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www.ietdl.org Published in IET Communications Received on 24th March 2010 Revised on 13th September 2010 doi: 10.1049/iet-com.2010.0234

ISSN 1751-8628

Genetic algorithm-assisted joint quantised precoding and transmit antenna selection in multi-user multi-input multi-output systems W.-H. Fang S.-C. Huang Y.-T. Chen Department of Electronic Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan E-mail: [email protected]

Abstract: This study presents a simple and efficient genetic algorithm-assisted approach for joint quantised precoding and transmit antenna selection based on the criterion of maximum capacity. The objective is to alleviate the effect of multi-user interference and to reduce hardware costs, such as the cost of radio frequency chains associated with antennas in the downlink of multi-input multi-output systems with limited feedback. To avoid the enormous search effort required by existing approaches, the authors propose a novel variant of the conventional genetic algorithm, called the hybrid genetic algorithm, in which each chromosome is divided into a bit string for precoding vector selection and an integer string for transmit antenna selection. In addition, new crossover and mutation operations are employed to accommodate these new chromosomes. The results of simulations show that the performance of the proposed approach is close to that of the exhaustive search method, but its computational complexity is substantially lower.

1

Introduction

Multi-input multi-output (MIMO) systems, which deploy multiple antennas at both the transmitter side and the receiver side, have the potential to meet the rapidly growing demand for better quality of service (QoS) under the limited spectral resources of wireless communication systems [1]. Some studies have shown that the QoS of MIMO systems scales with the number of antennas [1, 2]; however, deploying a large number of antennas in such systems has two drawbacks. First, it increases the cost of hardware, such as the radio frequency (RF) chains including the amplifiers and converters. Second, interference, such as multiple stream interference, becomes more pronounced and impacts on the system performance. An effective way to reduce the RF chain cost is through antenna selection, whereby only some of the available antennas are selected according to certain prescribed criteria [2]. This approach reduces the hardware cost, and maintains the full diversity of the original systems at the expense of a slight loss of array gain [3]. Transmit antenna selection and receive antenna selection have attracted considerable attention in recent years. However, the optimum antenna selection algorithm must conduct an exhaustive search of all possible combinations to find the optimum antenna subsets at either the transmitter side or the receiver side; thus, the computational overhead is very high. A number of suboptimal algorithms have been proposed to reduce the overhead, for example [2, 4], but they do so at the expense of some performance degradation. With regard to interference, preprocessing the data before transmission can alleviate the problem. This has led to the 1220 & The Institution of Engineering and Technology 2011

development of a number of linear multi-user MIMO precoding techniques [5] that increase the system capacity and simultaneously reduce the data error rate. Since most of the computational load is shifted to the base station (BS), precoding plays a key role in downlink systems. For example, Spencer et al. [6] proposed a block diagonalisation method that can prevent multi-user interference completely; however, the number of transmit antennas need to be larger than the rank of each interfering channel matrix. Choi and Murch [7] and Yang [8] used singular value decomposition (SVD) to decompose a multi-user MIMO channel into multiple parallel independent single-user MIMO channels; thus, the receivers originally proposed for single-user systems are also applicable. The drawback of above methods is that they need to have perfect channel state information at the transmitters, so they require a huge amount of feedback from receivers. To resolve this problem in wireless systems with limited spectral resources, another approach chooses the transmitter precoders based on a predetermined finite set of precoding vectors, referred to as the codebook, which is known to both the transmitters and the receivers. The receiver then chooses the optimal precoder from the codebook based on some prescribed criteria and sends the selected precoding vector back to the transmitter by using a limited number of bits to indicate its index [9, 10]. Since precoding can compensate for the loss of array gain during antenna selection, it is important to combine both techniques to enhance the overall system performance [11 – 13]. To this end, Chen et al. [11] proposed that the optimum transmit antenna subset based on the maximum of the minimum singular value of the channel matrix should IET Commun., 2011, Vol. 5, Iss. 9, pp. 1220–1229 doi: 10.1049/iet-com.2010.0234

