Path Based MIMO Channel Model for Hybrid

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communication (5G) and beyond, millimeter-wave bands will be considered to satisfy the customers demands on higher data rates [1]–[3]. While going to higher ...
GeMiC 2018 • March 12–14, 2018, Freiburg, Germany

Path Based MIMO Channel Model for Hybrid Beamforming Architecture Analysis Joerg Eisenbeis , Marius Krause, Tobias Mahler, Steffen Scherr, Thomas Zwick Institute of Radio Frequency Engineering and Electronics (IHE) Karlsruhe Institute of Technology (KIT) Engesserstr. 5, 76131 Karlsruhe, Germany  [email protected] Abstract—Hybrid beamforming communication systems allow the reduction of digital channels by employing an analogue beamforming network. This approach offers a large reduction in computational effort and energy consumption. Still there are only few practical realizations of such hybrid beamforming communication systems. Hence, there is a nearly infinite number of possible implementations of such an architecture. To analyze and compare the performance of different hybrid beamforming architectures, we present a path based multiple input multiple output (MIMO) channel model. This tool allows us to generate arbitrary propagation paths with distributions of direction of departure (DOD), direction of arrival (DOA) as well as phase and amplitudes. Therefore, an insight of the channel capacity and signal-to-noise ratio (SNR) can be given for different kinds of hybrid beamforming architectures. The results show that the newly developed path based channel model is beneficial for the analysis of hybrid beamforming architectures.

(a) Fully-connected hybrid beamforming receiver architecture.

I. INTRODUCTION Modern wireless communication systems utilize a MIMO configuration to improve capacity and signal coverage. Recent studies have shown that for the fifth generations of mobile communication (5G) and beyond, millimeter-wave bands will be considered to satisfy the customers demands on higher data rates [1]–[3]. While going to higher carrier frequencies to obtain larger bandwidths the resulting path loss has to be overcome, which leads to an increasing number of antenna elements and therefore digital channels (RF chains) to perform a precise beamforming. In theory, each digital channel enables the use of one particular spatial sub-channel, which might highly improve the maximum achievable data rate. However, each digital channel needs its own analogue front- or backend and a powerful analog-to-digital converter (or digital-to-analog converter respectively), which results in a large and expensive setup [4], [5]. Furthermore, the calculation effort in the digital signal processor increases with every added channel resulting in a high energy consumption of the system. One solution to decrease complexity and to improve energy efficiency is to build a hybrid beamforming system combining the digital beamforming with an analogue beamforming unit. The number of digital channels can thereby be reduced while conserving a precise beamforming. Different kinds of hybrid architectures have been proposed in literature so far [6]–[9]. For practical implementation so called subarray-based hybrid beamforming architectures seem the most promising, because of the modular setup and low circuit complexity [10]. Within

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(b) Subarray based hybrid beamforming receiver architecture. Fig. 1. The hybrid beamforming architectures are depicted here without loss of generality for a homodyne communication receiver. Both architectures consist of a digital precoding matrix W BB with Nbf output data streams, Ndig digital channels, an analogue precoding matrix W RF as well as a number of antennas Nant . For the subarray-based hybrid beamforming architecture the number of antennas Nant = Ndig · Nsub results out of the number of digital channels times the number of antennas per subarray Nsub .

the subarray-based architecture, each digital channel is connected to a particular group of antenna elements as shown in Fig. 1a. Due to separation into subarrays the degree of freedom for the digital precoding is lowered, leading to a reduction in performance. This can be compensated by connecting each digital channel with each antenna element as depicted in Fig. 1b, known as fully-connceted hybrid beamforming architecture. For practical realization that architecture does not seem reasonable, due to the high number of intersections and high circuit complexity. Nevertheless it serves as a benchmark for subarray-based hybrid beamforming architectures in the further simulations shown. To investigate and optimize subarray-based architectures channel modeling is needed to gain insights in the overall performance before starting with

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Fig. 2. Path based MIMO channel model for hybrid beamforming architectures.

