A reconfigurable iterative algorithm for the K-user MIMO interference channel

Signal Processing 93 (2013) 3353–3362

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A reconfigurable iterative algorithm for the K-user MIMO interference channel$ George C. Alexandropoulos n, Constantinos B. Papadias Athens Information Technology (AIT), 19.5 km Markopoulo Avenue, 19002 Peania, Athens, Greece

a r t i c l e i n f o

abstract

Article history: Received 11 August 2012 Received in revised form 24 May 2013 Accepted 27 May 2013 Available online 10 June 2013

Interference alignment (IA) is a recently proposed transmission technique for the K-user interference channel (IFC), which is proven to achieve a sum-rate multiplexing gain of K=2 at the high interference regime. Motivated by our recent work [1] that showed how the sum-rate scaling can range between K=2 and K for moderate-to-low interference conditions, in this paper we present a novel iterative algorithm for the K-user multipleinput multiple-output (MIMO) IFC with arbitrary number of transceiver antennas. The proposed algorithm automatically adjusts itself to the interference regime at hand, in the above sense, as well as to the wireless channel in order to achieve the appropriate sumrate scaling. Our reconfigurable algorithm combines the system-wide mean squared error minimization criterion with the single-user waterfilling solution to maximize each user's transmission rate according to the interference levels and channel conditions. Extensive computer simulation results for the sum-rate performance of the proposed reconfigurable algorithm over various Ricean fading channels are presented. It is shown that, in the interference-limited regime, the proposed algorithm reconfigures itself so as to achieve the IA scaling whereas, in the moderate-to-low interference regime, it chooses interference-myopic MIMO transmissions for all K communication pairs. & 2013 Elsevier B.V. All rights reserved.

Keywords: Interference alignment Multiple-input multiple-output systems Multiuser communications Sum-rate performance Transceiver design

1. Introduction Recent results on the characterization of the capacity region of the K-user interference channel (IFC) [2] have shown that the capacity of wireless systems can be substantially higher than previously believed [3–5]. Specifically, it has been shown in [5] that for K pairs of interfering users operating under symmetric Rayleigh fading channels with equal average powers, a sum-rate multiplexing gain of K=2 is feasible at the high

☆ This work has been supported by the European Union Future and Emerging Technologies (FET) Project HiATUS. The project HiATUS acknowledges the financial support of the FET programme, within the Seventh Framework Programme for Research of the European Commission, under FET-open Grant number 265578. n Corresponding author. Tel.: +30 6944692141. E-mail addresses: [email protected], [email protected] (G.C. Alexandropoulos), [email protected] (C.B. Papadias).

0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.05.027

interference regime. The feasibility of this astonishing result has been accomplished with interference alignment (IA) [5]. IA is a transmission technique that is based on appropriate linear precoding at the transmitters, aiming at complete post-receiver processing interference elimination, and requires only global channel state information (CSI) to be available at all participating transceivers. Since the introduction of the principle of IA, several research works exploited the space dimension offered by multiple-input multiple-output (MIMO) systems to perform IA and investigated the feasibility of IA solutions for the K-user MIMO IFC (see e.g. [6–10] and references therein). For the special case of K ¼3 a closed-form solution for the IA-achieving precoding matrices was presented in [5,9] that was further processed in [11–13] for increasing the sum-rate performance. The authors in [7] proposed a method for achieving IA in the K-user N  N constant MIMO IFC for the special case where K ¼ N þ 1. However, for K 4 3 MIMO communication pairs with

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G.C. Alexandropoulos, C.B. Papadias / Signal Processing 93 (2013) 3353–3362

arbitrary number of transceiver antennas closed-form solutions for IA are in general unknown and several centralized as well as distributed iterative algorithms have been recently proposed, e.g. in [12,14–26]. The vast majority of those algorithms targets at implicitly achieving IA through the optimization of one or more constrained objective functions. For example, Gomadam et al. [14] presented two distributed iterative IA algorithms that exploit the reciprocity of wireless channels and require only local CSI at each transceiver. The one algorithm tries to minimize the overall interference leakage in the IFC, whereas the second one attempts to maximize the signalto-interference-plus-noise ratio (SINR) for every transmitted data stream. The minimization of the overall interference in the IFC has been also considered in the centralized approaches proposed in [15–17], where alternating minimization procedures [27] were adopted for obtaining all transceiver filters. Furthermore, Peters and Heath Jr. [16] presented two additional IA-achieving iterative algorithms; one that was based on the system-wide minimum mean squared error (MMSE) criterion [28] and another one aiming at the maximization of a global SINRbased objective function that accounts for the SINRs of all transmitted data streams in the IFC. An iterative algorithm that jointly minimizes the total interference leakage and maximizes the sum rate was presented in [18]. In [20] the original IA problem [5] was reformulated as a rank constrained rank minimization problem. As it was shown, the rank minimization formulation guarantees that the interference spaces collapse to the smallest dimensional subspaces possible. A one-sided IA approach that does not require channel reciprocity and runs at the transmitters side only was presented in [22] which eliminates the need for synchronization between each communication pair. The authors in [23] considered the interfering MIMO broadcast channel and presented an iterative algorithm for maximizing the weighted sum-rate (WSR). That algorithm was based on the iterative minimization of a matrixweighted system-wide mean squared error (MSE). A linear transceiver design that maximizes WSR and utilizes deterministic annealing to track the WSR at any desired SINR was presented in [24]. Recently, a semidistributed iterative algorithm that maximizes the weighted sum of a utility of SINR values for each data stream in the K-user MIMO IFC was proposed in [26]. This algorithm is based on linear MMSE receive filters and utilizes semidefinite programming to compute the optimum transmit covariance matrices. Although IA attains the optimum sum-rate scaling at the high interference regime, there are certain combinations of interference levels and channel conditions where it does not [1,29–31]. For example, it was shown in [29] for the 2-user single-input single-output Gaussian IFC that the optimum sum-rate scaling in a regime with very weak interference is achievable by treating interference as noise. The authors in [30] characterized the sum capacity of the ðN þ 1Þ-user 1  N single-input multiple-output Gaussian IFC for the symmetric case where all intended links have the same signal-to-noise ratio (SNR) and all interference links have the same interference-to-noise ratio. It was shown that there are again certain regimes where it is