www.ietdl.org be selected first, and then the optimum precoding vector that maximises the system capacity can be determined. In [12], the optimum transmit antenna subset that maximises the largest eigenvalue of the channel covariance matrix is determined first, followed by the selection of the optimal beamforming vector from a codebook to maximise the capacity. However, since the transmit antenna subset and the precoding vectors are determined separately in the above works, these suboptimal approaches inevitably suffer some performance loss. To address the problem under limited feedback conditions, we propose to select the optimal quantised precoders and the transmit antenna subset simultaneously in the downlink of multi-user MIMO systems. Our objective is to alleviate the interference and reduce the hardware costs through a single feedback operation. The drawback of the joint approach is that it requires an enormous amount of search effort. To reduce the computational overhead, we propose an efficient genetic algorithm (GA) approach that locates the optimal precoding vectors and the transmit antenna subset simultaneously. The GA [14], devised by Holland, is an effective optimisation technique that emulates the crossover and mutation phases in the evolution of a chromosome. It has been applied successfully in various types of signal processing applications, such as control and wireless communications [15]. In the conventional GA, each chromosome is encoded by a bit string. However, if we follow this approach and use 1 in a chromosome’s bit string to denote that an antenna has been selected, the number of 1’s may vary during the chromosome’s evolution; thus, the number of antennas selected may not meet the prescribed target. To resolve the problem, we propose a variant of the GA, called the hybrid GA (HGA), in which each chromosome is divided into two parts: a bit string for quantised precoding vector selection and an integer string for transmit antenna selection. To accommodate the new chromosomes, the HGA employs novel crossover and mutation operations during the evolutionary process. The results of simulations demonstrate that the proposed joint scheme outperforms the methods that determine the transmit antenna subset and precoding vectors separately. Moreover, its performance is close to that of exhaustive search method, but with substantially lower computational complexity in the multi-user MIMO downlink.

The remainder of this paper is organised as follows. Section 2 introduces our system model. In Section 3, we describe the evolutionary process employed by the proposed HGA, and compare its complexity with that of existing approaches. In Section 4, we discuss the simulation results. Section 5 contains some concluding remarks.

2

Data model

The proposed model considers the downlink of a spatial multiplexing multi-user MIMO system with limited feedback. We assume that there are K active users, each of which receives data symbols from the BS concurrently via a Rayleigh fading channel. Each user employs N receive antennas, and M transmit antennas are deployed at the BS as shown in Fig. 1. A codebook, known to both the BS and the users, contains 2L precoding vectors, where L is the number of feedback bits. After precoding, the signal received by the kth user is expressed as yk = H k Fs + nk ,

k = 1, . . . , K

(1)

where yk is the N × 1 received vector of the kth user, Hk is the N × M flat Rayleigh fading channel matrix from the BS to user k and nk denotes the N × 1 complex Gaussian noise vector with zero mean and covariance matrix N0IN , where IN is an N × N identity matrix. In addition, the vector  √ √ √ T p2 s2 · · · pK sK p 1 s1 is the transmitted s= vector, where sk , 1 ≤ k ≤ K, is the independently identically distributed (i.i.d.) binary phase shift keying (BPSK) modulated symbol with the transmitted signal power pk and F ¼ [ f1 f2 . . . fK] denotes the M × K precoding matrix, where fk is the M × 1 precoding vector of the kth user. In this paper, the codebook design is based on the Grassmannian line packing criterion [9], which quantises the transmit beamforming vectors and maximises the minimum distance between any pair of 2L lines (or subspaces) in C M. In other words, the magnitude of the correlation between any two vectors in the codebook is as small as possible. Therefore the corresponding codebook matrix F isexpressed as F ¼ arg maxw d(W ) ¼ arg maxw  L min1≤i,j≤L 1 − |wH i wj |, where W ¼ [w1 w2 . . . w2 ] is

Fig. 1 Transmitter structure of the proposed hybrid scheme IET Commun., 2011, Vol. 5, Iss. 9, pp. 1220–1229 doi: 10.1049/iet-com.2010.0234

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www.ietdl.org an M × 2L matrix representing a pack of 2L lines, and wi is the ith line or vector in the pack. We assume that a linear receiver g˜ k is deployed at the kth user to improve the system performance. The output signalto-noise plus inference (SINR) for the kth user with the linear receiver can then be expressed as follows [10] SINRk =

pk |˜gk H k f k |2 SKi=1,i=k pi |˜gk H k

f i | + ˜gk  N0 2

2

(2)

where . denotes the Euclidean norm. In the considered systems, we assume that the corresponding channel matrix has been properly estimated and that each user can access the codebook. Each user computes the SINRs for all combinations of the transmit antennas and all permutations of the precoding vectors in the codebook, and sends the scalar SINRs along with the corresponding indices back to the BS. Then, based on all the feedback, the BS selects the optimum subset according to the prescribed criterion, which we discuss in the next section.

