Joint antenna subset selection for spatial multiplexing systems based on statistical and instantaneous selection criteria

Int. J. Electron. Commun. (AEÜ) 66 (2012) 715–720

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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.de/aeue

Joint antenna subset selection for spatial multiplexing systems based on statistical and instantaneous selection criteria Donghun Lee a,∗ , Saeid Nooshabadi b , Kiseon Kim c a b c

Electronic and Telecommunication Research Institute (ETRI), Yuseong-gu, 305-700, South Korea Department of Electrical and Computer Engineering, Michigan Technological University, Houghton, MI 49931, USA School of Information and Communication, Department of Nanobio Materials and Electronics, Gwangju Institute of Science and Technology (GIST), Gwangju 500-712, South Korea

a r t i c l e

i n f o

Article history: Received 26 October 2011 Accepted 9 December 2011 Keywords: Multiple-input multiple-output (MIMO) Spatial multiplexing (SM) Antenna subset selection (AnSS)

a b s t r a c t While the joint transmit/receive (Tx/Rx) AnSS with exhaustive search is the best solution for error rate minimization, its complexity makes it difficult for implementation on practical systems. To overcome this disadvantage, we propose a two-stage AnSS algorithm for the spatial multiplexing (SM) in the multiple input multiple output (MIMO) system, which employs both the statistical (i.e., average Euclidean distance, AED) and instantaneous selection criteria (i.e., modified instantaneous Euclidean distance, M-IED). The proposed algorithm reduces the computational complexity by decoupling the joint Tx/Rx selection into two separate selections of the numbers of Tx/Rx antennas and antenna subset, respectively. We show that the proposed AED criterion can be implemented through a simple look up table (LUT), thereby significantly reducing the computational complexity. Simulation results and computational complexity comparisons, prove that the proposed two-stage AnSS algorithm for the SM scheme reduces the hardware and computational complexity without any loss of the signal-to-noise ratio (SNR) and the diversity order, compared to the exhaustive search method. © 2011 Elsevier GmbH. All rights reserved.

1. Introduction The antenna subset selection (AnSS) technique in the multipleinput multiple-output (MIMO) system [1,2] is a powerful method to improve the reliability performance, while reducing the hardware cost by employing lesser number of radio frequency (RF) chains. Therefore, AnSS technique has been applied to the spatial multiplexing (SM) [3] and space-time block coding (STBC) MIMO systems [4,5] to reduce the cost. The AnSS in the MIMO system is used to serve two purposes: capacity maximization [6–10], and error rate minimization [11–16]. There are several published works on capacity maximization through transmit/receive Tx/Rx-AnSS algorithms by employing instantaneous criterion [6–9] and statistical criterion [10]. The work in [6] proposed an AnSS algorithm at the Rx side to maximize the capacity through the convex optimization, and showed almost an identical performance compared with the exhaustive search at both Tx and Rx sides of the channel. A low-complexity Rx-AnSS algorithm, which starts with the empty set of selected antennas and then adds on one antenna per step, was proposed

∗ Corresponding author. E-mail addresses: [email protected] (D. Lee), [email protected] (S. Nooshabadi), [email protected] (K. Kim). 1434-8411/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2011.12.008

in [7]. The work in [8] studied joint (Tx/Rx) AnSS for the capacity maximization. This approach reduces the computational complexity by decoupling the joint Tx/Rx selection into the two separate selections at the Tx and Rx sides. The work in [9] proposed an algorithm based on the mutual information criterion for the antenna selection. An analytical expression of the lower-bound capacity using the smallest eigenvalue distribution was derived in [10]. This work proposed a Tx-AnSS criterion using the average lower-bound for capacity enhancement with low complexity. AnSS techniques for error rate minimization were investigated in [11–16] through the instantaneous criterion with the exhaustive search method. The work in [11] studied Tx-AnSS for the Alamouti’s STBC scheme based on Euclidean norms and showed that it results in the highest signal-to-noise ratio (SNR) at the Rx side. The Tx-AnSS criterion using the largest minimum Euclidean distance was proposed for the SM scheme in [12]. Using this criterion, works in [13,14] investigated Tx-AnSS algorithms for the SM scheme with linear receivers and showed that the proposed algorithm provides the full spatial diversity as well as the array gain. A joint (Tx/Rx) AnSS scheme [15] was shown to provide the full spatial diversity without the array gain, which utilizes the antenna selection diversity at both Tx/Rx sides. The Tx-AnSS algorithm by employing threshold was studied in [16] for the SM scheme with the linear receivers. While reducing the computational complexity, this approach has the disadvantage of not

