Interference mitigation techniques for clustered multicell joint decoding systems
- Symeon Chatzinotas^{1}Email author and
- Björn Ottersten^{1, 2}
https://doi.org/10.1186/1687-1499-2011-132
© Chatzinotas and Ottersten; licensee Springer. 2011
Received: 25 March 2011
Accepted: 14 October 2011
Published: 14 October 2011
Abstract
Multicell joint processing has originated from information-theoretic principles as a means of reaching the fundamental capacity limits of cellular networks. However, global multicell joint decoding is highly complex and in practice clusters of cooperating Base Stations constitute a more realistic scenario. In this direction, the mitigation of intercluster interference rises as a critical factor towards achieving the promised throughput gains. In this paper, two intercluster interference mitigation techniques are investigated and compared, namely interference alignment and resource division multiple access. The cases of global multicell joint processing and cochannel interference allowance are also considered as an upper and lower bound to the interference alignment scheme, respectively. Each case is modelled and analyzed using the per-cell ergodic sum-rate throughput as a figure of merit. In this process, the asymptotic eigenvalue distribution of the channel covariance matrices is analytically derived based on free-probabilistic arguments in order to quantify the sum-rate throughput. Using numerical results, it is established that resource division multiple access is preferable for dense cellular systems, while cochannel interference allowance is advantageous for highly sparse cellular systems. Interference alignment provides superior performance for average to sparse cellular systems on the expense of higher complexity.
1 Introduction
Currently cellular networks carry the main bulk of wireless traffic and as a result they risk being saturated considering the ever increasing traffic imposed by internet data services. In this context, the academic community in collaboration with industry and standardization bodies have been investigating innovative network architectures and communication techniques which can overcome the interference-limited nature of cellular systems. The paradigm of multicell joint processing has risen as a promising way of overcoming those limitations and has since gained increasing momentum which lead from theoretical research to testbed implementations [1]. Furthermore, the recent inclusion of CoMP (Coordinated Multiple Point) techniques in LTE-advanced [2] serves as a reinforcement of the latter statement.
- 1.
the channel modelling of a clustered MJD system with IA as intercluster interference mitigation technique,
- 2.
the analytical derivation of the ergodic throughput based on free probabilistic arguments in the R-transform domain,
- 3.
the analytical comparison with the upper bound of global MJD, the Resource Division Multiple Access (RDMA) scheme and the lower bound of clustered MJD with Cochannel Interference allowance (CI),
- 4.
the comparison of the derived closed-form expressions with Monte Carlo simulations and the performance evaluation using numerical results.
The remainder of this paper is structured as follows: Section 2 reviews in detail prior work in the areas of clustered MJD and IA. Section 3 describes the channel modelling, free probability derivations and throughput results for the following cases: (a) global MJD, (b) IA, (c) RDMA and (d) CI. Section 4 displays the accuracy of the analysis by comparing to Monte Carlo simulations and evaluates the effect of various system parameters in the throughput performance of clustered MJD. Section 6 concludes the paper.
1.1 Notation
Throughout the formulations of this paper, $E\left[\cdot \right]$ denotes expectation, (·)^{ H }denotes the conjugate matrix transpose, (.)^{ T }denotes the matrix transpose, ⊙ denotes the Hadamard product and ⊗ denotes the Kronecker product. The Frobenius norm of a matrix or vector is denoted by ||·|| and the delta function by δ(·). I_{ n }denotes an n × n identity matrix, ${\mathbf{I}}_{n\times m}$ an n × m matrix of ones, 0 a zero matrix and ${\mathbf{G}}_{n\times m}~\mathcal{C}\mathcal{N}\left(0,{\mathbf{I}}_{n}\right)$ denotes n × m Gaussian matrix with entries drawn form a $\mathcal{C}\mathcal{N}\left(0,1\right)$ distribution. The figure of merit analyzed and compared throughout this paper is the ergodic per-cell sum-rate throughput.^{a}
2 Related work
2.1 Multicell joint decoding
This section reviews the literature on MJD systems by describing the evolution of global MJD models and subsequently focusing on clustered MJD approaches.
