A New Lattice Reduction Algorithm for LR-aided MIMO ... - CiteSeerX

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A New Lattice Reduction Algorithm for LR-aided MIMO Linear Detection Chiao-En Chen, Member, IEEE and Wern-Ho Sheen, Member, IEEE

Abstract— Lattice reduction (LR) has recently emerged as a promising technique for improving the performance of suboptimal multiple-input-multiple-output (MIMO) detectors. For LRaided MIMO detection, the Lenstra-Lenstra-Lov´asz (LLL) and Seysen’s algorithm (SA) have been considered almost exclusively to date. In this paper, we introduced a new LR algorithm for LR-aided linear detection (LD). In contrast to the LLL and SA, which are targeted to search for bases with relatively short basis vectors, the proposed algorithm has been designed to improve the minimum Euclidean distance of the LR-aided linear detector, thus exhibiting improved error rate at high SNR. The error-rate performance of the proposed algorithm as well as the required complexity has been demonstrated through extensive computer simulations. Index Terms— Lattice reduction, MIMO, Seysen’s algorithm

I. I NTRODUCTION

T

HE enormous demand for high data rate wireless telecommunication services has generated significant research interests in the multiple-input-multiple-output (MIMO) technologies. By deploying multiple antennas at the transmitter and receiver in a rich scattering environment, it has been shown that considerable capacity enhancement can be obtained [1]. Higher data rates can therefore be achieved economically by employing spatial-multiplexing, together with low-complexity MIMO detectors, such as zero-forcing (ZF), minimum-mean-square error (MMSE), and ordered successive interference cancellation (OSIC) detectors [2]. However, all these low-complexity detectors suffer from serious performance degradation from the optimal ML detector and do not achieve the full receive diversity of the MIMO fading channels. Recently, lattice-reduction (LR) has been introduced as a promising technique, which can improve the performance of many suboptimal MIMO detectors [3]–[8]. For LR-aided MIMO detection, the Lenstra-Lenstra-Lov´asz (LLL) algorithm [9] and the Seysen’s algorithm (SA) [10] have been considered almost exclusively to date. As the error rate of a system varies significantly with the type of detectors in use, the LLL and SA, which are designed to generate bases with relatively short basis vectors regardless of the receiver’s structure, may not provide the best performance. In this paper, we have proposed a new LR algorithm for LR-aided MIMO linear detection (LD). The proposed This work was supported by the National Science Council (NSC), Taiwan, under Grant Number NSC98-2221-E-194-020-MY2. Chiao-En Chen is with the Department of Electrical/Communications Engineering, National Chung Cheng University, Chiayi, Taiwan. (e-mail: [email protected]). Wern-Ho Sheen is with the Department of Information and Communication Engineering, Chaoyang University of Technology, Taichung, Taiwan. (e-mail: [email protected]).

algorithm consists of a standard reduction algorithm, such as LLL or SA, followed by a newly proposed C(p,k) reduction algorithm. The LLL or SA serves as an initialization stage, while the C(p,k) reduction algorithm is designed to improve the minimum Euclidean distance from the received signal vector to the decision boundary of an LR-aided linear detector. By taking the decision region into account, we showed that substantial performance improvement in the error rate, when compared with the conventional LR-aided linear detection, can be obtained with only moderate increase in the complexity. II. P RELIMINARIES A. System description Let us consider an uncoded MIMO system with Nt transmit and Nr receive antennas, where Nt ≤ Nr . The data stream is first de-multiplexed into Nt data substreams, mapped onto rectangular QAM symbols, and then transmitted over the Nt antennas simultaneously over a frequency-flat fading chanFor any given time instant, the received vector yc = nel. T c c can be represented by the well-known basey1 , . . . , yN r  T c band model y = Hc xc + wc , where xc = xc1 , . . . , xcNt c is the transmitted data  c vector, cHT is the Nr × Nt channel c is the noise vector. The matrix, and w = w1 , . . . , wNr Nt transmitted symbols {xck }k=1 are represented as drawn from a subset of complex-integers after some proper scaling and shifting [4, 11], while the wc is modeled as a zero-mean white 2 Gaussian random vector with covariance matrix σw INr . For subsequent notational convenience, an equivalent realvalued model y = Hx + w is introduced, where T x =  T Re(xc )T , Im(xc )T , y = Re(yc )T , Im(yc )T , w =  T Re(wc )T , Im(wc )T , and   Re(Hc ) −Im(Hc ) H= . (1) Im(Hc ) Re(Hc ) The dimension of H is set to be N × M , where N = 2Nr and M = 2Nt . B. LR-Aided MIMO detection In LR-aided MIMO detection, the noiseless received signal vector is interpreted as a point in the M dimensional lattice L (B), given by ( M ) X L (B) = cm bm , cm ∈ R, m = 1, . . . , M , (2) m=1

