Improved Channel Estimation in mmWave ... - IEEE Xplore

7th SEMINAR ON DETECTION SYSTEMS: ARCHITECTURES AND TECHNOLOGIES (DAT’2017) February 20-22, 2017, Algiers, Algeria.

Improved Channel Estimation in mmWave Communication System Irched Chafaa1 , Mustapha Djeddou2 1, 2

Communication Systems Laboratory Military Polytechnic School, Algiers, Algeria 1 [email protected] 2 [email protected]

Abstract The key element of enabling millimeter wave communications is the adoption of a hybrid beamforming. This hybrid architecture requires the development of new algorithms to estimate the mmWave channel. The main goal of this work is jointly estimating the mmWave channel parameters: the angle of departure (AOD), the angle of arrival (AOA) and the path’s gain using the sparsity of the mmWave channel in conjunction with a criterion test based on diagonality of data matrices. The results of the simulation, in terms of performance, show the superiority of the modified algorithm integrating the new criterion compared to the original one. keywords—Hybrid beamforming; Channel estimation; AOD; AOA

1

mmWave channel;

Introduction

We have been experiencing, these last years, an exponential growth in the mobile data traffic. This rapid growth is mainly due to the popularity of smart phones, social networks and the advanced multimedia applications such as 3D and ultra high definition videos. This explosion of the traffic, besides the congestion of the radio spectrum below 3 GHz, have made the mobile operators rethink the conception and the deployment of their cellular networks for the next generation (5G). The use of millimeter waves (20-300 GHz) is a promising technology for the future cellular networks [3], [8], [12], [15]. The main issue of using millimeter waves is the high path loss and the important attenuation caused by the rain [13]. Thanks to the reduced wavelengths of the millimeter band, we can use large antenna arrays [16] to produce a high beamforming gain in order to overcome the unfavorable path loss which characterizes the mmWave channel. According to the used beamforming architecture, the weighting coefficients employed to form the beam can be 978-1-5090-4508-2/17/$31.00 ©2017 IEEE

applied in either digital or analog domain. Nevertheless, hybrid analog/digital architecture [17], [6] will be inevitably adopted in the mmWave systems where the beams are formed by an analog beamforming while a digital one enables the application of the different precoding operations and the necessary signal processing techniques. The construction of the precoding matrix requires a complete knowledge of the communication channel, which is difficult to have in the case of the millimeter band because of the low values of the signal to noise ratio (SNR), before carrying out the beamforming operation, and the great number of antennas involved. Therefore, new channel estimation and precoding algorithms must be developed for the mmWave case using hybrid architecture. The authors, in [1], propose algorithms of low calculation complexity using the characteristic of sparsity of the mmWave channel in order to jointly estimate the different channel parameters in a mmWave system provided with large antenna arrays in both transmitting and receiving end. For a hybrid mmWave system, in [2], the authors proposed a channel estimation technique which does not require a feedback between the base station (BS) and the mobile station (MS). In [11], the authors developed a modified EM channel estimation algorithm that exploits the sparsity of the mmWave channel for multi-input multi-output systems (MIMO) equipped with one-bit analog-digital converters. An open loop estimator is proposed in [7] using orthogonal matching pursuit (OMP) algorithm and their modified versions called multi-grid OMP (MG-OMP). In [10], a new approach to estimate the mmWave channel parameters based on compressed sensing paradigm for a simplified hybrid architecture is proposed. This simplified architecture uses switchers instead of phase shifters on the receiver side. An algorithm to estimate the parameters of a multi-path mmWave channel is proposed [1] and thus allows the multiplexing of several data streams whereas the work of [4] and [9] is limited to the transmission with only one beam. In this paper, we introduce some modifications to the algorithm proposed in [1] in order to improve the quality of the estimation of a single path mmWave channel parameters. The remainder of this paper is organized as follows. In section II, we present the mmWave channel model and that of the data model. In section III, we feature the modified algorithm used to estimate the single path mmWave channel. Simulation results illustrating the performance of the modified algorithm are given in section IV, followed by a conclu-


limit our study to the case of an ULA. The channel matrix is given in a more compact form by: α )AH H = ARX diag(α TX

(3)

where: s α=

MN [α1 , α2 , ..., αL ]T ρ

ATX = [aTX (φ1 ), aTX (φ2 ), ..., aTX (φL )]

Figure 1: Block diagram of a hybrid BS-MS system.

ARX = [aRX (θ1 ), aRX (θ2 ), ..., aRX (θL )] sion in section V.

2.2 Data model

2 System model We consider a system with one cell that contains MS users who communicate with the BS via directing beams using large antenna arrays. It is supposed that the BS has M antennas and NRF RF chains while the MS is equipped with N antennas and NRF RF chains as shown in Fig. 1. The BS and the MS communicate via NS data streams such as NS ≤ NRF ≤ M and NS ≤ NRF ≤ N.