more preferable in terms of sum-rate performance to treat interference as noise. In [31] the low interference regime for the 2-user MIMO Gaussian IFC was studied and the authors analyzed conditions on the intended and interference links under which using Gaussian inputs and treating interference as noise at the receivers is sumcapacity achieving. Recently, the sum-rate performance results of [1] for the downlink of a K-user MIMO cellular network with asymmetric average powers and line-ofsight (LOS) conditions among the intended and the interference links demonstrated certain regimes where interference-myopic MIMO transmissions yield superior sum-rate performance than IA. Under those interferencemyopic MIMO transmissions each transmitter treats interference as noise and utilizes the transmit covariance matrix obtained from the waterfilling (WF) solution [32] for its intended single-user MIMO channel. From all the above it is obvious that to achieve the optimum sum-rate scaling for the K-user MIMO IFC one must devise a transmission scheme that is reconfigurable to the interference levels as well as to the wireless channel conditions. To the best of our knowledge, the majority of the available transmission techniques for the K-user IFC need to know a priori the interference levels so as to choose between the two extremes: treating interference as noise or performing IA. In addition, there are certain lowto-moderate interference levels where none of the latter techniques achieves the appropriate sum-rate scaling and/ or chooses the appropriate number of data streams that can be reliably transmitted in the IFC. In this paper we present a centralized1 iterative algorithm for the K-user MIMO IFC with arbitrary number of transceiver antennas. Inspired by the iterative algorithms of [14] and the findings in [1], we first investigate certain regimes where IA is suboptimal and interference-myopic MIMO transmissions yield superior performance. We then present a transceiver design that combines the system-wide MMSE criterion with the single-user WF solution to maximize each user's transmission rate according to the interference levels and channel conditions. Extensive computer simulation results are presented for the sum-rate performance of the proposed algorithm as well as its convergence characteristics. The remainder of this paper is organized as follows. Section 2 outlines the system and channel model. In Section 3 the IA conditions and representative algorithms for achieving them are described whereas, Section 4 presents the proposed reconfigurable iterative algorithm. In Section 5 computer simulated sum-rate performance results for all presented algorithms are demonstrated along with a relevant discussion. Finally, Section 6 concludes the paper. Notations: Throughout this paper vectors and matrices are denoted by boldface lowercase letters and boldface capital letters, respectively. The transpose conjugate and the determinant of matrix A are denoted by AH and detðAÞ, respectively. Moreover, ½Ai;j represents the (i,j)-element of A, spanðAÞ its column span, TrfAg its trace and AðnÞ denotes

1 A distributed version of the iterative algorithm presented in this paper has been recently presented in [33].

G.C. Alexandropoulos, C.B. Papadias / Signal Processing 93 (2013) 3353–3362

V1

U1 H1,1

Tx 1

H1,2

H1,K H2,1

H2,2

Tx 2

U2

HK,2 HK,1 HK,K

1=2

H UH k yk ¼ Uk Hk;k V k Pk sk

Rx 2

H2,K

VK

Tx K

represents the zero-mean complex additive white Gaussian noise (AWGN) vector with covariance matrix s2k In½k . R After signal reception each Rx k is assumed to process yk n½k d with a linear filter Uk ∈C R k as

Rx 1

V2

3355

þUH k

K

1=2

∑

ℓ ¼ 1;ℓ≠k

Hk;ℓ Vℓ Pℓ sℓ þ UH k nk :

ð2Þ

In this paper we are interested in the joint design of Vk 's, Pk 's and Uk 's that achieves the optimum sum-rate performance scaling at the whole SNR range. We first highlight some already well-known algorithmic designs for the K-user MIMO IFC in Section 3 and then present a novel iterative algorithm in Section 4.

UK

Rx K

Fig. 1. The K-user MIMO IFC.

the nth column of A. In addition, In denotes the n  n identity matrix, 0mn is the m  n zeros matrix and diagfag represents a diagonal matrix with vector a in its main diagonal. Notation ∥a∥ stands for the Euclidean norm of a and ∥A∥F is the Frobenius norm of A. The expectation operator is denoted as Efg whereas, notations X∼N ðμ; s2 Þ and X∼CN ðμ; s2 Þ represent a random variable X following the normal and complex normal distribution, respectively, with mean μ and variance s2 . 2. System and channel models We present in the following the K-user MIMO interference system model under consideration as well as the utilized flexible wireless channel model which is capable of describing various LOS conditions as well as asymmetric average power scenarios among the intended and the interference links.

2.2. Channel model A frequency-flat fading channel model is considered for which the channel gain matrix between Rx k and Tx ℓ is given by [1] ( H k;ℓ ; k¼ℓ Hk;ℓ ¼ ð3Þ αk;ℓ H k;ℓ ; k≠ℓ ½k

½ℓ

where parameter αk;ℓ ∈½0; 1Þ2 and H k;ℓ ∈CnR nT , which describes Ricean fading, is defined as [34] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ k;ℓ 1 LOS NLOS H k;ℓ ¼ H k;ℓ þ H : ð4Þ κk;ℓ þ 1 k;ℓ κk;ℓ þ 1 In (4) κk;ℓ is the Ricean κ-factor, defined as the power ratio of the specular to the random components of H k;ℓ . Furthermore, NLOS

½k

½ℓ

H k;ℓ ∈CnR nT is the scattered component of H k;ℓ such that NLOS

∀i ¼ 1; 2; …; n½k R

½H k;ℓ i;j ∼CN ð0; 1Þ LOS

½k

and

∀j ¼ 1; 2; …; n½ℓ T .

