Efficient Transmit Antenna Selection for Correlated MIMO Channels

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

Efficient Transmit Antenna Selection for Correlated MIMO Channels Hyungsoo Kim, Hyounkuk Kim, Namshik Kim and Hyuncheol Park School of Engineering Information and Communications University (ICU) 119 Munjiro, Yuseong-gu, Daejeon, 305-714, KOREA E-mail :{googii, hyounkuk, nskim, hpark}@icu.ac.kr

Seok Seo and Jinkyu Choi Smart Radio Research Team Electronics and Telecommunications Research Institute (ETRI) 138 Gajeongno, Yuseong-gu, Daejeon, 305-700, KOREA E-mail : {westone, jkchoi}@etri.re.kr

Abstract—While multiple-input multiple-output (MIMO) system gives many advantages to wireless communication, the complexity of multiple RF chains gives a burden to the system. Antenna selection is introduced as a technique to reduce the burden. On the transmit antenna selection scheme, the exhaustive search for optimal antenna set quickly becomes impractical as adopting more antennas. To reduce complexity for searching, we propose an efficient transmit antenna selection method. We first make a threshold for optimal antenna set by using Poincare separation theorem. Instead of exhaustive search, we select a transmit antenna subset that exceeds the threshold. On the exponentially correlated channel of array antennas, we give an order to search for reducing complexity further. We search from the most remotely located antenna subset which in generally less correlated to the closely located antennas which is more correlated. Finally, we show the BER performance and tradeoff relation between performance and complexity by Monte-Carlo simulations.

I. I NTRODUCTION MIMO wireless system employing multiple antennas at transmitter and receiver achieves higher capacity and diversity order than single-input single-output (SISO) systems. In the aspect of capacity, spatial multiplexing is a technique that obtains high spectral efficiencies by dividing the data into multiple substreams, and each substream is transmitted on each corresponded antenna. On the other hand, diversity methods improve the performance in terms of BER (bit error probability) by exploiting the multiple paths between transmit and receive antennas. Along with the advantages, many researchers explored them by analytical and simulation studies [1] and [2]. An important drawback in the implementing of systems with multiple antennas is complexity. By using an antenna selection technique, it is possible to reduce complexity, and we take many of the advantages of MIMO systems at the same time. For this reason, recently [3]-[5] are suggested antenna selection schemes that optimally select an antenna subset. However, complexity of the search for optimal antenna set grows rapidly as the number of antennas increases. So, various simplified selection methods have been introduced. In [8], some channel element is disregarded based on the mutual information. Similarly in [9], identical rows and highly correlated rows are disregarded. This method gives new smaller and less correlated matrix which makes it faster and easier to

Fig. 1.

Example of linear and circular array antennas

search. Another algorithms for reduce complexity of antenna selection are suggested in [10]-[12]. In this paper, we focus on transmit antenna selection technique in MIMO spatial multiplexing with zero-forcing (ZF) receiver [5]. We analyze the performance of MIMO spatial multiplexing in terms of vector symbol error probability (VSER). To select the optimal transmit antenna subset, the exhaustive search is required. As adopting more antennas, this exhaustive search becomes impractical. We propose a new efficient antenna selection method which uses both threshold and searching order. The searching order comes from the intuition that the subset of the most separated antennas such as left side of Fig. 1 which have less correlation than the subset of adjacent antennas like right side of Fig. 1. There are several kinds of multiple antenna arrays such as linear array and circular array. In this paper, we assumed using linear array for simplicity. The existing fast antenna selection methods introduced in [8]-[12] manipulate the channel matrix to make a channel matrix smaller for fast searching. After the channel manipulation, they use still exhaustive search for the optimal antenna subset. However, we do not manipulate the channel matrix but suggest an efficient searching algorithm. The proposed algorithm can be applied to the conventional fast antenna selection methods. This paper is organized as follows. In section II, we introduce the system and channel model. In section III, there is conventional optimum and suboptimum transmit antenna selection method for MIMO spatial multiplexing with linear receiver, based on [5]. Then, in section IV, we propose new two

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efficient transmit antenna selection methods which can reduce the complexity of conventional transmit antenna selection. In section V, we illustrate simulation results for performance and tradeoff relation of our proposed methods. After that, we conclude this paper in section VI.

source

Hp

Spatial Multiplexing 1: N

Linear Receiver

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II. S YSTEM AND C HANNEL M ODEL

r

We consider a spatial multiplexing MIMO system with Nt transmit antennas and Nr receive antennas with N transmit antenna selection as shown in Fig. 2. The channel is exponentially correlated Rayleigh flat-fading and slowly time varying. Then the full channel matrix Hcorr ∈ CNr ×Nt is given as 1

