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Low complexity lattice reduction scheme for STBC twouser uplink MIMO systems
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 76 (2011)
Abstract
Recently, a lattice reductionaided (LRA) multipleinput multipleoutput detection scheme has been proposed in junction with linear (as well as nonlinear) detectors. It is well known that these schemes provide a full diversity, and its complexity is comparable to that of linear detectors for the block fading channels. For the fast varying channels, however, the decoding complexity of LRA detection scheme is unreasonably high. This article proposes an efficient iterative lattice reduction (LR) scheme for an uplink system with two receive antennas at the base station and two users, each employing the Alamouti spacetime block code (STBC). By taking advantage of the certain inherent STBC structure of transmitted symbols from users, the proposed scheme provides the same performance as a conventional LR while saving about 80% computational complexity. We also show that it can be successfully extended to handle the scenario where another interfering user, who is also employing the Alamouti STBC, is present.
1 Introduction
Exploiting multipleinput multipleoutput (MIMO) in wireless communication systems has been proven to provide plenty of benefits in both increasing the system capacity and reliability of reception in rich scattering environments [1, 2]. To take advantage of these benefits, a spacetime block coding (STBC)oriented diversity scheme has been widely adopted in future wireless communication standards [3], such as 3GPP LTE, WiMax, etc. The STBC scheme, originally proposed by Alamouti in [4], achieves transmit diversity without channel knowledge. Even though Alamouti's STBC was originally designed for two transmit antennas and one receive antenna, this scheme has been generalized by Tarokh in [5] and extended to a system for four transmit antennas. One of those schemes, a double space time transmit diversity (DSTTD) scheme, which is also called "stacked STBC" [6], allows two STBC signals to be sent simultaneously. The theoretical performances of STBC and DSTTD were analyzed in [7–9]. There is also a lot of research to make STBC system work in the multiuser environment [10–14].
As systems with multiple antennas are adopted commercially and the number of multiplexed streams increase, a more efficient detection scheme is requested. Therefore, there is always tradeoff between complexity and performance, and a practical multiantenna system becomes limited by its complexity. The maximum likelihood detection (MLD) of a multiuser MIMO detection takes large complexity of O\left({\Lambda}^{{N}_{\mathsf{\text{t}}}K}\right), where Λ is the size of symbol constellation, N_{t} is the number of transmit antennas, and K is the number of users who access the base station (BS) simultaneously. Although there have been some attempts to reduce the complexity while achieving nearML performance, such as sphere decoding and modified MLD [15–17], these schemes still have large complexity compared with linear detection, with such a scheme being based on zeroforcing (ZF) or minimum mean square error criterion. Even though linear detection schemes have much lower complexity, its performance degradation is often excessively intolerable.
Recently, a lattice reductionaided (LRA) linear detection scheme has been proposed as an alternative, offering as many diversity orders as MLD [18, 19]. Moreover, its complexity is close to that of a linear detection scheme when the channel remains constant for a frame. Since lattice reduction (LR) should be performed at the beginning of the frameblock, the overload caused by the LR procedure can be negligible for large block length on slowly varying channels [20–22].