3

Proposed HGA approach

We assume that the BS has m RF chains demultiplexed into M antenna elements, where m ≤ M. To reduce the effect of multi-user interference and hardware costs in a multi-user MIMO downlink with limited feedback, we propose a joint scheme that selects the precoding vectors and the transmit antennas simultaneously to maximise the system capacity. The scheme can be formulated as follows arg

max

K 

 K {f k }K k=1 [F ,{H k }k=1 [H k=1

log2 (1 + SINRk )

(3)

where F is the set of all possible combinations of precoding vectors {f k }Kk=1 in the codebook, and H is the set of all  k }Kk=1 that contain only m possible channel matrices {H columns of Hk . Our objective is to choose the optimal K (out of 2L) precoding vectors and the optimal m (out of M ) transmit antennas simultaneously to maximise the channel capacity given in (3). 3.1

Proposed HGA

The joint scheme in (3) requires an enormous number of computations to search for all possible candidates in both F and H. To reduce the computational overhead, we propose a simple and efficient GA-based approach, called HGA, to solve (3). In the conventional GA, a chromosome is encoded as a bit string [14], where 1 denotes that the gene’s corresponding antenna elements have been selected. Consequently, the total number of transmit antenna elements selected may vary in the evolution of chromosomes and may not meet the prescribed target. To resolve the problem, similar to the approach in [16, 17], the proposed HGA divides each chromosome into two parts: a bit string for quantised precoding vector selection and an integer string for transmit antenna selection. The genes in the transmit antenna selection part form an integer string, which indicates the priority for selecting the desired transmit antennas. Meanwhile, like the conventional GA, the genes in the precoding vector selection part form a bit string, which indicates the binary index bits corresponding to a particular 1222 & The Institution of Engineering and Technology 2011

precoding vector in the codebook. To accommodate this new chromosome structure, we implement a hybrid mechanism that modifies the two key steps of the conventional GA, namely, the crossover operation and the mutation operation. Next, we describe the steps of the proposed HGA in which every solution is represented by a chromosome. Step 0: Initialisation: Given a population of size P, the algorithm first creates P chromosomes, called parent chromosomes, each of which is divided in a precoding vector selection part and a transmit antenna selection part. The precoding vector selection part comprises KL genes which are uniformly distributed 1s or 0s, and the transmit antenna selection part comprises M genes, which are a random permutation of {1, . . . , M}. In the latter part, the genes with a larger value have a higher antenna selection priority. Because of the initialisation scheme and the subsequent manipulations, the HGA ensures that the genes in the antenna selection part of a chromosome maintain a permutation of {1, . . . , M} throughout the evolutionary process. As a result, antennas can be selected based on the resulting gene values. For example, consider two parent chromosomes, a and b, as shown in Fig. 2, where M ¼ 5, m ¼ 4, L ¼ 4 and K ¼ 2. The parent chromosome a indicates that the transmit antennas {1,2,3,4,5} of a are with the corresponding priority values {4,3,5,2,1}. For this chromosome, antennas {1,2,3,4} are selected because they have larger gene values; therefore the new channel matrices  k }2k=1 are formed by the {1,2,3,4} columns of {H k }2k=1 . {H In addition, the precoding vector indices of the parent chromosome a are {0,1,1,1} and {0,0,1,1} for users 1 and 2, respectively. For the parent chromosome b, the selected transmit antennas are {1,3,4,5} and the corresponding precoding vector indices are {1,0,1,0} and {0,1,0,1} for users 1 and 2, respectively. Step 1: Evaluation: The fitness value, which measures a chromosome’s goodness of fit, is used to determine which chromosomes should survive. In each generation, the fitness values are computed for each of the P chromosomes in the current population by using the corresponding precoding  k }Kk=1 in the system vectors {fk }Kk=1 and channel matrices {H capacity in (3). The HGA algorithm takes the result as the chromosome’s goodness of fit. Step 2: Selection: To preserve the better chromosomes (solutions) in the population and obtain better offspring, we employ the truncated selection scheme [14], which only retains R parent chromosomes that have higher fitness values. We then reproduce them in a mating pool and randomly select two parent chromosomes for the following crossover step. Step 3: Hybrid crossover: The crossover between the two selected chromosomes is actually divided into two operations: conventional crossover [14] for precoding vector selection and priority crossover for antenna selection. The latter involves two steps: exchange-empty and sortingrefilling. The exchange-empty step begins by constructing an (M + KL) × 1 crossover mask sequence consisting of 1s and 0s generated with equal probability [14]. When the elements in the crossover mask are 1s, the genes of the two parent chromosomes in the corresponding positions will be exchanged. However, if they are 0s, the genes in the antenna selection part will be emptied, but those in the precoding vector selection part will remain unchanged as in the conventional crossover operation. The crossover operation in the precoding vector selection part, which is the same as that IET Commun., 2011, Vol. 5, Iss. 9, pp. 1220–1229 doi: 10.1049/iet-com.2010.0234