2.1.1 Global MJD
It was almost three decades ago when the paradigm of global MJD was initially proposed in two seminal papers [3, 4], promising large capacity enhancements. The main idea behind global MJD is the existence of a central processor (a.k.a. "hyper-receiver") which is interconnected to all the BSs through a backhaul of wideband, delayless and error-free links. The central processor is assumed to have perfect channel state information (CSI) about all the wireless links of the system. The optimal communication strategy is superposition coding at the UTs and successive interference cancellation at the central processor. As a result, the central processor is able to jointly decode all the UTs of the system, rendering the concept of intercell interference void.
Since then, the initial results were extended and modified by the research community for more practical propagation environments, transmission techniques and backhaul infrastructures in an attempt to more accurately quantify the performance gain. More specifically, it was demonstrated in [5] that Rayleigh fading promotes multiuser diversity which is beneficial for the ergodic capacity performance. Subsequently, realistic path-loss models and user distributions were investigated in [6, 7] providing closed-form ergodic capacity expressions based on the cell size, path loss exponent and geographical distribution of UTs. The beneficial effect of MIMO links was established in [8, 9], where a linear scaling of the ergodic per-cell sum-rate capacity with the number of BS antennas was shown. However, correlation between multiple antennas has an adverse effect as shown in [10], especially when correlation affects the BS side. Imperfect backhaul connectivity has also a negative effect on the capacity performance as quantified in [11]. MJD has been also considered in combination with DS-CDMA [12], where chips act as multiple dimensions. Finally, linear MMSE filtering [13, 14] followed by single-user decoding has been considered as an alternative to the optimal multiuser decoder which requires computationally-complex successive interference cancellation.
2.1.2 Clustered MJD
Clustered MJD is based on forming groups of M adjacent BSs (clusters) interconnected to a cluster processor. As a result, it can be seen as an intermediate state between traditional cellular systems (M = 1) and global MJD (M = ∞). The advantage of clustered MJD lies on the fact that both the size of the backhaul network and the number of UTs to be jointly processed decrease. The benefit is twofold; first, the extent of the backhaul network is reduced and second, the computational requirements of MJD (which depend on the number of UTs) are lower. The disadvantage is that the sum-rate capacity performance is degraded by intercluster interference, especially affecting the individual rates of cluster-edge UTs. This impairment can be tackled using a number of techniques as described here. The simplest approach is to just treat it as cochannel interference and evaluate its effect on the system capacity as in [15]. An alternative would be to use RDMA, namely to split the time or frequency resources into orthogonal parts dedicated to cluster-edge cells [16]. This approach eliminates intercluster interference but at the same time limits the available degrees of freedom. In DS-CDMA MJD systems, knowledge of the interfering codebooks has been also used to mitigate intercluster interference [12]. Finally, antenna selection schemes were investigated as a simple way of reducing the number of intercluster interferers [17].
2.2 Interference alignment
This section reviews the basic principles of IA and subsequently describes existing applications of IA on cellular networks.
2.2.1 IA preliminaries
IA has been shown to achieve the degrees of freedom (dofs) for a range of interference channels [18–20]. Its principle is based on aligning the interference on a signal subspace with respect to the non-intended receiver, so that it can be easily filtered out by sacrificing some signal dimensions. The advantage is that this alignment does not affect the randomness of the signals and the available dimensions with respect to the intended receiver. The disadvantage is that the filtering at the non-intended receiver removes the signal energy in the interference subspace and reduces the achievable rate. The fundamental assumptions which render IA feasible are that there are multiple available dimensions (space, frequency, time or code) and that the transmitter is aware of the CSI towards the non-intended receiver. The exact number of needed dimensions and the precoding vectors to achieve IA are rather cumbersome to compute, but a number of approaches have been presented in the literature towards this end [21–23].
