Interference alignment for a multiuser SISO interference channel
 Yasser Fadlallah^{1}Email author,
 Karine Amis^{1},
 Abdeldjalil AïssaElBey^{1} and
 Ramesh Pyndiah^{1}
https://doi.org/10.1186/16871499201479
© Fadlallah et al.; licensee Springer. 2014
Received: 16 October 2013
Accepted: 13 April 2014
Published: 15 May 2014
Abstract
Our work addresses the singleinput singleoutput interference channel. The goal is to show that although interference alignment is suboptimal in the finite power region, it is able to achieve a significant overall throughput. We investigate the interference alignment scheme proposed by Choi et al. (IEEE Commun. Lett. 13(11): 847849, 2009), which achieves a higher multiplexing gain at any given signal dimension than the scheme proposed by Cadambe and Jafar (IEEE Trans. Inform. Theory 54(8), 2008). Then, we try to modify the IA design in order to achieve enhanced sumrate performance in the practical signaltonoise ratio (SNR) region. Firstly, we introduce a way to optimize the precoding subspaces at all transmitters, exploiting the fact that channel matrices in the interference model of a singleinput singleoutput channel are diagonal. Secondly, we propose to optimize jointly the set of precoder bases within their associated precoding subspaces. To this end, we combine each precoder with a new combination precoder, and this latter seeks the optimal basis that maximizes the network sum rate. We also introduce an improved closedform interference alignment scheme that performs close to the other proposed schemes.
Keywords
1 Introduction
In most existing wireless multiuser communication systems, interference is avoided either by coordinating the users to orthogonalize the channel access or by treating interference from other transmitters as noise. However, until recently, the capacity region of the interference channel (IC) remained unknown, except for some special cases such as strong and very strong interference [1, 2]. In [3], Maddahali et al. have proposed a new approach in order to show that the Nantennas MIMO X channels can offer as much as $\frac{4N}{3}$ degrees of freedom (DoF). This new approach of interference management has been named IA.
The key idea of IA is to jointly design all transmitted signals such that interfering signals at each receiver overlap and remain distinct from the desired signal. This approach has been exploited by Cadambe and Jafar in [4]. The authors have shown that the maximum achievable DoF in the Kuser timevarying singleinput singleoutput (SISO) IC, in the n dimensional Euclidean space, is $\frac{K}{2}$ and is achieved, thanks to an IA scheme. Later on, Motahari et al. have addressed the achievable DoF of a quasistatic IC. They have extended the idea of IA from space/time/frequency dimensions to the signal level dimensions and have shown that based on the field of Diophantine approximation in number theory [5], the interference can be aligned in the rational spaces, achieving a maximum DoF of $\frac{K}{2}$.
The first IA scheme for SISO transmissions has been proposed in [4] for the time/frequencyvarying channel. This scheme has been designed to achieve the asymptotic capacity in the IC, i.e., when both the signaltonoise (SNR) and the signal dimensions tend to infinity. In contrast, Choi et al. have introduced another IA design that aims to achieve a higher multiplexing gain at any given signal dimension [6]. In this paper, we adopt an IA scheme for SISO transmission, and we try to modify the design in order to achieve higher sumrate performance in the practical SNR region. Most references, among which [7–9], deal with IA schemes for MIMO interference channels. However, all mobile communication standards still include a SISO transmission mode as for instance the LTE downlink transmission mode 1. This is the reason why in this paper, we try to define efficient IA schemes for SISO interference channels.
In our contributions, we firstly introduce a way to optimize the precoding subspaces at all transmitters, exploiting the fact that channel matrices in the IA model are diagonal. Two solutions are derived; the first is achieved iteratively using projected gradient descent method, the second is a closedform solution that avoids the numerical computation, thus, resulting in a very low computational complexity. Secondly, we propose to optimize the precoding vectors at each transmitter within its precoding subspace. To this end, we combine each IA precoder with a new combination precoder. The combination precoder seeks the optimal basis that maximizes the network sumrate assuming an individual transmit power constraint. However, a closedform solution does not seem trivial. Therefore, we apply an iterative process based on the simple gradient descent method, which converges to a local maximum due to the nonconcavity of the objective function.