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Fig. 2 Hybrid crossover operation, where M ¼ 5, K ¼ 2 and L ¼ 4

in the conventional algorithm, is called a uniform crossover operation. For example, the parent chromosomes a and b, along with the mask [1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1], will produce the offspring chromosomes a ′ and b ′ as shown in Fig. 2. Note that the crossover operation in the precoding vector selection part ceases at this point. Only the antenna selection part needs to proceed to the subsequent sortingrefilling step. In the sorting-refilling step, we refill the missing gene values sequentially in descending order in the antenna selection part of the chromosomes, beginning with the genes that had larger integer values before the exchange-empty step. In addition, the refilled genes are those integers that do not appear after the exchange-empty step. The refilling process ensures that genes with larger values (higher priority) will still have larger values after the crossover operation. This feature simply reflects the fitness-survival principle of GA [14]. For example, the values of the second and the fourth genes are missing in the antenna selection part of the offspring chromosome a ′ as shown in Fig. 2. The original values of those genes in the parent chromosome a were 3 and 2, respectively. Thus, we refill 4 and 1, respectively, at these two positions in the offspring chromosome a, as they do not appear after the exchange-empty step. Similarly, we refill the second and fourth genes in the antenna selection part of the offspring chromosome b with 2 and 3, respectively, since they do not appear after the exchangeempty step. It is noteworthy that, based on this new crossover, the genes in the antenna selection part of a chromosomes always contain a permutation of {1, . . . , M}; hence, antennas can be selected easily. This hybrid crossover operation is repeated until the size of the new population is the same as P, that is, the size of the original population. Step 4: Hybrid mutation: To increase the search space and prevent the solution from being trapped in a local maximum/minimum value, the GA includes a mutation operation [14]. Like the hybrid crossover operation, the hybrid mutation operation is divided into two operations: conventional mutation for precoding vector selection and priority mutation for antenna selection. First, for each chromosome, we create a (1 + KL) × 1 mutation mask sequence comprising 1s and 0s generated according to the IET Commun., 2011, Vol. 5, Iss. 9, pp. 1220–1229 doi: 10.1049/iet-com.2010.0234

mutation probability pm [14]. If the first element of the mask is 1, two randomly selected gene values in the antenna selection part of the chromosome will be exchanged. This ensures that the genes in the antenna selection part are still a permutation of {1, . . . , M}. However, if the first element is 0, the corresponding genes remain the same. The KL genes in the precoding vector selection part of the chromosome will toggle if the corresponding elements in the mutation mask are 1s and remain the same if they are 0s. For example, assume that the mutation mask sequence for the offspring chromosome a ′ in Fig. 2 is [1, 0, 0, 1, 0, 0, 1, 0, 0]. Since the first element of the mutation mask sequence is 1, two random genes in the antenna selection part of the offspring chromosome a ′ , say the third and fourth genes, will be exchanged, resulting in {5,4,1,2,3}, and the corresponding precoding vector selection part of the offspring chromosome a ′ will be {0,0,0,1,0,1,1,1}. Step 5: Repeat/end: The algorithm repeats Steps 1 to 4 until the number of the generations meets the prescribed number G, after which the chromosome with the maximum fitness value is chosen. The antennas corresponding to the largest m (integer) gene values in the antenna selection part of this chromosome are selected as the desired transmit antennas, and the (binary) gene values in the precoding vector selection part denote the indices of the precoding vectors for the respective users. The BS uses the feedback information to preprocess the data symbols before transmission. Note that the choice of the population size P and the generation G is application-dependent, but they must be large enough to yield a sufficient number of chromosomes and iterations to complete the evolutionary process. 3.2