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• • •

≤

≤

• • •

Fig. 1. Block diagram of the joint-AnSS SM scheme in the MIMO system.

providing both the full spatial diversity and array gain. Previous works on the AnSS based on error rate minimization have used the instantaneous selection criterion with the exhaustive search, at either the Tx or Rx sides or both, that employs upper bounds on the BER or effective SNR. The instantaneous selection criterion, however, suffers form the heavy computational complexity that grows exponentially with the number of antennas. To overcome the complexity issue, in this paper, we present a novel two-stage low-complexity joint Tx/Rx algorithm for the error-rate minimization in the SM scheme. The proposed two-stage algorithm employs both the average Euclidean distance (AED), and modified instantaneous Euclidean distance (M-IED) criteria. The algorithm first selects the numbers of Tx and Rx antennas that maximizes the statistical criterion of AED. The process of the antennas selections at the Tx and Rx sides are decoupled from each other in this first stage. Next, the best antenna subset for the number of antennas selected by the AED criterion, is adaptively chosen from the total number of available antennas through the M-IED criterion. We show that the proposed AED criterion can be implemented through a simple look up table (LUT), thereby significantly reducing the computational complexity. Further, the performance of the two-stage AnSS scheme is almost identical to the exhaustive search technique. This paper is organized as follows. Section 2 provides an overview of the SM scheme with joint AnSS. Section 3 highlights the statistical and instantaneous criteria and presents the proposed two-stage AnSS algorithm. Section 5 discusses bit-error rate (BER) performance and computational complexity of the proposed algorithm. Finally, Sections 6 concludes the paper.

[.]i,j (i, j)th element of [.] R No. of transmitted bits per transmission period B Modulated bits per symbol period As shown in Fig. 1, in the encoding and spatial multiplexing blocks, R bits are mapped onto AT independent streams. A resulting ]=IAT,k , can be expressed as signal vector sk with E[sk sH k sk = [sk1 , sk2 , . . . , ski , . . . , skAT,k ]T

where ski is the transmitted signal at the symbol time k by the ith Tx antenna, and where AT,k (= 1, 2, . . ., ≤ MT ) is the number of the selected Tx antennas at the symbol time k. IAT,k is an AT × AT identity matrix. In the RF-chain/RF-switch block, the signal vector sk is transmitted through the selected antenna subset, pT,k ∈ AT , where



AT =

MT AT = 1

  ,

MT AT = 2



,...,

MT AT = N ≤ MT



, is the set

 

n = Crn = n!/r!(n − r)! r denotes the binomial coefficient. The Tx antenna subset pT,k is adaptively selected through feedback information. This study assumes a flat Rayleigh fading channel and a very slowly varying channel. We also assume the availability of channel state information (CSI) at the Rx side, and no error in the feedback channel are assumed. At the Rx side, the received signal in the RF-chain/RF-switch block is passed by the antenna subset, pR,k ∈ AR , subject to AT ≤ AR , at the symbol time k, which is selected by the AnSS algorithm in the AnSS block. Then, the Rx signal vector at the symbol time k can be written as j

The SM scheme with joint AnSS is illustrated in Fig. 1. The notations used in this paper are defined as follows, where all vectors and matrices are in boldface. MT No. of Tx antennas MR No. of Rx antennas AT No. of Tx RF-chains AR No. of Rx RF-chains p Selected antenna subset [.]T Transpose operation E [.] Expectation operation [.]k kth symbol period of [.]



