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IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 7, JULY 2015

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Performance of Transmit Beamforming Codebooks with Separate Amplitude and Phase Quantization Alexis Dowhuszko and Jyri Hämäläinen

Abstract—A simple quantization scheme is proposed to adjust separately the amplitudes and phases of signals in a codebookbased Transmit Beamforming (TBF) scheme. The proposed quantization scheme is valid for both desired signal power maximization (egoistic TBF) and co-channel interference mitigation (altruistic TBF), and does not require an exhaustive search to find the optimal TBF vector. Performance results are presented when variable numbers of bits for amplitude and phase feedback resolutions are used for both egoistic and altruistic TBF schemes. Index Terms—Codebook design, interference mitigation, limited feedback terms, separate quantization, transmit beamforming.

I. INTRODUCTION

C

O-CHANNEL INTERFERENCE (CCI) puts hard restrictions on the data rate that a mobile network is able to achieve. In a macro cellular system, e.g., this impairment takes the form of macro-layer interference, degrading performance in cell edge areas of the network [1]. The CCI problem becomes even more challenging in Heterogeneous Networks, where conventional macro cells need to coexist with small cells of different sizes and characteristics, generating cross-layer interference that is even harder to control [2]. The management of CCI can be grouped into different categories, according to the degrees of freedom that are used to control this impairment. In this paper we focus on spatial-domain techniques, where convenient Transmit Beamforming (TBF) vectors are selected to control both intra-cell signal power and inter-cell CCI. For a Multiple-Input Single-Output (MISO) interference channel with 2 users, any Pareto optimal point of the data rate region is reached using a TBF strategy that combines both Maximum Ratio Transmission (MRT) and Zero-Forcing (ZF) beamforming [3]. Unfortunately, in presence of a FDD system with limited feedback, MRT and ZF beamforming vectors need to be quantized using e.g. codebook-based TBF schemes [4]. One alternative to design feedback messages is presented in [5], where a quantization strategy to optimize the received SNR is proposed. Similarly, the authors of [6] suggest a quantization criterion that minimizes the maximum correlation between any pair of valid TBF codewords. The use of Random Vector Quantization (RVQ) is also proposed in [7] as a simple codeManuscript received June 26, 2014; revised August 21, 2014; accepted October 26, 2014. Date of publication November 13, 2014; date of current version November 25, 2014. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Peter K. Willett. A. Dowhuszko and J. Hämäläinen are with the Department of Communications and Networking, School of Electrical Engineering, Aalto University, FI-00076 Aalto, Finland (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2014.2370762

book design with good asymptotic properties. Nevertheless, in all these cases, TBF codebook needs to be changed completely every time the number of feedback bits varies. Moreover, the most convenient TBF vector needs to be selected via an exhaustive search, which represents a computational burden as the size of TBF codebook grows [4]. One approach to tackle these limitations is presented in [8], where scalar quantization is used to adjust independently amplitudes and phases of the different Transmit (Tx) antennas. In [8], the relative channel phase information of the different Tx antennas is uniformly quantized (with respect to a reference antenna) and the order information of channel gains are informed separately. Nevertheless, when CCI minimization becomes the design criterion of a TBF codebook, the use of additional bits for the quantization of channel amplitudes is expected to provide notable performance improvements [9]. Among other advantages, the use of separate quantization schemes for both channel amplitudes and phases allows to adapt the structure of the TBF codebook on-the-fly, based on the feedback overhead that each user can generate. Moreover, this separate quantization approach is very appropriate when feedback bits need to be reported over a longer time span. In this situation, smarter signaling schemes that take into account the state transition of the channel can be implemented [10], and transmitter does not need to receive all the bits of the feedback word to update the TBF vector in transmission. In this paper, the most convenient way to share the feedback bits of a TBF scheme that quantizes channel amplitudes and phases separately is studied. Both egoistic and altruistic TBF schemes are considered, where goals are the maximization and minimization of the received signal power, respectively. The design methodology that is presented is generic, and can be used for any number of Tx and Receive (Rx) antennas, as well as for various kinds of fast fading channel models. Nevertheless, to keep the analysis tractable, we focus on a 2 Tx-1 Rx antenna scenario with independent block Rayleigh fading. Note that for egoistic TBF, the use of channel order information (i.e., coarse channel amplitude quantization) provides a performance close to the full Channel State Information (CSI) case [8]. Moreover, the number of Tx antennas does not have a notable effect on the interference mitigation capability of altruistic TBF with fixed feedback overhead [11]. The rest of the paper is organized as follows. Section II introduces the system model and presents the optimization problem to be solved. Section III divides the TBF codebook design into separate channel amplitude and phase quantizations. Section IV shows the obtained results for different feedback resolutions. Finally, conclusions are drawn in Section V. II. SYSTEM MODEL The general layout of the system model is illustrated in Fig. 1. The system contains a Base Station (BS) and a Mobile Station

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to the co-phasing factor applied in the second Tx antenna. Since quantization of relative channel amplitudes and phases is carried out independently, quantization set contains TBF vectors of the form , with and . Quantization regions and functions need to be defined for both relative channel amplitude and phase, to maximize (minimize) the received signal power. Fig. 1. General system structure to implement a TBF scheme that feeds back channel amplitude and phase information separately.

