2007 IEEE International Symposiumon Signal Processing and Information Technology

A Quantization Scheme for Lattice-Reduction Aided MIMO Detection

Tien Duc Nguyen, Tadashi Fujino, Xuan Nam TranDepartment of Information and Communication Engineering,

The University of Electro-Communications,Chofugaoka 1-5-1, Chofu-shi, Tokyo 182-8585, Japan.

E-mail: ntduc@ice.uec.ac.jp

Abstract- This paper aims at improving bit error rate (BER) quantization was used in previous works [3], [6]. The effect ofperformance of lattice reduction aided (LRA) linear detectors using suboptimal quantization is degraded BER performancefor multiple input multiple output (MIMO) wireless systems. as mentioned above.Specifically, we present an improved quantization scheme for The objective of this paper is to find methods to saveLRA detectors which can generate a set of candidate symbolsfrom the original LRA estimated symbol. From these symbols, this degraded Eb/No in LRA linear detectors. Based on thedecisions are made in term of the minimum Euclidean distance. idea in [9], we propose a modified list quantization schemeIt is shown by simulation that the method provides significant which can generate a set of candidate symbols from theBER performance improvement at the cost of small additional original LRA estimated symbols. From these listed symbols,complexity.

decisions are made according to the minimum Euclideandistance between the received and estimated points. It isshown by simulation that the methods provide significant

Keywords: ZF,MMSE, lattice reduction, MIMO, spa- BER performance improvement at the cost of small additionaltial multiplexing, lattice detection complexity, particularly, in the high Eb/NO region.

The remainder of the paper is organized as follows. InI. INTRODUCTION Sect. II, we present the signal model and conventional linear

Multiple-input multiple-output (MIMO) wireless systems detection methods (ZF,MMSE) for MIMO systems. Introduc-were shown to achieve enormous channel capacity in rich- tion of lattice reduction and its application to ZF,MMSE detec-scattering wireless environments [1]. As one of the practical tors are summarized in Sect. III. Explanation on the proposedapproaches to achieve the MIMO capacity, spatial multiplexing list quantization scheme is given in Sect. IV. Simulation resultswas demonstrated to have spectral efficiencies of 20-40 b/s/Hz are shown in Sect. V. Finally, the paper is summarized andin an indoor environment using the V-BLAST (Vertical-Bell concluded in Sect. VI.Labs Layered Space-Time) detector [2]. It is known that V-BLAST is a promising detector in terms of balancing its II. LINEAR MIMO DETECTIONBER performance and associated computational complexity.However, the BER performance of V-BLAST detectors is A. System Modelfar worse than that of the optimum detector using maximum A MIMO system with N transmit and M(M > N) receivelikelihood (ML) criterion. antennas is investigated in this paper as shown in Fig. 1. In

Recently, there has been a great interest in seeking de- the system, the signal is transmitted over a rich scatteringtectors with good BER performance but low computational flat fading channel. The fading channel from the nth transmitcomplexity. One effective approach is to apply lattice reduction antenna to the mth receive antenna is represented by the com-(LR) into the MIMO channel matrix to make it closer to plex gain hmn with n = 1, 2,. . . N, and m = 1, 2,... M.orthogonal before using linear detectors such as zero forc- These hmn is modeled as complex Gaussian random variablesing(ZF) or minimum mean square error (MMSE) to estimate with zero mean and unit variance. The noise samples Zm istransmitted symbols. This kind of detectors is often known independent complex Gaussian random variables with zeroas lattice reduction aided (LRA) linear detectors [6]-[8]. It mean and variance (O, or. The systemis interestingly noted from the literature that the LRA linear model of the considered system is expressed in matrix formdetectors allow to achieve the same diversity order of the ML asdetector but with some loss in average bit energy per noisevariance (Eb/No).We note that in the previous proposed LRA detectors, a y=Hs + z (1)

lattice reduction algorithm, such as the well-known Lenstra,Lenstra and Lovaisz (LLL) [16], is often used to transform the where Y= [Yi, Y2, ... ., YM] is the receive signal vector, H=original signal constellation into a lattice of parallel points. {h1, ..., hN} is the M x N MIMO channel matrix,In fact this transformation makes optimal quantization too s =[si, s2, . .. , sxiT is the transmit signal vector and z=complex, and thus the suboptimal unconstrained element-wise [z1, Z2,.. ., ZM]T iS the noise vector.

