IEICE Communications Express, Vol.2, No.2, 42–48
Quantization error reduction for MIMO detection based on orthogonal lattices Wei Houa) , Tadashi Fujino, and Toshiharu Kojima Department of Communication Engineering and Informatics, The University of Electro-Communications, 1–5–1 Chofugaoka, Chofu, Tokyo 182–8585, Japan a)
[email protected]
Abstract: Lattice-reduction is an effective technique to improve the performance of MIMO data detection. This paper presents a quantization error reduction scheme for MIMO detection based on orthogonal lattices. The motivation of the proposed detection is to solve the problem of degrading the performance due to the rounding quantization error in the signal estimate stage. The simulation results exhibit that the proposed detection can achieve the near-ML performance with a little additional complexity in the high Eb /N0 region. In addition, this quantization error reduction scheme is efficient even for the high modulation order. Keywords: Gram-Schmidt procedure, lattice-reduction, LLL algorithm, MIMO, quantization error Classification: Wireless Communication Technologies References [1] X. Ma and W. Zhang, “Performance analysis for MIMO system with lattice-reduction aided linear equalization,” IEEE Trans. Commun., vol. 56, pp. 309–318, Feb. 2008. [2] T. Fujino, “Gram-Schmidt combined LLL lattice-reduction aided detection in MIMO systems,” REV J. on Electron. and Commun., vol. 1, no. 2, pp. 106–114, April-June 2011. [3] T. Fujino, S. Wakazono, and Y. Sasaki, “A Gram-Schmidt based latticereduction aided MMSE detection in MIMO systems,” Proc. IEEE Global Communications Conf. (Globecom’09), Honolulu, USA, Dec. 2009. [4] C. M. Kinoshita, W. Hou, Y. Sasaki, and T. Fujino, “A low complexity lattice reduction aided detection based on Gram-Schmidt,” Proc. IEEE Signal Processing and Information Technology (ISSPIT), Luxor, Egypt, Dec. 2010. [5] H. Hassibi, “An efficient square-root algorithm for BLAST,” Proc. IEEE Int. Conf. on Acoustic, Speech, Signal Processing, Istanbul, Turkey, pp. II737–II740, June 2000. [6] D. W¨ ubben, R. B¨ ohnke, V. Kuhn, and K. D. Kammeyer, “MMSE extension of V-BLAST based on sorted QR decomposition,” Proc. IEEE. VTC 2003-Fall, Florida, USA, Oct. 2003. c
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DOI: 10.1587/comex.2.42 Received December 20, 2012 Accepted January 15, 2013 Published February 13, 2013
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1
Introduction
Lattice-reduction (LR) aided (LRA) detection has been introduced into MIMO detection to achieve sub-optimal bit error rate (BER) performance with a little additional complexity [1]. For the MIMO detection, the errors are mainly generated from the channel noise and the quantization errors in the signal estimate stage. In this paper, the Gram-Schmidt (GS) procedure is further applied after the lattice-reduction of the channel matrix. Using the orthogonal lattice basis should diminish the channel noise. The quantization operation in the LRA detection is simply performed by rounding. These rounding quantization errors often occur in the case that the decimal fraction of the estimate symbol is around 0.5. Hence, the proposed scheme is focused on solving this problem. In this paper, we first give another quantization candidate for the rounding of the estimate symbol, which candidate is usually obtained by +1 or −1 based on the rounding result. Then, we propose a threshold function based on the orthogonal lattices. This function is updated in each entry of the transmit signal. The proposed scheme may correct the quantization error to improve the BER performance. In addition, we also analyze the complexity for the proposed detection and compare with the conventional LRA detection.
