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[20] A. El Gamal, J. Mammen, B. Prabhakar, and D. Shah, “Throughput–delay trade-off in wireless networks,” in Proc. IEEE INFOCOM, 2004. [21] D. Gross and C. M. Harris, Fundamentals of Queueing Theory, 3rd ed. Hoboken, NJ: Wiley, 1998. [22] M. J. Neely and E. Modiano, “Capacity and delay tradeoffs for ad-hoc mobile networks,” IEEE Trans. Inf. Theory, vol. 51, no. 6, pp. 1917–1937, Jun. 2005.
Fixed-Complexity LLL-Based Signal Detection for MIMO Systems Yusik Yang and Jaekwon Kim, Member, IEEE Abstract—Recently, a hardware-friendly fixed-complexity Lenstra– Lenstra–Lovász (fcLLL) algorithm has been developed for use in the linear receivers of multiple-input–multiple-output (MIMO) systems. In this paper, we propose a novel fcLLL-based detection method that significantly outperforms conventional fcLLL-based linear receivers at a lower fixed complexity. The main idea is that the complexity of fcLLL can be significantly reduced by disregarding a single column of the channel matrix. We also propose column selection methods that consider both the error performance and complexity simultaneously. Simulations are performed to demonstrate the efficacy of the proposed method. Index Terms—Fixed complexity, lattice reduction (LR), Lenstra– Lenstra–Lovász (LLL), multiple-input–multiple-output (MIMO).
I. I NTRODUCTION Multiple-input–multiple-output (MIMO) techniques can increase data transmission speed or wireless channel reliability without using additional transmission power or spectral bandwidth. As a result, various multiple-antenna techniques have been adopted in recent wireless communication standards [1], [2]. High-speed data transmission is enabled by spatially multiplexed (SM) MIMO systems, for which the main challenge is the high signal detection complexity at the receiver. Fortunately, lattice reduction (LR) techniques, the most common of which is Lenstra–Lenstra–Lovász (LLL) [3], are able to reduce the complexity of SM MIMO receivers. In [4], it was shown that LLLassisted linear detection methods offer the same order of diversity as maximum-likelihood (ML) detection, which is widely considered to be computationally prohibitive. In [5]–[7], a complex LLL (CLLL) algorithm was proposed and was shown to require approximately half the complexity of a real LLL (RLLL) algorithm. In [8], to further reduce the complexity of CLLLassisted linear detectors, partial LR is performed, and the symbol vector is estimated using the partially reduced channel matrix and then allowing the LR to be terminated early if the estimated symbol vector is reliable. In [9], the temporal correlation between consecutive channel matrices is exploited to reduce the LLL complexity. The unimodular matrix obtained from the LLL reduction of a channel matrix is reused in the LLL reduction of the following channel matrix, thereby lowering the average complexity over a long time. Manuscript received June 27, 2012; revised September 10, 2012; accepted October 13, 2012. Date of publication October 22, 2012; date of current version March 13, 2013. This work was supported by the National Research Foundation of Korea through the Basic Science Research Program funded by the Ministry of Education, Science, and Technology under Grant 2010-0003922. The review of this paper was coordinated by Dr. T. Jiang. The authors are with the Computer and Telecommunications Engineering Division, Yonsei University, Wonju, Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/TVT.2012.2225856
