Massive MIMO for Crowd Scenarios: A Solution Based on Random Access

Transcription

1 Massive MIMO for Crowd Scenarios: A Soltion Based on Random Access Jesper H. Sørensen, Elisabeth de Carvalho and Petar Popovski Aalborg University, Department of Electronic Systems, Fredrik Bajers Vej 7, 9220 Aalborg, Denmark {jhs,edc,petarp}@es.aa.dk Abstract This paper presents a new approach to intra-cell pilot contamination in crowded massive MIMO scenarios. The approach relies on two essential properties of a massive MIMO system, namely near-orthogonality between ser channels and near-stability of channel powers. Signal processing techniqes that take advantage of these properties allow s to view a set of contaminated pilot signals as a graph code on which iterative belief propagation can be performed. This makes it possible to decontaminate pilot signals and increase the throghpt of the system. The proposed soltion exhibits high performance with large improvements over the conventional method. The improvements come at the price of an increased error rate, althogh this effect is shown to decrease significantly for increasing nmber of antennas at the base station. I. INTRODUCTION Mltiple-inpt mltiple-otpt (MIMO) has been identified as a key technology to improve spectral efficiency of wireless commnication systems and is finding its way into practical systems, like LTE and LTE-Advanced. The research in MIMO has recently took a trn, when the advantage of having a massive nmber of antennas at a base station (BS) was asserted in [1]. In [1], a massive MIMO system refers to a mlti-cell mlti-ser system with a massive nmber of antennas at the BS that serves mltiple sers. The nmber of sers is mch smaller than the nmber of BS antennas, defining an nderdetermined mlti-ser system with a massive nmber of extra spatial degrees of freedom (DoF). Exploiting those extra DoF and assming an infinite nmber of antennas at the BS, the mlti-ser MIMO channel can be trned into an orthogonal channel and the effect of small-scale fading and thermal noise can be eliminated. Based on those excellent properties, massive MIMO is acknowledged as a promising technology for very high system throghpt and energy efficiency [2]. When the nmber of antennas becomes massive, acqiring the channel state information (CSI) becomes a severe bottleneck. Downlink channel training reqires a training length that is proportional to the nmber of antennas at the BS and is ths impractical. One soltion promoted in [1] restricts massive MIMO operations to time-division dplex (TDD) for which channel reciprocity is exploited. As the downlink and plink The research presented in this paper was partially spported by the Danish Concil for Independent Research (Det Frie Forskningsråd) DFF Part of this work has been performed in the framework of the FP7 project ICT METIS, which is partly fnded by the Eropean Union. The athors wold like to acknowledge the contribtions of their colleages in METIS, althogh the views expressed are those of the athors and do not necessarily represent the project. channels are eqal, CSI is acqired at the BS based on plink training and then sed for downlink transmission. The benefit is that the training length is proportional to the nmber of sers, which is mch smaller than the nmber of BS antennas. As described in [1], CSI is acqired sing orthogonal pilot seqences, bt, de to the shortage of orthogonal seqences, the same pilot seqences mst be resed in neighboring cells, casing pilot contamination. This problem is considered as one of the major challenges in massive MIMO systems [3]. Mitigation of pilot contamination has been the focs of several works recently. These inclde [4], where it is tilized that the desired and interfering signals can be distingished in the channel covariance matrices, as long as the angle-of-arrival spreads of desired and interfering signals do not overlap. A pilot seqence coordination scheme is proposed to help satisfying this condition. The work in [5] tilizes coordination among base stations to share downlink messages. Each BS then performs linear combinations of messages intended for sers applying the same pilot seqence. This is shown to eliminate interference when the nmber of base station antennas goes to infinity. A mlti-cell precoding techniqe is sed in [6] with the objective of not only minimizing the mean sqared error of the signals within the cell, bt also minimizing the interference imposed to other cells. The