Generalized MMSE beamforming for multicell MIMO systems with random user geometry and channel feedback latency
© Sohn et al.; licensee Springer. 2014
Received: 10 October 2013
Accepted: 14 May 2014
Published: 24 May 2014
In this paper, we propose a generalized minimum mean square error (MMSE) beamforming for downlink multicell multiple-input multiple-output (MIMO) systems with local BS cooperations. Unlike the previous beamforming strategies which have been designed for an idealized multicell MIMO system model, we consider a complicated but realistic multicell MIMO system model. Our realistic multicell MIMO system model captures (i) different average SNRs of users due to random geometrical distribution of users within cells, (ii) channel feedback latency due to air propagation time and signal processing time, and (iii) different channel aging effects for intracell channel information and intercell channel information due to an additional backhaul latency during the exchange of information between pairs of neighboring cells. The key novelty of this paper is that we derive the closed-form beamforming expressions of the generalized MMSE beamforming using convex optimization technique. The closed-form beamforming expression gives insights on how the geometry of users and channel feedback latency affect the construction of optimal beamforming vectors. Numerical results verify that the proposed generalized MMSE beamforming outperforms previous beamforming strategies in both BER and average throughput performances.
With increasing demands of various multimedia service, the next generation wireless communication systems are expected to support much higher data rate than before. The demand prompts an increase of interests in spectrally efficient technologies and therefore multiple-input multiple-output (MIMO) has been extensively studied as a key technology for cellular systems. Recently, MIMO techniques are severely challenged by intercell interference. In [1–3], intercell interference has been shown to significantly degrade system performance especially in the reusability of the spectral resource, which results in limiting the overall spectral efficiency.
To mitigate the degradation, a promising approach is to allow the base stations (BSs) to cooperate. The BS cooperation was introduced in  and then extended to downlink cooperative MIMO systems [5, 6]. In downlink cooperative MIMO systems, multiple BSs exploit their multiple antennas and use beamforming techniques based on the sharing of all required information including user data and channel state information (CSI). The BS cooperation has been theoretically proven to improve the system performance in terms of cell coverage, cell-edge user throughput as well as average sum rate, and the improvement becomes much larger as the network size grows. However, the BS cooperation suffers from real-world constraints such as limited capacity and non-negligible latency of the backhaul network for information sharing and huge computational complexity of joint processing across all BSs.
Several pragmatic solutions for the problems have been suggested in [7–9]. The key idea behind these techniques is sharing CSI locally between neighboring cells without sharing user data. In multicell MIMO systems with local CSI sharing, each BS determines its beamforming vector based on CSI of both intracell and intercell links. In , a simple beamforming technique for maximizing user’s signal-to-noise ratio (SNR) has been introduced. And zero-forcing beamforming (ZFBF) has been extended to multicell MIMO systems in . A beamforming technique that computes the best beamforming vector by defining a new metric called signal-to-generating interference-plus-noise ratio (SGINR) has also been proposed for multicell MIMO systems in .
The aforementioned beamforming techniques, however, consider a simplified multicell MIMO system model assuming an ideal user distribution within a cell and an unlimited-capacity and zero-latency backhaul network. In the emerging cellular systems where an early form of coordinated multipoint transmission and reception (CoMP) is ready for commercial deployments , the following implementation issues should be carefully considered. Firstly, cellular users are randomly located within their cells and therefore experience quite different path loss [11, 12]. As a result, all users now have different values of average SNR. Secondly, when a mobile station (MS) receives a reference signal from a BS, it requires processing time to perform channel estimation and create CSIs for channel feedback. Similarly, a BS also requires processing time to collect CSI feedback and compute beamforming vectors after MSs feedbacks their CSIs. A typical time delay between the channel estimation and actual transmission via beamforming can increase up to 8 ms in commercialized long term evolution (LTE) systems. This channel aging effect due to the channel feedback latency significantly degrades beamforming performances as discussed in many related literatures. Finally, the backhaul network, which is specifically defined as X2-link in commercialized LTE systems, imposes some limitations for the exchanges of CSIs between pairs of BSs. The backhaul network introduces additional channel feedback latencies across cells, which further increases channel aging effects . Thus, the shared CSI collected from neighboring cells are normally more outdated than that of the self cell and thus provides relatively inaccurate information for the determination of beamforming vectors.