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This approach is efficient, particularly when the channels slowly fade. All of the previous works in this area [4]–[9] addressed the average complexity of the LLL LR. In implementing receivers as hardware, however, it is important to guarantee a fixed worst-case complexity to improve run-time predictability. In [10], it was shown that the worstcase complexity of both RLLL and CLLL is not bounded. In an effort to bound the worst-case complexity, the maximum number of possible LLL loops is fixed in [11], leading to the name fixed-complexity LLL (fcLLL). To mitigate the degradation of the error performance that is a consequence of the fixed complexity, the original-LLL algorithm was modified such that random channels are better conditioned after a fixed number of LLL loops. In this paper, we compare fcLLL and the original LLL in terms of the required number of floating-point operations (flops) rather than LLL loops to make the comparison more fair. We then show that the probability of complete LLL reduction by the fcLLL is higher than that by the original LLL when the complexity of the two methods is fixed to be higher than a certain threshold. As will be shown by numerical examples, sufficient error performance demands a higher fixed complexity than the threshold. Thus, we adopt fcLLL in developing LLL-based receivers. In this paper, we propose a novel fcLLL-based signal detection technique for MIMO systems. The main idea is that the complexity of fcLLL can be significantly reduced if we disregard a column of the channel matrix. We also propose column selection methods, considering both error performance and the worst-case complexity. It is shown that the proposed receiver significantly outperforms the conventional fcLLL-based zero-forcing (ZF) receiver in terms of both error performance and the worst-case complexity. II. R EVIEW OF THE F IXED -C OMPLEXITY L ENSTRA –L ENSTRA –L OVÁSZ Here, we briefly discuss the system model and review the fcLLL algorithm [11]. A. System Description We consider MIMO systems with NT transmit antennas and NR receive antennas. The transmitted signal vector is denoted as x = [x1 x2 · · · xNT ]T , where xi is the transmitted signal from the ith transmit antenna; y = [y1 y2 · · · yNR ]T denotes the received signal vector, where yj is the received signal at the jth receive antenna; H denotes the channel matrix with dimensions of NR × NT , of which entry hj,i is the channel gain between the ith transmit antenna and the jth receive antenna; and z = [z1 z2 · · · zNR ]T denotes the noise vector, where zj is the noise at the jth receive antenna and is assumed to be a zero-mean complex white Gaussian with a variance of σz2 . The MIMO system can then be mathematically described as follows: y = Hx + z =
NT
h i xi + z
(1)
i=1
where hi is the ith column of H. In this paper, we assume independent and identically distributed Rayleigh fading channel gains and ideal channel estimation at the side. receiver NT hi xi |xi ∈ Ω, i = 1, 2, . . . , NT } in (1) The combinations { i=1 can be viewed as a complex lattice with basis of {hi , i = 1, 2, . . . , NT }. Lattice basis reduction is based on the concept of transforming a given basis of a lattice into another basis of the same lattice so that the new basis vectors are nearly orthogonal and that the norm values of the basis vectors are not small [11]. When these two conditions are met by the transformed basis vectors, then the matrix that is composed of
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TABLE I COMPLEXITY OF fcLLL
Fig. 1. Comparison of the original LLL and modified LLL with H, HcN , T and Hcp [p is selected by (16)] in terms of the required number of flops for complete LLL reduction when δ = 1, NR = NT = 8.
The fcLLL algorithm, the modified LLL with a fixed number of LLL loops L = Niter (NT − 1), is summarized in Table I. In Table I, sizered[rk ] and colm-exch[rk−1 , rk ] denote the size reduction step and column exchange step, respectively. In the evaluation of the number of flops in Table I, we assumed six flops for a complex multiplication, two flops for a complex addition, two flops for a complex rounding, and one flop for all real operations such as multiplication, division, addition, max, and square root. The complexity or the required number of flops of these two steps is calculated as follows: flops (size-red[rk ]) =
k−1
(4 + 8j + 8NT )
j=1
= (k − 1)(4 + 8NT + 4k) Fig. 2. Comparison of original-LLL-ZF, fcLLL-ZF with various numbers of LLL loops and the proposed detection technique with various p selection methods in terms of VER performance when δ = 1, NR = NT = 8, and |Ω| = 16.