Computational complexity

In this section, we compare the computational complexity of the proposed algorithm with that of existing approaches in terms of the number of complex multiplications/additions (CMADs) required [18]. The transmit antenna selection scheme [3], which selects m out of M transmit antennas, must compute the SINRs in (2) K times. Since each SINR 1223

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www.ietdl.org Table 1

Computational complexity of the compared methods

Algorithm

Complex multiplications/additions (CMADs)

transmit antenna selection only [3]

2 CM m × [K (m + 1)N + KN]

precoding only [10]

PK2 × [K 2 (m + 1)N + KN]

separate scheme [11]

L

L

M Cm × [K (4N + 8m)m2 ] + PK2 × [K 2 (m + 1)N + KN]

joint scheme (exhaustive search) joint scheme (GA approach)

L

M Cm × PK2 × [K 2 (m + 1)N + KN]

P × G × [K 2(m + 1)N + KN]

M denotes the number of transmit antennas, m denotes the number of antennas selected, K donates the number of users, L donates the number of feedback bits, and P and G denote, respectively, the population size and the number of generations in HGA

Fig. 3 Convergence behaviour of HGA, where L ¼ 4 and SNR ¼ 6 dB

Fig. 4 Convergence behaviour of HGA, where L ¼ 6 and SNR ¼ 6 dB 1224 & The Institution of Engineering and Technology 2011

IET Commun., 2011, Vol. 5, Iss. 9, pp. 1220–1229 doi: 10.1049/iet-com.2010.0234

www.ietdl.org needs approximately K(m + 1)N + N CMADs, where K is the number of users (each equipped with N receive antennas), a 2 total of CM m × [K (m + 1)N + KN] CMADs are required. The precoding selection scheme [10], which selects K out of L the 2L precoding vectors in the codebook, requires PK2 × [K 2 (m + 1)N + KN ] CMADs to calculate the SINRs for all K users and all possible permutations. The suboptimal separate approach [11] first selects the best transmit antenna subset to maximise the minimum singular value of the N × m channel matrices for all users and the SVD involved calls for K × (4Nm 2 + 8m 3) CMADs. The subsequent determination of the precoding vectors needs K × [K(m + 1)N + N ] CMADs to compute the SINRs, L so a total of CmM × [K(4N + 8m)m2 ] +PK2 × [K 2 (m + 1)N +

KN ] CMADs are required. If we select the precoding vectors and the transmit antenna subset simultaneously, the exhaustive search needs to compute the L SINRs CmM × PK2 times to find an optimal solution; thus, L CmM × PK2 × [K 2 (m + 1)N + KN ] CMADs are required. In contrast, the proposed HGA approach only needs to calculate the SINRs P × G times, or equivalently, P × G × [K 2(m + 1)N + KN] CMADs. Since the values of the combinations and permutations grow rapidly as M, L or K increase, the overall complexity of the proposed HGA approach is substantially lower than that of the joint scheme. The analytic expressions for the computational complexity of the above approaches are summarised in Table 1.

Fig. 5 Performance comparison against the SNR for the 4-bit codebook a Comparison of the system capacity against the SNR b Comparison of the BER against the SNR, where L ¼ 4 IET Commun., 2011, Vol. 5, Iss. 9, pp. 1220–1229 doi: 10.1049/iet-com.2010.0234

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Simulation results and discussion

In this section, we discuss the simulations conducted to assess the performance of the proposed GA approach. We consider the downlink of a multi-user MIMO system with the data model described in (1), where the transmit antennas of the BS send i.i.d. BPSK modulated symbols to every user simultaneously with equal energy via the i.i.d. Rayleigh fading channels. We use two Grassmannian codebooks given in [19], corresponding to the number of feedback bits L ¼ 4 and L ¼ 6, where the number of precoding vectors is 16 and 64, respectively. The minimum mean square error receiver is employed for every user. The number of users, K, the number of the transmit antennas at the BS, M, and

the number of the selected transmit antennas, m, are 2, 16 and 4, respectively. The number of receive antennas for each user, N, is 2 in the first part of the simulations, but it varies in the second part. We compare the following five approaches in terms of the average system capacity, the bit error rate (BER) and the computational load: the transmit antenna selection only approach [3], the precoding only approach [10], the separate approach (transmit antennas are selected first followed by precoding vectors) [11], the joint selection of the transmit antenna subset and the precoding vectors based on the exhaustive search method and the proposed HGA. Under HGA, the crossover probability pc ¼ 0.5, the mutation probability pm ¼ 0.1 and the numbers of chromosomes selected for the mating pool, R,

