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Multi-Branch Tomlinson-Harashima Precoding for MU-MIMO Systems: Theory and Algorithms

K. Zu, R. C. de Lamare, M. Haardt

arXiv:1401.4753v1cs.ITmath.IT

TL;DR

THP performance depends on symbol ordering, but cooperative ordering is impractical for MU-MIMO users with multiple receive antennas. The paper proposes multi-branch THP algorithms for both decentralized and centralized structures, using LQ-derived MMSE filters and branch selection. Simulations report BER gains over conventional THP for both structures, with the reported complexity remaining reasonable.

  • Problem

    Ordering methods are impractical for MU-MIMO systems with multiple receive antennas because users are geographically distributed, and most ordering algorithms consider only one THP structure.

    Source: B. Prior Art · p. 2, A. Motivation of the Proposed MB-THP… · p. 4

  • Method

    The paper develops multi-branch THP for decentralized and centralized structures, using transmit patterns, branch selection, and MMSE filters derived with LQ decomposition.

    Source: C. Contributions · p. 2, B. Structure of the Proposed MB-THP · p. 4

  • Results

    At BER 10^-3, MB-MMSE-dTHP gains more than 3.6, 6, and 7 dB over conventional MMSE-dTHP with 2, 4, and 8 branches, respectively.

    Source: A. Perfect Channel State Information… · p. 9

  • Takeaways & Limitations

    The conclusion reports considerable improvement with only 2 or 4 branches and describes the MB-MMSE-THP filters as having reasonable computational complexity because they are derived using LQ decomposition.

    Source: §VI · p. 11

Abstract

from arXiv · show

Tomlinson-Harashima precoding (THP) is a nonlinear processing technique employed at the transmit side and is a dual to the successive interference cancelation (SIC) detection at the receive side. Like SIC detection, the performance of THP strongly depends on the ordering of the precoded symbols. The optimal ordering algorithm, however, is impractical for multiuser MIMO (MU-MIMO) systems with multiple receive antennas due to the fact that the users are geographically distributed. In this paper, we propose a multi-branch THP (MB-THP) scheme and algorithms that employ multiple transmit processing and ordering strategies along with a selection scheme to mitigate interference in MU-MIMO systems. Two types of multi-branch THP (MB-THP) structures are proposed. The first one employs a decentralized strategy with diagonal weighted filters at the receivers of the users and the second uses a diagonal weighted filter at the transmitter. The MB-MMSE-THP algorithms are also derived based on an extended system model with the aid of an LQ decomposition, which is much simpler compared to the conventional MMSE-THP algorithms. Simulation results show that a better bit error rate (BER) performance can be achieved by the proposed MB-MMSE-THP precoder with a small computational complexity increase.

I. INTRODUCTION

The paper addresses THP ordering and design challenges in MU-MIMO with geographically distributed users, developing multi-branch algorithms for both decentralized and centralized structures. Its system model assumes active streams with equal power loading and describes THP filters according to their functions and placement.

A. Background and Problem Formulation

  • MU-MIMO downlink precoding is motivated by the need to mitigate multiuser interference while reducing receiver effort, making cost-effective precoders important.The context identifies CSI at the transmitter as the condition for using these precoding techniques.Source: A. Background and Problem Formulation · p. 1
  • Channel-inversion precoding can require higher average transmit power on ill-conditioned channels, while BD can lose performance at low SNR and VP’s optimal perturbation search is NP-hard.VP is reported to achieve near-optimal performance, but its optimal perturbation problem can be computationally intractable.Source: A. Background and Problem Formulation · p. 1

B. Prior Art

  • Ordering methods for THP generally assume single-antenna receivers, making cooperative ordering impractical for geographically distributed MU-MIMO users with multiple antennas.The paper also notes that most existing ordering algorithms address only one THP structure.Source: B. Prior Art · pp. 1–2

C. Contributions

  • The work develops MB-THP for both basic structures, derives MMSE filters using LQ decomposition, and designs transmit patterns with a selection procedure.It also analyzes error covariance, sum-rate, and computational complexity and compares the proposed algorithms with existing precoders.Source: C. Contributions · p. 2

II. SYSTEM MODEL AND THP ALGORITHMS

  • The model assumes all data streams are active and equal power loading, with a square equivalent channel matrix formed when scheduling is needed.Power allocation is not the focus of the paper.Source: §II · p. 2

A. Two Basic THP Structures

  • THP uses feedback, feedforward, and scaling filters; its decentralized form places scaling at receivers, whereas its centralized form places scaling at the transmitter.The feedback filter cancels interference from previous streams, and the feedforward filter is implemented at the transmitter because distributed users cannot be processed jointly.Source: A. Two Basic THP Structures · p. 2, Figure 1 · p. 2

B. Review of THP Algorithms

THP uses LQ decomposition to obtain its filters and modulo processing to keep transmitted-symbol amplitudes within the modulation boundary. The modulo operation introduces power and boundary-related losses, which the paper neglects for moderate modulation sizes.