srvey of the related work indicates that the pilot contamination problem has been seen as an inter-cell problem that arises when the sers associated with two neighboring cells se the same pilot seqence. An implicit assmption associated with it is that the pilot seqences of the sers associated with the same cell are perfectly schedled, sch that no intra-cell pilot contamination occrs. These assmptions fall apart when one considers very dense, crowded scenarios as envisioned in 5G wireless scenarios [7]. In sch a setting, orthogonal schedling of the sers belonging to the same BS becomes infeasible, de to schedling overhead. In this work, we consider sch a crowd scenario, where the amont of sers and their access behavior make it infeasible to schedle the transmissions. Instead sers choose pilot seqences at random in an ncoordinated manner from a small pool shared by all sers. Since the sers are not coordinated, the pilot contamination problem can be cast as a random access problem. We identify two featres specific to massive MIMO: (1) asymptotic orthogonality between ser channels; and (2) asymptotic invariance of the power received from a ser over a short time interval. We se these featres in order /14/$ IEEE 352

2 Timeslot1 Timeslot2 Pilot Uplink Pilot Downlink Pilot Uplink Pilot Downlink t s Fig. 2. An example of a transmission schedle s 1 s 2 1, ,3 1,2 3 1,3 pilots time Fig. 1. A single cell crowd scenario. Fig. 3. An example of a pilot schedle. to formlate a pilot access protocol sing the framework of coded random access [8, 9]. In sch a framework, knowledge abot the pilot applied by each individal ser is not necessary a priori. This will be discssed in more detail in section III. The difference from existing approaches for coded random access is that the proposed protocol combines decoding of the data in the plink with estimation of the channel, which can be sed for downlink transmission. Moreover, the massive MIMO property of a stable norm makes it possible to apply the protocol in fading channels, which is not possible with existing approaches to coded random access. Overall, the soltion proposed in this paper is a radical departre from the sal treatment of the pilot contamination problem and introdces an important link to the area of random access protocols. II. SYSTEM MODEL In this work we denote scalars in lower case, vectors in bold lower case and matrices in bold pper case. A sperscript T denotes the transpose and a sperscript H denotes the conjgate transpose. We consider a random access system consisting of a single base station with M antennas and K sers with a single antenna, where M and K are in the hndreds or thosands. see Fig. 1. Commnication is performed on a time slotted basis, where each time slot consists of for phases; an plink pilot phase, an plink data phase, a downlink pilot phase and a downlink data phase, see Fig. 2. The channel between the k th ser and the base station in the n th time slot is denotedh nk = [h nk (1) h nk (2)...h nk (M)] T, where h nk (i) CN(0,1) i. It is assmed that h nk k are mtally orthogonal, which is jstified by the range of M. Moreover, it is assmed that channel coefficients in different time slots are i.i.d, while the channel power, h H nk h nk = h nk 2 remains constant within a period of β time slots. Note that the channel power varies de to path loss and shadowing effects, which cases it to vary mch slower than the channel coefficients. In each time slot, each ser is active with probability p a. If a ser is active, a random pilot seqence, s k = [s k (1) s k (2)...s k ()] T, is chosen among a set of size with mtally orthogonal pilot seqences. Note that mltiple sers may choose the same pilot seqence. See Fig. 3 for an example of a random pilot schedle with = 2 and K = 3. By A n, we denote all active sers in time slot n and by A j n, we denote the set of sers applying s j in the n th time slot. If Y p n denotes the plink pilot signal received in time slot n, we have Y p n = j=1 h nk s T j +Z p nj, (1) where Z p nj is a matrix of i.i.d. Gassian noise components, hence Z p nj (i,j) CN(0,σ2 n ) i,j. Any ftre instances of a vector z or matrix Z, with different sb- or sperscripts follow the same definition. All active sers transmit a message of length L in the plink data phase. The message from the k th ser is denoted x k = [x k (1) x k (2)...x k (L)]T. Denoting the received plink signal in time slot n as Y n, we then have Y n = h nk x T k +Z n. (2) k A n In the downlink phase we rely on channel reciprocity, sch that the plink channel estimate is assmed to be a valid estimate of the downlink channel. The base station transmits a precoded downlink pilot seqence, sch that the k th ser receives a downlink pilot signal, y pd nk, given by y pd nk = ht nk w nk s T j +zpd nk, (3) where w nk = [w nk (1) w nk (2)...w nk (M)] T is the precoding vector for ser k in the n th time slot. The base station is able to schedle the downlink messages, x d k = [ x d k (1) x d k (2)...xd k (L)] T, sch that the received signal in the downlink data phase is y d nk = ht nk w nk x d kt +z d nk. (4) 353