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System model and conventional LRA detection
2.1 System model Consider a multiple antenna system with M transmit and N (N ≥ M ) receive antennas. The signals are transmitted over a rich scattering flat fading channel. Assume that the receiver has perfect knowledge of the channel state c ]T ∈ CN ×1 is information (CSI). The received signal vector yc = [y1c , . . . , yN expressed as (1) yc = Hc sc + zc where ync is the receive signal at the n-th receive antenna. The transmit signal vector is denoted as sc = [sc1 , . . . , scM ]T ∈ SM ×1 , where each symbol scm at the m-th transmit antenna is chosen from a finite subset of the complex-valued integer set S. Let Hc = [hc1 , . . . , hcM ] denote the N × M channel matrix. We assume that the entries of Hc are of the i.i.d. complex Gaussian process with c ]T ∈ CN ×1 zero mean and unity variance. The noise vector zc = [z1c , . . . , zN is the additive white Gaussian noise (AWGN) vector, of which each entry is assumed to be zero mean and variance of N0 , the one-sided noise power spectral density. In this paper, we describe some MIMO detection schemes based on LR. These schemes estimate the transmit signals by the real-valued channel matrix and vectors. Hence, eq. (1) can be rewritten using the real representation as y = Hs + z (2) c
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DOI: 10.1587/comex.2.42 Received December 20, 2012 Accepted January 15, 2013 Published February 13, 2013
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where the equivalent real-valued channel matrix and vectors are given by Re[Hc ] −Im[Hc ] Re[sc ] 2N ×2M H , s ∈R ∈ Z2M ×1 , c c c Im[H ] Re[H ] Im[s ] (3) c] Re[yc ] Re[z y ∈ R2N ×1 , and z ∈ R2N ×1 c c Im[y ] Im[z ] The set of the real-valued signals is given by Z = {±1, ±3, . . . , ±(K − 1)} for K 2 -QAM (Quadrature Amplitude Modulation). In [5], Hassibi proposed an MMSE detector with the extend matrix form as y z H ¯ = ¯= ¯= , y , z H (4) O2M − γ −1 s γ −1 I2M where γ = Es /N0 with Es = E[s2 ]/2M . Here E[ · ] denotes the ensemble average operation. I2M is an 2M ×2M identity matrix, and O2M is an 2M ×1 vector with all zero entries. Then it holds instead of (2) that ¯ +z ¯ = Hs ¯ ≡ QRs + z ¯ y
(5)
¯ can be QR-decomposed as H ¯ = QR for where the extend channel matrix H the LR. Q is a unitary matrix with QT Q = I2M and R = [r1 , r2 , . . . , r2M ] is an 2M × 2M upper triangular matrix. R retains the property of the channel ¯ [6]. matrix H This MMSE soft estimate of the transmit signals is expressed as ¯ −1 H ¯ Ty ¯ T H) ¯ †y ¯ ≡ (H ¯ = (HT H + γ −1 I2M )−1 HT y ˜s(MMSE) = H
(6)
2.2 Conventional LRA detection The set of the column vectors {r1 , r2 , . . . , r2M } is called a basis of lattice. ˜ of The LR algorithm is to transform a given basis R into a new basis R which columns are near orthogonal, and create a transformation matrix T. The most popular LRA detection employs the LLL algorithm shown in [1]. ˜ Q ˜ and T, the system model in (5) Using the LLL lattice reduced matrix R, is rewritten as ˜ Rv ˜ +z ˜ R(T ˜ −1 s) + z ¯≡Q ¯ ¯ = QRs + z ¯=Q y
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(7)
˜R ˜ and the new transmit signal vector v is defined as v where QRT = Q −1 ˜ T s. R is the near orthogonal lattice basis to improve the decision region. The LRA detection applies its algorithm to estimate v instead of s [2]. In ˜ Ty ˜ −1 y , where y Q ¯. ˜ can be derived by v ˜ = R general, the estimate v −1 Since T contains integer elements, we perform the simply quantization vi ], i ∈ [1, 2M ]. Finally, the operations by rounding off each estimate vˆi = Q[˜ original signal vector is transformed back as s = Tv. The final decision ˆs is forced to the nearest symbol constellation points if they are lying outside the symbol constellation.