where the matrix B = [b1 , . . . , bM ] is called the basis of L (B). As a lattice can be generated by an infinite number

2

˜ = BT of bases, it can be shown that a different basis B also generates L (B), if and only if T is unimodular [12]. ˜1 , · · · , b ˜ M ], the noiseless ˜ = [b With the new lattice basis B ˜ where received signal vector Bc is also represented as Bz, T −1 c = [c1 , . . . , cM ] and z = T c. The idea of LR-aided MIMO detection is to first detect the ˜ and then transform the coordinate vector z in the new basis B result ˆ z back to the original basis B via ˆ c = Tˆ z. With the ˜ can be help of many existing LR algorithms, the new basis B designed to be near-orthogonal, which significantly improves the reliability of many low-complexity suboptimal detectors [3, 4, 13]. Two popular LR algorithms in the communications literature are the LLL [9] and the SA [10]. The LLL algorithm features polynomial time complexity and guarantees to find the shortest basis vector up to an exponential factor by using an upper bound on the orthogonality defect, defined by ˜ = δ(B)

QM

˜ m k2 kb  , ˜TB ˜ det B m=1 

(3)

˜ and while the SA simultaneously reduces the primal basis B its dual −1  ˜# , . . . , b ˜# ] = B ˜TB ˜ ˜ # = [b ˜ B (4) B 1 M

by minimizing the Seysen’s measure ˜ = S(B)

M X

2 ˜ m k2 kb ˜# kb mk

(5)

m=1

iteratively until a local minimum is found. Note that both the orthogonality measure and the Seysen’s measure attain their ˜ is orthogonal. For M ≤ minimum when the reduced basis B 31, it has been observed that the SA is capable of finding bases with shorter basis vectors, when compared with the LLL [8, 10].

C. Performance of LR-aided linear detection The performance of LR-aided MIMO detection based on LLL and SA has been studied and compared recently [8, 14]. It has been shown that with the assistance of LR, full receive diversity of the MIMO fading channels can be achieved with low-complexity suboptimal detectors. For the case of linear detection, SA has been shown to consistently outperform the LLL algorithm in the sense of generating more orthogonal bases, thus providing lower error rate [8, 14]. In [15], the author performed an analysis on the proximity factors and proposed to select the basis such that the minimum Euclidean distance from the received signal vector to the detector’s decision boundary is maximized. For linear detection, the decision region is simply the parallelotope defined by ( ) M X ˜ ˜ αm hm , |αm | ≤ 1/2 , P(x, H) = u + (6) m=1

˜1 , . . . , h ˜ M ] for ZF detection, ˜ =H ˜ = [h where u = Hx, H and T  (7) u = (Hx)T , 0TM , iT h i h ˜ ,...,h ˜ ˜ ,h ˜ = h ˜T H (8) 1 2 M = H , σ w IM

for MMSE detection [13, 16]. It follows that the Euclidean distance from the received signal vector u to the mth facet of ˜ is P(x, H)

1 ˜ (9) kh k sin θm , m = 1, . . . , M, 2 m ˜ and the hyperplane where θm denotes the angle between h m o n ˜ m−1 , h ˜ m+1 , . . . , h ˜ M . Using the ˜1 , . . . , h spanned by h dLD m =

union bound, the error rate for linear detection Pe can then be upper bounded by  LD  M X dm . (10) Q Pe ≤ 2 σw m=1