This paper is organized as follows. Section 2 describes the system model. Then, Section 3 presents the IA design in SISO IC. In Section 4, we propose to optimize the network sum rate through a diagonal matrix W. The precoding vector optimization within the IA subspaces is presented in Section 5. In Section 6, we present the convergence rate of the proposed iterative algorithms. Section 7 evaluates the sumrate performance of the proposed optimization. Finally, Section 8 concludes the paper.
Notations: boldface upper case letters and boldface lower case letters denote matrices and vectors, respectively. For the transpose, transpose conjugate, and conjugate matrices, we use (.)^{ T }, (.)^{ H }, and (.)^{∗}, respectively. I_{ p } is the p×p identity matrix, and 1_{ p } is the allone vector of length p.
2 System model
 1.
Users do not cooperate.
 2.
Nonprecoded user symbols are Gaussian continuously distributed and mutually independent.
 3.
The set of channel matrices H _{ k j } is entirely and perfectly known at all transmitters and all receivers.
 4.
All diagonal components of ${\mathit{H}}_{\mathit{\text{kj}}}\phantom{\rule{0.3em}{0ex}}\forall k,j\in \mathcal{K}$ are i.i.d. and continuously distributed, with absolute values upperbounded with a finite value.
where C(snr) represents the channel capacity.
3 IA design in a SISO interference channel
3.1 Precoding design
where U_{ k } is the decoding matrix at the k th receiver. In other words, the desired signal belongs to the subspace generated by the vectors of G_{ k }=U_{ k }H_{ k k }V_{ k }, while the interference is completely eliminated. The feasibility of the linear system in (3) is conditioned to the following proprieties: (i) the linear system has to be proper, i.e., the number of variables is more than or equal to the number of equations and (ii) the linear system has to be generic [10]. In some particular cases, the genericity is satisfied by providing a channel matrix with random and independent coefficients.
where M is a parameter depending on the user number, M=(K−1)(K−2)−1, d_{ i } is the DoF of the i th user, i.e., the number of transmitted symbols, and N the number of symbols in each IA vector is defined as N=d_{1}+d_{2}. In the particular scheme previously, the IA conditions can be satisfied by providing ${d}_{i}={d}_{3}\phantom{\rule{1em}{0ex}},{d}_{1}>{d}_{3},\phantom{\rule{1em}{0ex}}i\in \mathcal{K}\setminus \left\{1,3\right\}$. For example, in a threeuser SISO multiuser IC, we have d_{1}=n+1, d_{2}=d_{3}=n,N=2n+1, and n can be any nonnegative integer.
3.2 Linear decoding design
where ${\stackrel{\u0304}{H}}_{j}^{k}={\mathit{H}}_{\mathit{\text{kj}}}{\mathit{V}}_{j}$. We assume that the IA conditions are satisfied. Let ${\stackrel{\u0304}{H}}_{I}^{k}\in {\mathrm{\xe2\u201e\u201a}}^{N\times (N{d}_{k})}$ denotes the N−d_{ k } matrix spanning all interference subspaces, i.e., ${\stackrel{\u0304}{H}}_{j}^{k}$ for j∈{1,⋯,K}, j≠k are all spanned by ${\stackrel{\u0304}{H}}_{I}^{k}$. Before going further into the description, we introduce the following lemma:
Lemma 1
Let ${\mathit{A}}_{1}\in {\mathrm{\xe2\u201e\u201a}}^{N\times n}$ and ${\mathit{A}}_{2}\in {\mathrm{\xe2\u201e\u201a}}^{N\times n}(N>n)$, where rank (A_{1})=n and rank (A_{2})=m,(m≤n) and span (A_{2})⊂span(A_{1}). Then, for every ${\mathit{s}}_{2}\in {\mathrm{\xe2\u201e\u201a}}^{n\times 1},\phantom{\rule{1em}{0ex}}\exists \phantom{\rule{1em}{0ex}}{\mathit{s}}_{1}\in {\mathrm{\xe2\u201e\u201a}}^{n\times 1}$ such that A_{1}s_{1}=A_{2}s_{2}.