In most mobile wireless channel environments, however, LR should be often performed for every symbol to prevent performance degradation from the mismatch of channel variation, which could result in high complexity of LR. There were a few attempts to mitigate the LR complexity. In [23, 24], the LR method for complexlattice is introduced, which achieves an average complexity saving of nearly 50% (in terms of floatingpoint operations) without performance degradation. In [25–27], a low complexity LRA detection scheme has been proposed in temporally, spatially, and spectrally correlated MIMO channels, exploiting the channel correlation and unimodular property of the transformation matrix. Although these schemes reduce the detection complexity significantly, the complexity of LR still needs to be further reduced to make the LR scheme more attractive for MIMO detection in practical wireless environments.
In this study, we propose an efficient iterative LR scheme for an uplink system with two receive antennas at the BS and two users, each employing the Alamouti STBC. By taking advantage of the certain inherent STBC structure of transmitted symbols, the proposed scheme provides the same performance as a conventional LR while saving about 80% computational complexity. Furthermore, it is shown that the proposed scheme can be applicable with the whitening filtering preprocess for the interference environments.
The remainder of this article is organized as follows. In Section 2, we introduce the system model and establish the notation. In Section 3, an overview of conventional LR and the LRA ZF detector are reviewed. In Section 4, the proposed scheme for STBC multiuser uplink MIMO systems and simulation results are presented. Section 5 provides an application of the proposed scheme for an interference limited STBC multiuser uplink system. Finally, concluding remarks follow.
2 System model
We considered an uplink multiuser MIMO system with two receive antennas (N_{r} = 2) at the BS and two users (K = 2), each equipped with two transmit antennas (N_{t} = 2) employing the Alamouti STBC. (There are two receiving antennas at the BS and two transmitting antennas for each mobile station (MS), as shown in Figure 1.) The MIMO channels of each user are assumed to be homogenous without consideration of the user's geometry. If there are more than two users in the system, then we assume that a scheduling scheme would select two users for simultaneous transmission. However, this article focuses more on the receiver for the MIMO uplink detection processing rather than scheduling.
The received signal vector r at two receive antennas over two symbol time can be written as
where r_{ i } (t) is the received signal for the i th receive antenna at the symbol time of t, x_{ k } is a 2 × 1 transmitted symbol vector from the k th user with \mathsf{\text{E}}\left\{{x}_{k}^{H}{x}_{k}\right\}={E}_{s}, n_{ i } is a 2 × 1 additive white Gaussian noise with zero mean and variance of 0.5 per dimension for the i th receive antenna, and H_{ i, k } for 1 ≤ i, k ≤ 2 is an effective channel matrix from the k th user to the i th receiver antenna and has the following form from individual Alamouti's scheme [4] of each user:
where h_{ i, k } (j) is the channel coefficient from the j th transmit antenna of user k to the i th BS receive antenna, and (·)*, (·)^{T}, and (·)^{H}denote complex conjugate, transpose, and Hermitian transpose, respectively.
In this case, the equivalent channel model is the same as the DSTTD scheme, and analytic studies of their performances with the optimal MLD and suboptimal LRA ZF detector were analyzed in [8, 9]. (Note that the system model with uplink multiuser MIMO detection appears as a generalization of known singleuser MIMO concepts to the multiuser case.)
3 Conventional LLLbased lattice reduction scheme
3.1 LLL algorithm
In this section, we describe a LLL (Lenstra Lenstra Lovasz) [28]based LR (LLLLR) algorithm. Since the LLL algorithm treats real values, we first transform the received signals in (1) to the equivalent form as follows:
Where
where Re{·} and Im{·} denote the real and imaginary parts, respectively. Assuming the perfect channel estimation at the receiver, the LLLLR algorithm iteratively executes three functional processes: GramSchmidt orthogonalization (GSO), sizereduction, and basis swapping and finds the basisreduced matrix H'_{ R } = H _{ R } T, in which T is an unimodular integer matrix.
The reduced matrix H'_{ R } = {h'_{R ,1}, h'_{R, 2}, ..., h' _{ R },_{n}} has the following properties with parameter δ(1/ 4 < δ < 1) [28]:
and
and where {\widehat{{h}^{\prime}}}_{R,i}is an orthogonal basis vectors computed out of each channel vector h'_{ R,i }by GSO procedure, and correlation factor μ_{i+1,i}is defined as
