Over the last years, numerous equalization schemes for multiple-input/multiple-output channels have been studied in the literature. New low-complexity approaches based on lattice basis reduction are of special interest, since they achieve the optimum diversity behavior. Although the per-symbol equalization complexity of these schemes is very low, the initial calculation of the required matrices may impose an enormous burden in arithmetic complexity. In this paper, we give a tutorial overview and assess algorithms, which, given the channel matrix, result in the feedforward, feedback, and unimodular matrix required in lattice-reduction-aided decision-feedback equalization or precoding. To this end, via a unified exposition of the Lenstra–Lenstra–Lovász (LLL) algorithm, the LLL with deep insertions, and the reversed Siegel approach similarities and differences of these approaches are enlightened. A modification of the LLL swapping criterion, better matched to the equalization setting, is discussed. It is shown that using lattice-reduction-aided equalization/precoding better performance can be achieved at lower complexity compared to classical equalization or precoding approaches.
Communication using antenna arrays at transmitter and receiver, hence creating a multiple-input/multiple-output (MIMO) channel, is very interesting because of the very high spectral efficiencies and hence data rates which can be achieved, cf. [33], [13] and [27]. The multi-antenna interference present in such a scenario has to be combatted with some means of equalization.
The same situation occurs when considering multi-user transmission schemes, where multiple, usually non-cooperating users send their data streams over a common channel hence creating multi-user interference. Here, the task of a joint receiver is to separate the users, i.e., to equalize the channel, which again can be described by a MIMO channel matrix.
Taking the uplink/downlink duality [22] and [28] into account, instead of employing receiver-side equalization for the MIMO channel pre-equalization or precoding can be performed. In point-to-point transmission schemes transmitter- or receiver-side techniques can be used alternatively, whereas in a multi-user uplink (downlink) only receiver-side (transmitter-side) equalization is possible.
Over the last years, a number of equalization schemes for MIMO channels has been studied in the literature. Numerous techniques known from intersymbol-interference channels (e.g., linear (pre-)equalization, decision-feedback equalization (DFE),1 Tomlinson–Harashima precoding (THP), maximum-likelihood detection (MLD), vector precoding, cf. [7, Table E.1], see also [38]) have been transferred to the MIMO setting. However, new approaches based on lattice basis reduction, e.g., [37], [30], [32] and [35], are of special interest. For an overview on lattice reduction see [36]. Using these lattice-reduction-aided (LRA) techniques, low-complexity equalization achieving the optimum diversity behavior is enabled [26]. Although the per-symbol equalization complexity is very low when applying LRA techniques the initial calculation of the required matrices still imposes an enormous burden in arithmetic complexity.
In this paper, we study algorithms for calculating the matrices to be used in LRA DFE or LRA precoding. As the structure and the per-symbol processing of the equalization scheme itself is always the same, we restrict ourselves to the complexity required for performing the initial factorization task: given the channel matrix, what complexity is required to calculate the feedforward, feedback, and unimodular matrix. Conventional approaches perform the calculation of these matrices in two separated steps: first, a lattice basis reduction algorithm is applied to the channel matrix; second, a sorted QR decomposition of the reduced channel matrix is carried out.
We compare algorithms which performs these two steps in a joint fashion. To this end, complex-valued variants of the LLL algorithm—the original approach [16], the deep LLL [21], and the reversed Siegel algorithm [4]—whose internal results can be used readily (cf. [35]) are presented in a unified way. We particularly assess the tradeoff between computational complexity of these algorithms and the signal-to-noise ratio required to guarantee a desired error rate. It turns out that by appropriately adjusting the free parameter of the algorithms, an excellent tradeoff can be achieved—almost the performance of the optimum LRA schemes can be obtained with only marginally larger complexity than sorted QR decomposition.
The paper is organized as follows: in Section 2 the channel model and a brief review on lattice-reduction-aided decision-feedback equalization and precoding are given. Section 3 discusses algorithms from the literature in a unified way and presents some modifications. Numerical results are given and discussed in Section 4; Section 5 draws some conclusions.
In this paper, algorithms for joint lattice reduction and QR decomposition have been assessed. To this end, a unified exposition of the LLL algorithm, LLL with deep insertions, and the reversed Siegel approach has been given. It enlights the similarities and differences of the various approaches. Basically, the main operations (i) sorting, (ii) QR decomposition, and (iii) size reduction are carried out in different sequences and are based on different criteria. A modification of the LLL swapping criterion, better matched to the equalization setting, has been proposed.
All LLL variants show a good complexity-performance tradeoff, enabled by varying the free parameter, originally introduced to control the quality of the lattice reduction. The best tradeoff is obtained by the conventional LLL and the (reversed) Siegel algorithm. The optimum choice of the free parameter, when applying the algorithms in lattice-reduction-aided decision-feedback equalization and precoding, respectively, has been given. Moreover, it has been confirmed that operating directly on the complex-valued MIMO channel model requires approximately only half the number of multiplications as when starting from the equivalent real-valued model.
Finally, it should be noted that via lattice-reduction-aided precoding/equalization better performance is achieved at lower complexity compared to classical Tomlinson–Harashima precoding and the VBLAST approach. Hence, such schemes are highly recommendable in future downlink scenarios.