½ℓ

Moreover, H k;ℓ ∈CnR nT denotes the specular component of

2.1. System model A multiuser MIMO wireless system consisting of K pairs of communicating users is considered, as shown in Fig. 1. In particular, each transmitting user (Tx) k, where k ¼ 1; 2; …; K, equipped with n½k antennas wishes to T communicate with the n½k R -antenna receiving user (Rx) k. All K simultaneous transmissions of symbols sk ∈Cdk 1 , ½k with dk ≤minðn½k T ; nR Þ ∀k, are assumed perfectly synchronized and each Tx k processes its sk with a linear ½k precoding matrix Vk ∈CnT dk before transmission. In our system model we assume for each Vk that ∥VkðnÞ ∥ ¼ 1 ∀n with n ¼ 1; 2; …; dk . For the transmitted power 1=2 per Tx k it is assumed that Ef∥Vk Pk sk ∥2 g ≤Pk with Pk being the total power constraint per Tx and dk dk ðkÞ ðkÞ Pk ¼ diagf½P ðkÞ , where P ðkÞ denotes the n 1 P 2 …P dk g∈Rþ power allocated to the nth data stream at Tx k. Without loss of generality, it is assumed throughout the paper that for each Tx k: Pk ¼ P and Efsk sH k g ¼ Idk . The baseband received signal at Rx k can be mathematically expressed as

LOS H k;ℓ and is given by H k;ℓ ¼ ak ðθR ÞH aℓ ðθT Þ. The n½ℓ T -dimension -dimension vector a ðθ vector aℓ ðθT Þ and the n½k k R Þ denote the R

specular array responses at Tx ℓ and Rx k, respectively, with θT and θR being the angles of departure and arrival. For example, assuming linear arrays at both Tx ℓ and Rx k with xℓ and xk antenna spacings in wavelengths, respectively, yields ½ℓ

aℓ ðθT Þ ¼ ½1 ej2πxℓ cos ðθT Þ ⋯ ej2πxℓ ðnT

−1Þ cos ðθT Þ

;

ð5aÞ

ak ðθR Þ ¼ ½1 ej2πxk cos ðθR Þ ⋯ ej2πxk ðnR −1Þ cos ðθR Þ :

ð5aÞ

½k

ð1Þ

Setting αk;ℓ ¼ 0 in (3) ∀ k; ℓ results in interference-free communication pairs, whereas for αk;ℓ ¼ 1 ∀ k; ℓ the average powers of the intended and the interference links are equal. The latter case results in symmetric Ricean fading channels with equal average powers, which include the well-known symmetric Rayleigh fading scenario [3–5] as a special case, i.e. for κk;ℓ ¼ 0 ∀ k; ℓ. Various practical fading scenarios with asymmetric fading conditions and average powers among the intended and the interference links, e.g. as in macrocellular networks [35], can be effectively modeled with αk;ℓ ∈ð0; 1Þ.

where Hk;ℓ ∈CnR nT , with ℓ ¼ 1; 2; …; K, denotes the chan½k nel matrix between Rx k and Tx ℓ, and nk ∈CnR 1

2 It is noted that for αk;ℓ ¼ 1, (3) results in the one-branch expression Hk;ℓ ¼ H k;ℓ [1].

1=2

yk ¼ Hk;k Vk Pk sk þ ½k

K

∑

ℓ ¼ 1;ℓ≠k

1=2

Hk;ℓ Vℓ Pℓ sℓ þ nk

½ℓ

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G.C. Alexandropoulos, C.B. Papadias / Signal Processing 93 (2013) 3353–3362 ½k

Let tuple ðd1 ; d2 ; …; dK Þ consist of the numbers of the independently encoded Gaussian codebook symbols at all K transmitters. IA aims at jointly designing K Vk 's and K Uk 's to maximize the overlap of interference signal subspaces at each Rx k while ensuring that the desired signal vectors are linearly independent of the interference. According to [5] the IA conditions for Vk and Uk ∀ k ¼ 1; 2; …; K are mathematically expressed as UH k Hk;ℓ Vℓ ¼ 0dk dℓ ;

∀ ℓ≠k

ð6Þ

rankðUH k Hk;k Vk Þ ¼ dk :

ð7Þ

For Hk;ℓ ∀ℓ≠k drawn independently from a continuous Rayleigh distribution, i.e. given by (3) and (4) with αk;ℓ ¼ 1 and κk;ℓ ¼ 0 ∀k; ℓ, and for an IA-feasible allocation of spatial degrees of freedom (DoF) ðd1 ; d2 ; …; dK Þ, IA achieves the spatial DoF of the K-user MIMO IFC3. The feasibility of IA for this channel was firstly investigated in [6] and it was shown that for the antenna symmetric case, i.e. when n½k R ¼ nR , n½k T ¼ nT and dk ¼ d ∀k, IA is likely to be feasible if and only if nR þ nT ≥dðK þ 1Þ. Although, for the special case of K¼ 3 communication pairs some closed-form solutions for Vk 's and Uk 's satisfying (6) and (7) are known [5,9,11,13], for the general case of K 43 only iterative algorithms for designing Vk 's and Uk 's have been lately proposed [12,14–26], which implicitly achieve the IA conditions at certain interference-limited regimes. The vast majority of those algorithms requires the preassignment of the IA-feasible spatial DoF tuple ðd1 ; d2 ; …; dK Þ and allo1=2 cates pffiffiffiffiffiffiffiffiffiffiequal power at each Tx k data streams, i.e. Pk ¼ P=dk Idk ∀k, irrespective to the interference levels and channel conditions. We next summarize two representative of the iterative algorithms for IA whose sum-rate performance is comparable, to the best of our knowledge, to the majority of the iterative algorithms for the K-user MIMO IFC. 3.1. Interference leakage minimization (ILM) The ILM algorithm [14] tries to orthogonalize the desired signal and interference subspaces at each Rx k. To accomplished this, Vk 's and Uk 's are jointly designed to minimize the overall interference leakage in the IFC. As shown in [16], the ILM algorithm utilizes the procedure of [27] to iteratively solve the following global optimization problem: min

fVk gKk ¼ 1 ;fUk gKk ¼ 1

subject to

I WIL H VH k Vk ¼ Uk Uk ¼ Idk

∀ k ¼ 1; 2…; K:

ð8Þ

In (8) I WIL denotes the weighted system-wide interference leakage and is given by o P n H Tr Uk Sk Uk k ¼ 1 dk K

I WIL ¼ ∑

3

½k

where Sk ∈CnR nR is the interference covariance matrix at each Rx k which is obtained as

3. IA algorithms

ð9Þ

For the K-user MIMO IFC a sum-rate multiplexing gain of K=2 is feasible with IA at the high interference regime.