B. Review of THP Algorithms

  • THP can be implemented using an LQ decomposition of the channel matrix, whose factors are a lower-triangular matrix L and a unitary matrix Q.The resulting factors are used to obtain the THP filters.Source: B. Review of THP Algorithms · p. 3
  • The modulo operation restricts the amplitude of each channel symbol to the modulation boundary and is equivalent to adding a perturbation vector to the transmit data.The constellation is extended periodically, and the effective transmit symbols are drawn from the expanded set.Source: B. Review of THP Algorithms · p. 3
  • THP introduces power and modulo losses, but the paper neglects them for moderate constellation sizes.The passage states that power loss is not negligible for small modulation sizes and that modulo loss is more significant for small constellations.Source: B. Review of THP Algorithms · p. 3

III. PROPOSED MB-THP PRECODING ALGORITHM

The section explains why symbol ordering matters in THP and why cooperative ordering is impractical for MU-MIMO users with multiple receive antennas. It introduces multi-branch designs for both basic THP structures, using transmit patterns and branch selection.

A. Motivation of the Proposed MB-THP Algorithm

  • For dTHP, layer SNRs differ and the weakest-SNR layer dominates performance; cTHP instead has equal layer error covariance and can benefit from row reordering.The section relates these differences to the structures' feedback and scaling filters.Source: A. Motivation of the Proposed MB-THP… · pp. 3–4
  • Cooperative ordering methods are impractical for multi-antenna MU-MIMO users because users are geographically distributed, and most methods address only one THP structure.Source: A. Motivation of the Proposed MB-THP… · p. 4

III. PROPOSED MB-THP PRECODING ALGORITHM

B. Structure of the Proposed MB-THP

  • The two proposed structures are MB-cTHP and MB-dTHP, with the diagonal weighted filters placed centrally at the transmitter or decentrally at receivers, respectively.Source: B. Structure of the Proposed MB-THP · p. 4

C. Design of the Transmit Patterns

Transmit patterns combine ordering choices across users and their streams to form candidate branches. The design limits branch count to keep complexity reasonable, while requiring storage and a search procedure.

C. Design of the Transmit Patterns

  • Transmit patterns are designed to be effective and simple, using known permutations to produce different orderings and exploit extra transmit diversity gains.The design considers ordering changes across users and streams within each user.Source: C. Design of the Transmit Patterns · p. 4
  • The number of per-user ordering states is set using the maximum receive-antenna count, J = Maxk(Nk), though other choices for J are possible.Source: C. Design of the Transmit Patterns · p. 5
  • User and stream ordering patterns are packaged into transmit patterns that form candidate branches for minimum-error selection.For users with the same receive-antenna count, the packaging can use a Kronecker product; the general procedure preserves the sparse pattern.Source: C. Design of the Transmit Patterns · p. 5
  • The branch count is capped at K·J to maintain reasonable complexity, but storing branches at the transmitter and receivers requires extra memory and a search for each transmission.The passage says MB-cTHP and MB-dTHP can approach performance with LB branches using only 2 or 4 branches.Source: C. Design of the Transmit Patterns · p. 5

D. Selection Criterion for the MB-THP

The selection criterion chooses a branch based on error-related properties of the two THP structures. The selected branch's signal is then transformed back and decoded at the receiver.

D. Selection Criterion for the MB-THP

  • The minimum error selection criterion (MESC) selects a branch for both MB-cTHP and MB-dTHP using their branch-dependent error covariance properties.The passages relate the structures' layer SNRs to diagonal elements of their respective matrices.Source: D. Selection Criterion for the MB-THP · p. 5
  • After branch selection, the receiver can learn the selected pattern index from the transmitter or search for the best pattern itself.Source: D. Selection Criterion for the MB-THP · p. 5
  • The receiver reverses the ordering, applies modulo processing to remove the perturbation offset, and slices the symbols to constellation points.Source: D. Selection Criterion for the MB-THP · p. 5

E. Derivation of Filters for the MB-MMSE-THP

The paper derives MB-MMSE filters for both THP structures using an extended system model and LQ decomposition. This approach is presented as simpler than conventional MMSE-THP derivations, and the receive models are also described.

