CSI Map for Indoor Massive MIMO

Computing Conference 2017 18-20 July 2017 | London, UK

CSI Map for Indoor Massive MIMO Ahmad Abboud1, Ali H. Jaber2 2

Department of Statistics 2 Lebanese University 2 Nabatieh, Lebanon

Abstract- By implementing a massive number of antennas on the base station (BS) the performance of MIMO wireless systems will boost enormously in the dimension of (capacity per unit area, energy efficiency, and spectral efficiency). Unless considering multi-cell multi-user MIMO or Massive MIMO with division duplexing (TDD) wireless system, perfect channel estimation during the uplink session will become a challenge due to the limited number of orthogonal pilot sequences that can be generated in short coherence interval. This issue leads to pilot reuse across cells and hence uplink pilot contamination. This paper considers an indoor scenario of Massive MIMO systems and proposes a machine learning technique to mitigate uplink pilot contamination which increases system spectral efficacy, energy efficiency, and sum-rate. The proposed technique exploit the first stage of classical channel estimation to learn a memory Map of Quantized Channel State Information (QCSI) data. The learned CSI Map will be utilized in the second stage to predict the next UT’s channel instead of estimating it. A machine learning algorithm to create and update the CSI Map and two modified TDD format for learning stage and prediction stage is introduced. Simulation results promise a likely increase in performance compared to traditional Massive MIMO system in the dimension of sum-rate, spectral and energy efficiency. Keywords—Massive MIMO; Uplink Pilot Contamination; CSI Map; CSI Prediction.

I.

INTRODUCTION

Massive MIMO brings many motivations for the next 5th generation mobile technology. It offers a significant gain in the achievable sum-rate, energy efficiency, and spectral efficiency [1], [2]. However, considering Multi-cell scenario the performance of Massive MIMO will degrade due to uplink pilot contamination from neighbor cells [3]. As the number of antennas M on the BS increases without limits, the effect of Additive White Gaussian Noise (AWGN) and the fast fading will vanishes while the pilot contamination effect persists [4], [5]. Considering the indoor scenario, Massive MIMO can face the same challenges, unless the limited coverage region coped with low mobility speed UT’s and scattering objects permit to think in machine learning solutions to mitigate such challenge. Also, by measuring the entropy of each UT trajectory, one can find that UT mobility is predictable for high degree [6].

Jean-Pierre Cances 3, Vahid Meghdadi4 1,3,4

Department C2S2 Limoges University 1,3,4 Limoges, France

1,3,4

This paper introduces a novel technique to predict CSI rather than estimating it; this can be done in two stages. In the first stage, BS starts to learn the CSI matrices using classical linear estimation techniques e.g. (MRC, ZF, MMSE) and stores a quantized version of the learned CSI into a graph of nodes which is called the CSI Map. Thus, the CSI Map translates the geographical coverage map into a directive-connected graph using a special learning algorithm (will be introduced in section VI). The nodes of the graph represent the quantized CSI values, where the edges represent the transition from one CSI to another. In the second stage and after several learning epochs, the CSI Map will be considered matured to monitor UT mobility and to predict the next possible CSI matrix to be precoded. To benefit from this prediction, a new version of TDD format called predictive format that does not include pilots is introduced. The use of predictive format reduces the power transmission as well as the pilot contamination and increase the spectral efficiency (SE), energy efficiency (EE) and the sumrate of the system. A codebook of two parameters is used to easily store and retrieve quantized CSI (QCSI) from the map. At last, a Garbage Collection Algorithm (GCA) is also introduced to decrease the number of CSI nodes in the map by eliminating weakly connected nodes. The use of GCA will allow focusing on a highly-activated area in the map. Simulation results prove the added value of the CSI Map on the performance of EE, SE and system sumrate. Up to our knowledge, CSI Map had never been used in the literature to solve uplink pilot contamination in Massive MIMO. However, several tools used in our approach had been extensively studied. Uplink pilot contamination had been investigated by several researches [7]–[11]. In [12] authors tend to coordinate the use of pilots or adaptively allocate pilot sequences to different terminals in the network. This approach requires a cooperative BS scheme where each BS must be aware of pilot allocations in neighbor cells. The author in [13] deals with the problem of uplink pilot contamination by proposing a new channel estimation scheme that exploits interference cancellation and joint processing. Highly interfering users in neighboring cells are identified based on the estimation of large-scale fading and then included in the joint channel processing.