978-1 -4244-1 835-0/07/$25.00 ©C2007 IEEE 663

a (complex) lattice spanned by the generator (channel) matrix

H H. The set of column vectors {hi, h2, ..., hN} is called aYi basis of the lattice L. A lattice may have many different bases.

51SI \/ (j) Y1 . .............................Theaimof lattice basis reduction, or often simply referred2 2 . to as lattice reduction is to transform a given basis H into a

S2 \ >< / : Y2 new basis H' whose columns are close to mutually orthogonal.Y<2 The detection will be then carried out based on H', followed

* // \ X by a quantization operation. The most popular algorithm forN sV \ M ZM LR is the LLL algorithm introduced by Lenstra, Lenstra

SN YM and Lova'sz [16]. In fact, the whole LR algorithm can bedescribed by a linear transform H' = HT, where H' is theLLL-reduced matrix. The transform matrix T is an integerunimodular matrix with determinant det{T} = ±1, which

Fig. 1. System model of a MIMO system. is resulted from the LLL algorithm. Detailed steps of thealgorithm is presented in [3,16]. Further information can befound in [3,7, 8, 10, 16].

B. Linear DetectionIn linear MIMO detection, the receive signal is multiplied B. LRA Detectors

with the weight matrix and decided to the nearest point in The motivation of applying LR to linear detectors is tothe constellation. The weight matrix used for ZF and MMSE improve the decision region [6]. Using the transform matrixcriterion is given by [14] T obtained from a LR algorithm as described in Sect. III-A,

the system model (1) can be rewritten as [10]WZF (H (2) y = Hs +z = (HT)(T-1s) + z _ H'u +z, (7)

WMMSE

H 3 where H' A HT and uA T-ls are the new complex

channel matrix and transmitted signal vector, respectively.where Ht denotes the pseudo-inverse of H, p = o2/72. Based on this new model, LRA-ZF or LRA-MMSE detectorEstimates of the transmitted symbol s by the ZF detector are estimate u and then uses the transform matrix T to convertcalculated by u to s. Since columns of H' are closely orthogonal, linear

detection based on H' provides better BER performance [8].SZF = WY = Hty = s + Htz (4) A block diagram of LRA linear detector is illustrated in Fig.2.

Then, each element of SZF is forced to the symbol con- An important condition for LRA detectors is that s is takenstellation to get the original transmitted symbols. In [15], from a specified complex integer set, i.e. S CEN C 7N. SinceHassibi proposed a ZF-similar form of MMSE detector with T 1 contains only integers, u C T- 1N is also a complexthe introduction of the extended matrix integer vector. Therefore, after soft estimate iu is obtained

by LRA linear detection, a simple suboptimal quantizationH [ '$] and y=Yj (5) operation is used to round off the real and imaginary part

_V/j5INJ °2 of each element iin in ii to the nearest integer as iin =

then the MMSE method can be expressed in the similar form F i{i,}i + F3i{&l}j, where j = /71. The detected signalof the ZF as vector is then obtained by transforming iu to s by the transform

- ftt~matrix T as = Tit.SMMSE = WMMSEY = H Y (6) It is worth noting here that when s belongs to a QAM

That means if we replace H, y and W by H, and (Quadrature Amplitude Modulation) constellation, in order toW, respectively, then the MMSE detector can be described create a complex integer set sN C zN, proper shifting andequivalently as a ZF detector. We call this detector as the scaling are necessary. Detailed description of the these shiftingMMSE based on H or MMSE Ext detector. In the next section and scaling operations can be found in [10] and the referenceswe will introduce the concept about lattice reduction and LRA therein.detection.

IV. LIST QUANTIZATION LRA DETECTORS

III. INTRODUCTION OF LATTICE REDUCTION AND A. Principle of the list quantization methodLR-A-LINEARDETECTORS In this section, based on the list quantization proposed in

A. Principle of Lattice Reduction [9], we present an improved modified list quantization schemeThe basic concept of lattice reduction [16] is to consider for LRA detection. It is known from the previous section that

the noiseless receive signal vector in (1) the estimate of the transmit vector is given by

L2=HS =slhl+s2h2+..........+sNhN eCM(A, ......s=Tii. (8)

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empty, quit and return the estimate of LRA detection asthe final result.

