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Aspects of Favorable Propagation in Massive MIMO

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Aspects of Favorable Propagation in Massive MIMO ASPECTS OF FAVORABLE PROPAGATION IN MASSIVE MIMO Hien Quoc Ngo∗, Erik G. Larsson , Thomas L. Marzetta† ∗ ∗ Department of Electrical Engineering (ISY), Link¨oping University, 581 83 Link¨oping, Sweden † Bell Laboratories, Alcatel-Lucent Murray Hill, NJ 07974, USA ABSTRACT can simultaneously beamform multiple data streams to mul- Favorable propagation, defined as mutual orthogonality tiple terminals without causing mutual interference. Favor- among the vector-valued channels to the terminals, is one able propagation of massive MIMO was discussed in the pa- of the key properties of the radio channel that is exploited pers [4,5]. There, the condition number of the channel matrix in Massive MIMO. However, there has been little work that was used as a proxy to evaluate how favorable the channel is. studies this topic in detail. In this paper, we first show that These papers only considered the case that the channels are favorable propagation offers the most desirable scenario in i.i.d. Rayleigh fading. However, in practice, owing to the fact terms of maximizing the sum-capacity. One useful proxy for that the terminals have different locations, the norms of the whether propagation is favorable or not is the channel con- channels are not identical. As we will see here, in this case, dition number. However, this proxy is not good for the case the condition number is not a good proxy for whether or not where the norms of the channel vectors may not be equal. For we have favorable propagation. this case, to evaluate how favorable the propagation offered In this paper, we investigate the favorable propagation by the channel is, we propose a “distance from favorable condition of different channels. We first show that under fa- propagation” measure, which is the gap between the sum- vorable propagation, we maximize the sum-capacity under a capacity and the maximum capacity obtained under favorable power constraint. When the channel vectors are i.i.d., the propagation. Secondly, we examine how favorable the chan- singular value spread is a useful proxy to evaluate how fa- nels can be for two extreme scenarios: i.i.d. Rayleigh fading vorable the propagation environment is. However, when the and uniform random line-of-sight (UR-LoS). Both environ- channel vectors have different norms, this is not so. We also ments offer (nearly) favorable propagation. Furthermore, to ask whether or not practical scenarios will lead to favorable analyze the UR-LoS model, we propose an urns-and-balls propagation. To this end, we consider two extreme scenar- model. This model is simple and explains the singular value ios: i.i.d. Rayleigh fading and uniform random line-of-sight spread characteristic of the UR-LoS model well. (UR-LoS). We show that both scenarios offer substantially fa- vorable propagation. We also propose a simple urns-and-balls model to analyze the UR-LoS case. For the sake of the argu- 1. INTRODUCTION ment, we will consider the uplink of a single-cell system. Recently, there has been a great deal of interest in massive multiple-inputmultiple-output(MIMO) systems where a base 2. SINGLE-CELL SYSTEM MODEL station equipped with a few hundred antennas simultaneously arXiv:1403.3461v1 [cs.IT] 13 Mar 2014 serves several tens of terminals [1–3]. Such systems can de- Consider the uplink of a single-cell system where K single- liver all the attractive benefits of traditional MIMO, but at a antenna terminals independently and simultaneously transmit much larger scale. More precisely, massive MIMO systems data to the base station. The base station has M antennas can provide high throughput, communication reliability, and and all K terminals share the same time-frequency resource. high power efficiency with linear processing [4]. If the K terminals simultaneously transmit the K symbols , where E 2 , then the received One of the key assumptions exploited by massive MIMO x1,...,xK xk = 1 M 1 vector at the base station| is| × is that the channel vectors between the base station and the terminals should be nearly orthogonal. This is called favor- K able propagation. With favorable propagation, linear process- y = √ρ gkxk + w = √ρGx + w, (1) ing can achieve optimal performance. More explicitly, on the Xk=1 T CM×1 uplink, with a simple linear detector such as the matched fil- where x = [x1,...,xK ] , G = [g1,..., gK ], gk ter, noise and interference can be canceled out. On the down- is the channel