3 III. PILOT ACCESS PROTOCOL This section describes the proposed method of commnication in the system described in section II. The main focs of this work is the plink phase, however, a sbsection is dedicated to describing the operation in the downlink phase. A. Uplink From the plink pilot signals in (1), it is possible to estimate the channels between the sers and the base station. However, since mltiple sers may apply the same pilot seqence, it is only possible to estimate a sm of the involved channels. The least sqares estimate, φ nj, based on the pilot signal in time slot n from sers applying s j is fond as φ nj = ((s H j s j ) 1 s H j Y pt n ) T = h nk +z p nj. (5) where z p nj is the impairment of the estimate cased by the noise, z p nj. Any ftre instances of a vector z with a prime follow the same definition. The problem of interfering sers applying the same, or a non-orthogonal, pilot seqence is often called pilot contamination. If we proceed to detect the data in the plink phase sing a contaminated channel estimate, the reslt will be a smmation of data messages. If ψ nj is the data estimate based on the channel estimate φ nj, we have ψ nj = ((φ H nj φ nj ) 1 φ H nj Y n )T = h nk 2 h nj 2x k +z n. (6) Hence, a pilot collision leads to a data collision. In or system, one way to deal with this problem is to careflly select p a, sch that the probability of having one and only one ser applying a particlar pilot seqence in a particlar time slot is maximized. Hence, we have maximize p a Pr( A j n = 1) sbject to 0 p a 1 (7) This will maximize the nmber of non-contaminated channel estimates, and in trn maximize the nmber of sccessfl data transmissions. This approach is reminiscent of the framed slotted ALOHA protocol for conventional random access. We consider this a reference scheme in this work and refer to it as ALOHA. Note that a random access, i.e. nonschedled, scheme mst be considered as a reference, de to the assmption of a crowded scenario, where schedling is infeasible. ALOHA has been state-of-the-art for many years within random access protocols, bt recently a paradigm shift has started with the advent of coded random access [8, 9]. In this work, we view the problem of pilot contamination as a random access problem and apply newly developed tools in this area to solve the problem. Two featres from the massive MIMO scenario are essential to or soltion; near-orthogonality between ser channels and near-stability of channel powers. Throgh signal processing techniqes they allow s to resolve pilot collisions and thereby tilize otherwise wasted resorces. The soltion can be viewed as a two-stage processing approach: 1) Matched filter: The received plink pilot and data signals, in (1) and (2), are processed sing matched filters, which are constrcted from the contaminated estimates in (5). More specifically, we mltiply the received signals with φ H nj creating filtered signals, denoted f nj and g nj for data and pilots respectively. These signals contain linear combinations of the data and pilots transmitted by the sers contribting to the contaminated estimate, φ nj, see (8) and (9). The relationship between the variables we wish to estimate and the filtered signals can be viewed as a factor graph, see Fig. 4 and Fig. 5. 