Low Complexity SIC-Based MIMO Detection with List ... - IEEE Xplore

Low Complexity SIC-based MIMO Detection With List Generation In The LR Domain Jinho Choi

Huan X. Nguyen

School of Engineering Swansea University Singleton Park, Swansea SA2 8PP, UK Email: [email protected]

School of Engineering & Computing Glasgow Caledonian University Cowcaddens Road, Glasgow G4 0BA, UK Email: [email protected]

Abstract—In order to derive low complexity multiple input multiple output (MIMO) detectors, we combine two additional approaches, namely lattice reduction (LR) and list within the framework of the successive interference cancellation (SIC) based detection in this paper. The resulting detector, called the SICList-LR based detector, provides a near maximum likelihood (ML) performance with a significantly reduced complexity. For example, the signal-to-noise ratio (SNR) loss of the proposed detector compared to the ML detector is less than 0.3 dB for a 4 × 4 MIMO systems with 16-quadrature amplitude modulation (QAM) at a bit error rate of 10−3 while complexity is about half of that of the conventional LR based detectors.

I. I NTRODUCTION Signal detection in multiple input multiple output (MIMO) systems have recently drawn significant attention. If the maximum likelihood (ML) detection is used, the complexity grows exponentially with the number of transmit antennas. Thus, various approaches are devised to reduce the complexity. The successive interference cancellation (SIC) approach is employed in [1]. The relation between the SIC based MIMO detection and the decision feedback equalizer (DFE) is exploited in [2]. A probabilistic data association (PDA) algorithm, which was devised for the multiuser detection in [3], is applied to the MIMO detection in [4]. In [5], the partial maximum a posteriori probability (MAP) principle is derived to discuss the optimality of the SIC based detection. List detectors are also considered for the MIMO detection to obtain soft-decision in [6] and [7] based on [8]. In [9], a lattice reduction (LR) based MIMO detector used as a low complexity MIMO detector is first discussed. In [10], more LR based MIMO detectors are proposed. It is shown that the LR based MIMO detector using the minimum mean square error (MMSE)-SIC approaches the ML performance. An overview of LR based detection can be found in [11]. In [12], [13], and [14], it is shown that the LR based detection can achieve full diversity. It is noteworthy that a soft-decision can also be obtained from the LR based detection [15], [16], [17]. Although the Lenstra-Lenstra-Lovasz (LLL) algorithm [18], which is one of the LR algorithms, has a polynomial (average) complexity, the complexity increases relatively rapidly with the number of basis vectors. Thus, for a large MIMO

system, the computational complexity of the LR based detectors would still be high. To further reduce the complexity, we can decompose a large MIMO detection problem into multiple small MIMO sub-detection problems with SIC, as in [5]. However, this approach would suffer from error propagation which should be mitigated. In this paper, we propose an SIC based detector with list and LR approaches, called SIC-ListLR based detector. There are some distinguished differences in the proposed detector compared with conventional detection methods: i) different from [7], our approach produces a list in the LR domain, which results in a much more reliable list; and ii) while the list can be formed in the conventional LR based detector [17] to generate soft outputs, our approach uses the list technique to mitigate the error propagation within the SIC processing. In the SIC-List-LR based detector, as the LR technique is actually used for the sub-detection problem, the detector has a low complexity due to a smaller number of basis vectors. Having flexible choices of list length, the proposed detector can enjoy the trade-off between the complexity and performance, i.e, it has better mitigation against error propagation as the list length increases at the expense of higher complexity. Simulation results show that the SIC-List-LR based detector provides near-ML performance with reasonably low complexity. It is also shown that the list generated in the LR domain has a superior performance compared to the list conventionally generated in the original domain. II. S YSTEM M ODEL Suppose that there are K transmit antennas and N receive antennas. The N × 1 received signal vector, denoted by r, is given by r = Hs + n, (1) where H, s, and n are the N × K channel matrix, the K × 1 transmitted signal vector, and the N × 1 noise vector, respectively. We assume that n is a zero-mean circular complex Gaussian random vector with E[nnH ] = N0 I. We also assume that the elements of H are independent zero-mean complex Gaussian random variables with unit variance. Let S denote the signal alphabet for symbols, i.e., sk ∈ S, where sk denotes the kth element of s, and its size is denoted by M , i.e., M = |S|.