Fig. 6 Performance comparison against the SNR for the 6-bit codebook a Comparison of the system capacity against the SNR b Comparison of the BER against the SNR, where L ¼ 6 1226 & The Institution of Engineering and Technology 2011

IET Commun., 2011, Vol. 5, Iss. 9, pp. 1220–1229 doi: 10.1049/iet-com.2010.0234

www.ietdl.org are 40 and 18 for the 4-bit codebook and 6-bit codebook, respectively. First, we evaluate the convergence behaviour of the proposed HGA approach. The average system capacity against the number of generations for L ¼ 4 and L ¼ 6 when the signal-to-noise ratio (SNR) ¼ 6 dB are shown in Figs. 3 and 4, respectively. As expected, the capacity improves as the number of generations, G, or the size of the population, P, increases in both scenarios. The capacity remains the same when the number of generations exceeds 10 or when the population size exceeds 350. Therefore, in the following simulations, we use G ¼ 10 and P ¼ 350 for the 4-bit and 6-bit Grassmannian codebooks in HGA.

Next, we compare the average capacity and the BER against the SNR of the five approaches. The results of the average capacity and the BER against the SNR are shown in Figs. 5a and b, respectively, for the 4-bit codebook, and in Figs. 6a and b, respectively, for the 6-bit codebook. Except for the transmit antenna selection only scheme, the performance of the approaches improves when we use a larger codebook or when the SNR increases. The transmit antenna selection only scheme yields an error floor due to the impairment caused by interference in the multi-user MIMO downlink. Both of the capacity and the BER performance improve when both of the transmit antenna selection and precoding vector selection are taken into account as a whole, and the joint schemes based on the

Fig. 7 Performance comparison against the number of receive antennas for the 4-bit codebook a Comparison of the system capacity against the number of receive antennas b Comparison of the BER against the number of receive antennas, where L ¼ 4 and SNR ¼ 3 dB IET Commun., 2011, Vol. 5, Iss. 9, pp. 1220–1229 doi: 10.1049/iet-com.2010.0234

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www.ietdl.org exhaustive search or the proposed HGA outperform the suboptimal separate approach. Moreover, the performance of the proposed HGA is close to that of the exhaustive search method because of the effectiveness of the GA. Next, we compare the system capacity and the BER against the number of receive antennas, N, when SNR ¼ 6 and 3 dB, respectively. The results of the average capacity and the BER against the SNR are shown in Figs. 7a and b, respectively, for the 4-bit codebook, and in Figs. 8a and b, respectively, for the 6-bit codebook. Except for the transmit antenna selection only scheme, which is overwhelmed by interference, both of the capacity and the BER performance of all approaches improve as N increases, since the receive diversity increases. Once again, the joint schemes outperform the

precoding only and the separate schemes, and the performance of the proposed approach is close to that of the optimal exhaustive search method. In terms of the computational load, based on the analytic expressions of the complexity discussed in Section 3.2, the transmit antenna selection only approach, the precoding only approach, the separate approach, the joint scheme based on the exhaustive search and the proposed HGA approach require, respectively, 80080, 10560, 2340160, 19219200 and 154000 CMADs for the 4-bit codebook, and 80080, 177408, 2507008, 322882560 and 154000 CMADs for the 6-bit codebook. If we use more transmit antennas or feedback bits to enhance the performance, the computational savings are even more significant.

Fig. 8 Performance comparison against the number of receive antennas for the 6-bit codebook a Comparison of the system capacity against the number of receive antennas b Comparison of the BER against the number of receive antennas, where L ¼ 6 and SNR ¼ 3 dB 1228 & The Institution of Engineering and Technology 2011

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Conclusion

Aiming at mitigating the effect of multi-user interference and reducing the number of RF chains required in the downlink of multi-user MIMO systems with limited feedback, we have proposed a simple and efficient GA-based approach, called HGA, for simultaneously selecting the quantised precoding vectors and the transmit antenna subset. The computational complexity of HGA is significantly lower than that of the exhaustive search approach. The results of simulations demonstrate the efficacy of the proposed approach in finding an optimal solution.

6

Acknowledgments

We would like to thank the reviewers for many useful comments and suggestions which have enhanced the quality and readability of this paper. This work was supported by National Science Council of ROC under contract number NSC 98-2221-E-011-086.

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References

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