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More approaches to mitigate uplink pilot mitigation can be found in [14]. Quantized channel state information was studied by [15], the authors introduce a vector-quantization approach to channel state information encoding that requires modest feedback bit rate. The same authors published another patent on quantized channel information prediction in multiple antenna systems [16]. In this work, a quantized CSI (QCSI) based on vector quantization of geometric attenuation and shadow fading parameters is used. Prediction of information instead of estimating it is widely used in the literature, for instance [17] apply a technique to predict sensor data in order to reduce uplink transmission power. CSI prediction can be model-based (example [18]) or non-model based (example [19] using stochastic channel model). Authors in [19], proposes a decision-directed channel predictors for orthogonal frequency division multiplexing (OFDM) communications over time-varying channels. A robust technique to quickly retrieve stored data from memory is proposed by [20]. The multi-nary content addressable memory in [20], help to quickly search CSI Map for entries. Authors in [21] propose an adaptive codebook geodesic based channel prediction, where their simulation results prove that the proposed scheme effectively mitigate the feedback delay and clustering even with only a 4-bit codebook. CSI prediction had been recently proposed by [22] to reduce training overhead; the authors exploits the temporal correlation of UT channels to classify them into two groups. In each channel block, the BS select part of the UT’s for training, while prediction determines the other UT CSI’s. Nonetheless, the proposed technique in this paper allows the UT to decide whether to send its pilot or not. Also, it allows the BS to predict the most possible position of the UT channel within the CSI Map. Furthermore, UT mobility and trajectory can be easily predicted using CSI Map, which can be exploited for resources management (Frequency, Time, and Pilot) within the wireless network. The scope of this paper includes the system model and the performance on the dimension of spectral and energy efficiency, and we will leave CSI Map applications for future work for limitation of space. This paper is organized as follows: In Section II, we introduce the system model. Section III discusses the proposed TDD format. In Section IV, we present the Channel State Information Map. Section V and VI, Introduces the learning algorithm of the CSI map and the CSI quantization technique used respectively. In Section VII we introduce the Garbage Collection Algorithm. The sum-rate, SE, and EE are introduced in Section VIII. Then we introduce numerical results based on simulation in

Section IX. Finally, we conclude the paper in Section X. Notations: T(.), H(.) denote transpose and Hermitian transpose, respectively. (.)* denote the conjugate, det(A) denote the determinant of A, ⊙ denote Hadamard product, ‖A‖ denote the Frobenius norm of A and 𝔼{𝐴} denotes the expectation of A. II.

SYSTEM MODEL

Considering the uplink TDD session in a Massive MIMO system with L cells each contains one BS equipped with M antennas and serving K UT’s equipped with one antenna. Assuming a perfect synchronization among uplink session in all the L cells as in [4],[23] with a cell interference ratio 𝛾 knowing that they share the same frequency band. The 𝑀×1 received vector at the jth BS is given by: 𝐿

𝒚𝒋 = √𝑃𝑢 ∑

𝑮𝑗𝑙 𝒙𝒋 + 𝒏𝒋

(1)

𝑙=1

𝑮𝑗𝑙 is the M ×K channel matrix between the jth BS and the K UT’s in the lth cell, i.e. 𝑔𝑚𝑘 ≜ [𝑮]𝑚,𝑘 is the channel coefficient between the mth antenna of the BS and the kth user terminal (UT). 𝒙𝒋 is the K × 1 received symbol vector from the K UT’s in the jth cell and 𝒏𝒋 ~∁𝒩 (0, 𝑰𝑀 ) is the AWGN noise vector with i.i.d and unit variance. 𝑃𝑢 denotes the normalized uplink symbol SNR. Denotes by 𝑯𝑗𝑙 of dimension M ×K, the matrix of fast fading coefficients between the K users of cell l and the M antennas of the jth BS, i.e. ℎ𝑗𝑙𝑚𝑘 ≜ [𝑯𝑗𝑙 ]𝑚,𝑘 and 𝑫𝑗𝑙 is the K×K diagonal matrix, i.e. 𝛽𝑗𝑙𝑘 ≜ [𝑫𝑗𝑙 ]𝑘,𝑘 presents the large-scale fading between the jth BS and the kth user terminal of cell the lth cell. We assume a collocated antennas on the BS, where large-scale fading coefficient is independent of m as in [1],[2]. We can write the channel coefficient as: 𝑔𝑚𝑘 = ℎ𝑚𝑘 √𝛽𝑘 m=1, 2, …, M (2) The √𝛽𝑘 models the geometric attenuation and shadow fading which is assumed to be constant over many coherent time intervals and known prior. Not to lose generality, the channel matrix can be represented as: 𝑮 = 𝑯𝑫1/2 According to [1] as 𝑀 ≫ 𝐾 the following relation holds 𝐻 𝑮𝑗𝑙𝐻 𝑮𝑗𝑙 1/2 𝐇𝑗𝑙 𝐇𝑗𝑙 1/2 ( ) = 𝑫𝑗𝑙 ( ) 𝑫𝑗𝑙 𝑀 𝑀≫𝐾 𝑀 𝑀≫𝐾 1/2