2) Sccessive interference cancellation (SIC): The coefficients of the linear combinations in (8) and (9) are the two-norms, h nk 2, of the involved channels. In a massive MIMO system, these can be assmed slowly fading, contrary to the fast fading channel coefficients. Hence, sccessive interference cancellation can be applied on the filtered signals in order to redce the linear combinations to data signals from individal sers. This reqires knowledge abot the edges in the code graphs, i.e. what pilots have been applied by the individal sers and in which time slots. This information is not available a priori at the base station. However, it can be embedded in the plink data messages, sch that when a data message has been recovered, the base station is informed abot the pilot pattern chosen by the ser. In practice, this cold be realized by embedding the seed for a psedo random nmber generator. Note, that graph knowledge is not necessary to initiate SIC, since a data message can be recovered immediately when one and only one ser chose a particlar pilot in a particlar time slot. This provides the necessary graph information to proceed SIC sing belief propagation. The overhead reslting from embedding graph information is considered negligible. f nj = (φ H nj Y n )T = h nk 2 x k +z n (8) g nj = (φ H njy p n ) T = h nk 2 s j +z p nj (9) The prpose of the matched filters is to transform the received signals from linear combinations with fast fading coefficients (the channel coefficients) into linear combinations with slowly fading coefficients (the norms). Note that the signals only remain linear combinations, when the channels are orthogonal, and that the coefficients are slowly fading only when the norms are stable. Both are flfilled nder the conditions given by a massive MIMO scenario. We can ths see the filtered signals, f nj and g nj j and n = 1,2,...,β, as a code on which 354

4 h 1 2 g 11 g 11 = h 1 2 s 1 + h 2 2 p s 2 +z 11 x 1 f 11 h 1 2 g 11 x 2 f 12 h 2 2 g 12 h 2 2 g 12 p g 12 = h 3 2 s 3 +z 12 x 3 f 21 h 3 2 g 21 h 3 2 g 21 p g 21 = h 2 2 s 2 +z 21 f 22 g 22 g 22 p g 22 = h 1 2 s 1 + h 3 2 s 3 +z 22 Fig. 6. A graph representation of data and pilot collisions. x 1 Fig. 4. A graph representation of pilot collisions. f 11 f 11 = h 1 2 x 1 + h 2 2 x 2 +z 1 We introdce the variable c which acconts for accmlated noise components and estimation errors. Note, that the magnitde of the elements in c increases as processing progresses. This will be discssed in frther detail in section IV. Initially, we isolate the contribtion from ser 1, giving s h 1 2 x 1 +c = f 11 f 21. (11) x 2 f 12 f 12 = h 3 2 x 3 +z 1 Since the applied pilot seqence is known a priori by the base station, we can find the norm as h 1 2 +c = (s H 1 s 1 ) 1 s H 1 (g 11 g 21 ). (12) x 3 Fig. 5. f 21 f 22 f 21 = h 2 2 x 2 +z 2 f 22 = h 1 2 x 1 + h 3 2 x 3 +z 2 A graph representation of data collisions. iterative belief propagation can be performed. See Fig. 6 for a graph showing the inter-dependencies between f nj and g nj. Example: Consider the simple example already introdced in Fig. 3. We assme β = 2, sch that the reslting graphs after matched filtering are fond in Fig. 4 and Fig. 5. Note, that since the norms are assmed constant, we can omit the time index, sch that h k 2 = h nk 2 n. We then have f 11 = h 1 2 x 1 + h 2 2 x 2 +z 11, f 12 = h 3 2 x 3 +z 12, f 21 = h 2 2 x 2 +z 21, f 22 = h 1 2 x 1 + h 3 2 x 3 +z 22, g 11 = h 1 2 s 1 + h 2 2 s 1 +z p 11, g 12 = h 3 2 s 2 +z p 12, g 21 = h 2 2 s 1 +z p 21, g 22 = h 1 2 s 2 + h 3 2 s 2 +z p 22. (10) Finally, the estimate of the message from ser 1, ˆx 1, is ˆx 1 = ((sh 1 s 1 ) 1 s H 1 (g 11 g 21 )) 1 (f 11 f 21 ). (13) Similar operations can be performed for finding ˆx 2 and ˆx 3. B. Downlink In downlink we assme channel reciprocity, sch that the ser does not need to estimate each coefficient of h nk, which wold reqire a pilot signal for all M antennas. Instead, we let the receiver estimate the concatenated channel consisting of both the downlink precoder, w nk, and the actal channel. Denoting the concatenated channel, q nk, we have q nk = h T nkw nk, (14) where q nk is estimated throgh (3). In order to choose an appropriate precoder, the base station mst have an estimate of the crrent channel. The coded operation applied in plink does not garantee that sch an estimate is available. Uplink operation relies on SIC based only on knowledge of the norm. Hence, downlink transmission to a ser is only possible if that ser avoided collision dring the previos plink pilot phase, sch that an ncontaminated channel estimate is available. This incrs a delay in downlink transmissions, whose magnitde is analyzed in section III-C. C. Analysis The performances of the reference scheme and the proposed scheme are tightly connected with the factor node degree distribtion of the code graph. Here a factor degree, denoted as d nj, refers to the nmber of sers occpying the same resorce 355