As the bound is dominated by the minimum distance dLD min and is tight at high SNR, it is expected that the max-dLD min criterion shall provide near-optimal error rate performance for sufficiently high SNR. However, constructing an LR algorithm that achieves this goal remains an open problem [15]. III. A

NEW

LR ALGORITHM FOR IMPROVED LR- AIDED MIMO LINEAR DETECTION

A. New criteria for LR-aided linear detection Before deriving the proposed LR algorithm, we first present a new family of LR criteria for LR-aided linear detection. Consider the following sequence {GLD } defined by o n ˜ M sin θM k ˜ 1 sin θ1 k, . . . , kh (11) {GLD } = kh and let r be an extended real number. Then the generalized mean [17] of {GLD } of order r is defined as !1/r PM r ˜ (r) LD m=1 khm sin θm k µM ({G }) = , (12) M

which includes the minimum, harmonic mean, geometric mean, arithmetic mean, and the maximum as special cases corresponding to r = −∞, r = −1, r = 0, r = 1, and r = ∞, respectively. It follows that dLD min can be expressed as   1 ˜ m k sin θm = 1 µ(−∞) ({GLD }). dLD min kh min = 2 m=1,..., M 2 M (13) Inequalities that hold for the arithmetic, geometric, and harmonic means can also be considered as special cases of the following inequality [17]: (s)

(t)

µM ({GLD }) ≤ µM ({GLD }), for − ∞ ≤ s < t ≤ +∞, (14) ˜ 1 sin θ1 k = . . . = and the equality holds if and only if kh ˜ M sin θM k, or one of the kh ˜ m sin θm k’s is zero with s ≤ 0. kh

3

From the generalized-mean inequality (14) and the expression in (13), we come up with the idea1 of approximating 1 (−p) LD }) for some sufficiently large positive dLD min by 2 µM ({G (−p) number p. A lower bound for 21 µM ({GLD }) can be obtained as 1/p  M (−p)  (15) µM ({GLD }) =  PM 1 ˜ kp | sin θm |p m=1 kh m



 ≥ P =

1/p

M

˜ # kk kh M m ˜ kp−k | sin θm |p m=1 kh m

M

˜ # p+k kh ˜ kk m m=1 khm k

PM

(16)

 

!1/p

,

(17)

for any 0 ≤ k ≤ p. The inequality in (16) is due to the property [19] ˜# ˜ m kkh kh mk

1 ≥ 1, = | sin θm |

M X

#

˜ m kp+k kh ˜ m kk kh

˜ ′ = H(I ˜ + λi,j Ui,j ) H ˜ ,...,h ˜ ,h ˜′ , h ˜ = [h 1

˜ ′j h

(19)

j−1

˜j , ˜i + h = λi,j h

m=1

m

m

dLD min .

proximation of By choosing a proper (p, k) pair, we can optimize this approximation. It follows that the criterion of maximizing dLD min can now be well-approximated using the ˜ A more detailed discussion criterion of minimizing C(p,k) (H). on the selection of (p, k) is available in Section III-C. B. Proposed LR algorithm In this subsection, we present a two-stage LR algorithm, ˜ In which is targeted to minimize the measure C(p,k) (H). the first stage, a standard LR algorithm such as LLL or SA is performed on H, defined as H for ZF detection and [HT , σw IM ]T for MMSE detection, and then generates the ˜ 1 and unimodular matrix T1 . In the second reduced basis H ˜ 1 and stage, a new C(p,k) LR algorithm is performed on H ˜ generates the reduced basis H and unimodular matrix T. The C(p,k) algorithm is designed to search for bases with ˜ metrics, and hence, leads to the improved smaller C(p,k) (H) ˜ and performance. The proposed algorithm finally outputs H 1 It has been brought to authors’ attention that the idea is conceptually related to [18], where the precoding problem with peak-power reduction is considered.