of all possible Tx antenna subsets and

rk = [rk1 , rk2 , . . . , rk , . . . , rkAR,k ]T 2. Overview of joint-AnSS SM scheme

(1)

(2)

j

where rk is the Rx signal at the symbol time k, from the jth Rx antenna. AR,k (= 1, 2, . . ., ≤ MR ) is the number of the selected antennas at the symbol time k. The Rx signal vector is also expressed as follows:



rk =

Es H s + nk AT,k k k

(3)

where Es is the total transmitted energy, the Hk is an (AR,k × AT,k ) channel matrix with independent and identically distributed (i.i.d.) circular complex Gaussian elements with CN(0, 1), at the symbol j,i period k. The element hk ∈ Hk is the fading attenuation coefficient from the ith Tx antenna to the jth Rx antenna at the symbol period

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k. The additive white gaussian noise (AWGN) vector nk with CN(0, N0 ) is given by j

nk = [n1k , n2k , . . . , nk , . . . , nAk R,k ]T

(4)

j

where nk is the noise component at the jth Rx antenna at the symbol time k. IAR,k is an identity matrix of dimensions (AR × AR ). After passed by the antenna subset selection block, the Rx signal vector is decoded in the decoder block. In this paper, we consider maximum likelihood (ML) decoding and linear decoding, especially, zero-forcing (ZF). Also, note that for the reminder of this work we drop the symbol period subscript k from the formulation. 3. Selection criteria of the joint-AnSS SM scheme In this section, we present two selection criteria, instantaneous Euclidean distance (IED) and average Euclidean distance (AED), for the joint-AnSS for the SM scheme. At first, by providing the motivation for the work, we briefly present the IED selection criterion, which has already been employed in [12–14], and describe its complexity problem. Next, we describe an idea to simplify the complexity by using average concept of IED (i.e., average Euclidean distance, AED), and propose a modified instantaneous Euclidean distance (M-IED) selection criterion.

3.2. Instantaneous Euclidean distance (IED) Using the lower-bound of (6), works in [12–14] have defined the instantaneous Euclidean Distance (IED) as an instantaneous selection criterion:

  

Pjoint ≤ C

min

AT ,AR ,pT ,pR

Q



Es 2 d (AT , AR , pT , pR ) N0 min,RX



MAX

pT ,pR

(AT , R)] ,

where is the squared minimum Euclidean distance of the Rx signal defined in ([12], Eq. (3)), and C is a constant that depends on the detection technique. Based on the upper-bound error probability in (5). Works in [12–14] proposed the instantaneous antenna selection criterion employing 2 dmin,RX (AT , AR , pT , pR ) for SM scheme, which is further simplified through the Rayleigh-Ritz theorem [12,14] as,



IED := max max AT ,AR



2 2min (AT , AR , pT , pR ) dmin,TX

AT ≤ AR .

(7)

(5)

2 dmin,RX (AT , AR , pT , pR )

2 2 (dmin,RX (AT , AR , pT , pR ))MAX ≥ 2min (AT , AR , pT , pR )dmin,TX (AT , R)

λ

Fig. 2. CDF of max 2min for a given AT in the 4 × 4 MIMO system.

3.1. Motivation The error probability for the joint-AnSS SM scheme with ML and ZF detection techniques given in [12–14,17] is bounded as



(6)

2 (AT , R) is the squared minimum Euclidean distance of where dmin,TX

the Tx signal defined in [12], 2min (AT , AR , pT , pR ) is the squared minimum singular value of the channel matrix. Although using (6) as a selection criterion substantially reduces the computational complexity compared to (5), the cost of computation associated with 2min for each channel realization, is still relatively high. This is because in the exhaustive search the best antenna subsets pT and pR are chosen from the set of all possible Tx and Rx antenna subsets for 1 ≤ AT ≤ MT , 1 ≤ AR ≤ MR in the joint-AnSS scheme. Most existing works on AnSS [12–15] are based on the instantaneous selection criterion with the exhaustive search such as those in (5) or (6) at both Tx/Rx sides [15], or only at the Tx side [12–14] for the SM scheme. Even though providing the best AnSS strategy for error rate minimization, the existing works even when search is limited to the Tx side, entail high computational complexity. Limiting the search of antenna subsets to the specific numbers of Tx/Rx antennas, can significantly reduce the complexity of the joint-AnSS SM scheme. Thus, we propose a selection criterion which uses the statistical property of the channel matrix for limiting the numbers of Tx/Rx antennas to their best values ATs and ARs . In this study, we consider the joint-AnSS SM scheme with both ML and ZF detection techniques in the MIMO system.

the cumulative distribution functions (CDF) of Fig.