DOI: 10.1587/comex.2.42 Received December 20, 2012 Accepted January 15, 2013 Published February 13, 2013
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Gram-Schmidt procedure based on LR
Based on Fujino et al.’s previous work of the GS procedure based LR in MIMO systems [2, 3, 4], we introduce the GS orthogonalization algorithm ˜ shown in [2]. This algorithm transforms the LLL-reduced channel matrix R ˆ and the transformation matrix to create the GS-reduced channel matrix R ˆ with det{T} ˆ = 1. T ˆ with unity diagonal entries is invertible. The upper triangular matrix T ˆ = R ˜T ˆ are mutually orthogonal and The column vectors of the matrix R ¯ Using the span the same subspace as the columns of the original matrix H. ˆ and T, ˆ we have GS-reduced channel matrix R ˜ + z = (R ˜ Ty ˜ T)( ˆ T ˆ −1 v) + z ≡ Ru ˆ + z ¯ = Rv y Q
(8)
ˆ −1 v. Note that R ˜ Tz ˆ is the diagonal matrix and ¯ and u T where z Q ˆ T is an upper triangular matrix with unity diagonal entries and the others ˆ have almost of equal of non-integers. The orthogonal column vectors of R ˜ is simply calculated as length. Hence, the soft estimate of u ˆ −1 y or u ˜=R u ˜i = yi /ˆ ri,i , i ∈ [1, 2M ]
(9)
˜ are not integers, they cannot be quantized like the Since the entries of u ˜ is described LRA detection. One solution for the quantization scheme of u in [2]. This detection method based on LR and GS procedure is called the LR-GS detection.
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Quantization error reduction scheme
In the LRA detection, after the rounding quantization operations, the symbols are transformed to the original basis lattice as ˆs = TQ[˜ v]. Unfortunately, this quantization scheme may result in the quantization error. If this error occurs in the m-th row of the signals, it has to propagate to as many symbols as non-zero entries of the m-th column of T. These errors are mainly generated from the channel noise and the rounding error. Hence, we propose a quantization error reduction scheme based on LR and GS procedure for the MIMO systems. Using the orthogonal lattice basis, the channel noise should be further diminished. It is noted that the errors often happen in the case that the difference between the estimate value and the rounding value is around ±0.5. The property is the motivation to apply in the quantization error reduction scheme. The signal estimate of the proposed scheme is summarized as:
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IEICE 2013
DOI: 10.1587/comex.2.42 Received December 20, 2012 Accepted January 15, 2013 Published February 13, 2013
˜ is first obtained as u ri,i , i ∈ [1, 2M ]. (1) u ˜i = yi /ˆ ˆ is from the last entry down to the (2) The order of the signal estimate v u2M ]. first entry. The last entry symbol is firstly estimated as vˆ2M = Q[˜ For the rest of entries, we perform the cancellation 2M −1 ˆ ˜i − ti,j vˆj , i = 2M − 1 down to 1 vˆi = Q u j=i+1
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(3) Define the Euclidean distance in the v-domain as Λi = (1)
2M
vj j=i |˜
− vˆj |2 .
vi ] and Here we mark the original rounding symbol as vˆi = vˆi Q[˜ (2) give another quantization candidate as vˆi = vˆi + sgn{˜ vi − vˆi }. (4) The quantization error reduction scheme starts from the last entry. (l) Based on two quantization candidates to calculate Λ2M : l ∈ [1, 2], compare these with the threshold o2M defined as following, which is updated in each entry:
oi = min(Λi ) · 1 + RD /|ˆ ri,i |2 , i = 2M down to 1 (l) where |ˆ ri,i |2 is normalized as RD 2M ri,i |2 /2M . If Λ2M : l ∈ [1, 2] i=1 |ˆ (l) ri,i |2 is is less than o2M , the candidate vˆi is survived. Note that RD /|ˆ close to 1 and there are at most two retaining candidates in each entry. (5) Next, the quantization error reduction scheme is carried out in the (2M − 1)-th entry. Based on the number of retaining candidates in the previous entry, the cancellation operation is performed as step (2). If the number of retaining candidates is two, we have (l,l ) (l ) vˆ2M −1 = Q u ˜2M −1 − tˆ−1 v ˆ 2M −1,2M 2M , where l or l ∈ [1, 2] (l )
where the superscript l denotes the index of vˆ2M for the previous entry. (l,l ) Hence, there are four candidates at most to calculate Λ2M −1 . (6) The same operations are performed until the first entry. If the number of retaining candidates in the first entry is 1, ˆs = Tˆ v. Else, we have (l) ˆ , l ∈ [1, 2], to make the ML metric: two candidates: v ˆs = arg min |y − Hˆs(l) |2 l∈[1,2]
Here we define a threshold function in step (4) in order to retain another ˆ is a diagonal matrix, if |ˆ quantization candidate. Since R ri,i |2 is larger, the threshold oi becomes stricter. It implies that the quantization error with low probability occurs.
