A Fast And Optimal Deterministic Algorithm For NP-Hard Antenna Selection Problem

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1 A Fast And Optimal Deterministic Algorithm For NP-Hard Antenna Selection Problem Naveed Iqbal, Christian Schneider and Reiner S. Thomä Electronic Measurement Research Lab Ilmenau University of Technology P. O. Box 1565, D Ilmenau, Germany {naveed.iqbal, christian.schneider, Abstract Existing fast antenna selection algorithms have either low performance gains or they are efficient only for the selection of particular size sub-matrices. Our motivation to the study of NP-Hard antenna selection problem is two-fold. First, we intend to propose an optimal algorithm that is equally efficient for each possible subset cardinality of the transmit and/or receive antennas. Second, different from any of the antenna selection algorithms proposed so far, our algorithm deterministically increases the strength of the singular values during the selection of each column of the selected matrix. This increase in the singular values is defined by a very tight bound. This bound has previously been used to analyze the efficiency of low rank approximation algorithms. Hence, our antenna selection procedure is as efficient as that of a best algorithm for matrix low rank approximations. Keywords Multiple Input Multiple Output MIMO systems, Antenna subset selection, MIMO channel capacity I. INTRODUCTION Antenna selection AS in MIMO wireless communication systems is an application of column/row subset selection of the full matrix which is a NP-Hard problem [1]. Optimized brute force search over all possible antenna combinations have been proposed in [2]. However, these solutions require expensive singular value decomposition SVD/determinant operations for each possible subset. and decremental algorithms in [3], [4] and the global selection procedure in [5] add or remove antennas into or out of the selected channel matrix and update the matrix inverse through the Sherman-Morison formula. The Sherman-Morison formula is vulnerable to update the inverses of rank deficient matrices. Therefore, algorithms in [3] [5] are not valid for selection of MIMO sub-configurations which impose rank deficiency in algorithms. Authors in [6] proposed a fast incremental receive antenna selection and established that their algorithm is identical to the Classical Gram-Schmidt CGS orthogonalization for QR decomposition. The CGS orthogonalization has poor numerical properties and it causes severe loss in orthogonality [7]. Studies in [8] [1] have presented a modified antenna selection approach which is based on the MIMO Order Selection MOS followed by low complexity suboptimal AS techniques. The MIMO order defines the cardinality of the selected transmit antenna subset. The results in [9] show that the MOS approach is quite promising as it can offer the same or even better spectral efficiency than that of the optimal water-filling precoding [11]. Future 5G communication technologies such as massive and millimeter wave mmwave MIMO systems [12] consider very large number of antenna elements. The resulting MIMO channel matrix would then also have very large size. Therefore, existing AS algorithms would become impractical because either they are computationally expensive or they have low performance gains. Contribution: In order to select transmit and/or receive antennas, selection algorithms which are equally efficient for every possible MIMO channel sub-configuration are still required. We have proposed a fast, optimal and deterministic transmit antenna selection algorithm. With the best of our knowledge, a deterministic antenna selection algorithm have never been presented in wireless communications research community. In order to select receive antennas, the same selection procedure can be applied to the transpose of the channel matrix. Greedy antenna selection algorithms in [3], [4], [6] check each of the remaining non-selected antennas for their contributions into the channel capacity. This results in significant computational effort particularly for large MIMO configurations. Our criterion, however directly and efficiently detects the columns that should be permuted between the selected and non-selected parts of the MIMO channel matrix. Different from [3] [5], the proposed algorithm completely avoids the