K

Sk ¼

∑

P

H Hk;ℓ Vℓ VH ℓ Hk;ℓ :

ℓ ¼ 1;ℓ≠k dℓ

ð10Þ

The authors in [14] showed that I WIL reduces per algorithmic iteration, however, it was commented that convergence to the global minimum, i.e. I WIL ¼ 0, cannot be guaranteed in general. 3.2. Per stream SINR maximization (MaxSINR) The ILM algorithm makes no attempt at maximizing the signal power within the desired signal subspace and hence its sum-rate performance is poor especially at the noise-limited regime [14,16]. Mainly for this reason [14] proposed the MaxSINR algorithm that jointly designs VðnÞ 's k and UðnÞ 's for each nth data stream within the IFC to k maximize its received SINR. In particular, the alternating optimization procedure of [27] was utilized to sequentially solve the following maximization problems: γ kn

max

VðnÞ ;UðnÞ k k

∥VðnÞ ∥ ¼ ∥UðnÞ ∥¼1 k k

subject to

ð11Þ

where γ kn denotes the SINR of the nth data stream at Rx k which is given by γ kn ¼

ðnÞ P½UkðnÞ H Hk;k VðnÞ ½VkðnÞ H HH k;k Uk k

dk ½UðnÞ H Jk UðnÞ k k ½k

:

ð12Þ

½k

In (12) Jk ∈CnR nR comprises the covariances matrices of the noise, the inter-user and the inter-stream interference at each Rx k, and is obtained as Jk ¼

dℓ

K

∑

∑

P

ℓ ¼ 1;ℓ≠k i ¼ 1 dℓ

þ

dk

∑

P

j ¼ 1; j≠n dk

ðiÞ H H Hk;ℓ VðiÞ ℓ ½V ℓ  Hk;ℓ

2 Hk;k VðjÞ ½VðjÞ  H HH k;k þ sk In½k : k k

ð13Þ

R

It is clear that (11) consists of ∑Kk ¼ 1 dk optimization problems and as commented in [16] the MaxSINR algorithm has not resulted from the alternating optimization of a global objective function. Hence, its convergence to the global maximum, i.e. the point maximizing the SINR of all data streams in the IFC, is difficult to be proved. 4. A reconfigurable iterative algorithm We present in the following an iterative algorithm for the K-user MIMO IFC with arbitrary number of transceiver antennas that combines the system-wide MMSE criterion with the single-user WF solution to maximize each individual user rate performance. First, the motivation for the proposed algorithm is highlighted and next we present its mathematical formulation along with a discussion on its convergence characteristics. 4.1. Motivation As already mentioned in Section 1, the optimum sumrate scaling for the K-user MIMO IFC depends on the interference levels and channel conditions. Treating

G.C. Alexandropoulos, C.B. Papadias / Signal Processing 93 (2013) 3353–3362

interference as noise is preferable at the low interference regime [29–31] whereas, IA achieves the optimum sumrate scaling at the interference-limited regime [5]. To this end, choosing between the latter two strategies requires the a priori knowledge of the interference levels. However, as also described in Section 3, the vast majority of the algorithms for IA requires IA feasibility conditions to be met a priori [14–22,24,25]. For example, to achieve IA for the 3-user 4  4 MIMO IFC each Tx must be restricted to send at most 2 data streams to its intended Rx. On the other hand, the sum-rate performance results of [1] for the cellular channel model presented in [35] demonstrated several low-to-moderate interference scenarios where transmitting more data streams than those dictated by the IA feasibility conditions yields higher sum-rate performance than IA. For those scenarios interference-myopic MIMO transmissions were utilized, which are based on the single-user WF solution [2,32] and target at the individual user rate maximization. Finally, a typical feature of the majority of the algorithms for IA is the equal power allocation at each Tx's data streams. Inspired by the capacity-achieving strategy for single-user MIMO systems [32] and the sum-rate results for Ricean fading channels presented in [1], we intuitively expect that the equal power allocation per Tx will be suboptimal when the interference is weak and under strong LOS conditions. Motivated by all the above, we present in the following an iterative algorithm that jointly designs Vk 's and Pk 's for all K Tx's in the MIMO IFC to maximize each individual user rate according to the interference levels and channel conditions. The proposed algorithm implicitly chooses dk for each Tx k along with a rate-optimum power allocation, and also designs Uk 's for all K Rx's to minimize the system-wide MSE.

3357

end, the precoding matrix at each Tx k is designed as Vk ¼ Gk Fk

ð14Þ n½k dk T

dk dk

where Gk ∈C and Fk ∈C ∀ k are obtained iteratively from the system-wide MSE minimization and the single-user WF precoding, respectively, as follows. Setting 1=2 Ek ¼ Vk Pk in (2), the system-wide MSE is derived using [16, eq. (22)] as ( ! ) K

J MSE ¼ ∑ E ‖UH k Hk;k Ek sk þ k¼1

K

∑

ℓ ¼ 1;ℓ≠k

Hk;ℓ Eℓ sℓ þ nk −sk ‖2 :

ð15Þ The K Ek 's and K Uk 's jointly minimizing J MSE are obtained from the solutions of the following constrained optimization problem: min

fEk gKk ¼ 1 ;fUk gKk ¼ 1

subject to

J MSE TrfEk EH k g ≤P:

ð16Þ

In the sequel, each Tx k computes Gk shown in (14) as GðnÞ ¼ EðnÞ =∥EðnÞ ∥ ∀ n ¼ 1; 2; …; dk , and then utilizes the k k k ðK−1Þ MMSE-based filters Eℓ ∀ ℓ≠k satisfying (16) and the ðK−1Þ channel matrices Hk;ℓ ∀ℓ≠k to derive the rateoptimum transmit covariance matrix for its effective channel Hk;k Gk . In particular, each Tx k obtains Fk needed in (14) as well as its power allocation matrix Pk from the solutions of the well-known single-user rate maximization problem: Rk

max Fk ;Pk

subject to

H Fk Pk FH k ≽0 and TrfFk Pk Fk g ≤P

ð17Þ

where Rk denotes the instantaneous achievable rate at Tx k and is given by H H −1 Rk ¼ log2 ½detðIn½k þ Hk;k Gk Fk Pk FH k Gk Hk;k Q k Þ:

ð18Þ

R

n½k n½k R R

4.2. Algorithmic formulation The system-wide MSE metric [28] has been recently considered in the context of the K-user MIMO IFC and some iterative algorithms [16,19,21] for obtaining Vk 's and Uk 's via the alternating minimization [27] of this metric have been recently proposed. It has been shown that, when IA feasibility conditions are known a priori, the system-wide MMSE criterion balances the need for keeping the desired signal level above the noise with that of performing IA. For this case it was also shown through sum-rate simulation results that the above criterion is capable of achieving the appropriate sum-rate scaling at both the noise- and the interferencelimited regimes. The latter inherent capability of the systemwide MMSE was also investigated in [23,26,36,37], where various equivalences between versions of this criterion and the maximization of the WSR and the MaxSINR were shown. In this paper we present an algorithm that utilizes iteratively the system-wide MMSE with the single-user MIMO WF solution [32]. The system-wide MMSE criterion is used to account for the noise and interference levels within the IFC for fixed numbers of dk's whereas, the per user WF solution is used to obtain the rate-optimum transmit covariance matrices for the MMSE-based interference levels as well as the channel conditions. To this

represents the interference-plus-noise In (18) Q k ∈C covariance matrix at Rx k which is obtained as Qk ¼

K

∑

ℓ ¼ 1; ℓak

H 2 Hk;ℓ Gℓ Pℓ GH ℓ Hk;ℓ þ sk In½k :

ð19Þ

R

Similar to [16,19,21,28] we adopt an alternating minimization procedure [27] to obtain all Ek 's and Uk 's satisfying (16) and the WF solution [32] to derive Fk and Pk for each Tx k satisfying (17). After computing the final Vk 's using (14), i.e. those maximizing each individual user's rate, each Rx k obtains Uk minimizing its MSE. In the following we present an iterative implementation of our reconfigurable algorithm that exploits the reciprocity of wireless channels, such as e.g. in time-division duplexing (TDD) systems [38]. Similar to [14] we consider the implementation of our algorithm on reciprocal forward and reverse networks. 4.2.1. Forward network In the original forward network each Rx k computes Uk satisfying (16) given Ek and Pk ∀k ¼ 1; 2…; K. In particular, the nth column of Uk , with n ¼ 1; 2 ,..., dk , is obtained as ¼ UðnÞ k

ðnÞ B−1 k Hk;k Ek

ðnÞ ‖B−1 k Hk;k Ek ‖

ð20Þ

3358

G.C. Alexandropoulos, C.B. Papadias / Signal Processing 93 (2013) 3353–3362 ½k

½k

where Bk ∈CnR nR is given by

Reconfigurable Algorithm for the K-User MIMO IFC. 1:

K

H 2 Bk ¼ ∑ Hk;ℓ Eℓ EH ℓ Hk;ℓ þ sk In½k : ℓ¼1

ð21Þ

½k initialization: Set dk ¼ minðn½k T ; nR Þ ∀k ¼ 1; 2; …; K and start ½k

with arbitrary unit-column precoding Vk ∈CnT allocation Pk ¼ P=dk In½k

R

dk

and power

T

2:

4.2.2. Reverse network In the reverse network each Rx k utilizes Uk to transmit to its intended Tx k. Then, each Tx k computes Vk and Pk through the following two-step procedure. Step I: Given Uk and Pk ∀k ¼ 1; 2…; K, each Tx k in the reverse network computes its receive filter Ek so as to satisfy (16). To this end, the nth column of Gk is derived as ←

GkðnÞ

¼

ðnÞ ðBk Þ−1 HH k;k Uk ←

ðnÞ ‖ðBk Þ−1 HH k;k Uk ‖ ←

½k

where Bk ∈CnT ←

ð22Þ

n½k T

is given by

H Bk ¼ ∑ HH ℓ;k Uℓ Pℓ Uℓ Hℓ;k þ μk In½k :

ð23Þ

T

In (23) parameter μk is calculated so that the power constraint at Tx k is satisfied [28]. Step II: After obtaining all Gk 's from (22) and given Pℓ ∀ ℓ≠k, each Tx k computes Q k using (19) and obtains its rate-optimum Fk and Pk for its effective channel Hk;k Gk . In particular, the singular value decomposition (SVD) of Hk;k Gk after noise prewhitening is derived as −1=2

Qk

Hk;k Gk ¼ Wk Λk FH k n½k n½k R R

5: 6: 7: 8:

←

Compute Bk at each Tx k according to (23) Obtain each Tx k sum-MMSE-based filter Gk using (22) Step II: Compute Q k at each Rx k according to (19) −1

Perform SVD to each effective channel Q k 2 Hk;k Gk according to (24) and obtain Fk 9: Obtain each Tx k precoding Vk according to (14) 10: Compute Pk for each Tx k from the WF solution for its effective −1=2

K

ℓ¼1

3: 4:

Begin iteration Forward Network Compute Bk at each Rx k according to (21) Obtain each Rx k sum-MMSE-based filter Uk using (20) Reverse Network Step I:

ð24Þ n½k dk R

where Wk ∈C and Λk ∈C , and Fk represents the rate-optimum precoding matrix for channel Hk;k Gk to be utilized in (14). The rate-optimum power allocation Pk for each Tx k is finally derived from the WF solution [2,32] for −1 the prewhitened channel Q k 2 Hk;k Gk . The proposed reconfigurable iterative algorithm for the K-user MIMO IFC with arbitrary number of transceiver antennas is summarized in the top of the next column.

channel Q k Hk;k Gk 11: Repeat until ∑Kk ¼ 1 Rk using (18) converges, or until the number of iterations reaches a predefined limit 12: Obtain each Rx k MMSE-based filter Uk as Uk ¼ B−1 k Hk;k Vk

The reconfigurable algorithm is initialized with the maximum allowable number of data streams per Tx k, i. ½k e. dk ¼ minðn½k T ; nR Þ, and its inherent reconfigurability to the interference levels and channel conditions lies on the per user MIMO WF solution utilized in Step II. At each algorithmic iteration, Step I attempts to maximize the sum-rate performance for fixed dk per Tx k whereas, in Step II Pk reveals the optimum number of data streams resulting in the maximum Rk for each Tx k. Our numerous computer simulation results indicated that the sum of the instantaneous achievable user rates for our algorithm, K

given as ∑ Rk using (18), increases per iteration and k¼1

converges often to a maximum value. A thorough investigation of the convergence characteristics of the proposed algorithm is left as a future work.