In this paper, we consider the design of downlink beamforming technique for multicell MIMO systems with local CSI sharing. In particular, we follow the principle of MMSE beamforming that computes the best beamforming vectors by minimizing the error between the transmitted signal and received signal caused from interference and noise as in [14–16] and generalize it for multicell MIMO systems considering real-world implementation issues. The beauty of the proposed MMSE beamforming is that we can find the closed form beamforming expressions using convex optimization techniques even for the complicated realistic multicell MIMO system model . Numerical results verify that the proposed generalized MMSE beamforming outperforms other beamforming strategies in terms of both BER and average throughput.
The notations used in this paper are as follows: Boldfaces are used for vectors and matrices; denotes a mathematical expectation of random variables; I M denotes a M-by-M identity matrix; superscripts ∗ and ⋆ are used to denote complex conjugate and optimal value, respectively.
The rest of this paper is organized as follows: Section 2 describes our realistic multicell MIMO system model. In Section 3, we briefly provide an overview of three conventional beamforming strategies developed for downlink multicell MIMO systems with local CSI sharing. In Section 4, we formulate an optimization problem and derive a closed-form solution of the proposed generalized MMSE beamforming. Finally, we provide numerical results in Section 5 and conclude in Section 6.
2 System model
where xk,mis the data symbol of the k th user transmitted through M transmit antennas of the m th BS satisfying . hk,mand gk,m,jdenote the desired time-varying channel vector from the m th BS towards the k th user in the m th cell and the interfering time-varying channel vector between the k th user in the m th cell and the j th BS, respectively. xk,mand the elements of both hk,mand gk,m,jare assumed to be independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero mean and unit variance. wk,mis the unit-norm beamforming vector for the transmission to the k th user in the m th cell , and nk,mis the additive complex Gaussian noise with zero means and unit variance. ρk,mis the average signal-to-noise ratio (SNR) reflecting random geometry as , where dk,m, α, N0, and P denote the distance of the k th user in the m th cell from the m th BS, path loss exponent, noise power, and transmit power, respectively. Similarly, ηk,m,jis the average interference-to-noise ratio (INR) for the interference that the j th BS causes to the k th user in the m th cell, denoted as where dk,m,jis the distance of the k th user in the m th cell from the j th BS. In (1), the first term on the right implies a desired signal, the second term implies an intracell interference from the same cell, and the third term implies an intercell interference leaked from the neighboring cells.
where J0(x) is the zeroth-order Bessel function and f D is the maximum Doppler frequency in Hertz. As shown in , the maximum Doppler frequency can be represented by where vk,mis the velocity of the k th user in the m th cell, f c is the carrier frequency, and c is the speed of light, respectively.
3 Beamforming strategies for downlink multicell MIMO systems
In this section, three beamforming strategies are presented, which have been originally developed for the multiuser MIMO in a single-cell system model. Motivated by the analogies between mitigating intracell and intercell interference except the heterogeneity of interference links, three beamforming strategies are generalised to fit in our multicell system model with slight modifications. In addition, limitations of the conventional approaches are discussed in detail.
3.1 Maximum SNR (MAX-SNR) Beamforming Strategy
Note that the maximum SNR beamforming strategy has an advantage in enabling the decentralized network control without information sharing on the backhaul link. However, the system performance could be significantly degraded by intercell interference especially for near cell-edge users.
3.2 Minimum generating interference (MIN-GI) beamforming strategy
Note that the transmit beamforming vectors chosen in (7), called minimum generating interference (MIN-GI) beamforming, are recognized as an extension of zero-forcing beamforming (ZFBF) since they try to avoid interference both among users and between cells. Inheriting the drawbacks of ZFBF [20, 21], the MIN-GI beamforming strategy suffers from severe performance degradation especially in the time-varying channel, where reported CSI becomes quickly outdated.
3.3 Maximum signal-to-generating interference-plus-noise ratio (MAX-SGINR) beamforming strategy
Note that the numerator of (8) is the signal power at the desired user and the denominator consists of noise and interference to other users generated by the m th BS. In , it has been shown that in the simplified multicell-MIMO environment where all users have the same SNR and INR, MAX-SGINR beamforming outperforms both MAX-SNR beamforming and MIN-GI beamforming.