the basis vectors has a small condition number. The LLL algorithm is composed of two steps: the size reduction step, which transforms the basis to be nearly orthogonal, and the column exchange step, which tries to increase the norm values. In [11], it was analyzed that the fcLLL also fulfills the aforementioned purpose of the original-LLL algorithm. B. Comparison of fcLLL and the Original LLL The difference between the original LLL and fcLLL (or the modified LLL) is well shown in [11, Figs. 1 and 2]. In these two figures, the variable k is the column index, and M is the number of transmit antennas, i.e., NT in this paper. From [11, Fig. 1], we can observe that the column index k can either increase or decrease in the original LLL in a random manner until k = NT + 1; however, k in the modified LLL can only increase in a circular manner, as shown in Fig. 2. The required number of LLL loops for complete LLL reduction is random for both the original- and modified-LLL algorithms. When the number of loops was fixed, it was shown that more random matrices are LLL reduced by the modified LLL than by the original LLL.
flops (colm-exch[rk−1 , rk ]) = 71 + 28(NT + NR − k).
(2) (3)
From (2) and (3), we can see that the number of flops of these two steps depends on the column index k, and as k grows, flops (size-red[rk ]) increase, and flops (colm-exch[rk−1 , rk ]) decrease. Thus, we compare the complexity of LLL reduction algorithms in terms of the required number of flops rather than the number of LLL loops in this paper. The worst-case complexity of fcLLL occurs when the value μ in line 6 is nonzero and the condition in line 9 is satisfied for all k. The worst-case complexity can then be expressed as follows: max (flops (fcLLL[H, δ, Niter ])) = flops (qr[H])+Niter ·
NT
{flops (size-red[rk ])+flops (colm-exch[rk−1 , rk ])} . (4)
k=2
We note that the worst-case complexity for the original-LLL algorithm cannot be calculated because it is not bounded [10]. The input parameter δ in (4) tradeoffs between the complexity and the LLL reduction quality. C. Numerical Example Contemporary communication systems such as WirelessMANAdvanced systems [1] and LTE-Advanced systems [2] use up to eight antennas to increase the data transmission rate. Using parameter values
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where hp is the pth column of H, Hcp = [h1 · ·hp−1 hp+1 · ·hNT ], and xcp = [x1 · ·xp−1 xp+1 · ·xNT ]T . Let us consider the following signal detection method using (7). Step 1: Generation of candidate vector set, i.e., S=
ˆ |ˆ x xcp = D
Hcp
†
(y − hp x ˆp ) , x ˆp ∈ Ω
(8)
where (·)† denotes the pseudo inverse, Ω is the set of constellation points, and D{·} is an appropriate QAM slicer. Step 2: Search for the optimal vector in S, i.e., ˆ opt = arg min y − Hˆ x2 . x
(9)
ˆ ∈S x
Fig. 3. Comparison of original-LLL-ZF, fcLLL-ZF (L = 56), and the proposed detection with p selected by (16) in terms of the required number of flops when δ = 1, NR = NT = 8, and |Ω| = 16.
δ = 1 and NT = NR = 8, we compare the original-LLL, modifiedLLL, and fcLLL algorithms. In Fig. 1, it is shown that the complexity variance is much smaller in the modified LLL than in the original LLL. When the fixed complexity is set as smaller than 2.4 × 104 , the original LLL offers a higher probability of complete LLL reduction than the modified LLL does, but the reverse is true when the fixed complexity is larger than 2.4 × 104 . Fig. 2 compares these two LLL algorithms in terms of vector error rate (VER) when a 16-quadrature amplitude modulation (QAM) and ZF detection are used. As shown in Fig. 2, when the number of loops L = (NT − 1)Niter = 56 for fcLLL, the VER performance of fcLLL-ZF approaches that of the original-LLLZF. When [Q, R, T] = fcLLL[H, δ, Niter ] and the minimum distance of constellation points is 2, ZF detection is expressed as follows: ˆ ZF = 2T x
R1 QH y − T−1 (1 + j)1NT ) 2 + (1 + j)1NT (5)
where 1NT is the column vector of length NT , and all entries are 1. We note that the unitary matrix Q, the upper triangular matrix R, and the unimodular matrix T are the outputs of fcLLL. These three matrices are related with the channel matrix H as H = QRT−1 . The complexity of (5) is 12NT3 + 21NT2 + (8NR + 2)NT flops; the worstcase complexity of fcLLL-ZF detection is thus calculated as follows: max (flops(fcLLL-ZF)) = max (flops (fcLLL[H, δ, Niter ])) +12NT3 + 21NT2 + (8NR + 2) NT .