2.2.2 IA and cellular networks
IA has been also investigated in the context of cellular networks, showing that it can effectively suppress cochannel interference [23, 24]. More specifically, the downlink of an OFDMA cellular network with clustered BS cooperation is considered in [25], where IA is employed to suppress intracluster interference while intercluster interference has to be tolerated as noise. Using simulations, it is shown therein that even with unit multiplexing gain the throughput performance is increased compared to a frequency reuse scheme, especially for the cluster-centre UTs. In a similar setting, the authors in [26] propose an IA-based resource allocation scheme which jointly optimizes the frequency-domain precoding, subcarrier user selection, and power allocation on the downlink of coordinated multicell OFDMA systems. In addition, authors in [24] consider the uplink of a limited-size cellular system without BS cooperation, showing that the interference-free dofs can be achieved as the number of UTs grows. Employing IA with unit multiplexing gain towards the non-intended BSs, they study the effect of multi-path channels and single-path channels with propagation delay. Furthermore, the concept of decomposable channel is employed to enable a modified scheme called subspace IA, which is able to simultaneously align interference towards multiple non-intended receivers over a multidimensional space. Finally, the effect of limited feedback on cellular IA schemes has been investigated and quantified in [25, 27].
3 Channel model and throughput analysis
3.1 Global multicell joint decoding
where the n × 1 vector z denotes AWGN with $E\left[{\mathbf{z}}_{i}\right]=0$ and $E\left[{\mathbf{z}}_{i}{\mathbf{z}}_{i}^{H}\right]=\mathbf{I}$. The K n × 1 vector x_{ i }denotes the transmitted symbol vector of the i th UT group with $E\left[{\mathbf{x}}_{i}{\mathbf{x}}_{i}^{H}\right]=\gamma \mathbf{I}$ where γ is the transmit Signal to Noise ratio per UT antenna. The n × Kn channel matrix ${\mathbf{G}}_{i,i}~\mathcal{C}\mathcal{N}\left(0,{\mathbf{I}}_{n}\right)$ includes the flat fading coefficients of the i th UT group towards the i th BS modelled as independent identically distributed (i.i.d.) complex circularly symmetric (c.c.s.) random variables. Similarly, the term αG_{i, i+1}(t)x_{i+1}(t) represents the received signal at the i th BS originating from the UTs of the neighboring cell indexed i + 1. The scaling factor α < 1 models the amount of received intercell interference which depends on the path loss model and the density of the cellular system^{c}. Another intuitive description of the α factor is that it models the power imbalance between intra-cell and inter-cell signals.
where step (a) follows from [10, Eq.(34)]. □
3.2 Interference alignment
where y_{1} and y_{ M }represent the received signal vectors at the first and last BS of the cluster, respectively. The first UT group has to align its input x_{1} towards the non-intended BS of the cluster on the left (see Figure A), while the M th BS has to filter our the aligned interference coming from the M + 1th UT group which belongs to the cluster on the right. These two strategies are described in detail in the following subsections:
3.2.1 Aligned interference filtering
Assuming that the system operates in high-SNR regime and is therefore interference limited, the effect of the AWGN noise colouring ${\stackrel{\u0303}{\mathbf{z}}}_{M}=\mathbf{Q}{\mathbf{z}}_{M}$ can be ignored, namely $E\left[{\stackrel{\u0303}{\mathbf{z}}}_{M}{\stackrel{\u0303}{\mathbf{z}}}_{M}^{H}\right]={\mathbf{I}}_{K}$.
Lemma 3.1. The Shannon transform of the covariance matrix of QG_{ M,M }is equivalent to that of a K × K Gaussian matrix G_{ K×K }.
Based on this lemma and for the purposes of the analysis, QG_{ M,M }is replaced by G_{K × K}in the equivalent channel matrix.