3.4 Limitations of conventional multicell MIMO beamforming strategies
In [7–9], the conventional multicell MIMO beamforming strategies does not consider realistic cellular parameters such as the random geometry of the users and different temporal correlations due to the mobility of the users. In , it is proven that user’s different SNR and INR reflecting the distances from the BSs significantly affect system performance. Therefore, the beamforming vector for multicell MIMO systems should be effectively chosen according to the different user SNR-INR ratio. In addition, it has been shown that the accuracy of channel information dominates the beamforming performance of multicell MIMO systems , and thus channel feedback latency as well as user mobility play critical roles in determining beamforming vectors. The MMSE beamforming forms a favorable mathematical framework to capture aforementioned considerations as will be shown in the next section.
4 Generalized MMSE beamforming for downlink multicell MIMO systems
4.1 Problem formulation
where is the conditional expectation of x k,m , hw, and gw. Note that in (9), we modify the conventional MSE definition, , due to the difference between our system model and the conventional multicell MIMO system model [7–9], where, for simplicity, all average SNRs are assumed to be the same without considering the random geometry of users. However, our complicated system model considers the different received average SNR ρk,m, and INR ηk,m,j that capture the effects of both transmit power and path loss. Thus, the modified MSE definition reflecting the difference of those system models is used for the design of beamforming vectors.
Proof. See details in the Appendix.
4.2 Closed-form solution via convex optimization
where I M is the identity matrix of size M.
Based on the closed-form expression in (16), we can see how users’ different SNRs, INRs, temporal correlations, and channel feedback latencies influence the choice of optimal beamforming vectors. In particular, the ratio of INR to SNR, , is important factor to construct optimal beamforming vector. At the very-low INR regime (), the optimal beamforming vector, , converges to that of the conventional MMSE beamforming derived for single-cell multiuser MIMO systems as in . This confirms that the conventional MMSE beamforming for single-cell MIMO systems is a special case of the proposed generalized MMSE beamforming for multicell MIMO systems when INR ≪SNR. In addition, when all users’ SNRs and temporal correlations are also assumed to be the same, the optimal MMSE beamforming vector in (16) is simplified to similar expression as derived in [14–16] as , where and are the same average SNR and channel error of users. At the very-high INR regime (), the optimal beamforming vector is constructed largely dominated by the interference from neighboring BSs. In a nutshell, the generalized MMSE beamforming vectors are effectively determined and dynamically balanced considering the overall status of multicell MIMO systems.
where μ j is the j th singular value of matrix Φ and u j is the j th column vector of matrix U. The value of can be found solving (18). Now, all the beamforming vectors with Lagrange multipliers could be solved out.
4.3 Impacts of multiple receive antennas
Multiple receive antennas provide additional degree-of-freedom for beamforming construction, which enables several variants of receiver beamforming. For simplicity, we try to limit our interests to the case of the receivers with maximal ratio combining, i.e., matched filters. The matched filter, , combines received signals at multiple receive antennas, maximising received SNR. Since this SNR improvement is statistically independent with interference links, it reduces the contributions of the interference links in steering transmit beamforming vectors as seen from the analysis of the previous section. In the context of a multicell environment, increasing number of receive antennas can be effectively regarded as , i.e., diminishing INR; the intercell interference becomes negligible in the limit of receive antenna number. In short, increasing number of receive antennas has an impact on the proposed MMSE beamforming in a way that beamforming vector constructions become gradually robust against interference links.
5 Numerical results
Carrier frequency, f c
BS transmit power, P
10 ∼40 dBm
Number of BS antennas, M
Number of neighboring cells, L
Intracell feedback latency
Backhaul link delay
Pathloss exponent, α
Noise power, N0
MS mobility, v
In this paper, we have proposed a generalized MMSE beamforming for downlink multicell MIMO systems with local CSI sharing and derived a closed-form solution. As the key novelty of this paper, we have considered a complicated but realistic multicell MIMO system model including the users’ random geometries, different temporal correlations, and different channel feedback latencies. We have shown surprising improvements in terms of both BER and average throughput performances using the proposed generalized MMSE beamforming.
Proof of Theorem 1
where (a) uses the fact that , only when k = i and m = j, E[|xk,m∗xi,j|2] = 0 otherwise, and (b) computes the conditional expectation with respect to hw and gw after replacing hk,mand gk,m,jwith (2) and (3), respectively. In this computation, we also use the assumption that the elements of hw and gw are i.i.d. circularly symmetric complex Gaussian random variables with zero mean and unit variance, i.e., and . This finally gives (10).
This paper was supported by Samsung Electronics Co., Ltd. and the Brain Korea 21 Plus Project in 2014.
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