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IEICE 2013
DOI: 10.1587/comex.2.42 Received December 20, 2012 Accepted January 15, 2013 Published February 13, 2013
Numerical results
The computer simulations were carried out for 64QAM in the 4 × 4 MIMO system. The channel was assumed the typical flat Rayleigh fading. The performances of the different detections were measured by the BER and the complexity. As seen in Fig. 1, the BER curve of the conventional LRA detection can achieve the sub-optimal performance. At a BER of 10−5 , the LR-GS detection can further gain 1 dB over the conventional LRA detection. For the proposed detection, the BER curve totally approached the ML performance in the high Eb /N0 region. This fact also proved that the proposed quantization error reduction scheme was efficient even for the high modulation order. We evaluated the average number of retaining candidates (RC) in the i-th entry expressed as NRC (i) in Fig. 2. The proposed scheme starts from the last entry of new signal vector, and then the rest of entries are obtained 46
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Fig. 1. The Eb /N0 vs. BER characteristics in the 4 × 4 MIMO system: 64QAM.
Fig. 2. The Eb /N0 vs. the average number of retaining candidates for each entry.
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IEICE 2013
DOI: 10.1587/comex.2.42 Received December 20, 2012 Accepted January 15, 2013 Published February 13, 2013
by the iterations until the first entry. In this paper, we assumed that the number of retaining candidates in the last entry is equal to 2 as the curve (1). From the 7th entry down to the 2nd entry, there are six iterations with the same operations. Therefore, we counted up the average number of retaining candidates among these six entries as the curve (2). It is important for the average number of retaining candidates in the first entry, which decides the number of the estimate transmit signal to perform the ML metric in step (6). NRC (1) in the curve (3) is around 1 at Eb /N0 = 20 dB with the BER of 10−2 . Hence, the complexity of proposed detection may neglect the computing of ML metric in the high Eb /N0 region. According to the average number of retaining candidates, we presented the complexity in the signal estimate stage in Table I. In the high Eb /N0 region, NRC (i) ≈ 1 where i ∈ [1, 7]. The complexity of proposed detection in the signal estimate requires about 490 flops. However, if the transmit signal is detected by the conventional LRA detection
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Table I. The complexity of the proposed detection.
described in subsection 2.2, its complexity requires about 432 flops. Therefore, the proposed detection can achieve the near ML performance with lower complexity, which is much less than that of the ML detection.
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Conclusion
In this paper, we proposed a quantization error reduction scheme for MIMO detection based on orthogonal lattices. In order to reduce the quantization error, we gave another quantization candidate and defined a threshold function based on the orthogonal lattices. The BER performance of proposed detection is improved to decrease the rounding error compared to the conventional LRA detection. The numerical results exhibited that the proposed detection achieved the near-ML performance with the trivial complexity in the high Eb /N0 region. In addition, this quantization error reduction scheme was efficient even for the high modulation order.
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IEICE 2013
DOI: 10.1587/comex.2.42 Received December 20, 2012 Accepted January 15, 2013 Published February 13, 2013
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