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We assume that N ≥ K and consider the QR factorization of the channel matrix as H = QR, where Q is unitary and R is upper triangle. We have x = QH r = Rs + QH n.

(2)

Since the statistical properties of QH n are identical to that of n, QH n will be denoted by n. If N = K, there is no zero rows in R, otherwise the last N − K rows would be zero. Thus, the last N − K elements of x would be ignored for the detection if N > K. Accordingly, the first K rows of R would be considered. If there is no risk of confusion, hereafter, we assume that the sizes of x, R, and n are K × 1, K × K, and K × 1, respectively. III. SIC-L IST-LR BASED D ETECTION The complexity of the conventional LR based detector can grows significantly with the number of basis vectors. To avoid this problem, in this section, we propose an SIC-List-LR based detection method within the framework of the partial MAP detection in [5]. The main idea of this method is to break a high dimensional MIMO detection problem into multiple lower dimensional MIMO sub-detection problems. To perform the proposed LR and list-based detection, we consider the partition of x as follows:        R1 R3 s1 n1 x1 = + , (3) x2 0 R2 s2 n2 where xi , si , and ni denote the Ki × 1 ith subvectors of x, s, and n, i = 1, 2, respectively. Note that K1 + K2 = K. From (3), we can have two lower dimensional MIMO sub-detection problems to detect s1 and s2 . It is straightforward to extend the partition into more than two groups. However, for the sake of simplicity, we only consider the partition into two groups as in (3).

where List is a function that chooses the Q closest vectors ˜2 (1 ≤ Q ≤ M K2 ) in the LR domain. We will discus to c the list generation in Section III-C. S3) The list of candidates of s2 , denoted by S2 , can be converted from C2 . For convenience, denote S2 = (1) (2) (Q) {˜s2 , ˜s2 , · · · , ˜s2 }. S4) Once S2 is available, the LR-based detection of s1 can be carried out with SIC, i.e., (q)

(q)

˜1 = LRDet(x1 − R3˜s2 ), c (q)

where ˜s2 is the qth decision vector of s2 from list S2 . (q) (q) ˜1 in S5) Let ˜s1 denote the signal vector corresponding to c (q) T (q) T T (q) the LR domain and ˜s = [(˜s1 ) (˜s2 ) ] , the final decision of s is found as  2   ˜s = arg min x − R˜s(q)  . q=1,2,··· ,Q

In the following subsections, we will explain the proposed detection in detail. B. LR based Detection In this subsection, we describe the LR based detection that is used in Steps S1) and S4). Let C denote the set of complex integers or Gaussian integers, √ C = Z + jZ, where Z denotes the set of integers and j = −1. We assume that {αs + β|s ∈ S} ⊆ C, where α and β are the scaling and shifting coefficients, respectively. For example, for M -QAM, if M = 22m , we have S = {s = a + jb|a, b ∈ {±A, ±3A, . . . , ±(2m − 1)A}},  where A = (3Es /2(M − 1)) and Es = E[|s|2 ] denotes the symbol energy. Thus, α = 1/(2A) and β = ((2m − 1)/2)(1 + j). Note that the pair of α and β is not uniquely decided. Consider the MIMO detection with the following signal: y = Az + v,

(4) Ki

A. Algorithm Description In the proposed SIC-List-LR based detection, the subdetection of s2 is carried out first using the LR based detector. Then, a list of candidate vectors of s2 is generated. With the list of s2 , the sub-detection of s1 is performed with the LR based detector. The candidate vector in the list is used for the SIC to mitigate the interference from s2 . The proposed SIC-List-LR based detection is summarized as follows. S1) The LR-based detection of s2 is performed with the received signal x2 , i.e., ˜2 = LRDet(x2 ), c where LRDet(·) is the function of the LR detection operation, which will be discussed in Section III-B, and ˜2 is the estimated vector of s2 in the corresponding LR c domain. Note that there is no interference from s1 in detecting s2 . S2) A list of candidate vectors in the lattice-reduced domain is generated by c2 ), C2 = List(˜

is the signal where A is a MIMO channel matrix, z ∈ S vector, and v is a zero-mean Gaussian noise with E[vvH ] = N0 I. We scale and shift y as d = αy + βA1 = A(αz + β1) + αv = Ab + αv, T