≈ 𝑫𝑗𝑙

(3)

1

𝑯𝑇 𝑯∗ 𝑀 𝑗𝑙 𝑗𝑙

and = 𝑰𝐾 𝛿𝑗𝑙 , where 𝑰𝐾 is the 𝐾×𝐾 identity matrix and 𝛿𝑗𝑙 corresponds to the covariance factor of 𝐇𝑗𝑙 .

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During a training session of length 𝜏, the received signal at the jth BS in the lth cell will be presented as follows: 𝐿

𝐘𝑗 𝒑 = √𝜏𝑃𝑢 ∑ 𝑮𝑗𝑙 𝑿′𝑃𝑙 + 𝒏𝒋 𝒑

(4)

𝑙=1

with 𝑿′𝑙 = [𝑿𝑃𝑙 ⊙ 𝑺𝒍 ], where 𝑺𝑙 is a 𝐾×1 binary matrix of the lth cell with elements 𝑠𝑘 ∈ {0,1} as follows: 𝑠𝑘 = {

1 0

𝑖𝑓 𝑡ℎ𝑒 𝑘 𝑡ℎ 𝑢𝑠𝑒𝑟 𝑠𝑒𝑛𝑑𝑠 𝑎𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑎𝑚𝑒 𝑖𝑓 𝑡ℎ𝑒 𝑘 𝑡ℎ 𝑢𝑠𝑒𝑟 𝑠𝑒𝑛𝑑𝑠 𝑎 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑣𝑒 𝑓𝑟𝑎𝑚𝑒

𝑿𝑃𝑙 is the 𝐾×1 matrix with elements 𝑥𝑙𝑘 𝑝 each represents a pilot sequence of length 𝜏 uploaded from the kth UT of the lth cell, assuming that the same pilot sequences are reused once within the same time-slot in the same cell. Let 𝐾 = 𝐾 ′ + 𝐾" where 𝐾 ′ represents the number of UT’s that sends initiative frame and 𝐾 " the number of UT’s that sends predictive frame in one cell, then the probability to encounter pilot contamination in any 𝐾′

cell will be 𝛼 = where OP denotes the number of 𝑂𝑃 orthogonal pilot sequences in the system of L cells. Following this assumption, 𝐿′ = 𝑟(𝐿×𝛼) will represent the approximate number of cells that upload the same pilot sequence, where 𝑟(. ) is a function that rounds to the nearest integer. Note that 𝐿×𝛼 denotes the number of users per pilot, but since we assumed that every pilot sequence is used once per cell, 𝐿′ represents the approximate number of 𝑳′

𝑲′

contaminated cell. This means that ∝ and 𝑳 𝑶𝑷 hence, reducing the number of users that transmits their pilot proportionally equal to reducing the number of interfered cells. Using this proportionality, we can represent 𝐘𝑗 𝒑 statistically as: 𝐿′ 𝒑

𝐘𝑗 𝒑 = √𝜏𝑃𝑢 ∑ 𝑮𝑗𝑙 𝑿′𝑙 + 𝒏𝒋

(5)

𝑙=1

A linear processing MRC, ZF, and MMSE for uplink detection represented as follows: 𝑮 𝑮(𝑮𝑯 𝑮)−𝟏

𝑨= {

̅𝑗𝑗 is a diagonal matrix and the kth element where 𝑫 can be written as: 𝛽𝑗𝑗𝑘 ̅𝑗𝑗 ]𝑘𝑘 = [𝑫 ∑𝐿′ 𝑙=1 𝛽𝑙𝑗𝑘 ̂ , the linear By using the imperfect channel 𝑮 ̂ detector matrix is denoted as 𝑨. Following the work of [2], the achievable uplink rate with imperfect channel estimation of the kth UT can be written as