matrices to become rank deficient and therefore, it is equally efficient for each MIMO order. Notations: In the following, X, x and x correspond to the matrix, vector and scalar respectively. e i is the unitary vector corresponding to the i th column of the identity matrix. II. SYSTEM MODEL The MIMO transmission system consists of n T transmit and n R receive antennas. Let P be the total transmit power distributed equally among selected transmit antennas. Let s be the transmitted signal, then the receive signal vector y is P y = H p s + n 1 Where, H p C nr LT is the subchannel matrix of the full channel H C nr nt and n is the uncorrelated Additive /15/$ IEEE 911

2 White Gaussian Noise AWGN. The channel matrix normalization is done such that the Frobenius norm H 2 F =1.We assume that the channel state information CSI is available at the receiver which is used to select the MIMO order rank H and n R as in [9]. The receiver computes indices of the selected transmit antennas through the AS algorithm which are then fed back to the transmitter via a perfect feedback link. The total feedback overhead in this case would be of the order O log 2 n T bits. A. Channel model For the narrowband MIMO channel H, we considered the basic Kronecker model as H = R 1 2 R GR 1 2 T C nr nt. 2 Where, G is an independent and identically distributed i.i.d. MIMO channel. R T and R R are the transmit and receive correlation matrices with mean pairwise correlations ρ T and ρ R, respectively. It has been assumed that the channel spatial correlation statistics vary rapidly between different antenna pairs. Studies in [13], validate that in both single and dual polarized antenna configurations, high correlated antenna HC pairs have high frequency of occurrence. Therefore, for every new channel realization, we control the mean pairwise correlation in R T and R R with randomly positioned dedicated numbers of HC antenna pairs in the correlation range , while the remaining antenna pairs have low pairwise correlation. III. ANTENNA SELECTION ALGORITHMS A. Optimal Mutual information maximization MIM Given a channel matrix H C nr nt, select a subset H p of column-rank, among all the possible subsets p n T of H such that its channel capacity is maximum. C p =log 2 det I LT + E s Hp H H p 3 N E Note that for a given and SNR s N, the antenna selection problem depends only upon the volume det Hp H H p of the selected channel matrix H p. Therefore, the selected subset is optimal if arg max det H H p H p 4 p n T B. Deterministic subset selection By the interlacing property of SVD [14], we know that σ k H p σ k H 5 In 5, σ k corresponds to the k th singular value of the corresponding matrix H p or H. From 5, one may note that a selected matrix H p is optimal if its singular values σ k H p are closest to the singular values σ k H. Theoretically the performance of a selection algorithm can be measured with a low degree polynomial p n T,k that defines the degree of closeness of the singular values σ k H p to σ k H. σ k H p σ k H 6 p n T,k Such an algorithm is known as a deterministic algorithm. In next section, we propose a deterministic subset selection algorithm whose performance is bounded by a tight polynomial p n T,k= 1+k n T k. C. deterministic antenna selection Let s assume that the MIMO channel matrix H is composed of two submatrices H p and H q. H = H p H q 7 H p and H q correspond to the sub-matrices to be selected and to be discarded, respectively. Our objective is to permute columns between H p and H q such that H p is as non-singular as possible. We use the Pivoted Column QR decomposition PCQR [7] of matrix H. HΠ = QR 8 Where, Π is the permutation matrix. R is an upper triangular matrix and it can be written as, A B R = 9 C where, A C LT LT, B C LT nt LT and C C nr LT nt LT. The PCQR starts by selecting a maximum norm vector from H and then greedily selects columns such that after selecting each column from the channel matrix H, its projections are removed from each of the remaining columns of H. This gives us the first initial guess of H p as, A H p = Q 1 Considering R 1 = write Eq. 8 as A and R 2 = B, we can C HΠ = QR 1 QR 2 11 This implies that H p = QR 1 and H q = QR 2. Note that H and R have the same singular values, i.e. σ i H =σ i R, therefore the rank r of H and R is also same. r σi 2 H = r r L T σi 2 R = σi 2 A σ 2 i C 12 Note that the overall the product of singular values of H and R remains the same, therefore maximizing each σi 2 A would result in minimizing the singular values σ2 i C. From Eq. 1, we know that increasing σi 2 A results in an increase in the singular values of H p and therefore decrease in the singular values of H q. Note that if we permute columns i R 1 and j R 2, this corresponds to a similar permutation of columns i H p and column j H q. Gu and Eisenstat 912