. Step 3: Calculate the Euclidean distance from the re-ceived point to the LRA-based estimate point, update the

LRA-Detector distance for candidates in the list.. Step 4: Decide the best candidate which has the shortest

z=Hs dsanesy = Hs + z ZF/MMSE estimate: u u s distance.= (HT)(T-1s) + z Quantization: QQAM{S

= H'u + z u round {u} B. Complexity Analysisnew parameters: LLL algorithm In order to keep the additional complexity small, we take

u = -soutput :H',TH'= HT a further look into the main step of the scheme, i.e., the

Euclidian distance calculation. In most cases, the candidatesymbols from the list are different from the first LRA estimatesymbols by only one or two elements. Therefore, to calculatethe new distance, we only need to calculate the different part

Fig. 2. LRA detectors block diagram. of the term (y -H) inside the |. |. For example, if si and

s are different in the i-th element, the distance from ij canbe calculated as:

Thus an error in elements of ui will also cause an errorto s due to nonzero elements of the corresponding column D-= {hy-h (1) + ... + hN§l(N) + hs(i)} 2of T. It seems that when error happens the element-wise = {yi- hl(1) - -hN§(N)} - hi(i)§ 2difference between the optimum quantization and rounding where §i(i represents the ith element of i. We can seeoperation are mainly +1, -1, j or -j. Especially when there terese the ithcet of seo equan sthe noise level is low and the reduced channel matrix has that the term inside the bracket of the second equation isnearly orthogonal columns, the rounding error can not be already known when calculating the distance for s. Thea large number. With that kept in mind the modified list only calculation needed is the multiplication and additionq i is a b a follows. Ui operations when calculating his 1(i) and the norm. Moreover,quantization scheme iS explained briefly as flllows. Unlike

w a ersn h itnei h eldmi sn e

the method in [9], we suppose that error occurs in only one we can represent the distance in the real domain using newcomponent ili of u, then the transmitted signal element is variables H Hmainly expected to be iti = &i + sign(9{ifiii}- 9iij) or 1a{H}H R{H}] (10)vi = vi + jsign(3{fii - [3{&}i}] ). These candidate elementsare the nearest elements of the component Lii in the real and and the real-valued vectorsimaginary axis. If there are N transmit antenna elements, we [R{Y}I s* [_ 1(1obtain 2N+1 candidates of ui, including the original estimated [Q{Y}]SI - (1u. Based on (8), we have a list of 2N + 1 candidate symbols The real matrix H has the size m x n with m 2M ands. By removing the candidate symbols whose elements lying n = 2N. The real vectors y*, s* are 2N x 1 column vectors.outside the constellation, we can obtain a list of highly reliable Then the distance can be expressed in real domain as followscandidate symbols. Finally, the symbol nearest to the receivedpoint is chosen as the solution of the following equation D-={y - hts*(1) - .-h2NS*(2N)} - hi t(i) 2

s = arg min IIy - H 2 (9) It is clear that when comparing with hi§1(i), the number ofS required multiplication and addition operations in hi*s*(i) is

Since LRA detection with the LLL-reduced matrix usually reduced by half . For calculating the distance for the signalgives a good estimate, we may assume thats there is only point s we need (2N + 1)2M multiplications operations.one erroneous element in iu. Under this assumption, we can However, to update the distance D1 for the point sl, wegeneralize the list-quantization scheme for the LRA detection need only 4M multiplication operations. This means that ifas follows: the expected number of different elements in one candidate