3.2.2 Interference alignment
where ${\mathbf{G}}_{0,1}^{j}$ represents the fading coefficients of the j th UT of the first group towards the M th BS of the neighboring cluster on the left. Since the exact eigenvalue distribution of the matrix product ${\mathbf{G}}_{1,1}^{j}{\left({\mathbf{G}}_{0,1}^{j}\right)}^{-1}\mathbf{v}{v}_{j}$ is not straightforward to derive, for the purposes of rate analysis it is approximated by a Gaussian vector with unit variance. This approximation implies that IA precoding does not affect the statistics of the equivalent channel towards the intended BS.
3.2.3 Equivalent channel matrix
with ${\mathbf{\Sigma}}_{1}=\left[{\mathbf{I}}_{n\times K}\phantom{\rule{2.77695pt}{0ex}}\alpha {\mathbf{I}}_{n\times Kn}{0}_{n\times \left(M-2\right)Kn}\right]$ being a n × (M - 1)Kn + K matrix^{f}, ${\mathbf{\Sigma}}_{2}=\left[{0}_{\left(M-2\right)n\times K}{\stackrel{\u0303}{\mathbf{\Sigma}}}_{M-2\times M-1}\otimes {\mathbf{I}}_{n\times Kn}\right]$ being a (M - 2)n × (M - 1)Kn + K matrix and ${\mathbf{\Sigma}}_{3}=\left[{0}_{n-1\times \left(M-2\right)Kn+K}\phantom{\rule{2.77695pt}{0ex}}{\mathbf{I}}_{n-1\times K\phantom{\rule{0.3em}{0ex}}n}\right]$ being a n - 1 × (M - 1)Kn + K matrix^{g}.
Theorem 3.2. In the IA case, the per-cell throughput can be derived from the R-transform of the a.e.p.d.f. of matrix$\frac{1}{n}{\mathbf{H}}_{\mathsf{\text{IA}}}^{H}{\mathbf{H}}_{\mathsf{\text{IA}}}$.
and ${\mathcal{R}}_{\frac{1}{n}{\mathbf{H}}_{i}^{H}{\mathbf{H}}_{i}}$ given by theorem A.1.
where H_{1} = Σ_{1} ⊙G_{n×(M-1)Kn+K}, H_{2} = Σ_{2} ⊙ G_{(M-2)n×(M-1)Kn+K}and H_{3} = Σ_{3} ⊙ G_{n-1×(M-1)Kn+K}. Using the property of free additive convolution [30] and Theorem A.1 in Appendix A, Eq. (24) holds in the R-transform domain. □
3.3 Resource division multiple access
Eq. (28) follows from Eq. (42) with $q=\left(M-2\right)q\left(\stackrel{\u0303}{\mathbf{\Sigma}}\right)=\left(M-2\right)\left(1+{\alpha}^{2}\right)\u2215\left(M-1\right),\beta =K\left(M-1\right)\u2215\left(M-2\right)$ and theorem A.1. □
where the factor 2 is due to the doubling of the transmitted power.
The rest of this proof follows the steps of Theorem 3.4. □
3.4 Cochannel interference allowance
3.5 Degrees of freedom
This section focuses on comparing the degrees of freedom for each of the considered cases. The degrees of freedom determine the number of independent signal dimensions in the high SNR regime [35] and it is also known as prelog or multiplexing gain in the literature. It is a useful metric in cases where interference is the main impairment and AWGN can be considered unimportant.
Proof. Eq. (39) can be derived straightforwardly by counting the receive dimensions of the equivalent channel matrices (Eq. (3) for global MJD, Eq. (18) for IA, Eqs. (27) and (31) for RDMA, Eq. (34) for CI) and normalizing by the number of BS antennas. □
Remark 3.1. It can be observed that d_{IA}= d_{RD}only for single UT per cell equipped with two antennas (K = 1, n = 2). For all other cases, d_{IA} > d_{RD}. Furthermore, it is worth noting that when the number of UTs K and antennas n grows to infinity, lim_{ K,n }→ ∞ d_{IA} = d_{MJD}which entails a multiuser ga in. However, in practice the number of served UTs is limited by the number of antennas (n = K+ 1) which can be supported at the BS- and more importantly at UT-side due to size limitations.