3 [15] showed that the amount of volume change in A during the permutation of columns i and j in R can be expressed as, det Ã Ω i,j = det A = A 1 B 2 i,j + Ce j 2 2 e T i A i,j 13 where, à is the matrix formed after permutation of columns i and j in R, Ce j is the j th column of C and e T i A 1 is the i th row of A 1. The expression in Eq. 13 is remarkable as it can pre-determine the change in volume of a submatrix A due to permutation of two columns i and j in R before actually permuting them. These column permutations guarantee that once a column has been moved to R 1, it will not be moved back to R 2. It is important to note that no matter what is the size of H p, the matrix A is always a nonsingular square matrix. So permuting the columns between R 1 and R 2 will not create a rank deficiency problem. Hence, the resulting algorithm can be applied for subset selection of any arbitrary cardinality. Using Eqs. 8 and 13, we now propose Algorithm 1. The algorithm selects the columns in H p, such that each added column deterministically increases the singular values of H p in the same way as that of the strong RRQR factorization from [15] for the matrix low rank approximations i.e. σ k H p σ k H 1+k nt k 14 Importantly Eq. 13 also determines if a column permutation is required or not. The algorithm 1 terminates if there are no other column swaps in R n at the n th iteration to satisfy the condition Ω i,j max > 1 and returns Π. H p = H opt corresponds to the first columns of HΠ. Algorithm 1 antenna selection algorithm 1: procedure PROPOSEDH, 2: Compute QR decomposition with column pivoting for A B H as HΠ = QR with R = C 3: while There exists an entry Ω i,j max > 1 do 4: Permute columns i and j + in R n and in Π n. 5: Retriangularize : R n+1 = Q n R n 6: Modify A 1 B, Ce 2 2 and e T A 1 2 2, where Ce and e T A 1 corresponds to the arbitrary rows/columns of C and A 1. 7: end while 8: return Π 9: end procedure 1 Proof of optimality: Now we consider optimizing the Eq. 4. Taking the QR decomposition of H p, we write Hp H H p = A H Q H A Q 15 As Q is a unitary matrix, therefore we can write det Hp H H p =det A H det A = a i,i 2 16 where, a i,i R are the diagonal entries of the matrix A. In Eq. 13, if the detã > deta, then the ratio Ω i,j > 1. Therefore, a swap of columns i and j + will increase the det A. From the expression in Eq. 13 and the condition Ω i,j max > 1 in the algorithm 1, we know that the algorithm terminates only if there is no further increase in the volume det A. Therefore, at the termination of Algorithm 1 the volume of A is maximum. Hence, the resulting H p H will be an optimal subset that provides the maximum capacity. 2 Computational complexity: Let Ç represents the computational complexity of an algorithm or the mathematical operation in floating point operations flops. A flop is a floating point operation α β, where α and β are floating point numbers and is one of the +,,, operations. The cost of computing the PCQR at the step-2 of the Algorithm 1 is ÇPCQR =4n R n T 2L 2 T n R + n T L3 T 17 Main cost of initializing Ω is in the computation of A 1 B which takes L 2 T n T flops by using the forward substitution method [7]. We now discuss the case when Ω i,j max > 1 and the i th and j + th columns in R are interchanged. We use the Givens rotations [7] to zero out the elements below the main diagonal in R n. Let Q n be an orthogonal matrix formed as the product of the Givens rotation matrices. Let Q n = Q A Q C 18 where, Q A and Q C are used to zero out the elements below the diagonals in the matrices A and C respectively. R n+1 = Q n R n = Q A A n Q A B n + Q C C n 19 Note that after the permutation, A is an upper Hessenberg matrix, therefore the cost of computing both Q A A n and Q A B n is less than 3 2n T flops. The cost of computing Q C C n is around 4n R n T flops. Modifying A 1 B at the n th iteration requires 4 n T flops. Therefore, the total cost of modifying Ω n, Ce 2 2 and e T A is ÇΩ n 3 2n T +4n R n T +L 2 T 2 Ç Total ÇPCQR + L 2 T n T +t k ÇΩ n 21 where, t k corresponds to the total number of column swaps at the k th channel snapshot. It is important to note that the PCQR algorithm itself is near optimal in maximum volume subset selection. Our motivation to use the permutation results of the PCQR is to reduce the number of column swaps within the while loop. Therefore, as a result the Algorithm 1 requires very few and often no column swaps. Assuming that t k the computational cost of the Algorithm 1 reduces to Ç Total ÇPCQR + L 2 T n T

