}}_{{\mathrm{e}}_{({Q}_{m},m)}}^{{\left(p\right)}^{\mathrm{T}}}\end{array}\right]}^{\mathrm{T}},(37) 
where {\mathit{H}}_{{\mathrm{e}}_{(q,m)}}^{\left(p\right)}={\mathit{D}}_{q}^{{(p1)}^{\mathrm{T}}}{\mathit{H}}_{q,m} is the equivalent channel matrix for the q th user in the p th iteration.

Step 3: Calculate the precoding matrices {\mathit{F}}_{q,m}^{\left(p\right)}{\mathit{G}}_{q,m}^{\left(p\right)} (q=1,…,Q_{ m }) following Step 3 to Step 5 of Section 3.2 to achieve the suppression of the multiuser interference and the intrinsic interference.

Step 4: Update the decoding matrix for each user based on the realvalued equivalent channel matrix where the processing at the transmitter and the procedure of taking the real part of the receive signal are taken into account.

1.
For the q th user that is a cell interior user of the m th cell, its equivalent channel matrix {\mathit{H}}_{{\text{etx}}_{(q,m)}}^{\left(p\right)} is calculated as
{\mathit{H}}_{{\text{etx}}_{(q,m)}}^{\left(p\right)}=\text{Re}\left\{{\mathit{H}}_{q,m}{\mathit{F}}_{q,m}^{\left(p\right)}{\mathit{G}}_{q,m}^{\left(p\right)}\right\}.(38) 
2.
For the q th user that is a cell edge user, define a set {\mathcal{S}}_{q,m} that contains the indices of the cells that simultaneously transmit the same signals to the q th user. Then its equivalent channel matrix is expressed as
{\mathit{H}}_{{\text{etx}}_{(q,m)}}^{\left(p\right)}=\sum _{r\in {\mathcal{S}}_{q,m}}\text{Re}\left\{{\mathit{H}}_{{q}_{r},r}{\mathit{F}}_{{q}_{r},r}^{\left(p\right)}{\mathit{G}}_{{q}_{r},r}^{\left(p\right)}\right\},(39)
where q_{ r } represents the index of the q th user of the m th cell in the r th cell and q_{ m }=q following this definition. It is required that the base station of the m th cell is informed by the r th cell, r\in {\mathcal{S}}_{q,m} and r≠m, of the corresponding realvalued equivalent channel matrices after the precoding and the operation of taking the real part of the received signal. In a summary, the cooperation of the adjacent cells involves the knowledge of the signals for cell edge users. It also requires the exchange of these realvalued equivalent channel matrices that are used to compute the decoding matrix for each cell edge user, which can be achieved by adopting the two exchange mechanisms proposed in [19].

Afterwards, when a single data stream is transmitted to each user, the decoding matrix for the q th user in the p th iteration can be obtained by employing the maximal ratio combining (MRC) receiver or the MMSE receiver of the its equivalent channel matrix {\mathit{H}}_{{\text{etx}}_{(q,m)}}^{\left(p\right)}. On the other hand, when there exist users to which multiple data streams are transmitted, we propose to use the ZF receiver^{h}
{\mathit{D}}_{q,m}^{\left(p\right)}={\mathit{H}}_{{\text{etx}}_{(q,m)}}^{\left(p\right)}{\left({\mathit{H}}_{{\text{etx}}_{(q,m)}}^{{\left(p\right)}^{\mathrm{T}}}{\mathit{H}}_{{\text{etx}}_{(q,m)}}^{\left(p\right)}\right)}^{1}.(40) 
Step 5: Calculate the term {\xi}_{m}^{\left(p\right)} for the m th cell that measures the residual multiuser and the interstream interference for the p th iteration. When a single data stream is transmitted to each user, {\xi}_{m}^{\left(p\right)} is defined as
{\xi}_{m}^{\left(p\right)}={\u2225\text{off}\left({\mathit{D}}_{m}^{{\left(p\right)}^{\mathrm{T}}}\xb7\text{Re}\left\{{\mathit{H}}_{m}{\mathit{F}}_{m}^{\left(p\right)}\right\}\right)\u2225}_{\mathrm{F}}^{2},(41)
where off (·) indicates an operation of keeping all offdiagonal elements of its input matrix while setting its diagonal elements to zero. Moreover, {\mathit{D}}_{m}^{\left(p\right)}, H_{ m }, and {\mathit{F}}_{m}^{\left(p\right)} denote the block diagonal combined decoding matrix in the p th iteration, the combined channel matrix, and the precoding matrix in the p th iteration for the m th cell, respectively.