(5)

Ki

where 1 = [1 1 . . . 1] , and b = αz + β1 ∈ C . Let ¯ = AU where U is a unimodular matrix. Using any LR A algorithm including LLL algorithm [18], we can find U that ¯ shorter. It follows that makes the column vectors of A ¯ + αv, (6) d = AUU−1 b + αv = Ac where c = U−1 b. The MMSE filter to estimate c is given by ¯ − (c − c ¯)||2 ] Wmmse = min E[||WH (d − d) W

= (AAH α2 Es + |α|2 N0 I)−1 AU−H α2 Es , (7) ¯ = E[d] = βA1, c ¯ = E[c] = U−1 β1, and Cov(c) = where d 2 −1 −H |α| U U Es . The estimate of c is given by: ¯ ˜=c ¯ + WH (d − d). c mmse

In Table I, the signals and parameters for the LR based MMSE detection for each step are shown.

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TABLE I S IGNALS AND PARAMETERS FOR THE LR- BASED DETECTION Steps S1) S4)

y x2 (q) x1 − R2 ˆ s2

A R2 R1

z s2 s1

˜ c ˜2 c (q) ˜1 c

the list in the original domain converted from C2 as in step S3). Since no matrix-vector multiplications are required to generate C2 or S2 , we can use S2 as the list in the proposed detector to reduce computational complexity. Note that the list generated in the LR domain is much more reliable than the list generated in the original domain (this list is different from S2 ). This is illustrated by the simulation results shown in Fig. 2.

Ki K2 K1

C. List Generation in the LR Domain To avoid or mitigate the error propagation, the use of a list of candidate vectors of s2 in detecting s1 is crucial. Using the ML metric, we can find the candidate vectors in the list, S2 . Let (1)

(M K2 ) 2

(2)

||r − R2ˆs2 ||2 ≤ ||r − R2ˆs2 ||2 ≤ . . . ≤ ||r − R2ˆs2

|| ,

(q)

where ˆs2 denotes the symbol vector that corresponds to the qth largest likelihood. Therefore, an ideal list would be (1)

(2)

(Q)

S2 = {ˆs2 , ˆs2 , . . . , ˆs2 }.

(8)

However, this requires an exhaustive search, which results in a high computational complexity due to computing of R2 s2 for all s2 ∈ S K2 . To avoid a high computational complexity, we can find a suboptimal list in the LR domain with low complexity. Consider (5). According to Table I, let A = R2 , d = αx2 + βA1, ¯ = AU, we can see that and b = αs2 + β1. Then, since A the ML metric to construct the list is given by ¯ ||d − Ab|| = ||d − Ac||.

(9)

It is noteworthy that the metric on the right hand side in (9) is defined in the LR domain. Let ˜s2 be the signal vector in S K2 ˜2 and assume that ˜s2 is sufficiently close to corresponding to c (1) ¯ c2 . From this, the ML metric ˆs2 . Then, we can have d  A˜ (ignoring a scaling factor) for constructing the list in the LR domain becomes ¯ ¯ ¯ c2 − Ac|| = ||˜ c2 − c||A (10) ||d − Ac|| = ||A˜ ¯ HA ¯, √ where ||x||A = xH Ax is a weighted norm. The list in the LR domain becomes  C2 = {c  ||˜ c2 − c||A (11) ¯ HA ¯ < rA ¯ (Q)}, where rA ¯ (Q) > 0 is the radius of an ellipsoid centered at ˜2 , which contains Q elements in the LR domain. If the c ¯ or the basis vectors in the LR domain column vectors of A H¯ ¯ are orthogonal, A A becomes diagonal. Furthermore, if they ¯ ∝ I. Thus, for nearly orthogonal ¯ HA have the same norm, A basis vectors of almost equal norm, the list of c2 can be approximated as  C2  {c  ||˜ c2 − c|| < r(Q)}, (12) ˜2 which where r(Q) > 0 is the radius of a sphere centered at c contains Q elements. Since the LR provides a set of nearly orthogonal basis vectors for the LR-based detection, we can ¯ are nearly orthogonal, as see that the column vectors in A shown in Fig. 1, with a a two-basis system. Let S2 denotes