4 Assuming that the antenna array is quite large, we express computational complexity using the Big O notation as in [3] [6]. The complexity order of our algorithm is around O n T n R. IV. SIMULATION RESULTS In this section, the proposed method is compared with existing optimal mutual information maximization MIM [2] and sub-optimal [6] and [5] algorithms. In the Section III-A, we discussed that the antenna selection problem depends only upon the volume of the selected subchannel matrix. Therefore, we have considered a single SNR and focused on the performance evaluation during selection of different subchannel configurations. As channel coding gives a linear parallel shift in error rate performance [11] and this error rate scaling will remain same for each selection algorithm. Therefore, for simplicity we have considered an uncoded system with 64QAM modulation and a zero-forcing ZF detector at the receiver. Fig. 1 shows ergodic channel capacity analysis of different state of the art algorithms. As expected the performance of our proposed algorithm is optimal. Figs. 2a and 2b show symbol error rate SER analysis of different antenna selection algorithms. We observe that our algorithm outperforms existing and selection procedures in both correlation scenarios. We know that the algorithm is identical to CGS orthogonalization which due to its poor numerical properties leads to performance degradation. has been particularly designed for selecting a square submatrix from a short and wide parent matrix. This assumption is in general not always true because the matrix to be selected depends upon the number of available RF chains at the transmitter and the receiver. This does not always result in square MIMO configuration. We observe that when H p is not a square matrix, does not remain as efficient as shown in [5]. The primary reason for this performance degradations is that the selection criterion of fails to determine whether the selected subset is a maximum volume matrix. Computational complexity comparison: Computational order of the in the case of square matrices is same as that of Algorithm 1, i.e. O n T n R [5]. Computational cost for non-square matrices without any updating procedures not defined is expressed as { } Ç =t k +1 C H+ p H 23 Hence, the overall complexity of the is O n R L 2 T + n K Tn R flops. Let tavg = 1 K t k be the k=1 average number of column swaps over K = 2 channel realizations. The results in Fig. 3 show that the proposed algorithm needs very few column swaps as compared to the. In general t avg is upper bounded by t avg. However, due to the PCQR decomposition at the beginning, it is much likely that in most of the channel realizations not only a single column swap is required. This remark can be Ergodic capacity [Bits/Sec/Hz] MIM Optimal 6 1 Number of selected transmit antennas Fig. 1: Ergodic open loop capacity vs., 8 8 MIMO, independent Rayleigh flat fading channel, SNR=1 [db] TABLE I: Summary of computational complexity of different selection algorithms, where means that results are unknown Antenna selection Polynomial Algorithm Complexity p n T,j MIM Optimal [2] nt L O nr L 2 T T Decremental [3] O n 2 T n2 R [6] O n T n R [5] O n T n R O n T n R 1+k nt k easily verified from Fig. 3, where the average number of column swaps is t avg. The, however starts with a random initial guess of H p, and hence it needs more column swaps. After the first permutation both the and the proposed algorithm employ updating procedures. The update procedures for the can be derived only for the square matrix selection because they involve the Sherman-Morison formula for rank-1 update. In contrast to the, the updating procedures for our algorithm