where ${\stackrel{\u0304}{s}}_{k}=\left({\mathit{s}}_{1I}+\cdots +{\mathit{s}}_{(k1)I}+{\mathit{s}}_{(k+1)I}+\cdots +{\mathit{s}}_{\mathit{\text{KI}}}\right)$, ${\mathit{B}}_{k}\in {\mathrm{\xe2\u201e\u201a}}^{N\times N}$ is a full rank matrix that spans the union of the desired and the interference subspaces, and ${\stackrel{~}{\mathit{s}}}_{k}$ is the N×1 vector consisting of the d_{ k } desired streams and the N−d_{ k } interference streams. Equation 7 gives the mathematical formulation of a linear determined decoding problem, where an N length source data vector ${\stackrel{~}{\mathit{s}}}_{k}$ is mixed by a constant mixing matrix B_{ k } to produce a vector y_{ k } of N observations. Such a decoding problem can be resolved using classical criteria such as zeroforcing (ZF), minimum mean square error (MMSE), and maximum likelihood (ML).
4 IA precoding subspaces optimization
where ${\mathit{V}}_{k}^{\mathit{\text{IA}}}$ is the original matrix derived with respect to the IA conditions and W is any diagonal matrix, which satisfy the IA conditions. That is, the projection of all precoding matrices of the IA scheme on a common diagonal matrix W keeps the IA conditions respected.
The precoding subspaces can be optimized by judiciously selecting the components of W in (8). This diagonal matrix W determines the interference and the desired subspaces design, while maintaining the IA conditions at the receivers. We assume both MMSE and ZFbased detection schemes, widely used due to the simplicity of their implementation, and we derive two different optimized designs that maximize the network sum rate in both cases.
4.1 MMSEbased decoder  iterative solution
It is not obvious whether a closedform solution can be obtained or not; therefore, one can search for the solution iteratively. However, the convergence towards the global maximum is not guaranteed unless the objective function is concave. The proof of the concavity with respect to the variable vector $\stackrel{~}{w}$, requires the objective function to be twice differentiable and its Hessian matrix to be negative semidefinite [14]. Indeed, a similar problem has been treated in [15] for the threeuser IA scheme. The authors have demonstrated that a function having the form of (14) is concave if A_{ k } and B_{ k } are defined as in (12) (see Appendix 1 in [15]). In order to find the solution that achieves the optimum, we propose to use the projected gradient method with an optimized variable step size (details are given in Appendix 2). Other algorithms can also be used such as simple gradient descent method using Lagrange multipliers.
Remark 1
In [15], the diagonal matrix W has been optimized under an MMSE receiver assumption. However, the study has been restricted to the threeuser SISO interference channel case. Also, there is no description of the iterative algorithm used for achieving the solution. In this section, we have provided a general design for the Kuser SISO interference channel case. We have also described an iterative method (see Appendix 2) to converge towards the solution.
4.2 ZFbased decoder  closedform solution
In the previous subsection, we have proposed to optimize the precoding subspaces using iterative processing when an MMSE is applied at the receiver. In this section, we apply a ZF criterion at the receiver. Then, we propose a closedform solution for w that is asymptotically optimal. This solution is obtained from the network sumrate maximization problem approximation for very high SNR and under the hypothesis of a ZF applied at all receivers. It also avoids the need for a numerical solver that requires a matrix inversion at each iteration and increases the processing time and computational cost.
