Impact of EIRP Constraint on MU-MIMO ac Capacity Gain in Home Networks

Transcription

1 Impact of EIRP Constraint on MU-MIMO ac Capacity Gain in Home Networks Khouloud Issiali, Valéry Guillet, Ghais El Zein and Gheorghe Zaharia Abstract In this paper, we evaluate a downlink Multi-User Multiple-Input Multiple-Output (MU-MIMO) scenario, in which a ac access point with multiple antennas (up to 10) is transmitting to two receivers, each one with two antennas. Block diagonalization (BD) method is investigated under the Equivalent Isotropic Radiated Power (EIRP) constraint. This study shows that scaling the transmitted power according to the EIRP constraint can improve the multi-user (MU) sum capacity to single-user (SU) capacity ratio compared to the gain achieved under the transmitted power constraint. Keywords MU-MIMO capacity IEEE ac Home Networks EIRP 1 Introduction MU-MIMO techniques, proposed to increase the throughput, consist in applying a linear precoding to the transmitted (Tx) spatial streams. Thus, the antenna array pattern and gain are modified as functions of the user location and channel properties. This directly impacts the EIRP. In Europe, the EIRP is limited at 5 GHz band to 200 mw or 1 W depending on the propagation channels. This constraint may differ in other countries where it is rather based on the total Tx power. K. Issiali V. Guillet (&) Orange Labs, Belfort, France Valery.Guillet@orange.com K. Issiali Khouloud.Issiali@orange.com G.E. Zein G. Zaharia IETR INSA, UMR 6164, Rennes, France Ghais.El-Zein@insa-rennes.fr G. Zaharia Gheorghe.Zaharia@insa-rennes.fr Springer International Publishing Switzerland 2016 A. El Oualkadi et al. (eds.), Proceedings of the Mediterranean Conference on Information & Communication Technologies 2015, Lecture Notes in Electrical Engineering 380, DOI / _9 75

2 76 K. Issiali et al. The EIRP constraint is rarely evaluated for MIMO systems. Often, the packet error rate and the capacity value are evaluated based on the same total Tx power (P Tx ) which is a function of the Signal to Noise Ratio (SNR), commonly defined as the ratio of P Tx to the average noise power. The propagation channel is usually normalized to have an average path loss of 0 db. Few recent studies have focused on the capacity optimization problems under total Tx power constraint [1 3]. Sometimes, this optimization is performed on each subcarrier of the OFDM signal [1]. In [4], a new EIRP-based solution for IEEE power scaling is proposed. However, this study is dedicated to only one user system with a single spatial stream. The MU-MIMO linear precoding, like BD [2, 5], modifies dynamically the antenna array pattern and gain. This changes the EIRP of the Tx antenna array if Tx power remains unchanged. MU-MIMO and Transmit Beamforming (TxBF) are commonly associated with a large number of Tx antennas used to improve the antenna array gain and performance, as previously stated for narrowband i.i.d. Rayleigh SISO channels forming the MIMO channel in [3, 5]. In the case of the EIRP constraint, it may not be evident that TxBF and MU-MIMO linear precoding still improve the system performance. Therefore, this paper evaluates the impact of the EIRP constraint on ac MU-MIMO capacity gain using simulations. Two different power allocation schemes of the spatial streams are analyzed to optimize MU-MIMO capacity: equal and unequal power repartition under the same EIRP constraint. A typical indoor residential environment is evaluated based on the IEEE TGac correlated channels [6, 7]. Comparisons are given versus an i.i.d. Rayleigh channel. The rest of this paper is organized as follows. Section 2 describes the system model and briefly presents the BD algorithm. The problem formulation to compute the sum capacity for MU-MIMO system under the EIRP constraint is given in Sect. 3. Section 4 describes the simulation process and presents the simulation results with analysis. Finally, the conclusion is drawn in Sect. 5. Hereafter, superscripts ð:þ t, ð:þ and ðþdenote : transposition, transpose conjugate and complex conjugate, respectively. Expectation (ensemble averaging) is denoted by E(.). The Frobenius norm of a matrix is written as jj jj. 2 System Model The studied ac MU-MIMO system is composed of K users connected to one Access Point (AP), as shown in Fig. 1. The AP has n T antennas and each user k has n Rk antennas. We define n R ¼ P K k¼1 n R k. The L k 1 (where L k is the number of parallel symbols Tx simultaneously for the kth user) transmit symbol vector s k is pre-processed at the AP before being transmitted. For each ac OFDM subcarrier, the received signal at the kth receiver is:

3 Impact of EIRP Constraint on MU-MIMO ac Capacity 77 Fig. 1 Diagram of MU-MIMO system X K y k ¼ H k W k s k þ H k W i s i þ n k i¼1;i6¼k ð1þ where H k is the n Rk n T channel matrix for the kth receiver, W k is the n T L k BD precoding matrix resulting in an n T L ðl ¼ L 1 þ L 2 þ þl K Þ precoding matrix W ¼½W 1 ;...; W K Š, n k is the Gaussian noise vector (E n k n k ¼ r 2 n I ). nrk BD [3, 5] decomposes the MU-MIMO downlink into K parallel independent SU-MIMO downlinks. The BD consists first of perfectly suppressing the inter-user P interference IUI = H K k i¼1;i6¼k W is i in order to have parallel SU-MIMO systems. Then, a classic TxBF is applied to optimize the capacity for each user [8]. In this study, K = 2 and perfect knowledge of channel state information is assumed at the transmitter. The channel model specified for the ac standard within the TGac task group [7] is selected. This model takes into account realistictx and Rx correlations, contrary to an i.i.d. Rayleigh channel. It is based on a cluster model [6] amended by the TGac task group for the IEEE ac standard. The TGac modifications concern the power angular spectrum to allow MU-MIMO operation and are summarized as follows [7]: The TGn azimuth spread for each cluster remains the same for all users. For each user and for all taps, independent random offsets are introduced: between ±180 for the angles of arrival (AoA) and between ±30 for the angles of departure (AoD). In this study, a typical home network is evaluated by using the channel model TGac-B (15 ns RMS delay spread) for the 5.25 GHz frequency band. Rayleigh fading is exhibited for each one of the 9 uncorrelated taps, except for the Line Of Sight (LOS) tap which follows a Rice fading with a 0 db Rician factor. This study focuses on the TGac-B NLOS channel model. Similar results are obtained with the TGac-B LOS channel model as the 0 db Rician factor does not display significantly different results from the TGac-B NLOS channel model. For each ac OFDM subcarrier, the channel matrix is computed through a discrete Fourier transformation (size: 56 subcarriers) of the tap delay representation. For comparison, an i.i.d.

4 78 K. Issiali et al. Rayleigh channel is also evaluated. We apply the common normalization E jjhjj 2 ¼ n T n R for each subcarrier which means an average propagation loss equal to 0 db. 3 Problem Statement 3.1 Usual Definitions The MU-MIMO system is decomposed into K independent SU-MIMO systems by applying the BD algorithm. For each one of the ac OFDM subcarriers, the MU-MIMO sum capacity is expressed as follows [3]: C BD ¼ XK k¼1 X n R k i¼1 log 2 ð1 þ p ik r 2 l 2 ik Þ n ð2þ where p ik is the power dedicated to the ith antenna for the kth user, l 2 ik are the eigenvalues of the effective channel for the kth user after applying the IUI cancellation [2] and r 2 n is the noise power. The subcarrier index is not mentioned throughout this paper in order to simplify the notations, but since C BD is related to H, C BD depends on each OFDM subcarrier. For the corresponding SU-MIMO systems and for relevant comparisons with MU-MIMO, the number of antennas n T and n R remains unchanged. The considered SU-MIMO system applies a singular value decomposition and its capacity C SU is computed as detailed in [8] for each OFDM subcarrier. 3.2 EIRP in Linear Precoding For any receiver location, i.e. for any H matrix, the transmit antenna array pattern is modified by the W precoding matrix. We have used a linear array of omnidirectional 0 dbi gain antennas with regular spacing δ, typically δ = k=2: The transmitter antenna array manifold aðhþ, is a function of the h angle with the array axis: aðþ h t ¼ 1; e 2jpdcosðhÞ=k ; e 4jpdcosðhÞ=k ;...; e 2jpðn T 1ÞdcosðhÞ=k : ð3þ Due to the used TGac channel model, the antenna array pattern is simplified to a 2D problem. The average radiated power dðhþ in any direction θ relative to the antenna array direction is expressed as a function of the input signals x t =(x 1 ;...; x nt ):