(21) (22)

˜ # (I − λi,j Uj,i ) =H

= # ˜′ i h

=

# ˜# , . . . , h ˜# , h ˜′ , h ˜ # ], ˜# , . . . , h [h i 1 i−1 i+1 M # # ˜ i − λi,j h ˜j . h

(24) (25) (26)

ˆ i,j , such The algorithm then seeks for an integer number λ (p,k) ˜ ′ (p,k) ˜ that the decrement ∆(λi,j ) = C (H ) − C (H) is minimized (most negative), i.e. ˆ i,j = arg min ∆(λi,j ). λ λi,j

m=1

is minimized. It is also interesting to note that when p = ˜ ˜ is exactly the Seysen’s measure S(H). 0, k = 2, C(0,2) (H) This brings a new interpretation to the SA, which can now be viewed as a practical method that aims at minimizing a lower (0) bound of µM ({GLD }). From the above-mentioned discussion, we propose to use  P 1/p M ˜ kk ˜ # kp+k kh the quantity M/ kh , as an ap-

j

(20) ˜ j+1 , . . . , hM ],

where i, j ∈ {1, . . . , M }, i 6= j. It follows that the dual basis ˜ ′ can be expressed as of H  −1 ′# ′ ′T ′ ˜ ˜ ˜ ˜ (23) H =H H H

(18)

and the equality holds, if and only if k = 0 or θm = π2 for all m = 1, . . . , M . (−p) From (17), it is clear that the lower bound of µM ({GLD }) is maximized if the measure ˜ = C(p,k) (H)

T as the reduced basis and unimodular transformation matrix, respectively. The C(p,k) algorithm is described as follows: Initialization: The algorithm initializes with T = T1 and ˜ = H1 . H Iterations: Let Ui,j denote the M ×M matrix with a one at the (i, j)th entry and zero elsewhere. Then, in each iteration, we consider the following column operation:

(27)

˜ (19), the Using (20)-(26) and the definition of C(p,k) (H) calculation of ∆(λi,j ) can then be simplified as ˜ # − λi,j h ˜ kk kh ˜ # kp+k ˜ # kp+k + kλi,j h ˜ +h ˜ kk kh ∆(λi,j ) = kh i i j i j j p+k p+k ˜# ˜# ˜ j kk kh ˜ i kk kh − kh . − kh i k j k

(28)

If we only consider positive integers p and k, then ∆(λi,j ) corresponds to a polynomial function of λi,j with degree p+k. For the Seysen’s case (p = 0, k = 2), ∆(λi,j ) is simply a ˆ i,j can parabolic function, and hence, the best update value λ be easily obtained as [10]     T #T # 1 h ˜ ˜ ˜ ˜ h h h  i i j  j ˆ i,j =  (29) λ   # 2 − 2  ,  2 ˜

˜ h 

i

h j

where ⌊·⌉ denotes the operator that rounds the argument to the nearest integer. For another special case, where p = 2 and k = 0, ∆(λi,j ) is also a parabolic function, and hence, we can also obtain a closed-form solution for the best update value    #T # h ˜ ˜ hj  i ˆ i,j =  (30) λ  # 2  .

˜ 

h j 

For other general (p, k), there appears to be no closed-form ˆ i,j . Fortunately, this is not a problem, because solution for λ one can simply use (29) or (30) as an initialization and performs a one-dimensional integer grid search in the descent direction until a local minimum is reached. The local minimum

4

found is essentially the global optimum, because ∆(λi,j ) is a convex function of λi,j [20]. ˆ i,j and ∆(λ ˆ i,j ) are computed for all the candidate After λ pairs, one can select the pair (ˆi, ˆj) that gives the largest ˜ i.e. decrease in C(p,k) (H), (31)

˜ can then be updated accordingly via The matrices T and H T = [t1 , . . . , tˆj−1 , ˆ tˆj , tˆj+1 , . . . , tM ], ˆ tˆj = λˆi,ˆj tˆi + tˆj , ˜ = H

˜1 , . . . , h ˜ˆ , h ˜ˆ , . . . , h ˜ˆ′j , h ˜ M ], [h j−1 j+1

˜ ˜ˆ′j = λˆ ˆ h ˜ . h i,j ˆi + hˆ j

(32) (33)

0.58

(34)

0.56

(35)

(p,k)

The C algorithm repeats the iteration procedures until ˜ is ˜ is possible. As C(p,k) (H) no more reduction in C(p,k) (H) monotonically decreasing and bounded below, it is clear that the C(p,k) reduction algorithm is guaranteed to converge to at least a local minimum within finite number of iterations. A summary of the algorithm is given in Table I.