the design of such systems. The authors presented in [11] a channel model that is able to generate MIMO channel samples of arbitrary multipath components. It resolves the DOD and the DOA and the propagation path phases and amplitudes that together yield the angular power spectrum. To compare the performance of the hybrid beamforming architectures, the channel model is expanded to realize subarraybased and fully-connected architectures. For the implementation of hybrid beamforming architectures the used algrithms for separating the beamforming into the analogue and digital part are crucial. The knowledge about the channel properties allows a realistic estimation of the achieved channel capacity as well as an optimization of the employed algorithms. Moreover, this channel model empowers us to analyze different subarray-based hybrid beamforming realizations which serve as fundament for the system design. II. PATH BASED MIMO CHANNEL MODEL As presented in [11] the MIMO channel matrix used for our model H ∈ CNant ×Mant can be described by a sum of Np propagation paths consisting of an attenuation factor αp and an arbitrary phase ϕp of the p-th path between the Mant transmitting and Nant receiving antennas. The channel matrix  Np

and arrival ΩR,p represent the combination of the elevation angle θ and azimuth angle ψ and can be written in form of cartesian direction vectors s (ΩT,p ) at the TX and s (ΩR,p ) at the RX, respectively. As described above the overall beamforming at the TX and RX is separated into a digital beamforming and analogue beamforming, represented as precoding matrixes F BB and F RF at the TX and W BB and W RF at the RX. The complex channel matrix including transmitter and receiver beamforming can be written as H H H BF = W H BB W RF · H · F RF F BB = W HF

(2)

and is represented within the path based MIMO channel model depicted in Fig. 2. The received signal H H H y = WH BB W RF · H · F RF F BB · x + W BB W RF · n

(3)

H

where ( · ) denotes the Hermitian transpose, results as multiplication of the transmitted signal vector in baseband x with the beamforming channel matrix and an additive white Gaussian noise vector n, which entries follow an independent and identical distribution CN (0, σn2 ). The average transmit power is denoted by PTx . III. HYBRID BEAMFORMING ALGORITHMS

(1)

The hybrid beamforming algorithms used for this paper aim to maximize the achievable channel capacity     PTx H H H  (4) W HF F H W  C = log2  I Mant + 2 K · σn

results from the antenna elements radiation pattern matrices E T (Ω) at the transmitter (TX) and E R (Ω) at the receiver (RX), the array geometry in form of an antenna element position matrix AT at the TX and AR at the RX and the path influences as described above. The angles of departure ΩT,p

where F = F RF F BB and W = W RF W BB denote the combined beamforming matrices as introduced above. For simplicity the functional principle of the algorithms is only done for the TX beamforming matrices. For a hybrid beamforming setup, the precoding matrix has to satisfy several

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E R (ΩR,p ) · ej · k · AR · s(ΩR,p ) · αp · ej · ϕp

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· ej · k · AT · s(ΩT ,p )

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A. Fully-connected Architecture The algorithm proposed in [12] will approximate hybrid precoders for a fully-connected architecture based on matching pursuit algorithm. As initial information, an optimum precoder and a set of array response vectors are required. The optimum precoder F opt can be calculated by a singular value decomposition (SVD) of the channel matrix H. The array response vector contains the antenna weights corresponding to a given azimuth and elevation angle, towards which the array gain should be aligned. The array response vectors for an antenna array can be calculated as 1  · ej · k · dx · sin(θp ) · cos(ψp ) Mant   · ej · k · dy · sin(θp ) · sin(ψp ) · ej · k · dz · cos(θp )

(6)

where θp is the elevation and ψp is the azimuth angle of departure of the p-th path and dx , dy and dz contain the position of every antenna element in cartesian coordinates. Starting with the optimum precoder, the best array response vector is chosen regarding the projection on F opt . This vector is selected as the first column of the analogue precoder F RF . Now F BB is chosen to minimize ||F opt −F RF · F BB ||F . For the next iteration of the algorithm, the influence of the computed beam on the optimum precoder is removed, so the algorithm can proceed to the next vector. After computing all Mdig precoding vectors, the hybrid precoder F is obtained. B. Subarray-based Architecture The successive interference cancelation (SIC)-based hybrid precoding algorithm proposed in [13] computes the precoding matrices for a subarray-based setup. After selecting all entries of the co-variance matrix HH H in respect of the antennas of

ISBN 978-3-9812668-8-7

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looses degrees of freedom, while maintaining its outer shape. The analogue beamforming for each subarray can be described by weight vectors a ∈ CMsub ×1 . Due to the fact that, in the subarray setup, only one RF chain is connected to a fixed set of antennas, the role of the digital precoding matrix F BB is reduced to power allocation tasks and routing the signal to the desired subarrays. To satisfy the overall power constraint, the Frobenius norm of the combined precoding matrix must satisfy ||F RF · F BB ||2F ≤ K ≤ Mbf with K ∈ N representing the number of formed beams.