5 Impact of EIRP Constraint on MU-MIMO ac Capacity 79 dðhþ ¼ E aðhþ t x 2 : ð4þ With x = Ws, dðhþ is expressed as a function of B ¼ E SS t ¼ diagðpik Þ: dðhþ ¼ aðhþ WBW t aðhþ: ð5þ Considering all the subcarriers of the system, the total radiated power dðhþ is: and the EIRP is: d total ðhþ ¼ X subcarrier dðhþ; ð6þ EIRP ¼ max d total ðhþ h ð7þ If the total power P Tx is equally shared among N SS spatial streams, p ik ¼ P Tx NN SS ; where N = 56 is the number of subcarriers in ac. Thus: dðhþ ¼ P Tx aðhþ WBW t aðþ: h NN SS ð8þ 3.3 Optimization Problems In order to find the optimal value of the Tx power p ik compatible with the EIRP constraint, two power allocation schemes are evaluated: equal power allocation and unequal power allocation. This paper focused on the case where each subcarrier has the same allocated total Tx power. Furthermore, an unequal subcarrier power allocation may not have a favorable impact on the peak-to-average power ratio of the OFDM signal. The general optimization problem is thus expressed for each subcarrier as: max pik X K k¼1 X n R k i¼1 log 2 1 þ p ik r 2 l 2 ik and EIRP 23 dbm: n For the case with equally distributed powers, i.e. p ik ¼ P Tx NN SS, the problem has only one variable P Tx. It is simplified by seeking the maximum antenna array gain dðhþ P Tx and then scaling the power according to the EIRP limit. For the genaral case, the optimization is performed using a Matlab-based modeling for convex optimization ð9þ

6 80 K. Issiali et al. namely CVX [9].The case K = 1 uses the same optimization method for computing the SU-MIMO capacity for both equal and unequal power allocation under EIRP constraint. 3.4 Evaluated Systems and SNR Considerations ac MU-MIMO systems based on BD schemes are evaluated. The results are presented in Sect. 4 and compared to SU-MIMO systems relying on the same antennas and total power or EIRP constraint. Three capacity optimization techniques are evaluated and compared. The first one is the usual MIMO system (denoted basic), with a constant Tx power P Tx equally shared among the spatial streams. BD-basic and SU-basic denote the corresponding studied systems. For this case, the average signal to noise ratio is defined as SNR ¼ P Tx Nr This is the common n. 2 SNR definition. The second optimization labelled eirp-equal considers a 23 dbm EIRP constraint and a total Tx power equally shared among the spatial streams. A dynamic power scaling is applied, as a function of each channel matrix snapshot H. SUeirp-equal and BDeirp-equal denote respectively the corresponding SU and MU systems. The third one (eirp-unequal) considers a 23 dbm EIRP constraint and a total Tx power unequally and dynamically shared among the spatial streams. SUeirpunequal and BDeirp-unequal denote the corresponding systems applying this technique. For eirp-equal and eirp-unequal systems, the common SNR definition is biased as P Tx is no more constant, and depends on each channel matrix computation. Under EIRP constraint, we define SNR eirp = EIRP Nr for eirp-equal and eirp-unequal 2 n systems. Note that the maximum antenna array gain is n T : Since SNR ¼ P Tx Nr for a 2 n basic system, it implies that its corresponding SNR eirp value is upper bounded by n T P Tx Nr 2 n. 4 Simulation Results and Analysis The simulated system is composed of one access point equipped with multiple antennas (linear array of 0 dbi omnidirectional and vertically polarized antennas), and two receivers. Each receiver has two 0 dbi omnidirectional antennas. The antenna spacing is 0.5λ. A Matlab source code [10] was used to compute the ac TGac-B channel samples over a 20 MHz bandwidth. To have representative results, 100 couples of users (K = 2) are randomly drawn around the access point following the IEEE TGac recommendations [9]. For each drawing, we use a simulation length equal to 55 coherence times of the MIMO channel to simulate the