where ⟨·, ·⟩ and · denote inner product and Frobenius vector norm, respectively.
Equations 6 and 7 are sizereduction and basisswapping condition, respectively. If (6) is fulfilled, then the basis is called sizereduced or weakly reduced. In general, it will be assumed that δ is set to the 3/4 to ensure faster convergence [28]. If (7) is not satisfied, then corresponding two basis vectors are swapped and return to the GSO procedure; otherwise, the LLLreduction process will end. The complexity of the LLL algorithm, therefore, highly depends on the number of column exchanges within the algorithm.
3.2 LRaided ZF detection
By applying the LLL algorithm, the channel matrix H_{ R } is transformed into the reduced channel matrix H'_{ R } = H _{ R } T with nearorthogonal columns. Consequently, this nearorthogonal property of the reduced matrix enabled that the LRA linear receiver such as ZF achieves the same diversity orders as in MLD [20–22].
Considering the transformation unimodular matrix T into the system model (3), the received signal r_{ R } can be rewritten as
where the symbols to be detected become z_{ R } = T^{− 1}x_{ R }. By multiplying (H _{ R } T)^{−1} from left to r _{ R }, instead of H _{ R }^{1} for the ZF detector, we obtain
Since (H _{ R } T)^{−1} is also well conditioned, the noise enhancement and coloring is relatively small. In order to estimate the transmitted symbols, the following operation has to be applied [21, 22]
where d=\sqrt{6\u2215\left(M1\right)} is a given constant for MQAM, Q denotes the componentwise quantization with respect to the infinite integer space, and 1_{ n } is a n × 1 vector of ones.
4 Structured lattice reduction scheme
Since wireless channel often shows timeselectivity, the LR procedure should be performed as fast as the channel varies to offer the full diversity orders. Therefore, the complexity of LR needs to be further reduced to make the LR technique more practical for MIMO detection. In this section, the LLLLR scheme is modified to offer substantial complexity savings for Alamouti's STBCbased multiuser uplink MIMO signal detection, while providing the same performance as conventional one.
The structured lattice reduction (SLR) scheme, which exploits the inherent structure of multiuser STBC, is proposed to cut down the computational complexity further. As shown in Table 1, the proposed scheme consists of two stages, and each stage is operated by the orthogonal LR (OLR)block. The OLRblock consists of initial sorting, LLLLR, reordering, and remaining basis generation, which will be explained in the following subsections.
Assuming the given channel matrix of (1) as H = [h_{1}, h_{2}, h_{3}, h_{4}], the realvalued channel matrix H_{ R } of (4) can be rewritten as where
where
Since the transmitted symbols of each user have an orthogonal STBC structure, the matrix H and H_{ R } have an orthogonal property, such as
Using the orthogonal property above, the first stage of the SLR scheme breaks up the columns of channel matrix into the two matrices of half size, which are as follows:
Then, H_{ α } is transformed into realvalued form as in (4), and this is the input matrix of the OLRblock at the first stage of the SLR scheme
4.1 Initial sorting
A LLLLR scheme iterates the basis reduction until basis vector swapping does not occur, which means that the basis vectors are sorted to satisfy the condition of (7). In the consideration of (7), one heuristic and efficient method that reduces the iterations is to sort the basis vectors according to the magnitude of their norms before the LLLLR is applied [21, 23].
Let θ be the permutation order of basis vectors, then the input matrix H_{in, α}of OLRblock is sorted as follows:
Therefore, we obtain an ordered matrix S_{ α } = [s_{1}, s_{2}, s_{3}, s,_{4}], where s_{ i } = h_{α,θ (i)}for 1 ≤ i ≤ 4, and this is an input matrix of the LLLLR.
4.2 LLL lattice reduction
After all the basis vectors are sorted in ascending order of the magnitude, the conventional LLLLR is performed with ordered matrix S _{ α }, which gives a reduced channel matrix G_{ α } as its output
