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IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 5, MAY 2004
Reduced Complexity Schnorr–Euchner Decoding Algorithms for MIMO Systems Zhan Guo and Peter Nilsson, Member, IEEE
Abstract—A new reduced-complexity Schnorr–Euchner decoding algorithm is proposed in this letter for uncoded multi-input multi-output systems with -QAM ( = 4 16 . . .) modulation. Furthermore, a Fano-like metric bias is introduced to the algorithm from the perspective of sequential decoding, as well as an early termination technique. Simulation results show that these modifications reduce the algorithm complexity efficiently, with only a small degradation in bit error rate at high signal to noise ratios.
th transmit antenna to the th receive antenna. The channel matrix is assumed to be perfectly known to the receiver, and is assumed in the sequel. As in [2], the complex matrix (1) is transformed to its real , i.e., lattice representation
Index Terms—Lattice decoder, multi-input multi-output (MIMO) systems, Schnorr–Euchner (SE) decoder, sphere decoder.
where and denote the real and imaginary part of , are assumed to be i.i.d respectively. Since the elements of . Therefore, the complex Gaussian, has a full rank of can be considered as the lattice generated by set [4]. The rows of are called basis vectors for , is said , and the transmitted vector acts to be the dimension of as the coordinates of a lattice point. If the received vector is considered as a perturbed lattice point due to the Gaussian noise , the objective of MIMO detection is thus to find its closest , i.e., lattice point for a given lattice
I. INTRODUCTION
E
XPLOITING the limited radio spectrum plays an important role in wireless communication systems. It is well known that an extraordinary spectral efficiency can be achieved in multi-input multi-output (MIMO) systems [1]. However, the optimal maximum-likelihood (ML) decoder is infeasible for the MIMO system when a large number of antennas is used together with higher modulation constellations [2]. As an alternative, the sphere decoding (SD) algorithm, proposed in [2], reaches the ML performance with reasonable complexity. The Schnorr–Euchner (SE) decoding algorithm, a variant of SD, is more attractive due to its low complexity [3], [4]. In this letter, an improved SE algorithm with lower complexity is proposed. II. LATTICE REPRESENTATION OF MIMO SYSTEMS Consider a symbol synchronized and uncoded MIMO system transmit antennas and receive antennas. The basewith band equivalent model for this system is (1) where is the transmitted symbol vector, in which each component is independently drawn from a comis the plex constellation such as QAM, is an i.i.d. received symbol vector, and zero-mean complex Gaussian noise vector with variance per of the channel matrix are assumed dimension. The entries to be i.i.d. zero-mean complex Gaussian variables with variance 0.5 per dimension, representing the transfer function from the Manuscript received October 10, 2003. The associate editor coordinating the review of this letter and approving it for publication was Prof. D. P. Taylor. This work was supported by the INTELECT program under the Foundation for Strategic Research, Sweden. The authors are with the Department of Electroscience, Lund University, SE-22100 Lund, Sweden (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LCOMM.2004.827376
(2)
(3) where
is the set of real entries in the constellation, e.g., in the case of 16-QAM. III. SCHNORR-EUCHNER DECODING ALGORITHMS
In the SE algorithm, from the perspective of lattice, an dimensional lattice is decomposed into -dimensional sublattices. The algorithm calculates the orthogonal distance between two points in the adjacent sublattices, and tries to find the smallest possible accumulated disbetween the -dimensional sublattice and the tance one-dimensional sublattice. For a detailed discussion of the SE algorithm, the reader is referred to [4]. A. Modified SE Algorithm for the QAM Constellation It is not suitable in MIMO systems to employ the SE algorithm presented in [4] which uses an infinite lattice, since the finite lattice constellation is used in MIMO systems. A SE alconstellation is gorithm tailored to the -QAM proposed in [3]. To avoid an infinite loop or an incorrect result due to the finite constellation used, the algorithm in [3] adopts a searching method which allows an overflow on the constellation. However, only the lattice vector belonging to the constellation is kept. There are probabilities in this searching method where most of the elements in belong to , but the remained elements do not belong to . In such a case, the lattice vector is not kept and has to be recalculated, which increases the algorithm complexity.
1089-7798/04$20.00 © 2004 IEEE
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GUO AND NILSSON: REDUCED COMPLEXITY SCHNORR–EUCHNER DECODING ALGORITHMS FOR MIMO SYSTEMS
This reasoning motivates a new reduced-complexity SE algorithm for MIMO systems, in which only those lattice belonging to are investigated points and kept. The proposed reduced-complexity algorithm is thus sub-optimal compared to [3], and listed as below for completeness. The matrix is the inverse and transpose of matrix , . The matrix and are the upper triangular i.e., matrix and the orthogonal matrix in the QR-decomposition of , respectively. The other notations the channel matrix are in conformity with those used in [3]. Algorithm SE1: 1: , i.e., the size of 2: 3: 4: 5: 6: 7: 8: 9: loop 10: 11: if then then 12: if 13: for to do 14: 15: end for 16: 17: 18: 19: 20: 21: else 22: 23: 24: 25: 26: for to 2 do 27: 28: 29: if then 30: 31: goto 32: end if 33: end for 34: end if 35: else then 36: if 37: return 38: else 39: goto 40: end if 41: end if 42: end loop B. Fano-Like Metric Bias The number of searched sublattices, i.e., the number of evalin SE1, implies the algorithm complexity. uation on Line
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In the simulations that will be detailed in Section IV, it is observed that the complexity becomes excessively higher when signal-to-noise ratio (SNR) is lower, since the searching of the algorithm oscillates too frequently among the sublattices. On the other hand, the SE algorithm actually constructs a tree of levels, where the branches in the th level of the tree correspond to the lattice points in the -dimensional sublattice. If is considered as the metric of each branch, the objective of SE is to find the path with the smallest accumulated metric between the first and th level of the tree. This is essentially a perspective of sequential decoding. The proposed SE1 is, therefore, a metric-first and depth-first mixed sequential decoding algorithm [5]. Inspired by the Fano metric, which is derived from ML decoding for unequal length sequences in sequential decoding [5], a Fano-like metric bias may be devised for SE to alleviate the complexity problem mentioned above. With this Fano-like metric bias, the branches in higher levels should have a larger metric bias than those in lower levels, reflecting the fact that they are far away from the end of the tree and hence less likely to be part of the smallest path. Note from (2) and (3) that the average value of the smallest path is (4) It is, therefore, reasonable to choose as the metric bias for is a constant. The metric one level of the tree, where bias for the -level tree is simply the sum of the biases for the trees. Moreover, note that following -level the squared orthogonal distance is expressed as a proportion for -dimensional sublattice [4], the Fano-like metric bias of for SE is thus to be: (5) where as below.