1

Hcorr = Cr2 HCt2

(1)

where Ct ∈ CNt ×Nt and Cr ∈ CNr ×Nr are the correlation matrices for array antennas among the Nt transmit antennas and Nr receive antennas, respectively. H ∈ CNr ×Nt is matrix whose entries are i.i.d. complex Gaussian random variables with zero mean and unit variance. We consider uniformly distanced array antennas, so that the correlation matrices Ct and Cr are given by ⎤ ⎡ . . . ρtNt −1 1 ρt ⎢ ρt 1 . . . ρtNt −2 ⎥ ⎥ ⎢ Ct = ⎢ ⎥ and .. .. .. .. ⎦ ⎣ . . . . Nt −1 Nt −2 ρ ρt ... 1 ⎡ t ⎤ Nr −1 . . . ρr 1 ρr r −2 ⎥ ⎢ ρr 1 . . . ρN r ⎢ ⎥ Cr = ⎢ (2) ⎥, .. .. .. .. ⎣ ⎦ . . . . ρrNr −1

ρrNr −2

...

sink

1

where ρt and ρr are correlation coefficients of transmit and receive antennas, respectively. We assume that perfect channel state information (CSI) is available at the receiver. The receiver selects transmit antennas by using CSI. The information of the selected antennas is fed back into the transmitter. After that, the transmitter activates the N (N ≤ min {Nt , Nr }) transmit antennas based on the feedback. Let us consider that antenna index subset t p ∈ P where P is the set of all possible combinations of N N transmit antenna subsets. For example, if p is antenna index subset which consists of both first and last transmit antennas, then p = {1, Nt }. Hp is the sub-matrix of Hcorr , and it corresponds to a vector channel between activated N -transmit antennas and Nr -receive antennas. T The transmitted symbol vector s = [s1 , s2 , · · · , sN ] is E assumed to be normalized such as E[si 2 ] = Ns , i = 1, 2, . . . , N to make average of the total transmitted power Es . The si is a symbol from the i-th stream of spatial multiplexer at transmitter. The transmitted signal vector s is multiplied by an sub-matrix channel Hp before being added with an Nr -dimensional vector n which are i.i.d. complex Gaussian random vector with distribution n ∼ CN (0, N0 ). The received signal vector y = [y1 , y2 , · · · , yNr ]T where yi is a received

N

Compute Optimum Antenna

Channel Estimation

t

Feedback Link

Fig. 2.

Transmit antenna selection MIMO system model

signal of the i-th receiver antenna is given by y = Hp s + n

(3)

At the receiver, we use zero-forcing (ZF) linear MIMO detector to obtain an estimate of s as follows: ˆ s = Gy = GHp s + Gn.

(4)

where G ∈ CN ×Nr denotes ZF matrix. III. C ONVENTIONAL T RANSMIT A NTENNA S ELECTION The main idea of transmit antenna selection for MIMO spatial multiplexing is selecting antenna subset that results in the best performance in terms of vector symbol error rate (VSER) [5]. A VSER is dependent on minimum post-processing SNR among the Nr streams. The ZF linear MIMO detector matrix is pseudo-inverse of Hp which is G = (H∗p Hp )−1 H∗p . We use the ZF linear MIMO detector as (4), so that the postprocessing SNR of k-th stream is given as SNRZF p,k =

Es N N0 [H∗p Hp ]−1 kk

(5)

where [A]kk means the element of k-th row and k-th column for matrix A. By using (5), the minimum SNR for each antenna subset is compared to find the largest one. Then, the optimal selection criterion can be expressed as

ZF (6) arg max min SNRp,k . p

k

An lower bound of (5) is obtained by using Rayleigh-Ritz theorem [7] as −2 max[H∗p Hp ]−1 kk ≤ λmin (Hp ), k

(7)

where λmin (Hp ) denotes minimum singular values of matrix Hp . As a result, the lower bound of (5) can be expressed as 2 SNRZF p,min ≥ λp,min (Hp )

Es . N N0

(8)

This inequality (8) gives an intuition that the performance of each transmit antenna subset becomes better as increasing the smallest singular value of the channel. And approximately,


Transmit Antenna Selection, N = 2

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yit li ba obr P

R10 E B

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2x2 ZF 2x2 ML 3x2 Optimal 3x2 Sub-optimal 4x2 Optimal 4x2 Sub-optimal

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Fig. 3.