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constraints. Analogue beamforming is done exclusively in phase by reason of efficiency, resulting in equal amplitudes of all non-zero elements of F RF ∈ CMant ×Mdig . Depending on the chosen architecture, the inner and outer shape of F RF differs. In case of a fully-connected system, all elements can contain a non-zero complex weight in F RF . For subarray-based systems, the analogue beamforming matrix ⎤ ⎡ 0 a1 0 · · · ⎢ 0 a2 0 ⎥ ⎥ ⎢ F RF = ⎢ . (5) .. ⎥ . . . ⎣. . . ⎦

























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Fig. 3. 10 % outage capacity over SNR for hybrid beamforming architectures. As upper performance boundary a full digital architecture with 8 × 8 antennas is evaluated for the same channel realizations.

the used subarray, an SVD is applied to the resulting matrix. Now the optimal antenna weight vector v is selected. The extracted angles of v yield the phase shift on the corresponding antenna. Including the fixed power constraint for the analogue precoding matrix, the weight vector a is calculated as a = √

1 · ej · arg(v) . Mant

(7)

Note that the beamforming vector in the subarray-based case only contains Msub ≤ Mant non-zero elements. As a result, the used array gain per beam is much lower compared to other beamforming techniques. IV. HYBRID BEAMFORMING ARCHITECTURE EVALUATION To compare the architectures and corresponding algorithms the performance is evaluated statistically, using different channel realizations gained from the presented path based MIMO channel model. Within the simulations, the channel state information (CSI) is considered as known. As performance parameter serves the 10 % outage capacity, which is estimated from the cumulative density function including 4000 arbitrary channel realizations. The MIMO antenna constellation is supposed as a uniform planar array (UPA) with an antenna spacing of half a wavelength, reflecting a realistic mobile communication base station setup. Furthermore, the possible angular range for the DOD and DOA is limited in azimuth to 120◦ and in elevation to 45◦ , respectively. As element characteristic the radiation pattern of a patch antenna is used. For the analogue beamforming system digital controllable phase shifters with a 5 bit resolution are considered. At first the fully-connected and subarray-based hybrid beamforming architectures are compared using a 8 × 8 UPA with 2 × 2 = 4 digital channels at the TX and RX. For the subarray-based hybrid beamforming architecture also a TX

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beamforming architecture compared to the fully-connected hybrid beamforming and full digital approach, due to a reduced number of degrees of freedom. Furthermore, the results show that the number of formed beams should be adjusted to the available channel SNR to achieve maximum performance. For the subarray-based hybrid beamforming architecture a lower number of digital channels does not lead to a reduced channel capacity as long as the antenna array size is held constant and the number of formed beams is less than or equal to the available digital channels.

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Fig. 4. 10 % outage capacity over SNR for different subarray sizes at the TX. At the RX a full digital 4 × 4 UPA is used.

This work was supported by the European Union and the German Federal Ministry of Education and Research in frame of the ECSEL Joint Undertaking project TARANTO under grant number 16ESE0211. The authors would also like to acknowledge the support by the Helmholtz International Research School for Teratronics (HIRST). R EFERENCES

only case is evaluated. As performance reference serves a 8×8 full digital system. Fig. 3 shows the 10 % outage capacity over the SNR for different hybrid beamforming architectures forming either K = Mbf = 4 or K = 2 beams. As shown here, fully-connected hybrid beamforming systems are able to achieve the performance of full digital MIMO systems, if the maximum number of beams is not larger than half the available number of digital channels. The findings show for the subarray-based hybrid beamforming architecture, the reduction in degrees of freedom result in a reduction in capacity. Furthermore, Fig. 3 reveals that for low SNRs in the subarray-based case it can be advantageous to use a lower number of beams. For a deeper analysis of subarray-based hybrid beamforming architectures the subarray size is varied. For simplification the subarray-based approach is only applied at the TX while preserving a full digital 4 × 4 UPA at the RX. The antenna array of the subarray-based architecture is hold constant with Mant = 16 × 16 = 256 antenna elements. The subarray size results to Msub = 256/Mdig . The results in form of the 10 % outage capacity over SNR are shown in Fig. 4. It should be noted that for K < Mbf still all antenna elements are included into the beamforming process by subarray aggregation to enable a fair comparison. The curves with an identical K (K = 8: Mdig = 16, Mdig = 16 and K = 4: Mdig = 8, Mdig = 4) lie directly on top of each other. That means that as long as K ≤ Mdig holds, there exists no difference in capacity between different subarray sizes. Here too it becomes clear that for maximizing the outage capacity the number of formed beams K has to be adjusted to the channel SNR. V. CONCLUSION The presented path based MIMO channel model enables the comparison of different hybrid beamforming architectures and algorithms in a realistic wireless communication scenario. The results reveal the capacity loss of the subarray-based hybrid

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