The ML metric is calculated |Ω| times in (9) because |S| = |Ω|. We note that |Ω|NT times ML metric calculations are necessary ˆ ML , which is a number that is to find the ML solution vector x considered prohibitive. VER Performance: The vector error probability of the given detection method can be expressed as follows: / S) + P (x ∈ S)P (ˆ xopt = x|x ∈ S). (10) P (ˆ xopt = x) = P (x ∈ Since P (x ∈ S) < 1, S ⊂ ΩNT , and P (ˆ xopt = x|S = ΩNT ) = P (ˆ xML = x), the second term in (10) has an upper bound as follows: P (x ∈ S)P (ˆ xopt = x|x ∈ S) < P (ˆ xopt = x|x ∈ S) < P (ˆ xML = x).
By combining (10) and (11) and using the ML solution vector defˆ ML = arg maxx P (y|x), we have the following inequality inition x relationship: xopt = x) < P (x ∈ / S) + P (ˆ xML = x). (12) P (ˆ xML = x) < P (ˆ From the given inequalities, we can see that the probability of error xML = x) as P (x ∈ / S) approaches 0. P (ˆ xopt = x) approaches P (ˆ Consequently, it can be stated that the vector error probability of the detection method of (8) and (9) is dominated by the probability that the true vector is not included in the candidate vector set, that is, P (x ∈ / S). We now discuss two techniques that decrease P (x ∈ / S). A. Decreasing P (x ∈ / S) with fcLLL From the definition of S in (8), we have the following expressions:
ˆ cp = xcp |ˆ xp = x p P (x ∈ / S) = P x (6)
Since max(flops(fcLLL(H, 1, 8))) = 4.5×104 > 2.4×104 , the modified LLL is preferable to the original LLL. Fig. 3 compares the required number of flops between the original-LLL-based ZF and the fcLLLbased ZF (L = 56) for 50 random channels. In Figs. 2 and 3, we can conclude that fcLLL-ZF is almost identical to that of the original-LLLZF but requires significantly reduced worst-case complexity.
(11)
= 1 − P (ˆ xq = xq , ∀q = p|ˆ xp = x p )
NT −1
≈1 −
1−Q
2 † γ/ eTl Hcp
l=1
NT −1
≈
Q
2 † γ/ eTl Hcp
(13)
l=1
III. P ROPOSED F IXED -C OMPLEXITY L ENSTRA –L ENSTRA –L OVÁSZ -BASED S IGNAL D ETECTION Here, we propose a novel fcLLL-based signal detection method. The system equation (1) can be easily modified into the following equation: y − hp xp = Hcp xcp + z
(7)
√ ∞ where Q(x) = 1/ 2π x exp(−s2 /2)ds, vector el is the unit column vector of which the lth entry is 1, and the parameter γ reflects the modulation order and the signal-to-noise ratio (SNR). ˜ = [˜ If z z1 z˜2 · · · z˜NT −1 ]T = (Hcp )† z, then z˜l , l = 1, 2, . . . , NT − 1 are Gaussian random variables with zero mean and variance of eTl (Hcp )† 2 σz2 . The first approximation in (13) is due to the assumption that these noise components after the pseudo inverse are
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independent, and the last approximation is valid when the channel SNR is high. ˆ cp |ˆ xp = xp ) From (13), we can see that the probability P (xcp = x c depends on the condition number of the matrix Hp , whose dimension is NR × (NT − 1). In this paper, we propose to use the fcLLL in Section II to convert a random matrix Hcp into a better conditioned matrix Hcp,fcLLL = QR, where Q and R are the matrices after fcLLL, i.e., [Q, R, T] = fcLLL[Hcp , δ, Niter ]. The performance improvement by fcLLL can be expressed as follows:
NT −1
Q
† 2 γ/ eTl Hcp,fcLLL
TABLE II COMPLEXITY OF THE PROPOSED fcLLL-BASED DETECTION
l=1
NT −1
≤
Q
2 † T γ/ el Hcp . (14)
l=1
The inequality in (14) holds because fcLLL reduces the condition number of the matrix Hcp , and the equality holds when fcLLL does not change the condition number, that is, when T = INT −1 . B. Column Selection to Further Decrease P (x ∈ / S) We now try to select the pth column to further reduce the probability P (x ∈ / S), assuming that fcLLL is the algorithm used. We can select the pth column so that P (x ∈ / S) is minimized as follows:
NT −1
p = arg min i
Q
† 2 γ/ eTl Hci,fcLLL
disregarded in the fcLLL process of the proposed method. On the other hand, overheads of the proposed method exist, such as the candidate vector set generation in (8), the ML metric calculation in (9), and the p selection in (16). The worst-case complexity of the proposed method is calculated as follows:
l=1
≈ arg min max Q i
l
† 2 γ/ eTl Hci,fcLLL
† 2 = arg min max eTl Hci,fcLLL . i
l
max (flops(fcLLL-prop-detection))
l
2
+ 20NT3 − 17NT2
− 10NT + 20NR NT − 16NR + 13 (15)
However, evaluation of the objective functions in (15) is quite demanding, requiring fcLLL reductions for each i = 1, 2, . . . , NT . Instead, we propose to select column hp when it is most dependent upon Hcp , hoping that the smaller condition number of Hcp results in a smaller condition number of Hcp,fcLLL , which, in turn, reduces maxl eTl (Hcp,fcLLL )† 2 in (15). Such a p can be found rather simply using the inverse matrix concept. Thus p = arg max eTl H−1 .
= max flops fcLLL Hcp , δ, Niter
(16)
The removal of the most dependent column implies smaller offdiagonal entries of R, where Hcp = QR. Then, the fcLLL algorithm, which tries to reduce the off-diagonal entries, has fewer entries to reduce, which means a reduced number of flops for complete LLL reduction. As will be shown in Section IV, the computationally efficient p selection method in (16) causes a moderate performance loss when compared with the method in (15). We can fix p = NT to eliminate the complexity of the p selection; however, the error performance is also simultaneously degraded. Incorporating the two techniques in Section III-A and B that decrease P (x ∈ / S), we summarize the proposed fcLLL-based detection method in Table II. C. Complexity On one hand, the worst-case complexity of fcLLL in the proposed method is significantly lower than that in the conventional fcLLLZF detection in Section II-C because the most dependent column is
+ |Ω| 4NT2 + 19NT − 24 .
(17)
As will be demonstrated in the following section, the increased complexity by the overheads is less than the decreased complexity in fcLLL of the proposed method when compared with the conventional fcLLL-ZF; consequently, the worst-case complexity of the proposed method in (17) turns out to be lower than that of the fcLLL-based ZF detection in (6). If we disregard two columns instead of one column of the channel matrix, the complexity of fcLLL will be further reduced, and the error performance will be improved; however, the detection complexity dramatically increases. In addition to the complexity of the candidate vector generation, the complexity of optimal vector selection in (9) is 642 (4 × 64 + 19 × 8 − 24) = 1 572 864 flops when 64-QAM is used. IV. S IMULATIONS Here, we perform a set of simulations to confirm the efficacy of the proposed detection method. It is assumed that the number of antennas is NR = NT = 8, δ = 1 in the LLL reduction and that a 16-QAM constellation was used. We note that the conditions for the worst-case complexity in (4) imply that a smaller value of δ reduces the average complexity but not the worst-case complexity of fcLLL. Fig. 1 compares the required number of flops of the modified-LLL LR method considering various input matrices: H with dimensions of NR × NT , which is used in the conventional fcLLL-ZF, and Hcp with dimensions of NR × (NT − 1), which is used when p = NT or when selected by (16). As shown in Fig. 1, the complexity of the modifiedLLL reduction is dramatically reduced by disregarding the last column.