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comparison of SNRZF p,min can be substituted by comparing λmin (Hp ). In other words, the suboptimal selection method is to choose the transmit antenna subset which has the largest λmin (Hp ): p

(9)

This alternative selection method, based on only the channel, has the similar performance with the optimal selection method as shown in Fig. 3. IV. P ROPOSED A NTENNA S ELECTION M ETHOD Antenna selection scheme can reduce complexity and at the same time can exploit many of the advantages of MIMO systems. Even using transmit antenna selection scheme, the complexity for selecting transmit antenna subset increases rapidly as adopting more antennas. From optimal transmit antenna selection criterion (6), every SNR value should be needed to find max-min one. This exhaustive methods need t to Nr · N N SNR computation by using (5) for every p and k. Also there is suboptimal selection criterion (9) by using (8), t and this is also exhaustive method which needs N N singular value computations. As increasing number of antennas, the optimal and suboptimal methods quickly become impractical. To reduce complexity of transmit antenna selection scheme on exponentially correlated channel, we propose distanceaided transmit antenna selection method together with conventional method introduced in section II. From the fact that Hp is multiplying H by subset of the set of all Nt ×N unitary matrices [6], we can get an inequality by using the Poincare Separation Theorem [7] such as λN (H) ≥ λN (Hp ), for all p

ρt=0.7, ρρr=0.3

0.5

0.6

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1

α

Comparison of optimal and suboptimal antenna selection methods

arg max λmin (Hp ).

Correlated channel,

0

(10)

where λN (H) is the N -th largest singular value of H and λN (Hp ) is the minimum singular value of Hp , respectively. In our proposed method, we set an appropriate threshold by

Fig. 4.

Pr{λN (Hpopt ) ≥ α · λN (H)}

using (10) and check whether a minimum singular value for a certain antenna subset p exceeds the threshold or not. If the minimum singular value λmin (Hp ) exceeds threshold, then our method stops computing singular value for the rest of antenna subset. If the minimum singular value λmin (Hp ) does not exceed threshold, then our method keeps computing a minimum singular value for other antenna subset and checking t of recursively until finding exceeded one. When all N N minimum singular values do not exceed the threshold, then t we choose the largest one among the N N singular value. From the left term of (10), and we can make threshold by multiplying appropriate α (0 ≤ α ≤ 1) to λN (H). If α satisfies λN (Hpopt ) > α · λN (H) where popt is the optimum antenna set which is not known exactly, we select an antenna subset p which firstly satisfies the given condition as λN (Hp ) ∈ [α · λN (H), λN (H)] .

(11)

This method selects not only optimal antenna subset but also non-optimal antenna subset which satisfies threshold, so that the performance degradation may be happened. If α is too large or λN (Hpopt ) is too small so λN (Hpopt ) ≤ α · λN (H).

t there is no exceeded minimum singular value. Then all N N minimum singular values are computed, and this algorithm selects the largest λN (Hp ) based on (9). Before determining α for threshold, the tradeoff between performance and complexity should be evaluated. Smaller α makes the performance worse, because Pr{λN (Hp ) ≤ α · λN (H)} becomes larger. Therefore larger α makes the performance better. We can calculate empirically the probability, Pr{λN (Hp ) ≤ α · λN (H)} from Fig. 4. In the point


: activated antenna

Proposed Antenna Selection on Correlated Channel ρt = 0.7, ρr = 0.3

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4x4 ZF without AS α=0 α = 0.2 α = 0.4 α = 0.6 α = 0.8 α=1 Optimal AS 5

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Fig. 5. An example of grouping the antenna subsets for Nt = 4 and N = 2

Fig. 6. 8x4 (4-Tx antenna selected) BER performances of proposed method

of complexity, the smaller α finds a qualified antenna subset faster, so that complexity becomes lower. On the contrary, the larger α makes the complexity higher. The example of the tradeoff relation between the performance and complexity will be shown in next section. Even though selecting the non-optimal antenna subset by this method, the minimum singular value of the selected antenna subset is included in α · λN (H), λN (Hpopt ) . Thus, the performances are not degraded severely. For the exponentially correlated channel, the method can be extended. Generally, stronger correlation between the transmit antennas drops the diversity gain. So the more separated transmit antenna subset has higher probability to exceed a threshold than less separated transmit antenna subset, because the more separated transmit antenna subset has weaker correlation than less separated transmit antenna subset. It means that

- Step 2) Pick an antenna subset p according to the order from the most remotely located group to the most closely located group except picked one. - Step 3) Calculate λN (Hp ) and compare with the threshold from Step 1. - Step 4) If λN (Hp ) is larger than threshold, then p is selected. Otherwise, go back to the Step 2. If there is no qualified subset, then go to the next step. - Step 5) Select the subset which has the largest λN (Hp ).

