Simplified Maximum-Likelihood Precoder Selection for Limited Feedback Spatial Multiplexing Systems Jong-Ho Lee
Sung-yoon Jung
Daeyoung Park
Division of Electrical Electronic & Control Engineering Kongju National University, Korea Email:
[email protected]
School of Electronic Engineering Communication Engineering & Computer Science Yeungnam University, Korea Email:
[email protected]
School of Information and Communication Engineering Inha University, Korea Email:
[email protected]
Abstract—In this paper, we consider a precoder selection criterion for the maximum-likelihood (ML) detector in limited feedback spatial multiplexing systems. When the ML detector is employed, the exhaustive search for the optimal precoder selection requires very high computational complexity. As a good compromise, we propose a simple precoder selection scheme for the ML detector. It reduces the computational complexity significantly with negligible performance degradation. For the wideband OFDM systems, we extend the precoder selection by introducing a clustering concept in selecting the optimal precoder. Numerical results show that the proposed scheme achieves both the diversity order of the ML detector and the performance enhancement of the limited feedback spatial multiplexing. Moreover, the proposed precoder selection algorithm based on the clustering approach provides enhanced performance compared with conventional interpolation and clustering algorithms.
I. I NTRODUCTION Spatial multiplexing in multiple-input multiple-output (MIMO) system achieves high spectral efficiency to transmit multiple data substreams on different transmit antennas. However, it is quite sensitive to the rank deficiency of channel matrix. In order to overcome the ill-conditioning of the MIMO channel in spatial multiplexing system, recent research has focused on utilizing full or partial channel knowledge at the transmitter, which is fed back from the receiver [1]–[4]. Unfortunately, these schemes are optimized for linear receivers and the precoder selection criteria for linear receivers in [2] are not optimal solutions for maximum-likelihood (ML) detector. In addition, the problem of precoder selection for precoded MIMO-OFDM systems in the case of frequency selective channels has received great attention in the literature. lt is widely known that MIMO-OFDM signaling scheme can convert a frequency selective channel into a series of narrowband MIMO channels called subcarriers and makes it possible to apply efficient equalization [5]. Therefore, the precoding techniques developed for frequency-flat MIMO channels can be easily extended to MIMO-OFDM by treating each subcarrier as a narrowband MIMO channel. However, they may require precoder knowledge for all subcarriers, which may cause a large amount of feedback if not properly designed. Therefore, recent researches have focused on how to find out the set of precoder matrices for all of the subcarriers in a MIMOOFDM system with as low feedback overhead as possible. As
a solution, there are two main approaches, 1) interpolation and 2) clustering algorithms [6]–[10]. In this paper, we focus on the optimal precoder selection criteria for the ML detector and its extension to MIMOOFDM systems. The precoder selection criterion for the ML detection in [2] is impractical due to its high computational complexity. In order to find the minimum distance, it requires the exhaustive search for all the error vectors. Here, we propose a simplified precoder selection scheme tailored for the ML detector to reduce the computational complexity. The proposed scheme limits the search space considering the norm of the error vector, while it shows negligible degradation of performance. In addition, we also propose clustering approach which selects the optimal precoder to represent the corresponding cluster based on the precoder selection scheme for the MIMO-OFDM system. II. P RECODER S ELECTION C RITERIA Let M and N be the number of transmit and receive antennas, respectively. The N × 1 received signal vector y is given as y = HWs + n (1) where H is the N × M channel matrix and n is the N × 1 additive white Gaussian noise (AWGN) vector. The K × 1 transmit symbol vector s is predistorted by the M × K unitary precoding matrix W. At the receiver, the precoding matrix is selected among the precoder set W = {W1 , W2 , · · · , WL } and the index of the selected precoding matrix is fed back to the transmitter so that the overall channel matrix HW becomes favorable for signal transmission. A. Conventional Precoder Selection Criterion The linear detectors, such as the zero-forcing (ZF) and the minimum mean squared error (MMSE), turn a matrix channel into multiple scalar channels. Since the singular values of HW are directly related to the stream SNRs, it is desirable to make each singular value as high as possible. The conventional criterion to select a transmit antenna subset [1] or a precoder [2] is based on the maximization of the minimum stream SNR, i.e., (2) W = argmax SNRimin . Wi ∈W