vector between the base station and the∈kth ter- link, with linear beamforming techniques, the base station minal, and w is a noise vector. We assume that the elements of w are i.i.d. CN(0, 1) RVs. With this assumption, ρ has The work of H. Q. Ngo and E. G. Larsson was supported in part by the Swedish Research Council (VR), the Swedish Foundation for Strategic the interpretation of normalized “transmit” signal-to-noise ra- Research (SSF), and ELLIIT. tio (SNR). The channel vector gk incorporates the effects of large-scale fading and small-scale fading. More precisely, the Secondly, we consider a more relaxed constraint on the 2 2 mth element of gk is modeled as: channel G: constraint G F instead of gk . From (6), by using Jensen’s inequality,k k we get {k k } gm = β hm, k =1,...,K, m =1, . , M, (2) k k k K K 2 1 2 pm C log2 1+ ρ gk =K log2 1+ ρ gk where hk is the small-scale fading and βk represents the ≤ k k · K k k large-scale fading which depends on k but not on m. kX=1 kX=1 ρ K ρ 3. PRELIMINARIES OF FAVORABLE Klog 1+ g 2 =Klog 1+ G 2 , (7) ≤ 2 K k kk 2 K k kF PROPAGATION k=1 ! X In favorable propagation, we can obtain optimal performance where the equality in the first step holds when (3) satisfied, 2 with simple linear processing techniques. To have favorable and the equality in the second step holds when all gk are propagation, the channel vectors g , k =1,...,K, should equal. So, for this case, C is maximized if (3) holdsk andk g { k} k be pairwisely orthogonal. More precisely, we say that the have the same norm. The constraint on G that results in{ (7) is} channel offers favorable propagation if more relaxed than the constraint on G that results in (6), but the bound in (7) is only tight if all g have the same norm. 0, i,j =1,...,K, i = j { k} gH g = 6 (3) i j g 2 =0, k =1, . , K. k kk 6 3.2. Measures of Favorable Propagation In practice, the condition (3) will never be exactly satisfied, Clearly, to check whether the channel can offer favorable but (3) can be approximately achieved. For this case, we say propagation or not, we can check directly the condition (3) or that the channel offers approximately favorable propagation. (4). However, to do this, we have to check all (K 1)K/2 Also, under some assumptions on the propagation environ- possible pairs. This has computational complexity.− Other ment, when M grows large and k = j, it holds that 6 simple methods to measure whether the channel offers favor- 1 able propagation is to consider the condition number, or the gH g 0, M . (4) distance from favorable propagation (to be defined shortly). M k j → → ∞ These measures will be discussed in more detail in the fol- For this case, we say that the channel offers asymptotically lowing subsections. favorable propagation. The favorable propagation condition (3) does not offer 3.2.1. Condition Number only the optimal performance with linear processing but also represents the most desirable scenario from the perspectiveof Under the favorable propagation condition (3), we have maximizing the information rate. See the following section. GH G = Diag g 2, , g 2 . (8) {k 1k ··· k K k } 3.1. Favorable Propagation and Capacity We can see that if gk have the same norm, the condition { } H Consider the system model (1). We assume that the base sta- number of the Gramian matrix G G is equal to 1: tion knows the channel G. The sum-capacity is given by σmax/σmin =1, (9) H where σmax and σmin are the maximal and minimal singular val- C = log2 I + ρG G . (5) ues of GH G. Similarly, if the channel offers asymptotically favorable Next, we will show that, subject to a constraint on G, under favorable propagation conditions (3), C achieves its largest propagation, then we have 2 H possible value. Firstly, we assume gk are given. For G G D, M , (10) this case, by using the Hadamard inequality,{k k } we have → → ∞ where D is a diagonal matrix whose kth diagonal element is K βk. So, if all βk are equal, then the condition number is H H { } C = log2 I + ρG G log2 [I + ρG G]k,k asymptotically equal to 1. ≤ ! k=1 Therefore, when the channel vectors have the same norm Y K K (the large scale fading coefficients are equal), we can use = log [I +ρGH G] = log 1+ρ g 2 . (6) 2 k,k 2 k kk the condition number to determine how favorable the channel kX=1 kX=1 propagation is. Since the condition number is simple to evalu- ate, it has been used as a measure of how favorable the propa- We can see that the equality of (6) holds if and only if GH G gation offered by the channel G is, in the literature. However, is diagonal, so that (3) is satisfied.
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