0.54 min

(i,j)

0.52

d

ˆ i,j ). (ˆi, ˆj) = arg min ∆(λ

large. Furthermore, the value of p should be sufficiently large, (−p) because 21 µM ({GLD }) is a better approximation of dLD min for larger p. This observation is verified from the numerical simulations shown in Fig. 1, where the average dLD min ’s of the proposed LR-aided zero-forcing detectors, averaged over 105 channel realizations are plotted with different (p, k) combinations. It can be observed that dLD min tends to decrease as k increases, and tends to increase as p increases.

0.5 0.48 0.46

Input: real matrix H, nonnegative integers p and k. ˜ unimodular real matrix T. Output: real matrix H, 1: Perform standard LR algorithm such as LLL or SA on H and ˜ and unimodular matrix T1 . generates the reduced basis H 1 ˜ := H , T := T1 , f lag := 1. 2: Set H 1 3: while f lag = 1 do 4: for i := 1 to M do 5: for j := 1 to M , j 6= i do 6: if (p, k) = (0, 2) then ˆ i,j as (29). 7: Compute λ 8: else if (p, k) = (2, 0) then ˆ i,j as (30). 9: Compute λ 10: else 11: Use (29) or (30) as an initialization and perform a one-dimensional integer grid search for the minimum ˆ i,j for the (p + k)th degree polynomial (28). point λ 12: end if ˆ i,j ) as (28). 13: Compute ∆(λ 14: end for 15: end for 16: Determine the pair (ˆi, ˆ j) using (31). ˆˆ ˆ ) ≥ 0 then 17: if ∆(λ i,j 18: Set f lag := 0. 19: else ˜ via (32), (33), (34), and (35). 20: Update T and H 21: end if 22: end while

Remark: The selection method described in (31) corresponds to the so-called “greedy” selection in the literature. One can also come up with an algorithm using “lazy” selection, which simply chooses any available (i, j) pair that leads ˜ Our experiences as well as the to decrease in C(p,k) (H). empirical analysis for the SA [19] show that the greedy selection is generally capable of generating bases with similar performance as the lazy selection with less reduction steps. C. Discussion on the selection of (p, k): From (17), it can be noted that the bound becomes looser as k increases, and hence, the value for k should not be too

0

SA−C

(p,1)

SA−C

(p,0)

(p,2)

SA−C

(p,3)

SA−C

(p,1)

LLL−C

(p,2)

LLL−C

(p,3)

LLL−C

0.44

TABLE I S UMMARY OF THE PROPOSED LR-C(p,k) - REDUCTION ALGORITHM

(p,0)

LLL−C

2

4

6

8

10

p

(p,k) reduction Fig. 1. dLD min of the zero-forcing detector after LR − C (Nr = Nt = 8).

The choice of the parameters (p, k) is also subjected to implementation considerations. First, when implemented in a practical system, the value for p cannot be too large, because larger p requires larger dynamic range. Second, even numbers for p and k are more preferred because the computationally intensive square root operations can be avoided. In addition to these design considerations, one particular choice (p, k) = (2, 0) has the unique complexity advantage, because it alˆ i,j . As the most lows closed-form expression for computing λ advanced MIMO communications standards to date, such as IEEE 802.16m and 3GPP LTE-A only support up to eight spatial streams, our experiences as well as the simulation results show that at least for the case where Nt ≤ 8, (p, k) = (2, 0) stands out as an important tradeoff point. As a result, we decided to explore this particular choice in more detail in the following context. Remark: The proposed algorithm has a similar structure as that of the SA, and therefore, can be easily modified for complex bases as in [8]. Implementing the algorithm on the complex basis may have some complexity advantages as suggested by the study on complex LLL algorithm [5]. Detailed comparison study between the real and complex implementations is of practical interest and will be examined in our future work. IV. S IMULATIONS In this section, we provide the numerical simulations for the proposed LR algorithm. Throughout the simulations, a 8 × 8 MIMO channel with each element of the channel matrix