2.1 Channel model

y = WT H HFT s + WT H n

It is well established in the literature that the mmWave channel has a limited number of propagation paths. We adopt the geometrical model for the channel with L scatters [14]. Each scatter contributes with only one path of propagation between the transmitter and the receiver which makes the number of paths equal to the number of scatters [5]. Under this model, the channel H can be expressed by: s MN L H= (1) ∑ α` aRX (θ` )aTX (φ` )H ρ `=1 Where ρ represents the average path loss. θ` and φ` denote the AOAs/AODs of the l th path respectively. We only consider the azimuth for the AOD and the AOA. aTX and aRX represent the antenna array’s response vectors of the transmitter and the receiver respectively. αl represents the path loss. Using a uniform linear array ( ULA ), we have: 2π 2π 1 aTX (φ` ) = √ [1, e j λ d sin(φ` ) , ..., e j(M−1) λ d sin(φ` ) ]T M

In order to transmit a signal, the BS applies two consecutive precoding operations FT = FRF FBB , where FBB ∈ CNRF ×Ns represents a digital precoding performed in the base band and FRF ∈ CM×NRF is an analog precoder. Since FRF is implemented using analog phase shifters, its elements are of constant modules. At the receiver side, the combiner WT , composed of a RF combiner WRF and a digital one WBB , is used to extract the transmitted data from the received signal. After different signal processing operations, The signal is given by:

(2)

Where λ represents the wavelength of the signal and d is the distance between the antenna array elements. The response vector of the antenna array at the receiver is written in the same way. Although the algorithm presented in this work can be applied to antenna arrays of arbitrary forms, we

(4)

Where (.)H stands for the Hermitian, H ∈ CN×M is the mmWave channel matrix, s ∈ CNs ×1 is the vector of the transmitted pilot symbols such as E[ssH ] = NPSS INS and PS is the average transmitted power, and n ∼ N (0, σ 2 ) is the additive white Gaussian noise (AWGN) vector. We consider the channel estimation problem formulated in [1] according to the sparsity character of the mmWave channel. The MS uses a vector f among the possible vectors defined in the multi-resolution hierarchical codebook F [1] to transmit pilots s. The transmitted signal by one beamforming vector is given by: y=

√ PHfs + n

(5)

Where P represent the average power of the received signal. The BS applies all the possible cases of the vector w from the codebook W . The resulting signal can be written as : y=

√ ∗ Pw Hfs + w∗ n

(6)

This process is repeated for all the vectors f ∈ F . We obtain, after concatenating all various measurements, the data matrix Y, assuming all transmitted pilot symbols are equal, can be written as : Y=

√ PWH HF + N

(7)


Where N is the AWGN noise matrix. To go further, we need to reformulate (7) by applying successively the following two frequently used formulas: vec(XY Z) = (Z T ⊗ X)vec(Y ) vec(Xdiag(Y )Z) = (Z H ◦ X)Y Where ⊗ and ◦ denote the Kronecker and Khatri-Rao products respectively. In fact, we proceed by vectorizing the resultant matrix Y:

yv

= vec(Y) √ = Pvec(WH HF) + vec(N) √ = P(FT ⊗ WH )vec(H) + nQ √ α )ATX H ) + nQ = P(FT ⊗ WH )vec(ARX diag(α √ = P(FT ⊗ WH )A∗TX ◦ ARX α + nQ (8)

Using the quantized values of the AODs/AOAs, we get: y=

√ Ψz + n PΨ

(9)

Where y are the available measurements and Ψ can be viewed as known sensing matrix given by Ψ = (FT ⊗ WH )AD with AD is a dictionary matrix whose ith column is given by a∗TX (i) ⊗ aRX (i). This dictionary matrix contains all the possible quantized values of AOD/AOA. The vector z corresponds to the path’s gain of the quantized directions. Due to the sparsity of the mmWave channel, the vector z contains a limited number of non-zero elements. For a single path mmWave channel, it would have only one non-zero element. Using the sensing matrix Ψ , we can recover the nonezero element of the vector z which represents the estimated gain of the propagation path. the estimated AOD/AOA are the elements of the matrix AD which correspond to the nonzero element of the vector z.

3

Proposed method to estimate a single path mmWave channel

Considering the previous estimation problem, a single path channel implies that the vector z has only one non-zero element. Thus, estimating the channel means determining the position of this element which also defines the corresponding AOD/AOA. The module of this element determines the gain of the propagation path of the signal path under consideration. The algorithm proposed in [1] seeks to locate this element in an adaptive manner using the beamforming vectors of a dictionary already built in an off-line step. In the initial stage of the training phase , K beamforming vectors of the first level of the dictionary F and W are used to establish the communication between the BS and the MS.