Step 1: Create 2N candidate symbols from the estimate symbol is ndif then in order to update the distance we needof LRA detection. For i from 1 to N, add 2 candidates 4ndifdM multiplication operations. If the expected number ofto the list, changing element vii: candidate symbols in the list is Nlist we need 4NlistndifdM

multiplications for updating the distance, where -ndif is thevi=v + sign(9<{iii} - F93&}) number of different elements in average. The complexity of

and list quantization scheme is then (2N + 1)2M + 4Nl~iitdifM.&=a+ sign(:3{ii} - rC{v}r) Since the process in step 2 limits the number Nls and

rtdif to a small number, the additional complexity will be*Step 2: Remove those candidates lying outside the signal negligible compared to the total complexity. For example, withconstellation to achieve the list of candidate. If the list is 4 x 4 MIMO, at SNR= 10dB, the number value is around

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2forLRA-ZF detection and 3 or 4 for LRA-MMSEExt 100 -0--.............. Proposed list-quantization LRA-MMSE.

detection.(see figure 6). The complexity is then approximately .. Proposed list-quantizationLequal to complexity of calculating the ML distance for two ':: L

signal points. 101

V. SIMULATION RESULTS

Simulations were run for a 4 x 4 MIMO systems. The m 102channel was assumed to be the flat Rayleigh fading channelwith its complex gain generated using complex Gaussian 10-3 Xrandom variables and 4-QAM was used for modulation. X XBERs of the LRA-ZF/MMSE detectors with and without X

list-quantization are shown in Fig. 3 for 4 x 4 MIMO systems. 10 0 5 10 15 20In this figure, we compare BERs obtained using LRA detectors Eb/Nowith and without list quantization. It can be seen from thefigures that when the list quantization scheme is applied, the Fig. 3. BER performance of LRA-ZF/MMSE detection with and withoutBER performance of the LRA-ZF/MMSE detector is improved list-quantization method, MIMO 4 x 4, QAM modulationsignificantly. At the BER = 10-3, the list quantization LRA-ZFcan gain nearly 2.5dB over the conventional LRA-ZF, and the 10° L...l.l.. PFroposedLRA-ZFperformance gain of LRA-MMSE based on H is about 1.5dB.It is clear that the LRA-MMSE based on H with list quan- Proposed LRA-MMSE change 1,2 elements

tization can achieve nearly ML BER performance. In Fig.4, -0- P MLwe make another comparison of the list-quantization methodfor two cases. First, list of candidate is generated by changing z1 -2

only one element of the estimate. Second, list of candidate is mgenerated by changing one and two elements of the estimate.The improvement is almost the same in both cases. Therefore, lo-3 ..--it is enough to use the list of quantization by changing onlyone element. This fact also prove that in the MIMO systems iwith small number Tx,Rx antennas, the quantization error 10 2 4 6 8 10 12 14 16 18 2usually appears in only one element of the estimated vector ii. Eb/NOIn Fig.5 we can see the efficiency of list quantization moreclearly with the result for 16QAM modulation. The LRA- Fig. 4. BER performance with and without list-quantization method, MIMO

4 x 4 system, 4QAM modulation, list of candidates is created by changing 1ZF with list quantization even outperforms the conventional and 2 elementsLRA-MMSE. We can see from the figure that, in the cases wecreate the list by changing 1 and 2 elements, we lose morecomplexity but get almost the same performance. Finally, Fig.6 result shows that the method is efficient enough to generateshows the average number of candidate symbols in the list and the list of candidates by changing only one element of thethe number of different elements. Based on the complexity estimate. We also calculate the complexity needed for list-analysis from the previous section and the value in the figure, quantization scheme. We showed that the additional complex-the number of multiplication and addition operations is just ity is reasonable and becomes small in high SNR range.several multiples of number of transmit antenna elements. REFERENCESTherefore, if the number of antenna elements is not so large,only small additional complexity is required. With the above [1] I B. Telatar, "Capacity of multi-antenna Gaussian channels,"

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E /N

Fig. 5. BER performance of LRA-ZF/MMSE detection with and withoutlist-quantization method, MIMO 4 x 4, 16QAM modulation, list of candidatesis created by changing 1 and 2 elements

-U- Average of list size - LRA-MMSE9 -0- Average of list size - LRA-ZF

---Average number of different elements - LRA-MMSE8 -0--Average number of different elements - LRA-ZF

7 -

6

5

E

3-

2-

C

6 8 10 12 14 16 18 20

Eb/No

Fig. 6. The average size of the list and average number of differentelements,MIMO 4 x 4, QAM modulation

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