Fig. 1. List in different domains. (Left) C2 , generated in the LR domain, ˜2 ). (Right) S2 , converted into the is orthogonal (the black dot represents c original domain, is not necessarily orthogonal (the black dot represents ˜ s2 ).

D. Impact of List Length Q on Performance If Q increases, the error probability that S2 (C2 ) does not have the correct vector of s2 (c2 , resp.), denoted by Pe (S2 ) or Pe (C2 ), decreases. Thus, for a better performance, it is desirable to have a long list length or a large Q at the expense of higher computational complexity. Let s2 denote the transmitted signal vector and c2 is the corresponding vector in the LR domain. The error probability is given by / C2 ) = Pr (||˜ c2 − c2 ||A Pe (C2 ) = Pr(c2 ∈ ¯ HA ¯ > rA ¯ (Q)) . (13) To approximate rA ¯ (Q), consider a 2K2 -sphere in which there are Q lattice points in CK2 and denote by r¯A ¯ (Q) the radius of this sphere. Assume that the volume of this sphere is equal to the sum of Q volumes of the fundamental regions associated ¯ where V (A) ¯ denotes with the lattice points in C2 or Q×V (A), the volume of the Voronoi region with the generator matrix ¯ This assumption would be valid if Q is sufficiently large. A. The volume of an n-sphere with a radius r is given by ¯ and Vn (r) = (π n/2 rn )/(Γ( n2 + 1)). Letting Vn (r) = QV (A) 2 n = 2K2 , we can find the squared radius r¯A ¯ (Q) as follows:   K1 ¯ 2! 2 QV (A)K 2 . (14) r¯A ¯ (Q) = π Then, from (13), the error probability can be approximated as   2 Pe  Pr ||˜ c2 − c2 ||2A ¯A (15) ¯ >r ¯ HA ¯ (Q) , ˜2 ||2A Now, we need to find out the pdf of Z = ||c2 − c ¯. ¯ HA  From (6), letting c = c2 in (6), we have ¯ c2 ¯  + αv  A˜ d = Ac 2

(16)

˜2 is sufficiently close to the ML solution, as we assume that c ¯ c2 (note which has the minimum distance between d and A˜ that this assumption is used in building C2 in Subsection 3.3). It follows |α|2 N0 2 2 2 ˜2 ||2A χ2K2 , (17) ||c2 − c ¯ = |α| ||v|| = ¯ HA 2

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.

where χ2n denotes a chi-square random variable with n degrees of freedom. Then, the following approximation is obtained:

2¯ r2¯ (Q) . (18) Pe (C2 )  Pe (Q) = Pr χ22K2 > A2 |α| N0

According to Table I, A = R2 . Suppose that the elements of H in (1) are independent zero-mean circular-complex Gaussian random variables with variance unit. There exists a pair of the matrices, X and Q2 , that satisfies X = Q2 R2 , where X is an (N −K1 )×K2 random matrix whose elements are independent zero-mean circular-complex Gaussian random variables with variance unity and Q2 is an (N − K1 ) × K2 whose column vectors are orthonormal. Then, it is shown that H det(RH 2 R2 ) = det(X X),

−1

List generated in original domain

10

−2

10 BER

In the error probability in (18), there are two random variables. One is χ22K2 due to the background noise and the other is 2 r¯A ¯ (Q) due to fading. As we consider MIMO fading channels, ¯ ¯ depends on the A is a random matrix. From (14), V (A) ¯ ¯ ¯ random matrix A. Since A = AU and det(U) = ±1, V (A) becomes   ¯ H A) ¯ = det(UH AH AU) = V (A). ¯ = det(A V (A)