4.3. Convergence

5. Performance evaluation results and discussion

The proposed iterative algorithm capitalizes on the reciprocity of wireless channels, such as when TDD communication is used, to design Vk 's and Pk 's maximizing the individual user rates as well as Uk 's minimizing the MSE at each Rx k. At each algorithmic iteration and in Step I, (16) is considered in order to devise the appropriate sum-rate scaling for the interference levels caused by fixed number of dk's in the IFC. As our numerous simulation results showed and similar to [16,28], J MSE converges very fast at least to a local minimum. Then, at the same algorithmic iteration and in Step II, each Tx k solves (17) to obtain its rate-optimum transmit covariance matrix, according to the interference levels resulted from the minimization of J MSE in Step I as well as the channel conditions. More specifically, in Step II the proposed algorithm computes Fk and Pk for each Tx k within spanðHk;k Gk Þ, where Gk is obtained from Step I. The latter holds due the following property [12]:

This section presents computer simulation results for the sum-rate performance of the proposed reconfigurable iterative algorithm introduced in Section 4. For comparison purposes we have also simulated the sum rate of the ILM and MaxSINR iterative algorithms described in Section 3. In particular, we have obtained the ergodic sum-rate performance defined as [16] ( )

1=2

spanðHk;k Gk Þ ¼ spanðHk;k Gk Fk Pk Þ:

ð25Þ

K

−1 H ∑ log2 ½detðIn½k þ Hk;k Vk Pk VH k Hk;k Ck Þ

Rerg ¼ EH

k¼1

ð26Þ

R

where EH fg denotes the expectation over all channel ½k

½k

realizations Hk;ℓ ∀k; ℓ ¼ 1; 2; …; K and Ck ∈CnR nR represents the interference-plus-noise covariance matrix at each Rx k, which is obtained as Ck ¼

K

∑

ℓ ¼ 1; ℓak

H 2 Hk;ℓ Vℓ Pℓ VH ℓ Hk;ℓ þ sk In½k :

ð27Þ

R

The averaging in (26) was evaluated via Monte Carlo simulations for 200 independent channel realizations and

G.C. Alexandropoulos, C.B. Papadias / Signal Processing 93 (2013) 3353–3362

3−user 4x4 MIMO Interference Channel

90

Ergodic Sum Rate in bps/Hz

80 70 60

3−user 4x4 MIMO Interference Channel 40

Myopic with d=4 Myopic with d=3 Myopic with d=2 Myopic with d=1 Optimized IA Precoding Interference Leakage Minimization Per Stream SINR Maximization (d=2) Per Stream SINR Maximization (d=4)

35 Ergodic Sum Rate in bps/Hz

100

3359

50 40 30

30 25

Myopic with d=4 Myopic with d=3 Myopic with d=2 Myopic with d=1 Optimized IA Precoding Interference Leakage Minimization Per Stream SINR Maximization (d=2) Per Stream SINR Maximization (d=4)

20 15 10

20

5

10 0 −5

0

5

10

15

20

25

0 −5

30

0

5

½j the channels were normalized as Ef‖Hk;j ‖2F g ¼ n½k R nT ∀k; j ¼ 1; 2; …; K. Without loss of generality, for the channel model presented in Section 2.2 we have assumed that αk;ℓ ¼ α and κk;ℓ ¼ κ ∀k; ℓ. In addition, to compare the performance of all iterative algorithms with a closed-form perfect IA design, we have focused on the 3-user 4  4 MIMO IFC and also simulated the ergodic sum-rate performance of a linear precoder technique proposed in [12]. This technique, termed as optimized IA precoding in the figures, optimizes the closed-form IA precoding matrices [5] according to zeroforcing decoding and with respect to the individual user rate maximization. For all iterative algorithms Vk 's and Uk 's were randomly initialized with unit norm columns. For the considered ½k n½k T ¼ nR ¼ 4 ∀ k ¼ 1; 2 and 3 case, the reconfigurable algorithm was also initialized with dk ¼ d ¼ 4 and Pk ¼ ðP=4ÞI4 ∀ k whereas, for all other iterative algorithms IA feasibility conditions were set a priori, i.e. dk ¼ d ¼ 2 ∀ k, and Pk ¼ ðP=2ÞI2 ∀ k. In addition, for scenarios with low average powered interference links, we have simulated the ergodic sum rate of a genie-aided MaxSINR algorithm that utilizes dk ¼ d ¼ 4 ∀ k. A maximum of 1000 iterations was used per iterative algorithm and each algorithm was declared converged when the difference in its objective function between two successive iterations was less than 10−4 . Figs. 2–4 summarize the motivations for the proposed reconfigurable algorithm presented in Section 4.1. In particular, these figures depict Rerg versus the transmit SNR per Tx P for various precoding techniques including interference-myopic MIMO transmissions. For the latter transmissions each Tx k is assumed to possess Ck , treats interference as noise and utilizes selfishly the single-user WF solution for precoding at most d ¼1, 2, 3 and 4 data streams. As shown in Fig. 2 for Rayleigh fading channels with asymmetric average powers among the intended and interference links described by α ¼ 10−2 , interferencemyopic MIMO transmissions with either d ¼3 or 4 outperform all IA-achieving algorithms in the whole SNR range plotted. For this value of α, and certainly for lower ones,

15

20

25

30

Fig. 3. Ergodic sum-rate performance, Rerg , of various transmission techniques versus transmit SNR per Tx, P, for the 3-user 4  4 MIMO IFC over Ricean fading with κ k;ℓ ¼ 10 and αk;ℓ ¼ 1 ∀k; ℓ. 3−user 4x4 MIMO Interference Channel

80 70

Ergodic Sum Rate in bps/Hz

Fig. 2. Ergodic sum-rate performance, Rerg , of various transmission techniques versus transmit SNR per Tx, P, for the 3-user 4  4 MIMO IFC over Rayleigh fading with αk;ℓ ¼ 10−2 ∀k; ℓ.