5

H generated i.i.d. from a zero-mean circularly symmetric complex Gaussian process with unit variance is considered. In Fig. 1, it can be observed that the proposed LLL-C(2,0) and SA-C(2,0) provide substantial improvement in dmin , when compared with the conventional LLL or SA. As the error rates of LR-aided linear detectors at high SNRs are dominated by the dLD min ’s, the improvement in the minimum Euclidean distance results in lower bit-error-rates (BERs), as shown in Fig. 2 and 3, where the BER of an 8 × 8 MIMO system using LR-aided linear detection with 4-QAM and 16-QAM are plotted, respectively. 0

10

MMSE−OSIC ML −1

LLL−ZF SA−ZF

LLL−C(2,0)−MMSE

LLL−C(2,0)−ZF

SA−C(2,0)−MMSE

SA−C(2,0)−ZF

−2

10

−3

10

−4

10

−5

10

10

12

14

16

18

E /N (dB) b

0

Fig. 2. BER performances of various LR-aided linear detectors (Nr = Nt = 8 antennas with 4-QAM).

MMSE−OSIC ML

−1

10

LLL−ZF SA−ZF

LLL−MMSE SA−MMSE

LLL−C(2,0)−ZF SA−C(2,0)−ZF

LLL−C(2,0)−MMSE

Bit error rate (BER)

SA−C(2,0)−MMSE

Empirical cumulative distribution function (CDF)

Bit error rate (BER)

10

LLL−MMSE SA−MMSE

compared with the ML and MMSE-OSIC detectors, it can be observed that the LR-aided schemes tend to achieve similar diversity order as the ML as predicted from the theory [7], and thus provide performance gain at high SNR values over the MMSE-OSIC. The simulation results have also verified the observations in [21] that the LR-schemes provide more significant gain for higher order modulations, because the performance loss due to boundary effect in the quantization is negligible in large constellation cases. To study the performance and complexity tradeoff, we compared the time complexity of the LLL-C(2,0) and SAC(2,0) with that of the conventional LLL and SA (Fig. 4), demonstrating the empirical CDFs of the total number of iterations. The computational complexity was also compared (Fig. 5), showing the empirical CDFs of the total number of floating point operations (flops). The simulation results showed that the SA has the lowest time complexity, followed by the SA-C(2,0) , LLL, and LLL-C(2,0) , whereas the LLL has the lowest computational complexity, followed by the LLL-C(2,0) , SA, and SA-C(2,0) . 1 SA (2,0) SA−C LLL (2,0) LLL−C

0.8

0.6

0.4

0.2

0 0

100

200 300 Number of iterations

400

−2

10

Fig. 4. CDFs of the number of iterations for a variety of lattice reduction algorithms (Nr = Nt = 8 antennas) −3

10

V. C ONCLUSIONS

−4

10

14

16

18

20

22

E /N (dB) b

0

Fig. 3. BER performances of various LR-aided linear detectors (Nr = Nt = 8 antennas with 16-QAM).

In Fig. 2, the simulation results show that the LLL-C(2,0) and SA-C(2,0) provide roughly 0.39 and 0.49 dB performance gain, respectively, in the ZF case, when compared with the SA, and roughly 0.33 dB gain in the MMSE case. On the other hand, Fig. 3 shows that the LLL-C(2,0) and SA-C(2,0) provides roughly 0.31 and 0.52 dB performance gain, respectively, when compared with the SA in the ZF case, and roughly 0.26 and 0.34 dB gain, respectively in the MMSE case. When

In this paper, we proposed a new LR algorithm for LR-aided linear detection. The proposed algorithm uses the LLL or SA as an initialization stage, and then improves the minimum Euclidean distance of the associated LR-aided linear detector by the proposed C(2,0) algorithm. In addition, through computer simulations, we also demonstrated that the proposed algorithm is capable of improving the BER performance of LLL- or SAaided linear detection at high SNR with only moderate increase in time and computational complexity. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions that greatly improved the quality of this manuscript.

Empirical cumulative distribution function (CDF)

6

1 LLL (2,0) LLL−C SA (2,0) SA−C

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

2

4 6 8 10 12 14 16 5 Number of floating point operations (flops) x 10

Fig. 5. CDFs of the number of floating point operations for a variety of lattice reduction algorithms (Nr = Nt = 8 antennas)

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