When multiple data streams are transmitted to at least one of the cell edge users, we propose to express the term {\xi}_{m}^{\left(p\right)} as
On the righthand side of (42), the first term corresponds to the residual interstream interference of each cell interior user, while the second term represents the residual multiuser interference that it still suffers from. For each cell edge user, the third term on the righthand side of (42) measures its residual interstream interference. Here we take into account the fact that coordinated adjacent cells transmit the same signals to the cell edge user simultaneously. Recall that {\mathcal{S}}_{q,m} contains the indices of the cells that serve the q th user of the m th cell. Moreover, the fourth term corresponds to the residual multiuser interference that affects each cell edge user. If {\xi}_{m}^{\left(p\right)}<\epsilon, the convergence is achieved, and the iterative procedure terminates. Otherwise go back to Step 2.
As mentioned before, this coordinated beamforming scheme is designed based on the CoMP technique in [19]. Nevertheless, due to the fact that the intrinsic interference is inherent in FBMC/OQAM systems, we have proposed to incorporate the additional processing to suppress the intrinsic interference. Moreover, different choices of the stopping criterion are recommended for singlestream transmissions and multiplestream transmissions, respectively.
Finally, we summarize the proposed IIMCBF schemes in Table 2. These three algorithms are all based on the concept of jointly and iteratively computing the precoding matrix and decoding matrix. In addition, they mitigate the intracell and intercell interference as well as the intrinsic interference in a ZF fashion. Nevertheless, the three schemes are designed for different configurations and deal with the different types of interference. IIMCBF 1 is developed for symmetric singlecell FBMC/OQAMbased multiuser MIMO downlink settings. In this scenario, BD can still be employed to cancel the multiuser interference. Therefore, in IIMCBF 1 only the precoding matrix for each equivalent single user transmission (i.e., G_{ q } for the q th user) is computed via the iterative procedure to suppress the intrinsic interference. On the other hand, IIMCBF 2 is proposed to overcome the dimensionality constraint that {M}_{\mathrm{R}}^{\left(\text{tot}\right)} must not exceed {M}_{\mathrm{T}}^{\left(\text{BS}\right)} in the singlecell FBMC/OQAMbased multiuser MIMO downlink. The complete precoding matrix (i.e., for the q th user F_{ q }G_{ q }) that mitigates the multiuser interference and the intrinsic interference is computed iteratively and jointly with the decoding matrix. Unlike the first two schemes, IIMCBF 3, as a novel FBMC/OQAMbased CoMP technique, is devised to enable the joint transmission of adjacent cells and to mitigate both the intracell interference and the intercell interference. In the meantime, it is able to achieve the suppression of the intrinsic interference inherent in FBMC/OQAMbased systems. Although the three IIMCBF schemes exhibit satisfactory capabilities in alleviating the dimensionality constraint and in combating different types of interference, a complete elimination of the intrinsic interference that is inherent in FBMC/OQAMbased systems is not achieved. We revisit this issue in Section 5.2. It is worth noting that compared to the transmission strategy reviewed in Section 2.1, the residual intrinsic interference in case of the IIMCBF schemes is much smaller, which is reflected in the performance comparison in Section 5. Throughout this work, equal power allocation is considered. A direct implementation of optimal power allocation schemes developed for CPOFDMbased systems suffers from the loss of optimality due to the residual intrinsic interference existing in FBMC/OQAMbased multiuser MIMO downlink systems where the IIMCBF algorithms are employed. Although the power optimization issue is beyond the scope of this paper, it is of interest to devise power allocation schemes which are tailored for FBMC/OQAMbased multiuser MIMO downlink systems by taking into account the methods of the multiuser and intrinsic interference mitigation as well as possible residual interference.
5 Simulation results
In this section, we evaluate the BER and the sum rate performances of the proposed IIMCBF techniques in various simulation settings. For all examples, the number of subcarriers is 1,024 and the total bandwidth is 10 MHz. In the case of CPOFDM, the length of the CP is set to 1/8 of the symbol period. The ITU PedA channel or the ITU VehA channel [27] is adopted. Moreover, the PHYDYAS prototype filter [25] with the overlapping factor K=4 is employed. The data symbols are drawn from a 16 quadrature amplitude modulation (16 QAM) constellation. Perfect CSI is assumed at the transmitter and at the receiver.