Pr{λN (Hpf ) ≥ α · λN (H)} > Pr{λN (Hpc ) ≥ α · λN (H)} where pf ∈ P is the transmit antenna subset which is more separated than pc ∈ P. It is good example that pf = {1, N } and pc = {1, 2}. By using the fact that the more separated transmit antenna subsets has more chance to be included in α · λN (H), λN (Hpopt ) than the less separated antenna subsets, we classify the transmit antenna subsets into several groups according to the level of separation. For example of Nt = 4 and N = 2 in Fig. 5, we classify the antenna subsets into three groups. The group 1 is consist of the most separated transmit antenna subset, and the group 3 is the least separated. So, we check the transmit antenna subset exceeding the threshold α · λN (H) in the order from the group 1 to the group 3. We can get the qualified transmit antenna subset faster than random checking by using grouping the transmit antennas according to the separation level. We can summarize the procedure as below: - Step 1) Make a threshold by multiplying α to λN (H).

V. S IMULATION R ESULTS In this section, the proposed method is compared with the optimal transmit antenna selection by Monte Carlo simulations. Following the simulations, we consider a MIMO spatial multiplexing system with ZF linear receiver, Nt = 8 and Nr = N = 4. The wireless channel are assumed to be quasistatic, constant over a frame and varying from one frame to another. Also we assumed that the channel is exponentially correlated with ρt = 0.7 and ρr = 0.3. The system is uncoded, and modulation constellation is 16-QAM. Thus the spectral efficiency of our system is 16 bits/second/Hz. In Fig. 6, we plot the BER performances of the proposed method from α = 0 to α = 1. When α = 0 the BER performance is same as 4 × 4 ZF, because there is no diversity gain. The transmit antenna subset is selected randomly, as the minimum singular value of the first checked antenna subset is always larger than 0. As α increasing Pr{λN (Hp ) ≤ α · λN (H)} becomes smaller, then the BER performance goes better. The BER performance is same as optimal transmit antenna selection method when α = 1, Pr{λN (Hp ) ≤ α · λN (H)} is zero as Fig. 4. All because 8 = 70 of the singular values are computed and compared. 4 It is important that our proposed method reduce complexity with almost same BER performance of the optimal transmit antenna selection [5] when α = 0.8. As α increasing we finds that the proposed method shows increase of diversity order up


A tradeoff between complexity and performance, BER = 10-3

18

Proposed AS without order, Nt = 8, Nr = N = 4, 16-QAM Proposed AS, Nt = 8, Nr = N = 4, 16-QAM

16 3 0 1 = R E B r o f ) B d ( 0 N / b E y lt a n e P

R EFERENCES

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Fig. 7. A tradeoff between complexity and performance of proposed method

to Nt − N + 1, the diversity order of the optimal antenna selection, because the probability of selecting the optimal transmit antenna subset increases as α increasing. In Fig. 7, the tradeoff relation between complexity and performance of the proposed method is plotted. Each point is from α = 0 to α = 1 with step 0.1. At each step, we measure the required Eb /N0 for 10−3 in terms of BER and the average number of computations of the minimum singular values. The horizontal axis represents the complexity in terms of the ratio of the average number of minimum singular values computation for the proposed method and the optimal method. The vertical axis represents the penalty of the performance in terms of Eb /N0 (dB) when BER is 10−3 . The complexity becomes smaller as decrease of α, while the penalty increases. From Fig. 4, Pr{λN (Hpopt ) ≥ α · λN (H)} is almost 1 when α is below 0.6. So the penalty increase rapidly as decreasing α. The penalty from α = 0.6 to α = 1 is small. Also, there is the effect of the ordering search to reduce complexity. We can find that the proposed method shows reduction of complexity as α decreasing with performance penalty. The reduction of complexity is caused by increasing the probability of satisfying a threshold as α decreasing. VI. C ONCLUSIONS Our proposed method searches the transmit antenna subset which exceeds the threshold. While conventional method searches exhaustively for all the possible antenna subset, but the proposed method stops computation just after finding a qualified subset to reduce the complexity. Also, we make an order for searching according to the separation level among the antennas. This method uses the fact that the distance among the antennas relates the degree of correlation on exponentially correlated channel. With the simulations, we validate tradeoff between performance and complexity. Our proposed method gives flexibility between performance and complexity to implement an antenna selection scheme.

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