Hence, the components of w are obtained as ${w}_{i}^{\ast}=\sqrt{{\stackrel{~}{w}}_{i}^{\ast}}$ for all i. It is worth noting that beside maximizing the sumrate, the problem of maximizing the individual rate using the approximation in (16) has the same solution obtained in (22).
A major advantage of the proposed solution is the fact that it has an analytic simple expression making its implementation complexity very low. Indeed, the other algorithms proposed for sumrate maximization and interference power minimization in SISO and MIMO^{c} transmissions achieve the optimum using singular value decomposition (SVD) [11] and/or an iterative algorithm that requires hundreds to thousands iterations to converge [7–9, 16].
4.3 Complexity and sumrate performance
The computational complexity is a major bottleneck of practical implementation that is considered in system designs. In the following, we discuss the complexity of the precoding schemes proposed above.
The first optimized design that maximizes the sumrate assuming an MMSE detector is obtained using the projected gradient descent method. This iterative method requires at each iteration the computational cost of the firstorder derivative of the objective function. Looking at the expression given in (34), one can notice that the derivative is calculated using matrix multiplications and matrix inversions with dimensions N×N. Therefore, the computational complexity at each iteration can be considered of order $\mathcal{O}\left({N}^{3}\right)$. On the other hand, the design based on a closedform solution requires the computation of a Frobenius norm and N real division. Thus, the complexity order is $\mathcal{O}\left(N\right)$.
It is important to note that the proposed designs result from the optimization of the original designs proposed in [6] and not the optimal IA design that maximizes the sumrate. It explains why, when we compare the designs for different dimensions N, a higher sumrate is obtained for N=3 compared to the design for N=7. Furthermore, in order to have a fair comparison between the cases N=3 and N=7, the average sum rate performance is normalized by the dimension N, which means that the sum rate is divided by the number of symbols in one symbol vector as shown in (14).
5 Precoding vectors design within IA subspaces
The previous section has addressed the optimization of the IA precoding subspaces at once using a diagonal matrix W. However, there was no claim for the optimality of the precoding vectors within IA subspaces. In this section, we propose to maintain the IA subspaces design at the transmitters, and we aim to optimize the precoding vectors within each subspace. We consider both cases: MMSE and ZF criterion at the receiver, and we attempt to maximize the network sumrate in each case.
will modify the basis of V_{ k } within its own subspace without modifying the subspace itself. These variables can later be defined taking into account different criteria such as MSE, BER, sumrate, and average transmit power. Next, we show how to optimize the additional combination matrices so as to maximize the network sumrate.
5.1 MMSEbased decoder
or when a maximum number of iterations is reached, and ε is defined as a tolerance value. In our simulations, we assume ε=10^{−2}.
5.2 ZFbased decoder
where U_{ k } is the decoding matrix at the k th receiver. It is defined as the d_{ k }×N interference null space. The model defined in (27) is a typical MIMO single user model with channel matrix ${\stackrel{~}{H}}_{k}$ and precoding matrix C_{ k }. One optimized form of C_{ k } is the one composed of the right singular vector of the new channel matrix ${\stackrel{~}{H}}_{k}$. Such a precoding scheme achieves the channel capacity as described in [17]. Another form that requires less computational complexity is the one that orthonormalizes the columns of the original precoding matrix V_{ k }. In [18], the authors have shown that this last form gets close to the maximum information rate when the SNR becomes high.
5.3 Complexity and sumrate performance
The algorithm that optimizes the solution iteratively in Subsection 5.1 is based on the gradient descent method. At each iteration, the iterative algorithm requires the gradient of the objective function that needs itself inversion of N×N full rank matrices. Thus, the total computational complexity depends mainly on the number of iterations and on the precoding matrices dimensions. The complexity cost is of order $\mathcal{O}\left({\mathit{\text{nb}}}_{i}{N}^{3}\right)$ where n b_{ i } is the number of iterations.