(MS), which are equipped with 2 Tx and 1 Rx antenna, respectively. A codebook-based TBF scheme is applied at the BS to maximize (minimize) the power of the received signal, using finite rate signaling information that is reported from the MS to the BS via a separate feedback link. Based on this system model, received signal at MS is (1) is the channel gain vector (with zero-mean where complex Gaussian coefficients), and are the TBF vector and the information symbol of the BS, and is Additive . All TBF vecWhite Gaussian Noise (AWGN) with power and belong to a common TBF codebook tors verify that is shared between BS and MS in advance. A. Egoistic and Altruistic Transmit Beamforming If egoistic TBF is applied, aim is to maximize the received signal power from serving BS. To achieve this goal, MS finds (2) and then feeds back the corresponding index to the BS. When altruistic TBF is applied, aim is to minimize the power of the interfering signal. So, victim MS determines (3)

A. Quantization Scheme for Phase Information Uniform quantization can be used for the relative phase , since channel phases are uniformly distributed in . Thus, quantization function equals when lies in quantization region , for . In altruistic TBF, each should be rotated by radians. B. Quantization Scheme for Amplitude Information Aim is to determine quantization regions and quantized output values of quantization function , for , such that SNR gain of received signal (4) is maximized (minimized) for egoistic (altruistic) TBF. Assume that amplitudes of channels gains of are uncorrelated Rayleigh distributed Random Variables (RVs). Then, the probability that RV falls within quantization region can be easily computed using elementary probability theory; see (4) and preceding discussion in the Appendix. After expanding in the right-hand side of (4), the conditional SNR gain can be written in the form (5) where is the vector with quantized amis the correlation matrix of co-phased plitudes, and received signals conditioned on with elements

first and, after that, informs its index to the interfering BS. B. Separate Amplitude and Phase Quantizations In the TBF scheme of Fig. 1, the relative amplitudes and phases of the signals applied at the Tx antennas of serving (interfering) BS are adjusted such that the desired received signal (cochannel interference) power is maximized (minimized) at the target (victim) MS. In the procedure, the MS first estimates the complex channel gains from each Tx antenna and then, quantizes separately the relative channel amplitude and phase information with predefined resolutions. We note that independent quantization of both amplitude and phase information enables to report the corresponding feedback bits at separate time instants, allowing to spread them in a longer time window. So, for fixed feedback overhead per user, users with high mobility can track CSI with low resolution but more frequently, while users with low mobility have the chance to increase the accuracy of CSI reports with less frequent updates. III. TRANSMIT BEAMFORMING CODEBOOK DESIGN Let be a -bit quantization function for the relative , and let be a -bit channel amplitude quantization function that maps channel phase difference

(6) We note that phase adjustment using uniform -bit quantization scheme is done first. Thus, represents the phase quantization error, and is uniformly distributed in . So, . , and Conditional expectations are presented in (32)–(34), respectively. Egoistic TBF—Quantized Channel Amplitudes: Let . Then, optimal quantized channel amplitude vector (7) By using (5), (6) we find that the this optimization problem can be solved by determining the eigenvector that corresponds to , i.e., the maximum eigenvalue of matrix det Closed-form result is given by

(8) , where (9)

DOWHUSZKO AND HÄMÄLÄINEN: PERFORMANCE OF TRANSMIT BEAMFORMING CODEBOOKS

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TABLE I QUANTIZATION THRESHOLDS FOR THE AMPLITUDE INFORMATION IN CASE OF DIFFERENT FEEDBACK RESOLUTIONS ( BITS)

TABLE II QUANTIZED AMPLITUDE VECTORS FEEDBACK RESOLUTIONS (

FOR

DIFFERENT BITS)

Fig. 2. Coherent combining gain of egoistic TBF when different numbers of ) and channel phase bits (i.e., ) are used. In channel amplitude bits (i.e., all cases, point values (‘*’) were simulated to verify the analytical results.

Altruistic TBF—Quantized Channel Amplitudes: When the objective is the minimization of the received signal power, optimal quantized channel vector with Egoistic TBF—Amplitude Quantization Regions: Quantization thresholds that define the different quantization regions are determined using an algorithm that resembles the method of bisection. The basic idea consists in finding the best way to sub-divide each a priori quantization region to maximize the a posteriori SNR gain, every time the amplitude feedback resolution increases one bit. In practice, for each new quantization threshold to be determined in iteration , the following optimization problem needs to be solved: (10)

(11) Now the conditional SNR gain is the minimum eigenvalue of , i.e., det

(12)