After K 2 precoding-combining operations, the authors in [1] propose to compare the power of all K 2 received signals to determine the one with the maximum received power to be considered as the desired pilot signal. Since each beamforming vector is associated to a certain quantized AOD/AOA range, this first step divides the vector z to K 2 partitions. Therefore, the selection of the maximum power signal implies the selection of a certain partition of the vector z which corresponds to a range of AOD/AOA that contains the path of propagation. The method suggested in [1] is based on the selection of the maximum power received signal. In the case of millimeter waves, the received signal is completely submerged in the noise before carrying out the beamforming. This implies that the selected maximum power does not always correspond to the desired pilot signal. Consequently, there will be errors in the channel parameters estimation. To improve the performance of the channel estimation algorithm, we propose to modify the criterion used to choose the partition of the vector z which most likely corresponds to the desired propagation path. For each beamforming vector f p , p = 1, ..., K from the codebook F used by the MS to transmit its pilot symbols, the BS uses all the combining vectors w p , p = 1, ..., K of the corresponding level of the codebook W . Therefore, we get a data matrix Y whose rows y contain the received signals which are transmitted by all the beamforming vectors f p of the same level of the codebook F and decoded in the BS by the same combining vector w p using the data matrix Y. We form for each row vector y: yH y

Ψz)H (Ψ Ψz) = (Ψ ΨH Ψ )z = zH (Ψ

(10)

In order to recover the non-zero element of the vector z with the minimum of measures using the sensing matrix Ψ , the restricted isometric property requires the matrix Ψ H Ψ to be close to diagonal on average. Actually, a matrix X with a dominant diagonal verifies this property : N

|X j j | ≥

∑

|X jk |

(11)

k=1, j6=k

Hence, we apply the diagonally dominant matrix test on each matrix Ai = yH i yi where yi stands for a row vector of the matrix Y. We look for the matrices Ai that verify this criterion then among them, we choose the one with the highest diagonal value Aij j which means finding the two indices i and j such that: (i, j) = arg max [Aij, j ]

(12)

∀i, j=1,2,...,K

Where the index i corresponds to the row vector yi of Y and so it indicates the selected combiner wi . In the same


way, the index j indicates the used beamforming vector fi . Since each beamforming/combining vector corresponds to a specific range of AOD/AOA, the selected indices i and j correspond to the quantized estimated angles. These indices are then used to determine the set of beamforming/combining vectors of the next level of the codebook to be used on the next stage of the training stage. The BS informs the MS of the selected set of vectors. As the procedure moves from one stage to another, the resolution of the estimated angles keeps improving. Once the desired resolution is reached, the normalized value of the module of the last chosen element of the data matrix Y obtained in the final adaptive stage is the estimated gain of the propagation path. The steps of the procedure is resumed in algorithm 1. This improved technique can be extended to the case of multi-path, this part is left for future work. Input: F , W , K, N Initialization: S = logK N // number of adaptive stages initialize F and W for s ≤ S MS uses K beamforming vectors of level s of F ; BS uses K measurement vectors of level s of W ; BS Constructs the data matrix Y; BS Searches for the strongest beam using the diagonally dominant matrix test : for i ≤ K Ai = y H i yi search for Aij j such that |Aij j | ≥ ∑Nk=1, j6=k |Aijk | end (i? , j? ) = arg max∀i, j=1,2,...,K [Aii, j ]i, j BS Communicates i? and j? to MS to be used in level s + 1 of F and W ; end i? and j? of level S indicate the estimated AODs/AOAs ; Normalize the last chosen received signal amplitude .

Figure 2: Single path mmWave channel spectral efficiency width of 100MHz, and with path loss exponent n pl = 3. We evaluate the system’s spectral efficiency with low values of SNR for a perfect channel and for the estimated one. the number of beamforming vectors used in each stage of the training phase is set to K = 2. the spectral efficiency is given in [1] by : R

=

ρ −1 ? R WBB W?RF HFRF FBB NS n ×F?BB F?RF H ? WRF WBB |) log2 (|INS +

(13)

Where Rn = σn2 W?BB W?RF WRF WBB is the noise covariance matrix. Figure 2 shows that the spectral efficiency obtained by the modified algorithm is higher than the one obtained with the original algorithm and is closer to the one obtained with a perfect channel knowledge. This can be explained by the fact that the angle estimation by the modified algorithm has been improved and became more precise due to the modification of the criterion used to determine the AODs/AOAs range that contains most likely the propagation path. A better channel estimation yields into better performance because we approach those obtained with a perfect channel knowledge.

5 Conclusion 4

Preliminary simulation results

We consider a cellular cell with one user while neglecting the inter-cell interference. The BS has 64 antennas, 10 RF chains and uses a transmitted power of 37 dBm. The MS uses 32 antennas and 6 RF chains. Both the BS and the MS are equipped with ULA with an inter-element spacing of λ /2. The number of transmitted data streams is NS = 3. We use the geometric channel model with an average power gain of P¯R = 1 and a single path. The AoAs/AoDs are assumed to be uniformly distributed between 0 and 2π. The system is assumed to operate at 28GHz carrier frequency, has a band-

In this paper, we have presented an enhanced performance version of a recently proposed algorithm to estimate mmWave channel parameters. We have proposed a new method to choose the angle range that most likely contains the path of propagation for a single path mmWave channel using the diagonally dominant matrix test. The simulation results show that the introduced modification improves the performance of the estimation algorithm to approach more and more those achieved with a perfect channel knowledge.

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