𝑮 (𝑮𝑯 𝑮 +

𝑓𝑜𝑟 𝑀𝑅𝐶, 𝑓𝑜𝑟 𝑍𝐹, −𝟏 1 𝑰𝐾 ) 𝑃𝑢

𝑓𝑜𝑟 𝑀𝑀𝑆𝐸,

where 𝑨 is an M×K linear detector matrix which depends on 𝑮 and is used to detect the received signal by multiplying it with 𝑨𝑯 as follows: ̂ = 𝑨𝑯 𝒚 𝑿 (6) The MMSE estimation of the channel matrix 𝑮𝑗𝑗 given the received signal 𝒚𝑗 can be written as follows: 1 𝑃∗ ̂ 𝒋𝒋 = ̅𝑗𝑗 (7) 𝑮 𝑿′𝑗 𝐘𝑗 𝒑 𝑫 √𝜏𝑃𝑢

𝑅𝑘 = 𝔼 {log 2 (1 + 𝐻 𝑃𝑢|𝑎̂𝑘 𝑔̂𝑗𝑗𝑘 | 2

2

̂ 𝑘 𝐻 𝑔̂𝑗𝑗𝑖 | +𝑃𝑢‖𝑎̂𝑘 ‖2 ∑𝐾′ 𝑃𝑢 ∑𝐾′ 𝑖=1,𝑖≠𝑘|𝑎 𝑖=1

𝛽𝑖 +‖𝑎̂𝑘 ‖2 𝜏𝑃𝑢 𝛽𝑖 +1

)} (8)

̂ 𝒋𝒋 ̂ and 𝑮 where 𝑎̂𝑖 and 𝑔̂𝑗𝑗𝑖 are the ith columns of 𝑨 respectively. The contaminated estimated channel 𝑔̂𝑗𝑗𝑖 is given by: 𝑛̃𝑗𝑗𝑘 ℎ𝑗𝑗𝑘 + ∑𝐿′ 𝑙≠𝑗 √𝛾ℎ𝑗𝑙𝑘 + √𝜏𝑃𝑢 𝑔̂𝑗𝑗𝑖 = 1 (𝐿′ − 1)𝛾 + 1 + 𝜏𝑃𝑢 ( ) where 𝑛̃𝑗𝑗𝑘 in an i.i.d. with ∁𝒩 (0,1) noise and 𝛾 ∈ [0,1] represents the inter-cell interference factor. III.

TIME DIVISION DUPLEXING RECIPROCITY

UT’s are classified into two groups based on the type of TDD format they use which directly depends on their previous SNR (determined by test symbol) on the reverse link (see Fig. 1). Two TDD formats were introduced, the Initiative and the Predictive format. Initiative TDD format, used by K’ UT’s encounters low SNR or initially transmit their pilots, and it is supposed to be a conventional format to enable training for channel estimation at the BS. Predictive TDD format, used by K” UT’s that encounters high SNR at last forward link and UT’s belong to this group will skip sending their pilots where channel prediction at the BS will take place during the processing period. Each TDD format contains three parts within a time slot of sample duration T, (Reverse link, Processing period and Forward link) which will be discussed in a separate manner as follows:

Fig. 1. TDD protocol format

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A. At the forward link The BS will precode the downlink data and the test symbol applying linear precoder (e.g. MRT, MMSE or ZF) using a composite channel matrix with estimated and predicted elements. Each downlink field is tagged with a test symbol (known at both BS and UT’s) to allow UT’s easy compute their SNR and decide whether to use predictive or initiative format (referring to a specific threshold) at the next transmission session. Both initiative and predictive TDD formats use the same structure for the forward link. B. At the reverse link UT’s will either use predictive or initiative formats based on the measurement of the SNR of the test symbols from the previous forward link. The group of user terminals that use the predictive format is those who encounter high SNR according to a given threshold ϴ, and all other UT’s encounters test symbols SNR below ϴ will send an initiative formats (see Fig. 2). 𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑣𝑒 𝑖𝑓 𝑆𝑁𝑅 ≥ 𝛳 𝑆𝑤𝑖𝑡𝑐ℎ = { 𝐼𝑛𝑖𝑡𝑖𝑎𝑡𝑖𝑣𝑒 𝑖𝑓 𝑆𝑁𝑅 < 𝛳 The predictive format at reverse link starts with a padding of zeros, which supposed to be orthogonal to all pilot sequences occupying the same spectral period. The second part of the predictive TDD format is intended to be the same structure and spectral period with the initiative format, and it is exploited to send uplink data symbols. The initiative format at the reverse link is considered as classical training massive MIMO frame [10],[24] with an orthogonal sequence of pilots of length 𝜏 at the header of the data symbols. C. At the processing period After BS receive the reverse TDD, it will estimate the channels using linear channel estimation (e.g. MRC, MMSE, ZF) for UT’S that upload an initiative format. Furthermore, the BS will update the CSI Map by the new estimated CSI after quantization (learning algorithm will be discussed later in this paper). For UT’s uploading predictive format, the BS will use a special algorithm to retrieve the next possible channel information from the CSI Map. In both cases (prediction and learning), the BS will use a codebook to access the CSI Map for simplification and storage reduction issue. The BS can exploit the test symbol as a control flag to force UT’s to send their pilots, which is useful to increase the maturity of the CSI Map. This task can easily be done by reducing the transmission power of the test symbols. IV.

CHANNEL STATE INFORMATION MAP

CSI Map is a database of previously estimated QCSI represented by a directed graph with weighted

Fig.3. CSI node map

edges. The weight of the edges is proportional to the frequency of directed transition between the two connected nodes. In other words, an edge Ex,y issued from node Nx to node Ny means that there exist at least one transition from x to y. As a communication issue, this means that the CSI of one UT or more exhibit consecutive changes from CSI x to CSI y where CSI x and CSI y are the quantized versions of the CSI stored in the nodes (see Fig. 3). The intuition behind this map is to monitor the change in the CSI related to each position in the geographical map of the cell, independently from UT’s. Based on hypothesis one in [25], which was used for localization and proved by real measurements, CSI values maintain stability at a stationary location but exhibit variability between adjacent positions. The weight of 𝐸𝑥,𝑦 will increase in essence with the transition from 𝑁𝑥 to 𝑁𝑦 . If there is no transition for several TDD sessions, an edge will be created from the current node to itself e.g. 𝐸𝑥,𝑥 and its weight will be updated as much as there is no transition. The creation and update of the graph is done by applying a special learning algorithm that will be explained in the next section. After CSI Map get enough matured or converged (no updates for many learning epochs), it can be used to monitor UT’s CSI transitions in the cell and predict the most possible next node to be visited given the current node (see Fig. 5). V.

CSI MAP LEARNING ALGORITHM

To create and update the CSI Map a CSI learning algorithm represented by the flowchart of Fig. 6 is proposed. The learning algorithm will run on each CSI estimation of each UT and will decide either to update an existing edge weight or to add a new node and connect it to the CSI Map. After a finite number of learning iterations, the CSI Map will reach convergence (see Fig. 4.). This convergence can be defined by a specific large number of iteration done without encountering any update to the CSI Map. Touching on the flowchart of Fig. 6, steps 7 and 8 will be executed if there exist a previously visited node. The previous node is denoted as the QCSI that was stored in the CSI Map and previously visited by the user. One can look for the edge weights as the transition probability between two neighbor nodes.

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Fig. 2. Block diagram shows different UT’s and BS reciprocity operations

The 4 weight of the edges is updated using a modified version of the Erev and Roth algorithm [26]. Let denote by x the index of the last visited node and by y the index of the currently visited node and let denote by Ex,y the edge issued from node Nx to Ny with weight Wx,y>0. after the transition from Nx to Ny, the weight update of Ex,y is given by: 𝑊𝑥,𝑦 (𝑛𝑒𝑤) = 𝑊𝑥,𝑦 (𝑜𝑙𝑑) + Θ Subjected to Θ + 𝑊𝑐 ≤ 1. Let F denotes the set of all node indices connected to Nx by edges issued from Nx. For each edge Ex,c’, the weight update will be as follows: 𝑊𝑥,𝑐′ (𝑛𝑒𝑤) = 𝑊𝑥,𝑐 ′ (𝑜𝑙𝑑) − (∑𝐹

𝑊𝑥,𝑐′ (𝑜𝑙𝑑)