3.6 Complexity considerations
This paragraph discusses the complexity of each scheme in terms of decoding processing and required CSI. In general, the complexity of MJD is exponential with the number of users [36] and full CSI is required at the central processor for all users which are to be decoded. This implies that global MJD is highly complex since all system users have to be processed at a single point. On the other hand, clustering approaches reduce the number of jointly-processed users and as a result complexity. Furthermore, CI is the least complex since no action is taken to mitigate intercluster interference. RDMA has an equivalent receiver complexity with CI, but in addition it requires coordination between adjacent clusters in terms of splitting the resources. For example, time division would require inter-cluster synchronization, while frequency division could be even static. Finally, IA is the most complex since CSI towards the non-intended BS is also needed at the transmitter in order to align the interference. Subsequently, additional processing is needed at the receiver side to filter out the aligned interference.
4 Numerical results
Parameters for throughput results
Parameter | Symbol | Value | Range | Figure |
---|---|---|---|---|
Cluster size | M | 4 | 3 - 10 | 2, 3 |
α factor | α | 0.5 | 0.1 -1 | 4 |
UTs per cell | K | 5 | 2 - 10 | 5 |
Antennas per UT | n | 4 | 3 - 11 | 5 |
UT Transmit SNR | γ | 20dB | ||
Number of MC iterations | 10^{3} |
6 Conclusion
In this paper, various techniques for mitigating intercluster interference in clustered MJD were investigated. The case of global MJD was initially considered as an upper bound, serving in evaluating the degradation due to intercluster interference. Subsequently, the IA scheme was analyzed by deriving the asymptotic eigenvalue distribution of the channel covariance matrix using free-probabilistic arguments. In addition, the RDMA scheme was studied as a low complexity method for mitigating intercluster interference. Finally, the CI was considered as a worst-case scenario where no interference mitigation techniques is employed. Based on these investigations it was established that for dense cellular systems the RDMA scheme should be used as the best compromise between complexity and performance. For average to sparse cellular systems which is the usual regime in macrocell deployments, IA should be employed when the additional complexity and availability of CSI at transmitter side can be afforded. Alternatively, CI could be preferred especially for highly sparse cellular systems.
A Proof of theorem
and employing eq. (23), the proof is complete. □
Notes
^{a}The term throughput is used instead of capacity since the described techniques are suboptimal in the information-theoretic sense and lead to achievable sum-rates except for MJD which leads to MIMO MAC capacity.
^{b}The multiple antennas are assumed to be uncorrelated although the analytical results can be extended in the correlated case based on the principles described in
^{c}For more details on the modelling of the α parameter, the reader is referred to [37].
^{d}For the purposes of the analysis the variable $\stackrel{\u0303}{\gamma}$ is kept finite as the number of antennas Mn grows large, so that the system power does not grow to infinity.
^{e}Depending on the signal dimensions and the channel coefficients, more than one degree of freedom per user could be achieved. The feasibility of higher multiplexing gain has been studied in [21, 22]. More specifically, authors in [21] provide an algorithm which determines the achievable multiplexing gain by minimizing the interference leakage, while authors in
^{f}The structure of the first block of Σ_{1} originates in the Gaussian approximation of $\frac{1}{a}{\mathbf{G}}_{1,1}^{j}{\left({\mathbf{G}}_{0,1}^{j}\right)}^{-1}\mathbf{v}{v}_{j}$.
^{g}The structure of the last block of Σ_{3} is based on Lemma 3.1.
Declarations
Acknowledgements
This work was partially supported by the National Research Fund, Luxembourg under the CORE project "CO^{2}SAT: Cooperative and Cognitive Architectures for Satellite Networks".
Authors’ Affiliations
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