10

Transmit SNR per Tx in dB

Transmit SNR per Tx in dB

60 50

Myopic with d=4 Myopic with d=3 Myopic with d=2 Myopic with d=1 Optimized IA Precoding Interference Leakage Minimization Per Stream SINR Maximization (d=2) Per Stream SINR Maximization (d=4)

40 30 20 10 0 −5

0

5

10

15

20

25

30

Transmit SNR per Tx in dB

Fig. 4. Ergodic sum-rate performance, Rerg , of various transmission techniques versus transmit SNR per Tx, P, for the 3-user 4  4 MIMO IFC over Ricean fading with κ k;ℓ ¼ 10 and αk;ℓ ¼ 10−2 ∀k; ℓ.

interference-myopic MIMO transmissions with d¼4 achieve the optimum sum-rate scaling 3 at high SNRs whereas, all IA-achieving techniques appear to be lacking of an inherent mechanism to adjust to α, and are restricted to achieve a multiplexing gain of only 3/2. To this end, we have also plotted within this figure the Rerg of the genie-aided MaxSINR that runs with d¼4. Clearly, this algorithm performs similar to the best interference-myopic MIMO scheme at high SNR values, achieving the optimum sum-rate scaling 3. However, at low SNRs, the genie-aided MaxSINR yields poorer Rerg than the majority of the interference-myopic MIMO schemes and even the conventional MaxSINR algorithm. We intuitively believe that this behavior is mainly due to the equal power allocation assigned by both MaxSINR algorithms to each Tx k data streams. The findings of Fig. 2 are more pronounced in Figs. 3 and 4, where we have considered Ricean fading channels with κ ¼ 10 as well as

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Ergodic Sum Rate in bps/Hz

45 40

40

Optimized IA Precoding Interference Leakage Minimization Per Stream SINR Maximization (d=2) Per Stream SINR Maximization (d=4) Reconfigurable Precoding

35 Ergodic Sum Rate in bps/Hz

50

3−user 4x4 MIMO Interference Channel

3−user 4x4 MIMO Interference Channel

55

35 30 25 20 15

30

Optimized IA Precoding Interference Leakage Minimization Per Stream SINR Maximization (d=2) Per Stream SINR Maximization (d=4) Reconfigurable Precoding

25 20 15 10

10 5

5 0 −5

0

5

10

15

20

25

0 −5

30

0

5

Fig. 5. Ergodic sum-rate performance, Rerg , of the reconfigurable algorithm versus transmit SNR per Tx, P, for the 3-user 4  4 MIMO IFC over Rayleigh fading with αk;ℓ ¼ 1 ∀k; ℓ.

80

20

25

30

3−user 4x4 MIMO Interference Channel

80

Optimized IA Precoding Interference Leakage Minimization Per Stream SINR Maximization (d=4) Reconfigurable Precoding

Optimized IA Precoding Interference Leakage Minimization Per Stream SINR Maximization (d=4) Reconfigurable Precoding

70 Ergodic Sum Rate in bps/Hz

Ergodic Sum Rate in bps/Hz

90

15

Fig. 7. Ergodic sum-rate performance, Rerg , of the reconfigurable algorithm versus transmit SNR per Tx, P, for the 3-user 4  4 MIMO IFC over Ricean fading with κk;ℓ ¼ 10 and αk;ℓ ¼ 1 ∀k; ℓ.

3−user 4x4 MIMO Interference Channel 100

10

Transmit SNR per Tx in dB

Transmit SNR per Tx in dB

70 60 50 40 30

60 50 40 30 20

20

10

10 0 −5

0

5

10

15

20

25

30

Transmit SNR per Tx in dB

Fig. 6. Ergodic sum-rate performance, Rerg , of the reconfigurable algorithm versus transmit SNR per Tx, P, for the 3-user 4  4 MIMO IFC over Rayleigh fading with αk;ℓ ¼ 10−2 ∀k; ℓ.

α ¼ 1 and α ¼ 10−2 , respectively. It is again obvious that all considered algorithms designed for IA are incapable of adjusting to low values of the SNR and/or α. In addition, as shown in Fig. 3 for α ¼ 1 and very low SNR values, interference-myopic MIMO transmissions with either d¼3 or 4 result in the best Rerg performance. Moreover, it is clear from this figure that the genie-aided MaxSINR does not achieve the optimum sum-rate scaling 3/2 offered by IA at high SNRs. Finally, Fig. 4 shows that for α ¼ 10−2 the Rerg gap between the best interference-myopic MIMO schemes and the genie-aided MaxSINR is larger and for a wider SNR range compared with that in Fig. 2. This shows that the impact of equal power allocation in the Rerg performance is more severe when there exist LOS components in the IFC. Both Figs. 3 and 4 reveal that the optimized IA precoding, ILM and both versions of the MaxSINR algorithm are failing to adjust to strong LOS channel conditions. The sum-rate performance of the proposed reconfigurable iterative algorithm is illustrated in Figs. 5 and 6 for Rayleigh

0

−5

0

5

10

15

20

25

30

Transmit SNR per Tx in dB

Fig. 8. Ergodic sum-rate performance, Rerg , of the reconfigurable algorithm versus transmit SNR per Tx, P, for the 3-user 4  4 MIMO IFC over Ricean fading with κk;ℓ ¼ 10 and αk;ℓ ¼ 10−2 ∀k; ℓ.