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
In order to reduce the computational complexity for evaluating the SNR of each of K streams, we utilize the fact that the minimum stream SNR is bounded by the minimum singular value of HWi and we get W = argmax λmin {HWi },
(3)
Wi ∈W
where λmin {HWi } denotes the minimum singular value of HWi . Thus, in (3), the precoding matrix W is selected so that the minimum singular value of the overall channel matrix HW is maximized. Even though the selection criterion in (3) improves the performance of the linear detectors, it may deteriorate the performance of the ML detectors. B. Precoder Selection for the ML Detector The conventional criterion that maximizes the minimum stream SNR is well suited for linear detectors, but is not necessarily optimal in the sense of the error probability. We may consider another criterion that directly minimizes the error probability for the ML detection. Suppose we choose the precoding matrix Wi and transmit a symbol vector s1 . The pairwise error probability (PEP) is the probability that the decoder erroneously selects s2 as an estimate of s1 , for s2 = s1 , i.e., ⎞ ⎛ 2 HW (s − s ) i 1 2 ⎠ , (4) Pr(s1 → s2 |H, Wi ) = Q ⎝ 2N0 where is the Gaussian tail function Q(x) = ∞ Q(x) −t2 /2 √1 e dt. The frame error probability (FER) is upper2 x bounded by the union bound that is summation of all pairwise error probabilities [11]. For given channel realization H, we may choose the precoding matrix Wi that minimizes the FER. However, due to its high complexity, we instead minimize the worst-case PEP for all possible transmit sequence pair (s1 , s2 ). Since the Gaussian tail function Q(x) is monotonically decreasing, minimizing the PEP is equivalent to maximizing the modified minimum distance and thus we choose the precoding matrix Wi based on W = argmax μ(Wi , H)
(5)
Wi ∈W
where the metric μ(Wi , H) is given by μ(Wi , H) = min HWi (s1 − s2 )2 . s1 =s2
(6)
It is difficult to analytically find the optimal Wi that minimizes the metric in (6) because the modified minimum distance heavily depends on both the modulation order and the channel realization. Moreover, it is necessary to consider all possible error vectors, given by se ≡ s1 − s2 to search for the optimal W. This exhaustive search has a prohibitive complexity for practical implementation.
C. Simplified Precoder Selection for QPSK Modulation For a good compromise between the performance and the complexity, the search space is limited to the following error vectors: Only one or two entries are nonzero in error vectors and their norm is less than 4. If the search space includes error vectors that have 3 or more nonzero entries or that have norm of 5 or higher, it brings forth a slight performance improvement with higher computational complexity. We classify the error vectors according to their norm and derive the selection criterion. se (1) s¯e (2) · · · s¯e (K)]T be the normalized Let ¯se = [¯ error vector defined by ¯se = se /dm , where dm denotes the minimum separation distance among the given constellation. We now consider the error vectors that has only one nonzero entry. In this case, there is only one k such that s¯e (k) ∈ {±1, ±j, ±1 ± j} and s¯e (l) = 0 with l = 1, 2, · · · , k − 1, k + 1, · · · , K. Under this condition, we have H gi,k |¯ se (k)|2 HWi¯se 2 = gi,k
(7)
where HWi = [gi,1 , gi,2 , · · · , gi,K ]. Since the minimum of |¯ se (k)|2 is 1, the minimum distance can be given as H m1 (Wi , H) = min gi,k gi,k . 1≤k≤K
(8)
So, for a given channel realization H and a candidate precoding matrix Wi , this distance metric finds the smallest-norm column vector in HWi . It covers 8 ( K 1 ) = 8K error vectors, while it just searches the minimum value among K candidates. This indicates that the size of the search space in this distance metric is K. Then, we consider the error vectors that has two nonzero entries. There are four possible patterns for two nonzero entries. se (l)|) The absolute values of two nonzero entries (|¯ se (k)|, |¯ are one of (1,1), (2,1), (1,2), (2,2). We first consider the case of (1,1). In this case, there are k and l (k = l) such that s¯e (k) ∈ {±1, ± j}, s¯e (l) ∈ {±1, ± j}, and s¯e (m) = 0 with m = k and m = l. Then, we get H H H HWi¯se 2 = gi,k gi,k + gi,l gi,l + 2Re{¯ se (l)¯ s∗e (k)gi,k gi,l }. (9) The last term of the right-hand side of (9) can have 4 different s∗e (k) possible values depending on the value of s¯e (l)¯ H H H Re{¯ se (l)¯ s∗e (k)gi,k gi,l } ∈ ±Re{gi,k gi,l }, ±Im{gi,k gi,l } . (10) So, it is bounded below by the minimum of 4 possible values, i.e., H s∗e (k)gi,k gi,l } ≥ Re{¯ se (l)¯
H H gi,l } , Im{gi,k gi,l } . − max Re{gi,k
(11)
If we combine (9) and (11) H H HWi¯se 2 ≥ gi,k gi,k + gi,l gi,l
H H − 2 max Re{gi,k gi,l } , Im{gi,k gi,l } ,
(12)
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
