Intrinsic interference mitigating coordinated beamforming for the FBMC/OQAM based downlink
 Yao Cheng^{1}Email author,
 Peng Li^{1} and
 Martin Haardt^{1}
https://doi.org/10.1186/16876180201486
© Cheng et al.; licensee Springer. 2014
Received: 6 December 2013
Accepted: 19 May 2014
Published: 9 June 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.
Keywords
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
Coefficients c_{ iℓ } representing the system impulse response determined by the synthesis and analysis filters[7]
n−3  n−2  n−1  n  n+1  n+2  n+3  
k−1  0.043ȷ  −0.125  −0.206ȷ  0.239  0.206ȷ  −0.125  −0.043ȷ 
k  −0.067  0  0.564  1  0.564  0  −0.067 
k+1  −0.043ȷ  −0.125  0.206ȷ  0.239  −0.206ȷ  −0.125  0.043ȷ 
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
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
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
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
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.
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.
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.
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
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.
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.
Acronyms of the proposed IIMCBF schemes and the corresponding scenarios
Acronyms  Configurations  Interference type 

IIMCBF 1  Multiuser MIMO downlink,${M}_{\mathrm{T}}^{\left(\text{BS}\right)}={M}_{\mathrm{R}}^{\left(\text{tot}\right)}$  Intrinsic, multiuser 
IIMCBF 2  Multiuser MIMO downlink,${M}_{\mathrm{T}}^{\left(\text{BS}\right)}<{M}_{\mathrm{R}}^{\left(\text{tot}\right)}$  Intrinsic, multiuser 
IIMCBF 3  CoMP downlink  Intrinsic, intra/intercell 
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
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.
5.2 CoMP downlink
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.
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.
Declarations
Acknowledgements
The authors gratefully acknowledge the financial support by the European Union FP7ICT project EMPhAtiC (http://www.ictemphatic.eu) under grant agreement no. 318362.
Authors’ Affiliations
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