5.1 Singlecell multiuser MIMO downlink
In the first experiment, a symmetric singlecell multiuser MIMO downlink setting is considered, where the base station is equipped with {M}_{\mathrm{T}}^{\left(\text{BS}\right)}=8 transmit antennas, each of the two users has four receive antennas, and the number of spatial streams transmitted to each user is 3. The BER performances of the three schemes for FBMC/OQAMbased systems are presented and also compared to that of a CPOFDMbased system in Figure 2. It can be found that IIMCBF 1 described in Section 3.1 yields the best performance. The multiuser interference is first mitigated by using the BD algorithm [14], and the precoding matrix and the decoding matrix are computed jointly and iteratively for each of the resulting 4×4 equivalent pointtopoint transmissions. Note that the decoding matrix is initialized as (25). In this case, we observe that it takes only two iterations to converge. On the other hand, the transmission scheme that is a straightforward extension of BD [14] for the CPOFDM case as described in Section 2.1 relies on the assumption that the channel frequency responses remain the same across adjacent subcarriers. As the ITU VehA channel exhibits frequency selectivity and such an assumption is therefore violated, the performance of this scheme degrades especially in the high signaltonoise ratio (SNR) regime. We can also observe that in this symmetric scenario, the BDbased precoding algorithm^{i} proposed in [13] performs much worse compared to the other schemes.
In the second and the third examples, we consider singlecell multiuser MIMO downlink settings where the base station equipped with {M}_{\mathrm{T}}^{\left(\text{BS}\right)}=8 transmit antennas simultaneously serves two users and three users, i.e., Q=2 and 3, respectively. Each of the two users in the second scenario has five receive antennas, and the number of data streams transmitted to each user is 3. On the other hand, the three users in the third scenario are equipped with three receive antennas each, and two data streams are transmitted to each of the three users. Note that for such {M}_{\mathrm{R}}^{\left(\text{tot}\right)}>{M}_{\mathrm{T}}^{\left(\text{BS}\right)} configurations, the transmission strategy proposed in [13] and briefly reviewed in Section 2.2 cannot be employed. Figures 3 and 4 show the BER curves for these two scenarios, respectively. Two techniques are considered for the FBMC/OQAMbased system, i.e., IIMCBF 2 presented in Section 3.2 and a direct extension (cf. Section 2.1) of the LoCCoBF algorithm [24] originally designed for the case of CPOFDM. For the purpose of comparison, we also present the BER performance of a CPOFDMbased system with the same configuration where LoCCoBF is employed. For both the proposed IIMCBF 2 scheme and LoCCoBF, ε for the stopping criterion is set to 10^{−5}, and the maximum number of iterations^{j} is 50. It can be observed that the performance of the FBMC/OQAMbased multiuser MIMO downlink system where IIMCBF 2 is employed is slightly better than its CPOFDMbased counterpart due to the fact that no insertion of the CP is required. The other transmission scheme for the FBMC/OQAMbased system suffers from a performance degradation due to the frequency selectivity of the channel. By assuming that the channel stays constant across the neighboring subcarriers, the multiuser interference and the intrinsic interference cannot be eliminated even for high SNRs.
In addition, the convergence behavior of IIMCBF 2 for the singlecell multiuser MIMO FBMC/OQAM downlink system where {M}_{\mathrm{R}}^{\left(\text{tot}\right)}>{M}_{\mathrm{T}}^{\left(\text{BS}\right)} is investigated. The complimentary cumulative distribution function (CCDF) of the number of iterations required to achieve the convergence of the IIMCBF 2 scheme presented in Section 3.2 and LoCCoBF for the case of CPOFDM is illustrated in Figure 5. The threeuser scenario used for Figure 4 is considered. By comparison, we also plot the same set of results for a fouruser case, i.e., Q=4. The base station has eight transmit antennas, and each of the four users is equipped with three receive antennas and is sent to two data streams. Therefore, the total number of the receive antennas of the users {M}_{\mathrm{R}}^{\left(\text{tot}\right)}=12 and all spatial degrees of freedom are exploited.
It can be observed that for the threeuser scenario, IIMCBF 2 converges within six iterations in almost all of the cases. As the number of users and consequently the total number of receive antennas increase, the number of iterations needed for the convergence becomes slightly larger. Nevertheless, the convergence is achieved within ten iterations. Moreover, we can see from the comparison of the proposed IIMCBF 2 technique for the FBMC/OQAMbased system and LoCCoBF for the case of CPOFDM that the number of iterations required for the convergence for both schemes is similar. Hence, compared to the CPOFDMbased multiuser MIMO downlink setting, employing such an IIMCBF technique in the FBMC/OQAMbased system where {M}_{\mathrm{R}}^{\left(\text{tot}\right)}>{M}_{\mathrm{T}}^{\left(\text{BS}\right)} does not result in an increased number of iterations for the convergence. Only the processing dedicated to the elimination of the intrinsic interference contributes to a slight additional complexity.