Remark 2.
The proposed optimization of IA precoding subspaces cannot directly be extended to MIMO interference channels, as the channel matrices are no longer diagonal in the MIMO system model. However, the proposed optimization of the precoding vectors within each IA subspace can be used for MIMO optimization design since the IA conditions at all receiver are always maintained.
6 Convergence rate of the iterative solutions
In Subsections 4.1 and 5.1, we have proposed two iterative solutions, one aims to optimize the IA subspaces and the other optimizes the precoding vectors within each IA subspace without modifying the subspace itself.
The first iterative solution to the problem in (14) for the IA subspaces optimization is reached using the projected gradient method. We have mentioned that the objective function is concave, thus, the convergence towards the global optimum is guaranteed. On the other hand, the iterative solution proposed for the IA precoding vector optimization is reached using an algorithm based on the gradient descent method for a multivariable objective function. Thereby, the objective function changes at every iteration yielding a nonconvex optimization problem. However, as long as the iterative method is based on the gradient descent and the variable follows the direction of the gradient using an optimized step size, a convergence towards a local optimum is guaranteed.
7 Comparison of the proposed optimized designs to the state of art schemes

? OWZF : the proposed IA design with the closedform solution derived in Subsection 4.2 that uses orthogonal precoding vectors

? CWMMSE: the IA design with the two iterative proposed optimization in Subsections 4.1 and 5.1

? IAIter: the IA design obtained with the distributed algorithm proposed in [8]

? MaxSINR: the beamforming design proposed in [7] that maximizes the signaltointerferenceandnoise ratio (SINR) of all streams
On the other side, compared to the orthogonal interference mitigation techniques such as time division multiple access (TDMA), the optimized IA scheme (i.e., CWMMSE) achieves similar performance in the low SNR region. However, when the SNR increases, i.e., beyond 15 dB, the gain between both optimized IA designs (i.e., OWZF and CWMMSE) and the TDMA starts to get wider, and achieves 2 bits/s/Hz/dim over the TDMA at 30 dB. This is due to the suboptimality of the IA design in the low SNR region and its tendency to optimality in the high SNR region [4].
8 Conclusions
In this paper, we have introduced three optimized designs for the IA scheme in a Kuser SISO IC. The first and the second try to optimize the precoding subspaces at the IA transmitters through a common diagonal matrix assuming an MMSE and ZF linear detector, respectively. The third assumes an MMSE linear detector and seeks the optimal precoding vectors within a predefined subspace at each transmitter. The first and the third designs referred to as WMMSE and CMMSE, respectively, require iterative algorithms to converge to their optimum, whereas the second design referred to as WZF is obtained from a closedform solution. Comparing to other IA distributed designs, the proposed designs show a significant sumrate performance improvement and much less computational complexity when the closedform solution is applied. To enhance the sumrate performance, we have introduced an orthogonalization of the precoding vectors in the WZF design, which enables to achieve a tradeoff between complexity and data rate.
Endnotes
^{a} This hypothesis is very optimistic, but it is taken by many research works in the literature.
^{b} It is important to note that the Cholesky decomposition, originally defined for a positive definite matrix, can be extended to the positive semidefinite case.
^{c} The IA schemes proposed for MIMO transmission can also be used in SISO systems.
Appendix 1
Appendix 2
with ε is the tolerance factor for stopping the iterations or a maximum number of iterations is attained. In this algorithm, the step size μ is a determining factor to ensure a faster convergence, thus, it must be judiciously selected. In [14], two line search methods are proposed: exact line search and inexact line search methods. In practice, most line searches are inexact, and many methods have been proposed. One is the backtracking method, which is employed for our design. It is very simple to implement and quite effective. Besides, the step size is updated at each iteration to satisfy ${\stackrel{~}{w}}_{i}>0$ for all i.