16−QAM

0

10

−3

10

List generated in LR domain

−4

10

Q=4 Q = 12 −5

10

4

6

8

10 Eb/N0 (dB)

12

14

16

Fig. 2. Performance of the SIC-List-LR based detector with the list generated in different domains. 16−QAM, K = N = 4, K = K = 2 1

0

2

10

List from (12) List from (11)

(19)

Eb/N0 = 4 dB

−1

2 e

where XH X is a Wishart matrix. For convenience, define

Error Probability P (C )

10

¯ = det(XH X)1/(2K2 ) . Z = V 1/K2 (A)

(20)

In addition, we can show that the conditional error probability is given by    1 γ (K2 , τ Z) φ(Z) = Pr χ22K2 > τ Z  Z = 1 − Γ(K2 ) x where Γ(x) is the Gamma function, γ(n, x) = 0 tn−1 e−t dt is the lower incomplete Gamma function, and τ is τ=

12(QK2 !)1/K2 Es 2(QK2 !)1/K2 = . 2 |α| N0 (M − 1) N0

(21)

It can be observed from (21) that τ is proportional to Q1/K2 . Thus, increasing Q would improve the performance of the detector and this is shown in Fig. 3. However, as τ decreases with M , a longer list length Q is required for higher order modulation schemes to sustain a satisfactory performance. From our simulation observations, the choice of Q ≥ 6, 12, and 16 guarantees a near-ML performance for 4-QAM, 16QAM and 64-QAM, respectively (see simulation results in Figs. 4, 5, and 6). IV. S IMULATION RESULTS In this section, we present simulation results. We mainly focus on the case of K = 4, particularly the case of K1 = K2 = 2. This case is particularly interesting as the Gaussian reduction, which can find the two shortest vectors in 2-basis systems [9], can be used for LR. The elements of H are independent zero-mean complex Gaussian random variables with unit variance.

Eb/N0 = 8 dB −2

10

Eb/N0 = 12 dB

−3

10

−4

10

0

Fig. 3.

2

4

6 List Length (Q)

8

10

12

Error probability with the list of c2 for various list lengths.

A. List Generation: LR Domain versus Original Domain Bit error rate (BER) performance comparison between the proposed SIC-List-LR based detector (which has the list generated in the LR domain) and the conventional list based detector [7] is considered for the case of K = N = 4 and 16QAM signalling. To have a fair comparison, the conventional detector also employs the LR algorithm as the subdetector with the list, however, being conventionally generated in the original domain. The results in Fig. 2 show a superior performance of the SIC-List-LR based detector in two cases of Q = 4 and Q = 12. This justifies the benefits of generating the list in the LR domain. B. Impact of Q In the proposed SIC-List-LR based detection, list length Q plays a key role in the tradeoff between complexity and performance. In general, it is desirable that the list has the

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4−QAM

0

10

16−QAM

0

10

MMSE Q=1 Q=3

−1

10

Q=4

−1

10

Q=6 ML −2

10

−2

BER

BER

10

−3

10

−3

10

MMSE Q=1

−4

10

Q=4

−4

10

Q=8 Q = 12 ML

−5

0

2

4

6 E /N (dB) b

8

10

12

−5

10

4

0

Fig. 4. BER performance for a 4 × 4 MIMO system with 4-QAM signaling.

true transmitted vector of c2 . If not, the proposed detector will have an incorrect decision. If Q increases, the error probability that the correct vector of s2 (c2 ) is not in the list S2 (C2 ), which is denoted by Pe (S2 ) or Pe (C2 ), decreases. Error probability Pe (C2 ) is considered for the MIMO system with 16-QAM signalling, and N = K = 4. Simulation results are shown in Fig. 3, where the error probabilities are shown with two different lists in (11) and (12). As the list in (12) is suboptimal, the performance is worse. However, this performance degradation is not significant as the column ¯ are nearly orthogonal. vectors of A

6

8

10 Eb/N0 (dB)

12

14

16

Fig. 5. BER performance for a 4×4 MIMO system with 16-QAM signaling.