2 presents  max 2min for three antenna configurations of AR for given AT =1 and 2 for a 4 × 4 MIMO system. Looking at the cross section of the CDF plots along the line for an arbitrary CDF value to the

(parallel  x-axis) we observe that the corresponding max 2min increases 2 as AR increases for a given AT . Consequently, since dmin,TX (AT , R) is independent from AT , the IED selection criterion increases as AR increases for given values of R and AT . Further, comparing CDF curves for a given AR as shown in Fig. 2, we can see that the corresponding max 2min value decreases as AT increases for a given value of AR , approaching zero as the number of 2 Tx antennas increases. On other hands, for a given value of R, dmin,TX increases as the number of active Tx antennas AT increases. From these observations, we conclude that

for  given2values of R and AR , there is a trade-off between max 2min and dmin,TX as a function of AT .

3.3. Modified IED (M-IED) employing average Euclidean distance (AED) The problem of the IED criterion is its huge search set in selection of best antenna subset (pTs , pRs ) over all possible values of AT and AR . To decouple decision metrics, (i) AT , AR and (ii) pT , pR , to reduce the search space of antenna selection to their best values ATs and ARs , we consider an average concept of IED. The reasoning for the use of the average IED metric is two-fold: (1) simplicity, because this will allow us to change the joint Tx/Rx selection into decoupled selection of the numbers of Tx/Rx antennas (AT and AR ) and antenna subset (pT and pR ), respectively and (2) effectiveness, as it still uses proper selection of antenna subset within the limited search space. To implement the average concept of IED, we take one assumption that squared minimum singular values are inde2 pendent. Then, AED = [EpT ,pR 2min (AT , AR , pT , pR ) dmin,TX (AT , R)].

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There are still other alternatives to reduce search space of antenna selection, which is not within the scope of this paper. Rather than optimizing the new rule, we will check performance degradation of the newly proposed rule, to find effect of the independent assumption. 4. Proposed joint-AnSS algorithm The discussion in the preceding section sets the stage for the proposed AED and M-IED selection criteria. Since the selections of AR , (ATs and A the best values of AT and  Rs ), are decoupled from each other, and the fact that E 2min (AT , AR ) increases as the number of AR increases for a given AT , we can set ARs = MR to reduce the computational complexity to the selection of the number of antennas at the Tx side. Thus, the AED criterion for selecting the number of Tx RF-chain, can be written as

 Fig. 3. CDF of IED for various joint-AnSS SM options in the 4 × 4 MIMO system.

ATs = arg max

 D 

AT

Consequently, we can decouple the joint selection into two separate selections, that is, (i) the numbers of Tx/Rx antenna selection by AED as [ATs , ARs ] = arg[max AED] and (ii) the antenna subset selecAT ,AR

tion by the M-IED as [pTs , pRs ] = arg[max[2min (ATs , ARs , pT , pR ) pT ,pR

2 dmin,TX (ATs , R)]]. Fig. 3 shows the CDF of the IED for several AR antenna configurations for given AT =1 and 2, where AT ≤ AR . From the figure we observe that for a fixed CDF value IED increases as AR increases for a given AT , subject to the condition AT ≤ AR . It means that the proposed statistical criterion, AED, is maximized when AR = MR . Therefore, we may simplify the rule as,





arg max [AED(AT , AR = MR )] , AT

AT < MR .