As mentioned in Section 3.1, the basis vectors are exchanged if swapping condition in (7) is not satisfied. Therefore, the order of basis vectors is changed, when swapping event occurs. Let π be the permutation order of basis vectors caused by the swapping event, so that we can express s'_{π (i)}= g _{ i }, where s'_{(i)}is the reduced basis of s'.
4.3 Reordering
The proposed scheme begins with the LLLLR for submatrix having half columns of channel matrix, with which the whole reduced matrix will be constructed by taking advantage of the known STBC structure. Therefore, the corresponding order of columns in G_{ α } of (18) and H_{in, α}of (16) must be the same to maintain the STBC structure. In order to keep track of the column swaps in the initial sorting and the LLLLR, the inverse permutations of θ and π should be applied.
First of all, reordering is executed to return the order back to the original order incurred by LLLLR.
Then, we also perform the reordering procedure to correct the mixed order of basis vectors caused by the initial sorting and obtain a reordered reduced matrix H'_{in,α}as follows:
4.4 Remaining basis generation
Finally, the reduced matrix of the remaining basis vectors, {{H}^{\prime}}_{\mathsf{\text{in}},\beta}=\left[{{h}^{\prime}}_{\beta ,1},{{h}^{\prime}}_{\beta ,2},{{h}^{\prime}}_{\beta ,3},{{h}^{\prime}}_{\beta ,4}\right], is simply generated from H'_{in,α}. According to the known STBC structure of (2), the column vector of H'_{in,β}can be obtained from the following relationship:
where h'_{ α, l }(j) denotes the j th low entry of h'_{ α, l }. Accordingly, the resulting matrices H'_{in,α}and H'_{in, β}wind up satisfying the STBC structure, which are as follows:
and this is the output matrix at the first stage of the SLR scheme.
In the second stage, we update the partial matrices H'_{R,α}and H'_{ R,β } as (23) and (24), where H'_{ R,α } is a combination of the 1st and 4th (correspondingly 5th and 8th)related column vectors of the first stage output H' _{ R }, so that all the channel vectors are jointly involved:
By applying the same procedure with (23) as was done in the first stage of the SLR from (16) to (21), we can find the basisreduced matrix {{H}^{\u2033}}_{\mathsf{\text{in}},\alpha} and generate the remaining matrix {{H}^{\u2033}}_{\mathsf{\text{in}},\beta}, Eventually, the proposed scheme ends up with the basisreduced matrix {{H}^{\u2033}}_{R} as follows:
where {{h}^{\u2033}}_{R,l} and {\stackrel{\u0304}{{h}^{\u2033}}}_{R,l} are the reduced basis from {{h}^{\prime}}_{R,l} and {\stackrel{\u0304}{{h}^{\prime}}}_{R,l} of H' _{ R }.
Equations 1521 show that the size of the input matrix of the LLL algorithm is reduced by half from 8 to 4. The other half matrix is directly computed by (21) without the LR process, which is the dominant factor in complexity reduction. The whole algorithm of the SLR is shown in Table 1. To make it easier to understand the proposed scheme, an example of the first stage of SLR is depicted in Figure 2. In the case of Figure 2, the permutation orders of initial sorting and LLLLR are exemplified by {θ(1) = 1, θ(2) = 3, θ(3) = 2, θ(4) = 4}, and {π(1) = 3, π(2) = 1, π(3) = 2, π(4) = 4}, respectively.
4.5 Iterative scheme of structured lattice reduction
The SLR scheme reduces the complexity by decomposing the common LR into two stages of LRs where each stage uses a half size matrix. However, it may happen that the basis swapping at the second stage may affect the orthogonal structure obtained by the first stage. Therefore, we propose an iterative SLR (ISLR) scheme to further reduce the basis vectors, in this subsection. As shown in Figure 3, the ISLR is executed with the 1st and 3rd column of H in the first stage of the OLRblock and the 2nd and 4th column of H in the second stage of the OLRblock, which is the same as the SLR scheme. If the swap event occurs at the second stage of the SLRblock, then the iteration begins again with the updated partial input matrix {H}_{R,\alpha}:=\left[{{h}^{\u2033}}_{R,1},{{h}^{\u2033}}_{R,3},{\stackrel{\u0304}{{h}^{\u2033}}}_{R,1},{\stackrel{\u0304}{{h}^{\u2033}}}_{R,3}\right] from (25). Otherwise, the output matrix {{H}^{\u2033}}_{R} is determined as the finally reduced matrix, and the process will end. It is noted that this iterative approach still exploits the STBC structure.
4.6 Simulation results