64−QAM

0

10

−1

10

−2

10 BER

10

−3

10

MMSE Q=1 Q=4 Q=8

−4

10

Q = 12

C. BER Performances

Q = 16 ML −5

10

8

10

12

14 Eb/N0 (dB)

16

18

20

Fig. 6. BER performance for a 4×4 MIMO system with 64-QAM signaling.

16−QAM

−1

10

LR based MMSE−SIC detector LR based list detector, Q = 12 ML detector

−2

10

BER

The BER performance of the proposed SIC-List-LR based detection for a 4×4 MIMO system with 4-QAM, 16-QAM and 64-QAM signaling is shown in Fig. 4, 5 and 6, respectively. A near ML performance can be achieved when Q is increased to a reasonable value (e.g., Q = 6, 12, 16 for 4-QAM, 16-QAM, 64-QAM, respectively). In this cases, the signal-to-noise ratio (SNR) loss of the proposed detector, compared with that of the ML detector, is less than 0.5 dB at a BER of 10−3 . Since τ decreases with M according to (21), we need to have a longer list length when M increases to keep a reasonable performance. Note that a full diversity may not be achieved by the proposed detector with a fixed list length, as shown in Figs. 4, 5 and 6. Thus, investigating a variable list length can be a future topic for the SIC-List-LR based detection. We also compare the proposed detector with the LR based MMSE-SIC detector which is the best LR based detector among the LR based detectors proposed in [10]. The BER performance results are shown in Fig. 7. It is shown that the proposed detector can provide a performance that is better by about 1 dB than the LR based MMSE-SIC detector at a BER of 10−2 . Again, we can confirm that employing the list detection in the LR domain can improve the performance of the MIMO detector and is an effective means to approach the ML performance.

−3

10

−4

10

4

5

6

7

8

9 E /N (dB) b

10

11

12

13

14

0

Fig. 7. BER performance comparison for a 4 × 4 MIMO system with 16QAM signaling.

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.

16−QAM

13

12

x 10

result, the proposed detector provides a near ML performance while the complexity is significantly reduced compared with the conventional LR-based detectors.

LR based (LLL) MMSE−SIC LR based list detector ML sphere detector 11

R EFERENCES

Estimated Flops

10

9

8

7

6

5

4

5

6

7

8 Eb/N0 (dB)

9

10

11

12

Fig. 8. Complexity comparison of a 4 × 4 MIMO system with 16-QAM signaling.

D. Complexity Comparison First, we compare the proposed detector with the LRbased MMSE-SIC detector. We consider the upper bound on the average number of LLL iterations in [19], which ¯cs = K 2 log K/(N − K + 1). We ignore is given by N some minor terms to simplify the comparison. For 4 × 4 ¯cs = K 2 log K/(N − K + 1) = MIMO channels, we have N ¯cs = 16 log 4 for the LR-based MMSE-SIC detector and N 2 2(K/2) log((K/2)/((N/2) − (K/2) + 1) = 8 log 2 for the proposed detector. This shows that the complexity is reduced by more than half in terms of LLL iterations. Note that the proposed detector has additional complexity to build a list, which may offset the complexity advantage of the proposed detector over conventional LR based detectors [10]. Thus, to fully examine the whole complexity (i.e., including complexity of building the list), simulations are considered and the results are shown in Fig. 8 where the estimated flops using MATLAB execution time were obtained over all operations for each detector under the same environment. The execution time is averaged over hundreds of thousands of channel realizations. We also include the ML sphere (Schnorr-Euchner algorithm [20]) detector for comparison. The LLL-reduced algorithm with reduction factor δ = 3/4 [18] is chosen for the LRbased MMSE-SIC detector, which is the same as that in Fig. 7. No limitation on the number of iterations is imposed for any LR algorithm. The proposed LR-based list detector clearly requires the lowest number of flops. We can also see that the number of flops of the proposed detector is slightly higher than half of that of the LR-based MMSE-SIC detector where the LLL-reduced algorithm is used. V. C ONCLUDING R EMARKS

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We derived an SIC-List-LR based detector for the MIMO detection using two additional techniques, namely LR and list detection, within a framework of SIC based detection. By generating the list in LR domain, a more reliable list detection is obtained to facilitate the SIC detection. As a

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