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conventional and the proposed detection methods when MMSE is used instead of ZF after the LR. We note that when we multiply the received signal vector with an MMSE weight matrix, we are estimating T−1 x instead of x; thus, we need to use covariance matrix E{T−1 xxH T−H } instead of E{xxH } = INT . When MMSE is applied, the R−1 in (5) of the conventional method is replaced by (RH R + σz2 TH T)−1 RH . The pseudo inverse in (8) of the proposed method can be similarly modified, reflecting the new covariance matrix. As shown in Fig. 4, MMSE offers about 1-dB SNR gain for both the conventional and proposed ZF-based methods. V. C ONCLUSION
Fig. 4. Comparison of fcLLL-ZF, fcLLL-MMSE, ZF-based proposed detection, and MMSE-based proposed detection. The p selection method (16) is used for the proposed methods. Parameters δ = 1, NR = NT = 8, and BPSK are used.
In this paper, we have proposed a novel signal detection method based on the fcLLL algorithm. In the proposed method, a channel matrix column was disregarded in the fcLLL process, thereby reducing the complexity dramatically. We also proposed column selection techniques, taking into account both error performance and complexity. With the aid of computer simulations, the proposed method was shown to achieve better error performance at a lower worst-case complexity than the conventional fcLLL-ZF detection. R EFERENCES
The required complexity is further reduced by selecting p as in (16), i.e., by removing the most dependent column. Fig. 2 compares the VER performance of the conventional fcLLLZF and the proposed methods. Although not shown in Fig. 2, having L larger than 18 (i.e., Niter = 3) does not improve the VER performance of the proposed methods. As shown in Fig. 2, the proposed method with p selected by (15) offers the best error performance of the four methods. The proposed method with p selected by (16) approximately offers a 4-dB SNR gain when compared with the conventional fcLLLZF at a VER of 10−4 . When p = NT , the proposed method achieves less than a 2-dB SNR gain when compared with the conventional method. Given that p selection by (15) requires NT fcLLL reductions and that the performance of the method with p = NT incurs significant performance penalties, the proposed method with p selected by (16) is the most preferable choice of the four methods, considering both complexity and performance. The performance degradation of the conventional and the proposed methods occurs due to the matrix inversions in (5) and (8), respectively. We argue that the improved performance of the proposed method is mainly due to smaller size of the matrix to be inverted. Fig. 3 compares the conventional fcLLL-ZF and the proposed method in terms of complexity for random channels. Fig. 3 shows that the complexity variation of the proposed method is much lower than that of the conventional method, which is due in part to the smaller L in the proposed method. The worst-case complexity values of the conventional and the proposed methods are 53 360 and 30 756, respectively. Even when |Ω| = 64, which is the largest constellation size used for contemporary communication systems, the worst-case complexity of the proposed method in (17) is 49 188 flops. When the parameter value L = 28 is used for the conventional fcLLL-ZF, then the complexity is reduced to 32 444 flops, which is comparable to the complexity of the proposed method with L = 18; however, the reduced complexity comes at the degraded performance, as shown in Fig. 2. Therefore, we can state that the proposed method is superior to the conventional fcLLL-ZF method in terms of both performance and complexity. Fig. 4 compares the conventional fcLLL-ZF, the proposed detection method using (16), and the ML detection method. In this simulation, we used BPSK instead of 16-QAM due to the prohibitive complexity of ML detection. Fig. 4 also compares the performance of the
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