𝑐′=1,𝑐′≠𝑐 𝑊𝑥,𝑐′ (𝑜𝑙𝑑)

)Θ

Subjected to 𝑊𝑥,𝑐′ (𝑛𝑒𝑤) ≥ 0 where 𝑐′ ∈ 𝐹 and 𝑐′ ≠ 𝑦. The weight of the first issued edge from each node is initiated to one, and this value is conserved for all future issued edges such that ∑𝐹𝑐′=1 𝑊𝑥,𝑐′ = 1. The factor 0 < Θ < 1 is used to control the learning speed where assigning Θ = 0 will stop the learning and Θ = 1 will disconnect all issued edges except the last transition edge. After convergence, the BS can predict the next possible QCSI of any monitored UT from the CSI Map. Given the place of any UT at node Nx, the next possible visited node Ny can be found by 𝑁𝑦 = argmax 𝑊𝑥,𝑦 , where y ∈ F. 𝑦

The predicted QCSI stored by 𝑁𝑦 can be exploited in the next time-slot data receiving and precoding.

Fig. 5. Conception of CSI Map representing an indoor scenario

Fig. 4. Simulation results showing hit ratio of finding the new CSI in the MAP vs. number of TDD session, applyed in an indoor scenario of area 300m2 with UT’s moving randomly.

VI.

CSI QUANTIZATION

The parameters of QCSI are stored in a two-part codebook 𝑍 and 𝑅 , where the first part stores a finite set of shadow fading parameters and the second part stores a finite set of distances between BS and each UT. The channel between the jth BS and the kth UT of the lth cell are modeled as 𝑔𝑗𝑙𝑘 = √𝛽𝑗𝑙𝑘 ℎ𝑗𝑙𝑚𝑘 , i.e. ℎ𝑗𝑙𝑚𝑘 ≜ [𝑯𝑗𝑙 ]𝑚,𝑘 . Based on (3), as M >> K the effect of fast fading can be ignored and simply write: 𝑔𝑗𝑙𝑘 ≅ √𝛽𝑗𝑙𝑘 The large-scale fading 𝛽𝑗𝑙𝑘 =

𝑧𝑗𝑙𝑘 𝛿 𝑟𝑗𝑙𝑘

, where

𝑧𝑗𝑙𝑘

represents the shadow fading of lognormal distribution with standard deviation 𝜎𝑆ℎ𝑎𝑑𝑜𝑤 and 𝛿 is the path loss exponent with r representing the distance from BS to the kth UT. Followed a technique used in [15], [27], the space of all possible channel realization of Z and R can be divided into 𝑉, 𝑊 vector length respectively. Let 𝑧̂𝑣 = {𝑍: |𝑧𝑣 2 − 𝑍| < |𝑧𝑗 2 − 𝑍| 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 ≠ 𝑣 } and 𝑧𝑣 is a scalar representing region 𝑧̂𝑣 , and let 𝑟̂𝑤 = {𝑅: |𝑟𝑤 2 − 𝑅| < |𝑟𝑗 2 − 𝑅| 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 ≠ 𝑤 } where 𝑟𝑤 scalar representing region 𝑟̂𝑤 . A classical non-uniform quantizer algorithm found in [15] is used to design Z and R. The parameters of Z and R is acquired from the estimated CSI by minimizing the root mean square error as follows:

Fig. 6. Flowchart of CSI Learning Algorithm

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1 arg 𝑚𝑖𝑛𝑧𝑣,𝑟𝑤 [ 𝑉𝑊

𝑉,𝑊

∑

2 √𝑔̂𝑗𝑙𝑘 −

𝑣=1,𝑤=1

𝑧𝑣 ] (9) 𝑟𝑤𝛿

Where the vth and the wth elements of Z and R respectively are 𝑧𝑣 , 𝑟𝑤 and 𝑔̂𝑗𝑙𝑘 is the estimated version of CSI related to user k. VII. GARBAGE COLLECTION ALGORITHM The huge amount of nodes in the CSI map after convergence can be controlled by applying the following Garbage Collection Algorithm (GCA): For each Node Ni in the Graph If ( ∑𝐸𝑐"=1 𝑊𝑐",i < 𝑇ℎ ) Then Delete ( Ni ) Next