7 Impact of EIRP Constraint on MU-MIMO ac Capacity 81 fading. By setting the Fading Number of Iterations in the Matlab channel model to 512, 488 interpolated channel samples are collected for each couple of users to simulate 10 fading periods. 4.1 Results for Equal Power Allocation Figures 2 and 3 display the MU-MIMO to SU-MIMO capacity ratio for basic and eirp-equal systems. Average values, 10 and 90 % quantiles (q 10 and q 90 ) are represented to estimate the confidence intervals. SNR = 20 db and n T varies from 4 to 10. They show that the MU to SU-MIMO capacity ratio increases with n T for TGac-B and Rayleigh channels. The ratio changes from 1.2 to 1.77 for the eirpequal system in a residential environment, which is more than 50 % of capacity gain. Note that the gain without the EIRP constraint is around 45 %. It has been shown in [3, 5] that increasing n T favorably impacts the capacity gain on an i.i.d. Rayleigh channel under SNR constraint. We have been able to prove that this result holds even under the EIRP constraint and with correlated channels as in TGac models. The difference q 90 q 10 decreses if n T increases. This shows that fading has less impact on the capacity values. The basic and eirp-equal comparisons are biased. In fact, a system relying on a total Tx power does not satisfy a constant EIRP constraint since it may have an increasing EIRP as n T increases. We could expect that for a basic system, the MU-MIMO to SU-MIMO capacity ratio increases more rapidly in function of n T than for an eirp-equal system, but simulations prove the opposite. SU-MIMO takes advantage of the power P Tx when the system is not under EIRP constraint. For instance in our simulated case (K = 2) where N SS = 4 for MU-MIMO and N SS =2 for each one of the single users, the EIRP reached by the MU-MIMO system Fig. 2 MU to SU capacity ratio for an IEEE TGac-B channel (residential) MU to SU capacity ratio q 10 (basic) mean(basic) q 90 (basic) q (eirp-equal) 1 10 mean(eirp-equal) q (eirp-equal) Number of transmit antennas

8 82 K. Issiali et al. Fig. 3 MU to SU capacity ratio for an i.i.d. Rayleigh channel MU to SU capacity ratio q (basic) 10 mean(basic) 1.2 q (basic) q (eirp-equal) 10 mean(eirp-equal) 1 q (eirp-equal) Number of transmit antennas Fig. 4 Capacity value achieved by the basic and eirp-equal systems Channel capacity in bits/s/hz 28 MU-basic 26 MUeirp-equal SU-basic 24 SUeirp-equal Number of transmit antennas (EIRP MU Þ is expressed as EIRP MU ¼ P Tx NN SS ½max aðhþ WW t aðhþ Š and is upper h P bounded by n Tx T 4 : P Similarly, for the same P Tx, the single user EIRP is upper bounded by n Tx T 2. This means that under the same EIRP constraint, the allocated power tends to be lower for SU-MIMO than for MU-MIMO. For the basic system, the allocated power is the same for SU-MIMO and MU-MIMO. Figure 4 shows the average capacity value for MU and SU. It is well observed that the SU capacity increases rapidly with n T. 4.2 Impact of Power Allocation Strategy The probability when the MU-MIMO capacity is lower than the SU-MIMO capacity is illustrated in Fig. 5 for eirp-equal and eirp-unequal schemes versus n T. We also observe that the MU-MIMO capacity gain for unequal repartition is slightly greater than the one observed for a fair power distribution. Nevertheless, the

9 Impact of EIRP Constraint on MU-MIMO ac Capacity 83 Fig. 5 ProbaðC BD =C SU 1Þ versus the number of transmit antennas Proba (C BD /C SU <1) TGac-B eirp-equal Rayleigh eirp-equal TGac-B eirp-unequal Rayleigh eirp-unequal Number of transmit antennas gain is not significant: we have around 3 % of capacity gain by contrast to high computational complexity. The probability is almost 0 ð1%þ for n T ¼ 6 and is equal to 0 for n T 8. These results can be explained by examining the overall system: for n T ¼ 4, the MU-MIMO system is composed of 4 antennas in the transmit and the receive sides with 4 spatial streams. This gives no diversity possibilities, N SS < n T when n T increases. As a result, the system takes benefit from transmit diversity with probabilities which tend to 0. 5 Conclusion This paper has analyzed the capacity optimization for ac MU-MIMO systems with multiple spatial streams for each user under the EIRP constraint. A typical home environment and correlated ac channels were considered. Two transmit power allocation methods have been evaluated: equal and unequal repartition based under the same EIRP constraint. These two strategies were compared with the more common MU-MIMO under the total Tx power constraint. It is shown under EIRP constraint, that the number of transmit antennas must be larger than the total number of spatial streams to guarantee a MU-MIMO capacity gain over SU-MIMO. References 1. Saavedra, J.: Multidiffusion et diffusion dans les systèmes OFDM sans fil. Thèse de doctorat d état, Univ. Paris-Sud, Paris, France, (2012) 2. Zhao, L., Wang, Y., Charge, P.: Efficient power allocation strategy in multiuser MIMO broadcast channels. PIMRC, , (2013) 3. Spencer, Q.H., Swindlehurst, A.L., Haardt, M.: Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels. IEEE Trans. Sig. Proc. 52(2), (2004)

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