In this section, we compare the empirical complexity and the quality of reduced matrix for each LR scheme. At first, we discuss the empirical complexity in terms of an average number of required column exchanges within each LR algorithm. The complexity of the LLL algorithm depends on the size of input matrix and the number of column exchanges. Even though the proposed schemes have additional processes, all the processes of the SLR and ISLR scheme consist of a half size input matrix compared with the conventional one. In order to investigate the impact of this half size matrix, Table 2 shows the average number of required column exchanges (\stackrel{\u0304}{c}) for each LR scheme.
A common LLLLR scheme requires an average number of 7.03 column exchanges. By applying the proposed schemes, an average number of column exchanges are significantly reduced by a factor of 10 to 0.70 for the SLR, and to 0.73 for the ISLR. About 90% of the complexity can be saved by the proposed schemes. For a fair comparison, we also investigate the effects of the common LLLLR with initial sorting, where it helps to reduce the complexity of common LLLLR from 7.03 to 3.07. In this case, the saving of the proposed scheme over LLLLR with initial sorting is about 77%.
Next, we can verify the degree of LR in terms of the bit error rate (BER). Figures 4 and 5 show that the SLRaided ZF detector (SLRAZF) has almost the same performance as a conventional LLLLRaided ZF detector (LLLLRAZF) when the required BER is less than 0.005. Moreover, the ISLR aided ZF detector (ISLRAZF) provides the full diversity and the same performance as the LLLLRAZF. Note that this performance comes at almost no cost, because the complexity measured by column exchanges is comparable to that of SLRAZF.
From Table 2 and Figures 4 and 6, it is noted that the reduced matrices obtained by the conventional LLLLR and ISLR schemes provide almost the same performance when it is applied to the detection of a spatially multiplexed STBC signal. The complexity saving obtained does not come with any loss in performance. We can also observe that although the SLR scheme does not always provide full diversity, its output {{H}^{\u2033}}_{R} is well conditioned compared to a nonreduced channel matrix H _{ R }.
5 LRA linear detection scheme in the interference limited STBC multiuser uplink systems
In this section, a proposed scheme is extended to handle the scenario where another interfering user (also employing the Alamouti STBC) is present. We examine the feasibility of the proposed SLR scheme with whitening filter in the interference environment.
Suppose there is a cochannel user in a multicell environment, the detection capability of the receiver could be significantly degraded by the interference from that user. In this scenario, a received signal at the BS can be written by
where H_{ i, j } is the 2 × 2 interference channel matrix, which also has the same STBC structure, and x_{ i } is interferences signal. The optimum preprocessing requires prewhitening filtering against interference.
5.1 Conventional LRA detection with whitening in the interference limited channel
In order to overcome the performance degradation caused by the correlated interference, we can apply the whitening filter before the LRA detection. The interference whitening procedure can be done as follows:
where
and {\sigma}_{i}^{2}=\mathsf{\text{E}}\left\{{x}_{i}^{H}{x}_{i}\right\}\u2215\mathsf{\text{E}}\left\{{x}_{1}^{H}{x}_{1}\right\}=\mathsf{\text{E}}\left\{{x}_{i}^{H}{x}_{i}\right\}\u2215\mathsf{\text{E}}\left\{{x}_{2}^{H}{x}_{2}\right\}, and I_{ m } denotes the m × m identity matrix. By applying the conventional LRA detection scheme to the above whitened effective channel matrix H_{ W } in (28), we can recover the transmitted symbol of two intended cochannel users with interference whitening.
5.2 Proposed ISLRaided detection with whitening in the interference limited channel
As shown in Section 4, the SLR and ISLR scheme can reduce the complexity for the LR procedure, if the channel matrix has the known STBC structure. In order to apply the proposed iterative scheme to LRA detection, the input channel matrix should have a known orthogonal structure. Therefore, if only the whitened effective channel matrix H_{ W } = W^{− 1/2}H keeps the known orthogonal structure, the proposed iterative scheme can be applied successively.