Furthermore, we consider again the fouruser singlecell multiuser MIMO downlink system (as used in Figure 5) and investigate the impact of the residual CFO. The IIMCBF 2 scheme for the FBMC/OQAMbased system and LoCCoBF for CPOFDM are considered. In addition, it is assumed that the residual CFO (normalized by the subcarrier spacing) ranges^{k} either from 0 to 0.1 or from 0 to 0.15. Figure 6 illustrates the corresponding BER performances of these two schemes in the presence of the residual CFO. Notice that the CPOFDMbased system suffers from a severe performance degradation. In contrast, the FBMC/OQAMbased system with IIMCBF 2 significantly outperforms its CPOFDMbased counterpart, and it exhibits a greater robustness against the synchronization errors in the frequency domain.
5.2 CoMP downlink
In the sequel, the performance of the FBMC/OQAMbased CoMP downlink is assessed. Instead of BER, we choose to use the sum rate performance as the evaluation approach. We start with a twocell scenario consisting of five users in total. Each cell equipped with four transmit antennas serves three users each with two receive antennas, which forms a 4×6 multiuser downlink setting. Among the three users, two are cell interior users, and one is a cell edge user. A single data stream is transmitted to each cell interior user, and two data streams are transmitted to the cell edge use, i.e., for each cell full spatial multiplexing is considered. The path loss of the transmission to the cell edge users is assumed to be ten times bigger than that for the cell interior users [19]. To calculate the sum rate for both the FBMC/OQAM and the CPOFDM cases, we first compute an instantaneous signaltointerferenceplusnoiseratio (SINR) on each data stream for each subcarrier. In case of the FBMC/OQAMbased multiuser MIMO downlink where IIMCBF 2 is employed, the decoded signal is written as
where
Note that as mentioned in Section 2, for the case where (ℓ+i) is even, the signal d_{ ℓ }[ i] on the ℓ th subcarrier and at the i th time instant is real, while the corresponding coefficient c_{ i ℓ } is pure imaginary. On the contrary, when (ℓ+i) is odd, d_{ ℓ }[ i] and c_{ i ℓ } are pure imaginary and real, respectively. This explains the definitions of {\stackrel{\u0304}{c}}_{\mathrm{i\ell}} and {\stackrel{\u0304}{\mathit{d}}}_{\ell}\left[\phantom{\rule{0.3em}{0ex}}i\right] shown above. Let us further write (43) as
where
To calculate the SINR denoted by γ_{r,k} for the r th data stream on the k th subcarrier, where r=1,…,d, the entry (r,r) of {\mathit{H}}_{k}^{\left(\text{eff}\right)}\left[\phantom{\rule{0.3em}{0ex}}n\right] determines the effective gain for the r th data stream, while the entries (r,u) (u=1,…,d,u≠r) of {\mathit{H}}_{k}^{\left(\text{eff}\right)}\left[\phantom{\rule{0.3em}{0ex}}n\right] determine the residual multiuser and interstream interference. In addition, the elements of the r th row of {\mathit{H}}_{\ell}^{\left(\text{inf}\right)}\left[\phantom{\rule{0.3em}{0ex}}i\right] provide a measure of the residual intrinsic interference. Notice that only when D_{ k }[ n]=D_{ ℓ }[ i] (ℓ=k−1, k, k+1, i=n−3,…,n+3, and (ℓ,i)≠(k,n)), {\mathit{H}}_{\ell}^{\left(\text{inf}\right)}\left[\phantom{\rule{0.3em}{0ex}}i\right] is rendered as zero, and the intrinsic interference is completely eliminated. Due to the fact that the channel is not flat fading, the aforementioned condition is not fulfilled, leading to the existence of the residual intrinsic interference. The noise component can be extracted from the covariance matrix of {\stackrel{~}{\mathit{n}}}_{k}\left[\phantom{\rule{0.3em}{0ex}}n\right]
Therefore, γ_{r,k} is computed as
The sum rate is then approximated as
where N_{c} is the total number of subcarriers. For the case of CPOFDM, the sum rate can be computed similarly, but the residual intrinsic interference term does not exist. Moreover, the Re{·} operation is not used. On the other hand, when a transmission scheme that is a straightforward extension of the CPOFDM case (cf. Section 2.1) is adopted, the residual intrinsic interference results from the violation of the assumption that the channels of adjacent subcarriers are the same. Therefore, the effective channel that determines the intrinsic interference caused by the signal on the ℓ th subcarrier and at the i th time instant to that on the k th subcarrier and at the n th time instant, where (ℓ,i)≠(k,n), is expressed as
Then, the sum rate for this case can be calculated accordingly. It is worth mentioning that in case of the downlink of the CoMP system, the fact that adjacent cells send the same signal to each cell edge user should be taken into account in the sum rate calculation, while the rest resembles the aforementioned case of the multiuser MIMO downlink system.