(8)

Note that in (8) we use the condition AT < MR . That is because AT = MR corresponds to the non-AnSS scheme, which exhibits a degradation in the diversity order compared to the AnSS-based scheme [14]. Then, we may use the PDF of 2min (AT = 1, 2, . . . , ≤ MR , AR = MR ), fAT , given as [10,18], fAT (x) = e−AT x

D 

cn,AT xn ,

x := 2min

(9)

n=0





the independent assumption, the term E 2min (AT , AR = MR ) can be derived as follows:



E 2min (AT , AR = MR )



∞

=

xf AT (x)dx



0 D

=



=

 n=0

∞

cn,AT

xn+1 e−AT x

0

n=0 D

cn,AT

(n + 1)! An+2 T

(10)

.

AED =

D  n=0

cn,AT

2 dmin,TX (AT , R)

forAT < MR . (12)

Since the AED can be pre-computed, then the first stage of selecting number of Tx antennas can be implemented by a LUT. The AED simplification through a LUT results in the complexity-reduced MIED criterion to find the best antenna subset pTs as,



pTs = arg max pT



2min (ATs , ARs

=



2 MR , pT )dmin,TX (ATs , R)



.

(13)

Note that in (13) we have removed the optimization parameter pR , as with optimum choice of ARs = MR the Rx side only contains one subset. Let ATs be the all possible transmitted antenna sets corresponding to ATs by (12) and ARs = MR . Based on above two selection criteria, the flow chart of the proposed two-stage AnSS algorithm is presented in Fig. 4, and the proposed algorithm is summarized as follows: Stage 1: Find the best number of Tx antennas, ATs , maximizing AED Initialize R, MT , MR Find ATs , ARs = MR from (12) Stage 2: Find the antenna subset, pTs , maximizing M-IED with ATs Initialize ATs corresponding to ATs by (12) and ARs = MR Find pTs for pT ∈ ATs from (13)

(n + 1)! An+2 T



2 dmin,TX (AT , R)

forAT < MR .

This section presents the BER performance of the proposed two-stage AnSS algorithm through the Monte-Carlo simulation. It also presents the complexity of the proposed scheme in terms of required maximum number of computations and RF chains. The industrial standards such as IEEE 802.16e and 3GPP Rev7 consider configurations with up to 4 antennas, and especially have adopted the 4 × 4 multiple antenna system. For this reason, our study mainly focuses on MT = MR = 4 antenna configuration for the practical MIMO system. 5.1. Simulation parameters

Based on this reasoning, we may devise a AED decision metric as function of AT and R, as:



n=0

An+2 T



5. Performance and complexity discussion

where D = AT (MR − AT ) is the degree of the polynomial part of fAT (x) and cn,AT is a positive constant coefficient of xn for a given AT . Under



cn,AT

(n + 1)!

(11)

In the SM scheme, the constellation size for each Tx antenna is T 2R/A . For instance, the SM scheme with the data rate R = 4 and AT = 1 employs a 16-QAM modulation type. For the case of AT = 2, the SM scheme employs a QPSK modulation type. We assume that energy is normalized to unity for various modulation types. We evaluate performance figures under the same data-rate condition for the fair

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Fig. 5. BER performance of the proposed scheme with ML in the 4 × 4 MIMO system with R=4, ATs =2, ARs =4.

SNR gain of 6 dB, compared to the scheme in [15]. This because the scheme in [15] does not use the optimum value for AT with R = 4. 5.3. BER performance of the proposed scheme with ZF for R = 4

Fig. 4. Flow chart of the proposed algorithm.

BER comparison. BER performance is measured by averaging over 2 million transmission periods.

Fig. 6 shows BER performance of the proposed scheme with the ZF detection in the MIMO system with MT = MR = 4. We consider the same simulation condition as in the ML case. The scheme in [14] which selects Tx antennas by employing IED selection criterion, varies modulation types, 16-QAM for AT = 1, QPSK for AT = 2 and BPSK for AT = 4. Next, the antenna subset selection scheme in [14] is is employed to select subset pT from the set of all possible Tx antenna subsets for 1 ≤ AT ≤ MT . Plots in Fig. 6 show that all of the schemes, except the SM without AnSS, show full spatial diversity, and identical BER performance. At the BER value of 10−5 the proposed scheme provides a SNR difference within 0.5 dB with respect to the exhaustive search with the ML detection.