At first, the whitening filter W^{− 1/2}in (27) can be written as follows: [see Appendix]
where {H}_{i}={\left[{H}_{1,i}^{T}{H}_{2,i}^{T}\right]}^{T}, and α, β, and γ are real numbers. Moreover, the submatrix {H}_{2,i}{H}_{1,i}^{H} can be written as
where
and
Therefore, the whitening filter of (30) can be expressed as
By applying the whitening filter W^{1/2} of (34) to the effective channel matrix H, the whitened effective channel matrix H_{ W } can be rewritten as follows:
In (35), it is shown that the whitened effective channel matrix H_{ W } has the known orthogonal structure as the original effective channel matrix H. Therefore, we can apply the proposed SLRaided detection scheme to the whitened effective channel matrix H _{ W }. As shown in Section 4, the proposed scheme has the same performance as a LLLLR scheme while saving about 80% complexity.
5.3 Simulation results
The simulations are performed using a STBC multiuser uplink MIMO system with a BS of two receive antennas and two cochannel users equipped with two transmit antennas. It is assumed that there is a STBCcoded interference user. Figure 6 shows four groups of BER curves for three different values of signaltointerference ratio. In each group, the upper two curves indicate the LLLLRAZF and the ISLRAZF without prewhitening, and the bottom two curves indicate the LLLLRAZF and the ISLRAZF with prewhitening. It is shown that the whitening procedure before the ISLRAZF keeps the same performance in comparison to the conventional one while reducing the complexity.
6 Conclusion
In this article, an efficient (iterative) structured LR scheme for uplink twouser STBC is proposed, so that it can reduce the complexity of the LR significantly by exploiting the orthogonal structure of the STBC. The proposed scheme is shown to provide the same performance as a conventional LRAZF with the reduction of complexity. The numerical results show that the proposed scheme achieves about a complexity reduction of 80%. It is also shown that the SLR can still be applicable with a prewhitening filter in the interference environment.
Appendix
A Derivation of whitening filter W ^{1/2} in (30)
Assuming the interference symbols have Alamouti's STBC structure, the effective matrix of interference H_{ i } can be written as
where {H}_{1,i}{H}_{1,i}^{H}={\sigma}_{1}^{2}{I}_{2}, and {H}_{2,i}{H}_{2,i}^{H}={\sigma}_{2}^{2}{I}_{2}. Therefore, in (22), the matrix W can be written as
where {\sigma}_{1,i}^{2}={\sigma}_{1}^{2}{\sigma}_{i}^{2} and {\sigma}_{2,i}^{2}={\sigma}_{2}^{2}{\sigma}_{i}^{2}. Let the W^{1/2} as
where
From the conditions of (39)(41), we can obtain
Therefore, if we choose the real number b such as {b}^{2}\ge \left({\sigma}_{2,i}^{2}{\sigma}_{1,i}^{2}\right), then we can treat the matrix W^{1/2} as a square root of W (where a and c are also the real numbers).
Using the matrix inversion property in [29], the inverse matrix of W^{1/2} can be written as follows:
where
Abbreviations
 BER:

bit error rate
 MS:

mobile station
 BS:

base station
 DSTTD:

double space time transmit diversity
 GSO:

GramSchmidt orthogonalization
 LLL:

Lenstra Lenstra Lovasz
 LR:

lattice reduction
 LRA:

lattice reduction aided
 MIMO:

multipleinput multipleoutput
 MLD:

maximum likelihood detection
 OLR:

orthogonal LR
 SLR:

structured lattice reduction
 ISLR:

iterative structured lattice reduction
 STBC:

spacetime block coding
 ZF:

zeroforcing.
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Acknowledgements
This study was supported by the Technology Innovation Program, 10035389, funded by the Ministry of Knowledge Economy (MKE, Korea). The authors gratefully appreciate Ms. Tina and Miri for their helpful comments.
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An, Ch., Lee, T., Yang, J. et al. Low complexity lattice reduction scheme for STBC twouser uplink MIMO systems. J Wireless Com Network 2011, 76 (2011). https://doi.org/10.1186/16871499201176
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DOI: https://doi.org/10.1186/16871499201176
Keywords
 uplink MIMO
 spacetime block coding
 lattice reduction
 low complexity
 cochannel interference