To this end, we use Figure 7 to illustrate the sum rate performances of two transmission schemes for the FBMC/OQAMbased CoMP downlink systems described at the beginning of this section. Here \text{SNR}={P}_{\mathrm{T}}/{\sigma}_{n}^{2}, where P_{T} is the transmit power of each cell. The CPOFDMbased system where Extended FlexCoBF^{l} is employed is also considered for comparison. It can be seen that IIMCBF 3 leads to a slightly better performance of the FBMC/OQAMbased system in contrast to the CPOFDMbased system. The reason is that the former has a higher spectral efficiency compared to the latter, since no CP is required. This observation also implies the effectiveness of the proposed transmission strategy IIMCBF 3 in mitigating the intracell and intercell interference as well as the intrinsic interference. By comparison, when a transmission strategy originally designed for CPOFDM, here Extended FlexCoBF, is straightforwardly extended to the FBMC/OQAM system and the channel is frequency selective, the performance is much worse than that of the proposed IIMCBF 3 scheme due to a much higher level of the residual interference.
Moreover, Figure 8 depicts the CCDF of the number of iterations required for the IIMCBF 3 technique to converge. For all the evaluated schemes, ε used for the stopping criterion is set to 10^{−5}, and the iterative procedure is manually terminated when the number of iterations reaches 50. Two types of the frequency selective channel, ITU PedA and VehA, are considered. Recall that in the initialization step of IIMCBF 3, the decoding matrices calculated for the previous subcarrier are used to initialize the decoding matrices for the current subcarrier. Therefore, the lower the frequency selectivity of the channel is, the more effective becomes such a choice of the initial decoding matrices, which leads to a smaller number of iterations required for the convergence of the coordinated beamforming algorithm. Consequently, we observe in Figure 8 that compared to the case of the ITU VehA channel which is more frequency selective, the number of iterations that are needed by the coordinated beamforming schemes to converge in case of the ITU PedA channel is slightly smaller. However, even when the ITU VehA is considered as the propagation channel, the proposed IIMCBF 3 technique converges within 15 iterations in almost all of the cases.Finally, a threecell CoMP downlink scenario is considered. Each cell equipped with four transmit antennas serves three users each with two receive antennas. This leads to a 4×6 multiuser downlink setting. Among the three users, two are cell interior users, and one is a cell edge user. A single data stream is transmitted to each cell interior user, and two data streams are transmitted to the cell edge user. The numerical results with respect to the sum rate performance are shown in Figure 9. Similar observations are obtained as in Figure 7.
6 Conclusions
We have developed three IIMCBF based transmission schemes for the downlink of FBMC/OQAM based multiuser MIMO systems and CoMP systems. The first algorithm that is called IIMCBF 1 is designed for a multiuser MIMO FBMC/OQAM downlink system where {M}_{\mathrm{T}}^{\left(\text{BS}\right)}={M}_{\mathrm{R}}^{\left(\text{tot}\right)}. We have employed an iterative procedure to jointly compute the precoding matrix and the decoding matrix of each equivalent singleuser transmission that results from the elimination of the multiuser interference. On the other hand, the IIMCBF 2 technique has been proposed as a solution to the problem that the stateoftheart transmission strategies for the downlink of FBMC/OQAMbased multiuser MIMO systems fail to work when {M}_{\mathrm{R}}^{\left(\text{tot}\right)}>{M}_{\mathrm{T}}^{\left(\text{BS}\right)}. Moreover, we have conducted an investigation of FBMC/OQAMbased CoMP downlink systems for the first time. With a focus on the case of partial cooperation of adjacent cells, the scheme, IIMCBF 3, has been designed to enable the joint transmission of base stations in adjacent cells and combat both the intracell and the intercell interference. It is worth noting that in addition to the suppression of the multiuser interference, these three proposed IIMCBF schemes are effective in mitigating the intrinsic interference that is inherent in FBMC/OQAMbased systems without assuming that the propagation channel is almost flat fading. To demonstrate the advantages of the three IIMCBF algorithms, their BER and sum rate performances have been shown in different settings. Via the simulation results, it has been shown that the FBMC/OQAMbased multiuser MIMO and CoMP downlink systems where IIMCBF 1, IIMCBF 2, or IIMCBF 3 is employed achieve a similar performance compared to their CPOFDMbased counterparts but with a higher spectral efficiency and a greater robustness against misalignments in the frequency domain. In addition, we have numerically analyzed the convergence of the IIMCBF techniques. It leads to the conclusion that the additional complexity is quite acceptable as the price of alleviating the aforementioned dimensionality constraint.