5 block, i.e. applies the j th pilot seqence in the n th time slot. A ser is active and applying pilot seqence j with probability p a /, sch that the degree probability distribtion is Pr(d nj = d) = ( K d ) (pa ) d ( 1 p a ) K d. (15) For the ALOHA scheme, we fond that the optimal performance is achieved when the probability of having d nj = 1 is maximized. Differentiating Pr(d nj = 1) with respect to p a and finding the roots satisfying or conditions, we get that p a = K maximizes the performance of the ALOHA scheme. We can not se the same approach for optimizing the proposed scheme, since resorce blocks with d nj > 1 may be sefl. Instead we mst seek a well performing degree distribtion which favors the iterative belief propagation. Several works have stdied this, e.g. in [10, 11], however, in this work we can not freely tailor the degree distribtion. We are limited to the binomial distribtion as expressed in (15), with only the freedom to choose a proper p a. Similar limitations were considered in [9] with focs on choosing an average degree, d, which was optimized nmerically. In or context, we have d = p ak. (16) A nmerical optimization of d and thereby in trn p a for a specific pair of K and will be performed in section IV. As described in section III-B, downlink transmissions experience a delay de to lack of channel knowledge. We denote the delay for ser k, k. This delay is eqal to the nmber of time slots ntil ser k is active and avoids a collision dring the plink pilot phase. Denoting the probability of a ser being active and avoiding collision, p a, we have ( p a = p a 1 p ) a K 1. (17) The probability distribtion of k follows the negative binomial and is therefore given by Pr( k = δ) = p a (1 p a )δ 1. (18) The expected vale, E[ k ], of the delay is then fond as E[ k ] = (1 p a ) p. (19) a There is a natral tradeoff between optimizing p a for high plink throghpt and optimizing it for limiting the delay in the downlink phase. Sch a joint optimization is otside the scope of this work. In the nmerical evalations in section IV, we will solely be concerned with the plink throghpt. IV. NUMERICAL RESULTS The proposed scheme is simlated and compared to framed slotted ALOHA in terms of plink throghpt and block error rate. Framed slotted ALOHA does not tilize SIC, bt optimizes performance throgh a maximization of degree one nodes in the code graph, see (7). The proposed scheme is based on an assmption that the channel coefficients in different time slots are i.i.d, while their two-norms remain constant within a period of β time slots. In the nmerical evalations, we will challenge these assmptions by simlating with fading channels. A rich scattering environment is assmed, sch that h nk (m) can be modeled sing Clarke s model [12], hence h nk (m) = 1 N s e j2πf dnt s cosα i+φ i, (20) Ns i=1 where N s is the nmber of scatterers, f d is the maximm Doppler shift, α i and φ i is the angle of arrival and initial phase, respectively, of the wave from the i th scatterer. Both α i andφ i are i.i.d. in the interval[ π,π) andf d = v c f c, where v is the speed of the ser, c is the speed of light and f c is the carrier freqency. An overview of the simlation parameters is given in Table I. Note that β = 1.2K/ is chosen in order to ensre a 20% srpls of resorce blocks relative to K, sch that the iterative belief propagation performs well. All simlation reslts are averages over 10, 000 iterations. Initially, in Fig. 7 we present reslts for the normalized throghpt of the proposed scheme as a fnction of the average degree of a resorce block, which is directly related to the activation probability, as seen in (16). We define normalized throghpt as the total nmber of sccessflly decoded messages divided by β, i.e. the total amont of resorce blocks. It is evident that an average degree of approximately 2.5 shold be aimed for in the considered range of K, which is confirmed by the reslts from [9]. All other simlations are performed sing an average degree of 2.5 regardless of K. Note that improved performance cold be achieved by optimizing the activation probability to a particlar vale of K. Fig. 8 shows normalized goodpt, i.e. throghpt with erroneos messages discarded, as a fnction of the nmber of sers accessing the base station. The proposed scheme clearly otperforms the conventional method of framed slotted ALOHA. The improvement increases with K, since the proposed scheme benefits from a larger nmber of messages to code across. An increase in K can be viewed as an increase in the block length, which improves coding