fading channels with average powers among the intended and interference links modeled by α ¼ 1 and α ¼ 10−2 , respectively. Assuming Ricean fading with κ ¼ 10, Figs. 7 and 8 depict the same performance for α ¼ 1 and 10−2 , respectively. As shown from all these figures the proposed algorithm yields the best Rerg performance irrespective of the values for the SNR, α and κ. In particular, it is shown that for α ¼ 1 and high SNR values the proposed algorithm achieves the optimum sum-rate scaling 3/2, i.e. that of IA, for all considered κ values. Moreover, it outperforms the optimized IA precoding, ILM and both the MaxSINR algorithms. In addition, the proposed algorithm adjusts itself to the case where α ¼ 10−2 and achieves the optimum sumrate scaling 3 at high SNRs for every value of κ. For the latter case the proposed algorithm yields the same or higher Rerg than the genie-aided MaxSINR. As also depicted in Figs. 5–8, the reconfigurable algorithm outperforms all others and in the noise-limited regime. In fact as κ increases, i.e. LOS conditions become stronger, the Rerg

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3−user 4x4 MIMO Interference Channel

3361

Reconfigurable Algorithm

70

50

d=4

d=4

45

Ergodic Sum Rate in bps/Hz

d=4 50

Instantaneous Sum Rate in bps/Hz

60

SNR=20 dB d=3 d=2

40 Optimized IA Precoding Per Stream SINR Maximization Reconfigurable Precoding

30

20

d=3

d=3

d=2

d=3

40 35 30 25 20 15 10

0 dB 5 dB 15 dB 25 dB

10

SNR=5 dB 0 −3 10

−2

−1

10

5 0

0

10

10

1

5

10

α

Fig. 9. Ergodic sum-rate performance, Rerg , of the reconfigurable algorithm versus α for the 3-user 4  4 MIMO IFC over Rayleigh fading channels with transmit SNR per Tx P ¼ 5 and 20 dB.

25

30

35

40

Fig. 11. Convergence of the instantaneous achievable sum rate of the proposed reconfigurable algorithm and for the 3-user 4  4 MIMO IFC over Rayleigh fading with αk;ℓ ¼ 1 ∀k; ℓ.

Reconfigurable Algorithm 40

d=3

40

SNR=20 dB

d=3

35 Instantaneous Sum Rate in bps/Hz

d=3

Ergodic Sum Rate in bps/Hz

20

3−user 4x4 MIMO Interference Channel

50

d=2 30

20

15

Number of Algorithmic Iterations

d=2

10

0 −3 10

SNR=5 dB

d=2

d=2

d=2

Optimized IA Precoding Per Stream SINR Maximization Reconfigurable Precoding −2

25 20 15 10 0 dB 5 dB 15 dB 25 dB

5 −1

10

30

10

0

10

α

0

1

5

10

15

20

25

30

35

40

Number of Algorithmic Iterations

Fig. 10. Ergodic sum-rate performance, Rerg , of the reconfigurable algorithm versus α for the 3-user 4  4 MIMO IFC over asymmetric Ricean fading with κ k;ℓ ¼ 10 ∀k ¼ ℓ and Rayleigh fading with κk;ℓ ¼ 0 ∀k≠ℓ, and with transmit SNR per Tx P ¼ 5 and 20 dB.

Fig. 12. Convergence of the instantaneous achievable sum rate of the proposed reconfigurable algorithm and for the 3-user 4  4 MIMO IFC over Ricean fading with κ k;ℓ ¼ 10 and αk;ℓ ¼ 1 ∀k; ℓ.

gains offered by the reconfigurable algorithm increase. This behavior lies in the inherent power allocation mechanism of the proposed algorithm that implicitly chooses dk for each Tx k together with its rate-optimum power allocation. To further illustrate how our reconfigurable algorithm chooses dk for each Tx k according to the interference levels and channel conditions, we plot Rerg versus α in Figs. 9 and 10, respectively, for an IFC with symmetric Rayleigh faded links and an asymmetrically faded IFC for which the intended links are subject to Ricean fading with κ ¼ 10 and the interference links are subject to Rayleigh fading. For these two figures we have simulated the Rerg of the MaxSINR, the genie-aided MaxSINR and for an additional version of the MaxSINR algorithm that utilizes dk ¼ d ¼ 3 ∀k, and plotted the Rerg of the MaxSINR version that resulted in the higher Rerg performance. As shown the proposed algorithm appears to choose the dk for each Tx k that yields the higher Rerg performance. Furthermore, it is clear that the proposed algorithm obtains power allocation

schemes, especially at low SNR values, that result in higher Rerg than the best version of the MaxSINR algorithm. All in all, by comparing Figs. 2–4 with Figs. 5–8, we conclude that for high interference levels the proposed algorithm reconfigures itself so as to achieve the IA scaling whereas, for low-to-moderate interference it performs similar to the interference-myopic MIMO transmissions. Some representative examples for the convergence of the proposed reconfigurable iterative algorithm for the case where α ¼ 1 are included in Figs. 11 and 12 for Rayleigh and Ricean with κ ¼ 10 fading channels, respectively. In these figures the instantaneous achievable sum-rate performance, obtained as ∑Kk ¼ 1 Rk using (18), is plotted for each algorithmic iteration and for various values of the Tx SNR P. As it is shown from these results, the instantaneous sum rate converges fast to a maximum value independently of the Ricean factor κ. However, the specific value of P seems to influence the speed of the sum-rate convergence. The same trend has been also observed from additional experiments

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for varying values of α. In particular, we have noticed that as α-0 and P-0 less algorithmic iterations are needed for the convergence of our algorithm. It is finally noted that, from our numerous computer simulations, we have observed that the convergence of the proposed algorithm is on average faster than that of the MaxSINR [14].

[14]

[15]

6. Conclusions

[16]

In this paper a novel reconfigurable iterative algorithm for the K-user MIMO IFC with arbitrary number of transceiver antennas was presented. The proposed algorithm combines the system-wide MMSE criterion with the single-user MIMO WF solution to maximize each user's transmission rate according to the interference levels and channel conditions. Extensive computer simulations for the sum-rate performance of the reconfigurable algorithm over various Ricean fading conditions were presented and compared with those of already well-known transceiver techniques for IA. As shown, in the interference-limited regime, our algorithm reconfigures itself so as to achieve the IA scaling whereas, in the low-to-moderate interference regime, it leads itself towards interference-myopic MIMO transmissions. Furthermore, it was shown that for all investigated test cases the sum-rate performance of the proposed algorithm is always higher than that of all considered transmission techniques.

[17]

[18]

[19]

[20]

[21] [22]

[23]

[24]

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[25]

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