Endnotes
^{a} Parts of this paper have been published in ICASSP 2014 [28] and ISCCSP 2014 [29].
^{b} Here we only provide the formulas of the channel matrices, precoding matrices, and data vectors on the k th subcarrier and at the n th time instant explicitly due to limited space. In case of the ℓ th subcarrier and the i th time instant, the corresponding expressions can be obtained by replacing k and n with ℓ and i, respectively.
^{c} For the case where (k+n) is odd, the desired signal on the k th subcarrier and at the n th time instant is pure imaginary, while the intrinsic interference is real providing that the prototype pulse satisfies the perfect reconstruction property [7, 12]. As the two cases are essentially equivalent to each other, we only take the case where (k+n) is even to describe the proposed algorithm in this paper. In addition, each entry in the data vector corresponds to either the inphase component or the quadrature component of a QAM symbol that is assumed to have unit energy.
^{d} In IIMCBF 3 for the CoMP downlink, F_{ q } suppresses both the intracell interference and the intercell interference.
^{e} Other receivers, such as zero forcing or maximal ratio combining, can also be employed in this coordinated beamforming algorithm.
^{f} These two stopping criteria require different thresholds for the convergence. In this work, we only present numerical results obtained when the stopping criterion as in (34) is employed, and the threshold ε is set to 10^{−5}.
^{g} In this example, we assume that the indices of each cell edge user in Cell 1 and Cell 2 are the same to facilitate the description of the scenario.
^{h} In such scenarios where multiple data streams are transmitted to at least one user, by employing the MMSE or the MRC to update the decoding matrices in IIMCBF 3, convergence cannot be achieved.
^{i} In the implementation of this algorithm, after the cancelation of the multiuser interference by using BD, only the ZF based step that ensures the cancelation of the intrinsic interference is considered. The remaining part of the transmit processing (spatial mapping) and the receive processing (MMSE receiver) are chosen to be the same as the other schemes for the purpose of a fair comparison. Note that the precoding algorithm proposed in [13] is dominated by the BD and ZF based steps.
^{j} Via numerical simulations, it has been observed that there exist rare cases where it takes a large number of iterations for the stopping criteria of the IIMCBF schemes to be fulfilled. Therefore, a maximum number of iterations is set to handle these cases. The iterative procedure is manually terminated if the stopping criteria are not fulfilled when the number of iterations reaches this maximum number. Taking IIMCBF 2 as an example, its stopping criterion corresponds to the residual interference. We have observed that when the algorithm is manually terminated, though the residual interference is above the threshold, its value is already so small that the performance is not affected much. Except for this simple way of setting a maximum number of iterations, the termination of the iterative procedure can also be determined based on the variation of the residual interference (in case of IIMCBF 2 and IIMCBF 3) or the change of the precoding matrix (in case of IIMCBF 1) as the number of iterations increases.
^{k} In the simulations, the residual CFO is drawn uniformly from the range (0,0.1) or (0,0.15).
^{l} In the implementation of Extended FlexCoBF [19], we adopt the same mechanism of initializing the decoding matrices as in the LoCCoBF algorithm [24] such that the correlation of the channels of adjacent subcarriers is exploited, and consequently the number of iterations required for the convergence is reduced.
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The authors gratefully acknowledge the financial support by the European Union FP7ICT project EMPhAtiC (http://www.ictemphatic.eu) under grant agreement no. 318362.
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Cheng, Y., Li, P. & Haardt, M. Intrinsic interference mitigating coordinated beamforming for the FBMC/OQAM based downlink. EURASIP J. Adv. Signal Process. 2014, 86 (2014). https://doi.org/10.1186/16876180201486
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DOI: https://doi.org/10.1186/16876180201486