Altruistic TBF—Amplitude Quantization Regions: Sub-division of quantization regions at each iteration is now obtained by solving (13)

where takes odd values between 0 and . The initial conditions are and . The quantization thresholds for even values of between 0 and are taken from the for thresholds obtained in previous iterations, i.e. . Objective function in (10) is continuous and . So, it can be solved using traditional unimodal in methods, like golden section search. A summary of this procedure is presented as Algorithm 1. Algorithm 1 Amplitude quantization thresholds and weights 1: Initialization: Set and 2: for to do 3: for to do 4: Set 5: end for 6: for to do 7: Set 8: Compute from optimization problem (10) 9: end for 10: end for 11: for to do 12: Compute with the aid of (9) 13: end for

where takes odd values between 0 and . Again, the initial and . Similarly, quantization conditions are for even values of are taken from the previous thresholds for . iterations, i.e. IV. SIMULATION RESULTS For illustrative purposes, quantization thresholds and quan. tization amplitudes are given in Table I and II for These values were obtained following the procedure described in Section III-B. Similar results can be obtained for other numbers of feedback bits to quantize channel amplitudes. It is interesting to observe that quantization regions and amplitude vectors are the same for both egoistic and altruistic TBF schemes. Only the mapping function needs to be changed. Fig. 2 shows the effect of feedback resolution when aim is to maximize the performance of an own cell user (i.e., egoistic TBF). As expected, the performance in terms of SNR gain improves as the amount of both channel amplitude and phase feedback information grows. Nevertheless, saturation is observed and . Then, the use of additional bits to when increase feedback resolution does not provide notable improvement because, in this situation, SNR performance is already less than 0.2 dB away from the 3 dB upper bound.

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and its Cumulative Distribution Function (CDF) is (24) Therefore, (25) Let us first compute the conditional expectation of RV when , i.e., (26) With the aid of Bayes’ theorem [12], conditional PDF Fig. 3. Interference mitigation gain of altruistic TBF when different numbers ) and channel phase bits (i.e., ) are used. of channel amplitude bits (i.e., In all cases, point values (‘*’) were simulated to verify the analytical results.

Fig. 3 shows the effect of feedback resolution on the interference suppression performance of altruistic TBF. In this case performance saturation is only observed if the amount of feedback information grows only with respect to phase or amplitude grows for fixed , or when grows for fixed (i.e., when ). If the number of channel amplitude and phase quantization bits admit a proper linear relation, then there is no performance bound for interference suppression. As a rule of thumb, very good interference suppression results are obtained when . With , performance is close to the when . bound obtained for fixed

(27) where

(28) Then, combining (26)–(28) with (25), we obtain the formula that appears in (32). For the conditional expectation of RV , formula (33) can be derived following similar steps. Finally, we derive the conditional expectation

(29) To achieve this goal, integration formulas

V. CONCLUSION Codebook-based TBF provides a good alternative to improve the data rates in FDD mobile networks. Multiple transmit antennas can be used to increase signal power of own cell users (egoistic TBF), or reduce CCI power of users in adjacent cells (altruistic TBF). In this paper, a simplified TBF codebook design was analyzed, where both channel amplitudes and phases were quantized and reported separately. Performance results were presented for different numbers of phase and amplitude feedback bits. For egoistic TBF, most of the available performance gain was achieved with a relative low number of feedback bits. In case of altruistic TBF, on the other hand, the interference suppression performance increased with the number of feedback bits with no saturation (provided that feedback bits are properly shared between the quantizations of channel amplitude and phase parts of channel gains).

(30) and

(31) from [13] are sequentially used. The resulting closed-form expression appears in (34).

(32) APPENDIX I The PDF of RV , where and joint Probability Density Function (PDF)

are two RVs with , is (22)

(33)

When both and are independent and identically distributed (i.i.d.) exponential RVs with unitary mean, (23)

(34)

DOWHUSZKO AND HÄMÄLÄINEN: PERFORMANCE OF TRANSMIT BEAMFORMING CODEBOOKS

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[7] W. Santipach and M. L. Honig, “Capacity of a multiple-antenna fading channel with a quantized precoding matrix,” IEEE Trans. Inf. Theory, vol. 55, no. 3, pp. 1218–1234, Mar. 2009. [8] J. Hämäläinen, R. Wichman, A. A. Dowhuszko, and G. Corral-Briones, “Capacity of generalized UTRA FDD closed-loop transmit diversity modes,” Wireless Pers. Commun., vol. 54, pp. 467–484, Aug. 2010. [9] A. A. Dowhuszko, M. Husso, J. Li, J. Hämäläinen, and Z. Zheng, “Performance of practical transmit beamforming methods for interference suppression in closed-access femtocells,” in Proc. Future Network and MobileSummit: Workshop on Broadband Femtocell Networks, Jun. 2011, pp. 1–12. [10] J. Hämäläinen and R. Wichman, “Closed-loop transmit diversity for FDD WCDMA systems,” in Proc. Asilomar Conf. on Signals, Systems and Computers, Oct. 2000, vol. 1, pp. 111–115. [11] A. A. Dowhuszko, J. Hämäläinen, A. R. Elsherif, and Z. Ding, “Performance of transmit beamforming for interference mitigation with random codebooks,” in Proc. Int. Conf. Cognitive Radio Oriented Wireless Networks, Jul. 2013, pp. 190–195. [12] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York, NY, USA: McGraw-Hill International, 1991. [13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. ed. New York, NY, USA: Academic , 2007.