Declarations
Acknowledgement
The authors would like to thank the ‘Institut Carnot Télécom et Société Numérique’ for funding the work of this paper.
Authors’ Affiliations
References
 Sato H: On degraded Gaussian twouser channels. IEEE Trans. Inform. Theory 1978, IT24: 637640.View ArticleGoogle Scholar
 Carleial A: Interference channels. IEEE Trans. Inform. Theory 1978, 24: 6070. 10.1109/TIT.1978.1055812MathSciNetView ArticleMATHGoogle Scholar
 MaddahAli M, Motahari AS, Khandani AK: Communication over MIMO X channels: interference alignment, decomposition, and performance analysis. IEEE Trans. Inform. Theory 2008, 54(8):34573470.MathSciNetView ArticleMATHGoogle Scholar
 Cadambe VR, Jafar SA: Interference alignment and degrees of freedom of the Kuser interference channel. IEEE Trans. Inform. Theory 2008, 54(8):34253441.MathSciNetView ArticleMATHGoogle Scholar
 Motahari AS, Gharan SO, MaddahAli MA, Khandani AK: Real interference alignment: exploiting the potential of single antenna systems. arXiv:0908.2282Google Scholar
 Choi SW, Jafar SA, Chung SY: On the beamforming design for interference alignment. IEEE Commun. Lett 2009, 13(11):847849.View ArticleGoogle Scholar
 Gomadam K, Cadambe VR, Jafar SA: Approaching the capacity of wireless networks through distributed interference alignment. In Proc. of IEEE Global Communications Conference, GLOBECOM. IEEE, New Orleans, USA; 2008.Google Scholar
 Peters SW, Heath RW: Interference alignment via alternating minimization. Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2009, 24452448.Google Scholar
 Santamaria I, Gonzalez O, Heath R, Peters S: Maximum sumrate interference alignment algorithms for MIMO channels. In Proc. of IEEE Global Communications Conference, GLOBECOM. IEEE, Miami, USA; 2010.Google Scholar
 Yetis C, Gou T, Jafar SA, Kayran AH: On feasibility of interference alignment in MIMO interference networks. IEEE Trans. Signal Process 2010, 58(9):47714782.MathSciNetView ArticleGoogle Scholar
 Sung H, Park S, Lee K, Lee I: Linear precoder designs for Kuser interference channels. IEEE Trans. Wireless Commun 2010, 9(1):291300.View ArticleGoogle Scholar
 Serre D: Matrices: Theory and Applications, Second Edition. Springer, New York; 2010.View ArticleGoogle Scholar
 Higham NJ: Analysis of the Cholesky Decomposition of a SemiDefinite Matrix. Oxford University Press; 1990.MATHGoogle Scholar
 Boyd S, Vandenberghe L: Convex Optimization. Cambridge University Press, New York; 2004.View ArticleMATHGoogle Scholar
 Kim D, Torlak M: Optimization of interference alignment beamforming vectors. IEEE J. Selected Areas Commun 2010, 28(9):14251434.View ArticleGoogle Scholar
 Shen H, Li B, Tao M, Luo Y: The new interference alignment scheme for the MIMO interference channel. In Wireless Communications and Networking Conference (WCNC), 2010 IEEE. Sydney, Australia; 2010:16.Google Scholar
 Wolniansky PW, Foschini GJ, Golden G, Valenzuela RA: Vblast: an architecture for realizing very high data rates over the richscattering wireless channel. In Signals, Systems, and Electronics, 1998. ISSSE 98. 1998 URSI International Symposium On. IEEE, Pisa; 1998:295300.Google Scholar
 Shen M, HostMadsen A, Vidal J: An improved interference alignment scheme for frequency selective channels. In Proc. of IEEE International Symposium on Information Theory, ISIT. Toronto, ON; 2008:611.Google Scholar
 Magnus J, Neudecker H: Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, Chichester; revised version 2007.MATHGoogle Scholar
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