only need to re-triangularize R and then update the Ω i,j accordingly. They are valid for any MIMO configuration and cost a total of O t k n T n R flops. Table I and the results in Fig. 2 show that our algorithm operates at a significantly lower computational cost while maintaining the performance level of the optimal algorithm. V. CONCLUSION We have proposed an optimal algorithm, which works at the low computational cost for every possible MIMO order and correlation setup. With each selected antenna, our algorithm deterministically increases the strength of singular values of the selected channel matrix. In future, we would like to extend the underlying approach to user selection in massive MIMO broadcast channels. 914

5 SER SER MIM Optimal Number of selected transmit antennas a SNR= 21 [db], ρ T =.5,ρ R =.3 MIM Optimal Number of selected transmit antennas b SNR= 3 [db], ρ T =.9,ρ R =.6 Fig. 2: SER vs., 8 8 MIMO system, 14 HC at Tx and 6 HC at Rx ACKNOWLEDGMENTS This work has been performed in the framework of the FP7 project ICT RESCUE Links-on-thefly Technology for Robust, Efficient and Smart Communication in Unpredictable Environments which is partly funded by the European Union. The authors would further like to thank the administration and members of Thüringer Innovationszentrum Mobilität ThIMo and the VISTA4F ProExellenz research group funded by the Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft. Furthermore, authors would also like to thank Prof. Stanley C. Eisenstat from Yale University, U.S. and Dr. Ali Çivril from Melikşah University, Turkey for their valuable comments. tavg Number of selected transmit antennas, ρ T =.5, ρ R =.3, ρ T =.5, ρ R =.3, ρ T =.9, ρ R =.6, ρ T =.9, ρ R =.6 Fig. 3: Comparison of average # of column swaps t avg vs., for results in Figs. 2a and 2b REFERENCES [1] A. Çivril and M. Magdon-Ismail, On selecting a maximum volume sub-matrix of a matrix and related problems, Theor. Comput. Sci., vol. 41, no , pp , 29. [2] J. Heath, S. Sandhu, and A. Paulraj, Antenna selection for spatial multiplexing systems with linear receivers, IEEE Communications Letters, vol. 5, no. 4, pp , april 21. [3] A. Gorokhov, Antenna selection algorithms for MEA transmission systems, in IEEE International Conference on Acoustics, Speech, and Signal Processing ICASSP, vol. 3, 22, pp. III 2857 III 286. [4] L. Zhou, Low complexity optimum transmit antenna selection algorithms in spatial multiplexing systems, in IEEE 21st International Symposium on Personal Indoor and Mobile Radio Communications PIMRC, 21. [5] B. H. Wang, H. T. Hui, and M.-S. Leong, Global and fast receiver antenna selection for MIMO systems, IEEE Transactions on Communications, vol. 58, no. 9, pp , September 21. [6] M. Gharavi-Alkhansari and A. Gershman, Fast antenna subset selection in MIMO systems, IEEE Transactions on Signal Processing, vol. 52, no. 2, pp , 24. [7] G. H. Golub and C. F. Van Loan, Matrix Computations 3rd Ed.. Baltimore, MD, USA: Johns Hopkins University Press, [8] N. Iqbal, C. Schneider, and R. S. Thomä, Efficient MIMO order and transmit antenna subset selection in correlated channels, in 17th International ITG Workshop on Smart Antennas WSA, 213. [9] C. Schneider, N. Iqbal, and R. S. Thomä, Low complexity MIMO order and transmit antenna subset selection in correlated channels, in Proceedings of 19th European Wireless Conference EW, 213. [1] N. Iqbal, C. Schneider, and W. Ahmad, Efficient matrix volume control to maximize correlated mimo channel capacity, in IEEE International Conference on Communication, Networks and Satellite COMNETSAT, Nov 214, pp [11] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge University Press, 23. [12] F. Boccardi, R. Heath, A. Lozano, T. Marzetta, and P. Popovski, Five disruptive technology directions for 5G, IEEE Communications Magazine, vol. 52, no. 2, pp. 74 8, February 214. [13] C. Schneider, N. Iqbal, and R. S. Thomä, Performance of MIMO order and antenna subset selection in realistic urban macro cell, in IEEE 24th International Symposium on Personal, Indoor and Mobile Radio CommunicationsPIMRC 13 - Fundamentals Track, London, United Kingdom, Sep [14] R. Horn and C. Johnson, Topics in Matrix Analysis. Cambridge University Press, [15] M. Gu and S. C. Eisenstat, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM J. Sci. Comput., vol. 17, no. 4, pp , Jul