then we can get the minimum distance m2 (Wi , H) =
min
1≤k,l≤K k=l
H H gi,k gi,k + gi,l gi,l −
(1) 2Γk,l
(13)
where
(1) H H gi,l } , Im{gi,k gi,l } . Γk,l = max Re{gi,k
(14)
Moreover, we also consider the (2,2) case. In this case, the error vectors can be given as s¯e (k) ∈ {±1 ± j}, s¯e (l) ∈ {±1 ± j}, and s¯e (m) = 0 with m = k and m = l. Then, we obtain H H H gi,k +2gi,l gi,l +2Re{¯ se (l)¯ s∗e (k)gi,k gi,l }. HWi¯se 2 = 2gi,k (15) The last term of the right-hand side can also have 4 different s∗e (k) shown as values according to s¯e (l)¯ H H H s∗e (k)gi,k gi,l } ∈ ±2Re{gi,k gi,l }, ±2Im{gi,k gi,l } . Re{¯ se (l)¯ (16) In (15) and (16), it is obvious that the distance evaluated in the (2,2) case is just the twice of the distance in the (1,1) case so that we do not need to consider the error vectors in the (2,2) case as the candidates to compute the minimum distance. Therefore, the distance metric in (13) covers 2 · 4 · 4 · ( K 2 )= 16K(K − 1) error vectors in both the (1,1) and (2,2) cases, while it only deals with 4·( K 2 ) = 2K(K −1) values. Thus, the size of the search space in this distance metric is 2K(K − 1). Secondly, we consider the case of (2,1) and (1,2). In this case, the norm of the error vectors is 3. As derived in the Appendix, the minimum distance in this case can be obtained as
m3 (Wi , H) = where Θmin k,l Θmax k,l and (2) Γk,l
= max
min
1≤k,l≤K k=l
(2)
max 2Θmin k,l + Θk,l − 2Γk,l
(17)
H H = min gi,k gi,k , gi,l gi,l , H H = max gi,k gi,k , gi,l gi,l H H gi,l } + Im{gi,k gi,l }|, |Re{gi,k H H |Re{gi,k gi,l } − Im{gi,k gi,l }|
(18) .
(19)
This distance metric covers 2 · 4 · 4 · ( K 2 ) = 16K(K − 1) error vectors, while it only deals with 2·4·( K 2 ) = 4K(K−1) values. Thus, the size of the search space in this distance metric is 4K(K − 1). From (8), (13) and (17), we propose the metric for a simplified precoder selection for the i-th precoder Ci as follows μ(Wi , H) = min {m1 (Wi , H), m2 (Wi , H), m3 (Wi , H)} . (20) Note that the error vectors in the form of s¯e (k) ∈ {±1, ± j}, s¯e (l) ∈ {±1, ± j}, s¯e (m) ∈ {±1, ± j} and s¯e (n) = 0 have the same norm with the error vectors considered in m3 (Wi , H). In this case, there are three nonzero entries in error vectors. We could include these types of error vectors in
the proposed selection criterion for the performance enhancement. However, we exclude them because we found that they are much less probable to yield the dominant error event. This is the reason why we only focus on the error vectors with one or two nonzero entries 1 . The size of the search space in the proposed precoder selection is 6K 2 − 5K, which is order of O(K 2 ). Since the size of the search space in the exhaustive search is O(eK ), the proposed precoder selection reduces the computational complexity significantly. III. P RECODER S ELECTION FOR THE MIMO-OFDM S YSTEM In the last section, we proposed a simplified precoder selection scheme for the ML detector in narrowband MIMO systems. Now, we extend the proposed scheme to the MIMOOFDM systems with the aid of the clustering concept. Since we select one representative precoding matrix per each cluster, clustering may reduce the amount of feedback and simplify the implementation complexity. For the MIMO-OFDM channels, each subcahnnel becomes frequency flat and the received signal is given by y(f ) = H(f )W(f )s(f ) + n(f ).
(21)
Suppose that we choose the precoding matrix Wi as the representing precoder and transmit a symbol vector s1 (f ) for subcarrier f belonging to the c-th cluster. By using the union bound, the PEP of the subcarrier f is expressed as Pr(s1 (f ) → s2 (f )|H(f ), Wi ) = ⎛ ⎞ 2 H(f )W (s (f ) − s (f )) i 1 2 ⎠. Q⎝ 2N0
(22)
For given set of channel realizations {H(f )}f ∈Cluster c 2 , we choose the precoding matrix Wi that minimize the worst-case PEP for all possible transmit sequence pair (s1 (f ), s2 (f )), which is equivalent to maximizing the modified minimum distance. So, one may choose the precoder matrix so that the summation of the modified minimum distances is maximized, i.e., W(c) = argmax Wi ∈W
f ∈Cluster c
min
s1 (f )=s2 (f )
H(f )Wi (s1 (f )−s2 (f ))2 . (23)
Since the computational complexity of the exhaustive search is quite high, we may use the derived metric for the simplified precoder selection (20) according to the given constellation. 1 For
higher order modulation, additional metrics can be found in [12]. order to provide the channel realization of all the subcarriers in practical scenario, we firstly obtain the estimated channel in pilot subcarriers. Then, by interpolating the subcarriers based on the pilot subcarrier’s channel, we can obtain the entire estimated channel realizations. 2 In
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
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Fig. 1. Probability of symbol vector error versus the number of precoders in the codebook (L) transmitting 2 streams in 2 × 2 MIMO using QPSK and ML detector at Eb /N0 =15dB .