Fig. 8. Spectral Efficiency Vs. Number of antennas

where C” is the index of the edge pointing towered Ni and E is the number of those edges. GCA works in a periodic manner to delete nodes that have weakly connected edges. In other words, all rarely visited nodes will be removed from the map. Consider increasing the threshold 𝑇ℎ, CSI Map will shrink to represent most frequent visited positions in the Map. VIII. SPECTRAL AND ENERGY EFFICIENCY A lower bound on uplink spectral and energy efficiency can be modeled based on [2] as follows: 𝑇−𝜏 Spectral Efficiency 𝑆𝐸 = 𝐾𝑅𝑘 (10) 𝑇 where the sum-rate 𝑅𝑘 = 𝑙𝑜𝑔2 (1 + 𝑆𝐼𝑁𝑅) (11) moreover, the signal to interference ratio for ZF and MRC receiver are given below: 𝑆𝐼𝑁𝑅 = 𝜏(𝑀−1)𝑃𝑢2 2 𝜏(𝐾𝐿̅′ −1+𝛾(𝐿̅′−1)(𝑀−2))𝑃𝑢2 +𝐿̅′(𝐾+𝜏)𝑃𝑢+1 𝜏(𝑀−𝐾)𝑃𝑢 2 2 2 ̅ ̅ ̅ ′ ′ { 𝜏𝐾(𝐿 −𝛾𝐿 +𝛾−1)𝑃𝑢 +𝐿′(𝐾+𝜏)𝑃𝑢+1 ′ ′ ̅ where 𝐿 ≜ (𝐿 − 1)𝛾 + 1.

𝑓𝑜𝑟 𝑀𝑅𝐶 𝑓𝑜𝑟 𝑍𝐹

A lower bound on the energy efficiency can be modeled as in (12).

Fig. 7. Sum-rate Vs Number of antennas

𝑇−𝜏 ′ 𝐸𝐸 = ( 𝐾 + 𝐾") 𝑅𝑘 𝑇 IX.

(12)

NUMERICAL RESULTS

Considering the parameters in (TABLE I), the simulated scenario shows the performance of CSI Map over conventional Massive MIMO with increasing the number of antennas M at the BS. With the assumption that the user terminals are distributed uniformly in the cells, and they share the same time slot and bandwidth. This assumption brings a worstcase scenario to focus on the spatial multiplexing performance of the system. Referring to Fig. 7, it is evident that the performance of CSI Map with prediction increases the sum-rate dramatically with both MRC and ZF receivers among increasing the number of antennas. The increase in sum-rate is due to the reduction in uplink pilot contamination. Fig. 8 and 9 show the performance of prediction using CSI Map in spectral and energy efficiency respectively. The number of antennas is increased and keep on using the parameters in Table I, where it is evident from Fig. 8 that the performance in spectral efficiency can be scaled up by a factor of three with MRC and ZF receiver even with M=60. The performance in energy efficiency compared to the conventional technique is more interesting,

Fig. 9. Energy Efficiency Vs. Number of antennas

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where with the same parameters the performance scaled up by a factor of four with ZF receiver and M >100 (see Fig. 9). X.

[11]

CONCLUSION

This paper introduces a novel technique to map the CSI in an indoor cell region independently from user terminals. An algorithm to build and update the CSI Map is introduce with the exploitation of another Garbage Collection Algorithm to control the number of nodes in the CSI Map. This paper also introduces a simple algorithm to predict the next possible CSI of any UT given his current CSI. At the end, simulation results show the performance of the CSI Map on increasing the number of antennas at the BS. The results on sum-rate, spectral efficiency and energy efficiency outperform the conventional Massive MIMO system with a scale factor. The introduced CSI Map can be further used to model the user mobility in an indoor scenario. CSI Map is a useful tool that can be used in many applications including query out critical what-if scenarios to take an optimal decision considering wireless resources management. We look forward to implement CSI map in future researches on outdoor Massive MIMO.

[12]

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ENDNOTES 1Ahmed

A. Abboud (2nd Oct 1985) was born in Mazraat Mechref, South Lebanon. Received a technical diploma in communication & computer engineering from University Institute of Technology (UIT) Jwaya, South Lebanon and the Master of Science in Computer Science & Communication from Arts, Sciences & Technology University in Lebanon, in 2008 and 2012 respectively. He is a Ph.D. student at the University of Limoges since October 2014. His research interests are on applying artificial intelligence algorithms on 5G wireless communication systems.

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