. With this metric bias, the SE1 can be modified
Algorithm SE1_FM: • In Algorithm SE1, replace Line if
with:
then
The value of can be determined by simulations. When , there would be actually no Fano-like metric bias for SE1_FM. , on the other hand, there would be the lowest comWhen plexity of SE1_FM with the most degradation in the bit error rate (BER) performance. Therefore, is simply set to be 0.5 in our simulations as a trade-off between the algorithm complexity and the BER performance. C. Early Termination The (4) also implies that the loop in SE1 can be terminated is early as soon as the currently smallest distance smaller than a pre-calculated distance
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(6)
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IEEE COMMUNICATIONS LETTERS, VOL. 8, NO. 5, MAY 2004
Fig. 1. Average BER of SE for MIMO systems, modulation.
M=N
= 4, 16-QAM
where is a noise level dependent constant that needs to be estimated for each SNR point. The algorithm complexity can thus be further reduced. The pre-calculated distance is simply at each SNR the average value of the smallest distance point, which can be determined by simulating SE1 or the algorithm in [3]. With this early termination criteria, the SE1 can be modified as below. Algorithm SE1_ET: • In Algorithm SE1, insert the following lines between and : Line then return if end if Furthermore, with both the Fano-like metric bias and the early termination criteria considered, the algorithm SE1 can be improved to be SE2 which combines the algorithm SE1_FM and SE1_ET. IV. SIMULATIONS antennas are used in our simulations. 100k independent channel realizations (packets) of 40 uncoded 16-QAM symbols are transmitted with ten symbols from each antenna. The average energy per bit is fixed to , and the variance of the AWGN is adjusted by , where is the average symbol energy of the -QAM when [2]. The BER for the proposed algorithms is plotted in Fig. 1 as a function of SNR, in which the algorithm SE_REF presented in [3] is also shown as a reference. Fig. 1 shows a BER of 10 can be attained at dB by all the algorithms, while the performance difference between SE_REF and the proposed algorithms is minor. It is also clear from Fig. 1 that SE_REF , since outperforms SE1 by about 1 dB when SE1 is sub-optimal as stated above. However, the performance difference between SE_REF and SE2 is about 0.5 dB when , which shows that a suitable early termination criteria could improve the BER performance of SE1 at high SNR.
Fig. 2. Algorithm complexity of SE for MIMO systems, 16-QAM modulation.
M
=
N
= 4,
The algorithm complexity is defined as the average number of searched sublattices per symbol vector, which is shown in Fig. 2 for the proposed algorithms and SE_REF as the function of SNR. It is clear from Fig. 2 that SE1 reduces the complexity of SE_REF significantly at low and moderate SNR. Moreover, the algorithm complexity is further reduced in SE2, which combines SE1_FM and SE1_ET, but the effects of SE1_FM and SE1_ET are different on the complexity reduction. The Fano-like metric bias of (5) is more effective at low SNR than is too small at high SNR to at high SNR, since the value of affect the path metric of the higher level of tree. On the other hand, the complexity reduction due to the early termination only is almost invariant within the investigated range of SNR, as shown in Fig. 2. V. CONCLUSION A new reduced-complexity SE algorithm is proposed in this letter for uncoded MIMO systems with -QAM modulation. Furthermore, a Fano-like metric bias and an early termination technique dependent on SNR are proposed to improve the algorithm complexity. The simulation results show that the proposed algorithms can reduce the implementation complexity of SE efficiently, with only a small degradation in BER performance at high SNR. REFERENCES [1] G. J. Foschini, “Layered space-time architecture for wireless communication in fading environments when using multiple antennas,” Bell Labs. Tech. J., vol. 2, Autumn 1996. [2] M. O. Damen, A. Chkeif, and J.-C. Belfiore, “Lattice code decoder for space-time codes,” IEEE Commun. Lett., vol. 4, pp. 161–163, May 2000. [3] G. Rekaya and J.-C. Belfiore, “Complexity of ML lattice decoders for the decoding of linear full rate space-time codes,” IEEE Trans. Wireless Commun., to be published. [4] E. Argell, E. Eriksson, A. Vardy, and K. Zeger, “Closest point search in lattices,” IEEE Trans. Inform. Theory, vol. 48, pp. 2201–2214, Aug. 2002. [5] J. B. Anderson and S. Mohan, “Sequential coding algorithms: A survey and cost analysis,” IEEE Trans. Commun., vol. COM-32, pp. 169–176, Feb. 1984.
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