efficiency. The coding gain comes at the price of an increased error rate. As mentioned in section III, whenever SIC is performed, noise and estimation errors are accmlated, which may lead to errors. At higher K, it is more common to see high degrees in the code graph, even if the average degree remains constant. Moreover, SIC is performed across a larger time span, which leads to larger errors in the norm estimation. As a reslt, we experience an increased error rate for increasing K, which is illstrated in Fig. 9. It also shows that the error rate drops significantly, as the nmber of base station antennas increases. The reason is that the norm stabilizes for increasing M, making the assmption of a constant norm increasingly valid. V. CONCLUSIONS We presented a soltion for the pilot contamination problem in crowded scenarios, where sers within a single cell mst share a small set of pilot seqences. We view intracell pilot contamination as a random access problem and 356

6 TABLE I SIMULATION PARAMETERS Parameter Vale Description f c 1.8 GHz Carrier freqency v 3 km/h User mobility N s 20 Nmber of scatterers σn Relative noise power 5 bits Length and nmber of pilot seqences t s 0.01 s Length of a time slot L 1000 bits Length of plink data messages β 1.2K/ Nmber of time slots Normalized plink goodpt Coded M = 100 Coded M = 200 Coded M = 500 Coded M = 1000 Coded M = 2000 ALOHA K 0.75 Fig. 8. Throghpt as a fnction of the nmber of sers. Normalized plink throghpt K = 50 K = 100 K = d = (p a K)/ Block error rate K = 50 K = 100 K = 200 Fig. 7. Throghpt as a fnction of the average degree of a resorce block. draw on newly developed ideas from this area of research. The massive MIMO setting provides two essential properties; near-orthogonality between ser channels and near-stability of channel powers. These properties allow s to view a set of contaminated pilot signals as a graph code on which iterative belief propagation can be performed. The proposed soltion proves highly efficient, comfortably otperforming the conventional ALOHA approach to random access. The price to pay is an increased error rate, de to accmlation of estimation errors in the belief propagation algorithm. However, this downside is shown to significantly diminish as the nmber of base station antennas increases. REFERENCES [1] T. Marzetta, How mch training is reqired for mltiser mimo?, in Signals, Systems and Compters, ACSSC 06. Fortieth Asilomar Conference on, pp , Oct [2] F. Boccardi, R. Heath, A. Lozano, T. Marzetta, and P. Popovski, Five disrptive technology directions for 5g, Commnications Magazine, IEEE, vol. 52, pp , Febrary [3] F. Rsek, D. Persson, B. K. La, E. Larsson, T. Marzetta, O. Edfors, and F. Tfvesson, Scaling p mimo: Opportnities and challenges with very large arrays, Signal Processing Magazine, IEEE, vol. 30, pp , Jan [4] H. Yin, D. Gesbert, M. Filippo, and Y. Li, A coordinated approach to channel estimation in large-scale mltiple-antenna systems, Selected Areas in Commnications, IEEE Jornal on, vol. 31, pp , Febrary Fig M Block error rate as a fnction of the nmber of base station antennas. [5] A. Ashikhmin and T. Marzetta, Pilot contamination precoding in mlticell large scale antenna systems, in Information Theory Proceedings (ISIT), 2012 IEEE International Symposim on, pp , Jly [6] J. Jose, A. Ashikhmin, T. Marzetta, and S. Vishwanath, Pilot contamination and precoding in mlti-cell tdd systems, Wireless Commnications, IEEE Transactions on, vol. 10, pp , Agst [7] A. Osseiran, F. Boccardi, V. Bran, K. Ksme, P. Marsch, M. Maternia, O. Qeseth, M. Schellmann, H. Schotten, H. Taoka, H. Tllberg, M. Usitalo, B. Tims, and M. Fallgren, Scenarios for 5g mobile and wireless commnications: the vision of the metis project, Commnications Magazine, IEEE, vol. 52, pp , May [8] G. Liva, Graph-Based Analysis and Optimization of Contention Resoltion Diversity Slotted ALOHA, Commnications, IEEE Transactions on, vol. 59, pp , febrary [9] C. Stefanovic, P. Popovski, and D. Vkobratovic, Frameless ALOHA protocol for Wireless Networks, IEEE Comm. Letters, vol. 16, pp , Dec [10] M. Lby, LT Codes, in Fondations of Compter Science, Proceedings. The 43rd Annal IEEE Symposim on, pp , [11] J. Sørensen, P. Popovski, and J. Østergaard, Design and analysis of lt codes with decreasing ripple size, Commnications, IEEE Transactions on, vol. 60, pp , november [12] R. Clarke, A statistical theory of mobile-radio reception, Bell system technical jornal, vol. 47, no. 6, pp ,