5.2. BER performance of the proposed algorithm with the ML detection for R = 4 Fig. 5 shows BER performance of the proposed scheme with the ML detection in the MIMO system with MT = MR = 4. For the exhaustive search, the antenna subset of pT is chosen from the set of all possible Tx/Rx antenna subsets for 1 ≤ AT ≤ MT , 1 ≤ AR ≤ MR , subject to AT ≤ AR . The exhaustive search varies modulation types, 16-QAM for AT = 1, QPSK for AT = 2 and BPSK for AT = 4. The scheme in [15] uses the selection diversity for Tx/ Rx sides, which is identical to AT = AR = 1 employing 16-QAM. In the proposed scheme, we set ATs = 2 according to AED selection criterion, in the stage one of the algorithm. Therefore, the modulation type for the proposed scheme is QPSK. In stage two the subset pTs is chosen from the set of all possible antenna subsets for ATs = 2. Plots in Fig. 5 show that all the schemes except for the SM without AnSS exhibit full spatial diversity order as observed in the high SNR regions. This is because Tx-AnSS with the ML detection provides for the full spatial diversity of MT MR compared to diversity order of MR for the SM scheme without the AnSS [19]. Even though having the full spatial diversity. The scheme in [15] exhibits a BER performance that is worse than the proposed scheme. At the BER value of 10−5 the proposed scheme provides a

Fig. 6. BER performance of the proposed scheme with ZF in the 4 × 4 MIMO system with R = 4, ATs =2, ARs =4.

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5.4. Complexity

Acknowledgements

For complexity analysis, we consider a Tx data rate of R = 4, and a system with MT = MR = 4. From the viewpoint of computational complexity, we first determine the maximum number of required comparisons for the proposed scheme and exhaustive search in the joint AnSS-SM scheme. For the proposed scheme   the maximum MT = 4 number of required computations is =6. On other hand, ATs = 2 the respective values search the scheme in   for the exhaustive   and  MT = 4 MR = 4 MT = 4 [14] are =127 and =15, respecAT AR AT tively. Based on this result, we know that the proposed scheme reduces the computational complexity with respect to the exhaustive search and the scheme in [14] by about 95% and 60%, respectively. This is because the proposed scheme searches is limited to the antenna subsets for ATs number of Tx antenna. Further, the number of ATs and ARs are predetermined by a LUT and fixed to MR , respectively. Furthermore, the number of Tx RF-chains AT , in the proposed scheme is less than the number of physical antennas MT , a significant reduction in the hardware cost, and power dissipation at the Tx side. Although we have shown the simulation results for a 4 × 4 MIMO system the arguments presented here for AnSS are also valid for other MIMO configurations. This is because the AED based statistical selection using LUT always chooses a value of AT < MT . Consequently, we know that the proposed scheme can reduce the complexity in terms of the hardware cost and the number of comparisons, compared to exhaustive search regardless of the configuration.

This research was supported by the World-Class University Program funded by the Ministry of Education, Science, and Technology through the National Research Foundation of Korea (R31-10026), and by Leading Foreign Research Institute Recruitment Program through the National Research Foundation of Korea(NRF) (K20902001632-10E0100-06010).

6. Conclusion This paper presented a low-complexity joint AnSS algorithm for the SM scheme. We proposed a two-stage statistical AED and instantaneous M-IED selection criteria AnSS algorithm for the SM scheme. The proposed algorithm first selects the numbers of Tx and Rx antennas that maximizes the AED. After the numbers of Tx/Rx antennas are determined by the AED, the antenna subset that maximize the M-IED is adaptively chosen. From the simulation and the analysis of computation, we verify that the proposed two-stage joint AnSS algorithm reduces complexity in terms of the hardware cost and the number of comparisons without the performance degradation, compared to exhaustive search techniques regardless of antenna configurations.

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