is given in (20). Since where μ(Wi , H(f )) mins1 =s2 H(f )Wi (s1 − s2 )2 ≤ μ(Wi , H(f )), the optimal W(c) for (24) may not be necessarily the optimal precoding matrix for (23). So, we reduce the computational complexity at the cost of performance degradation.
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In summary, the simplified precoder selection criterion for the ML detector based on the clustering scheme is
μ(Wi , H(f )) W(c) = argmax Wi ∈W f ∈Cluster c (24) (c) W(f ) = W , ∀f ∈ Cluster c,
(a)
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IV. N UMERICAL R ESULTS −5
The proposed precoder selection scheme was simulated to illustrate the performance enhancement for the ML detector in the limited feedback spatial multiplexing system. In Fig. 1, we present the probability of symbol vector error according to the number of precoders in the codebook (L) in 2 × 2 MIMO transmitting 2 data streams. A DFTbased codebook in [13] is used and each entry of MIMO channels is assumed to follow independent and identically distributed (i.i.d.) complex Gaussian distribution. We observe that the precoding with right singular vectors in [2] and the conventional precoder selection in (2) do not perform well with ML receiver. The precoding with right singular vectors shows worse performance than no precoding and the conventional precoder selection in (2) provides worse performance as L increases. Note that the proposed scheme outperforms the conventional precoder selection in all ranges of L. Fig. 2 shows the probability of symbol vector error in 2 × 2 and 4 × 4 MIMO systems transmitting 2 and 4 streams using QPSK. A 3-bit DFT-based codebook is employed [13]. As shown in Fig. 2(a), the proposed scheme in (20) with the ML detector achieves about 4dB SNR gain at the probability of symbol vector error of 10−3 compared with no precoding,
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(b) Fig. 2. Probability of symbol vector error comparison of conventional and proposed selection schemes using QPSK and ML detector. (a) 2 × 2 MIMO transmitting 2 streams, (b) 4 × 4 MIMO transmitting 4 streams.
while the conventional precoder selection gives only 0.5dB SNR gain. Moreover, it is seen that the proposed scheme provides exactly the same performance of the exhaustive search in (6) for QPSK. For a 4 × 4 MIMO in Fig. 2(b), the proposed scheme also outperforms the conventional precoder selection. However, there is a performance loss compared with the exhaustive search at high SNR region, because the error vectors with one or two non-zero entries are considered. This loss can be made smaller if the error vectors with more entries are included for the precoder selection. Next, we simulated the performance of the proposed clustering algorithms in the MIMO-OFDM case. We used 2 × 2
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(‘No Precoding’) which shows the worst performance, full subcarriers precoding (‘Full Precoding’) which shows the best performance, the interpolation precoding (‘Geodesic’) and the cluster precoding (‘Karcher mean’) algorithms [10]. It is seen that the proposed clustering algorithm outperforms other interpolation and clustering schemes.
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V. C ONCLUSION
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In this paper, we proposed a new simplified precoder selection criterion for the ML detector. The proposed scheme reduces the computational complexity significantly by limiting the search space to most probable error vectors that may yield the worst case minimum distance. For the extension to OFDM system, we also proposed clustering approach which selects one precoder per each cluster by using the simplified precoder selection scheme. Numerical results showed the performance enhancement in the limited feedback spatial multiplexing systems.
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ACKNOWLEDGMENT This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NO. R01-2008-000-20333-0).
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(b) Fig. 3. Probability of symbol vector error according to subcarrier clustering (subcarrier spacing for interpolation algorithms) in MIMO-OFDM system. (a) 8-subcarrier clustering, (b) 16-subcarrier clustering.
MIMO transmitting 2 streams using QPSK with the ML detector based on precoder selection scheme. The total number of subcarriers is 256. In order to generate the frequency selectivity of the channel, we adopted a T -tap channel which has the exponentially decaying power delay profile. Each tap follows i.i.d. complex Gaussian distribution and T is set to 12 for simulations. Fig. 3 shows the probability of symbol vector error when the number of subcarriers in a cluster is 8 and 16 (the corresponding pilot subcarrier spacing for interpolation algorithm is 8 and 16) with 4-bit DFT-based codebook. All the subcarriers in a cluster are compromised to select the precoder. For the comparisons, we included no precoding
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