E. Derivation of Filters for the MB-MMSE-THP

IV. PERFORMANCE ANALYSIS

The error-covariance analysis compares conventional and multi-branch THP, using branch selection to assess the expected BER effects of changing the transmit order.

A. Performance Analysis of the Error Covariance Matrix

  • ZF-dTHP is expected to have slightly better BER performance than ZF-cTHP, consistent with a simulation result reported in.The comparison is motivated by the relationship between BER and the error covariance matrix.Source: A. Performance Analysis of the Error… · p. 6
  • The paper derives error-covariance expressions for MMSE-dTHP and MMSE-cTHP to compare structures whose relationship had not been analyzed or simulated in prior literature.For MB-MMSE-cTHP, changing transmit-signal order yields different perturbation vectors, and the algorithm selects the branch with minimum σ(l)^2.Source: A. Performance Analysis of the Error… · pp. 6–7
  • The trace of the MB-MMSE-dTHP error covariance is upper-bounded by that of conventional MMSE-dTHP.The bound follows from the MESC branch-selection criterion.Source: A. Performance Analysis of the Error… · p. 7
  • The analysis predicts that MB-dTHP and MB-cTHP can each achieve better BER than their respective original counterparts.Source: A. Performance Analysis of the Error… · p. 7

B. Sum-Rate Performance Analysis

The sum-rate analysis relates branch SNR and selection to the expected performance of the multi-branch MMSE schemes.

B. Sum-Rate Performance Analysis

C. Complexity Analysis

The complexity analysis counts floating-point operations and compares the proposed algorithms with existing precoders across system dimensions.

C. Complexity Analysis

V. SIMULATION RESULTS

Under perfect CSI, multi-branch MMSE-THP improves BER over conventional MMSE-THP, with gains depending on structure and branch count; its sum-rate behavior also differs between cTHP and dTHP. The simulations use an (2, 2, 2, 2) × 8 MU-MIMO system and compare precoders under QPSK and 16-QAM.

V. SIMULATION RESULTS

  • The simulations use 8 transmit antennas, four 2-antenna users, uncoded QPSK and 16-QAM, and i.i.d. complex Gaussian channels.The branch counts are 2, 4, 6, and 8; each simulation uses 10^6 trials and packet length 102 symbols.Source: §V · p. 8

A. Perfect Channel State Information Scenario

  • For 16-QAM, ZF-dTHP slightly outperforms ZF-cTHP in BER, whereas the MMSE-THP comparison reverses; ZF-VP and MMSE-VP achieve maximum transmit diversity.SO-THP is slightly better than ZF-cTHP at low Eb/N0, but is nearly the same as ZF-dTHP and ZF-cTHP at high Eb/N0.Source: A. Perfect Channel State Information… · pp. 8–9, Figure 4 · p. 9
  • At BER 10^-3, MB-MMSE-cTHP gains more than 2 dB, 3 dB, and 3.4 dB over conventional MMSE-cTHP with 2, 4, and 8 branches, respectively.With 4 branches, its 16-QAM performance is 2 dB behind MMSE-VP; for QPSK it is better than MMSE-VP at low Eb/N0 and very close at BER 10^-3 with much lower computational complexity.Source: A. Perfect Channel State Information… · p. 9, Figure 5 · p. 9
  • At BER 10^-3, MB-MMSE-dTHP gains more than 3.6 dB, 6 dB, and 7 dB over conventional MMSE-dTHP with 2, 4, and 8 branches, respectively.Fig. 6 reports the (2, 2, 2, 2) × 8 MIMO, 16-QAM case.Source: A. Perfect Channel State Information… · p. 9, Figure 6 · p. 10
  • The proposed cTHP and dTHP algorithms have transmit diversity between VP and conventional MMSE-THP, with substantial improvement already at 2 branches and 4-branch BER approaching 8-branch performance.The authors attribute the diversity to constructing a branch list and selecting its best candidate; 4-branch MB-MMSE-cTHP is not far from MMSE-VP with much less computational complexity.Source: A. Perfect Channel State Information… · p. 9
  • MB-MMSE-cTHP sum rate improves as branch count increases, reaching almost the 8-branch performance at 4 branches, while MB-MMSE-dTHP sum rate is nearly unchanged across branch counts.At low Eb/N0, cTHP has higher sum rate than dTHP; at high Eb/N0, dTHP performs very well, and SO-THP exceeds cTHP.Source: A. Perfect Channel State Information… · pp. 9–10

B. Correlated Channel State Information Scenario

The proposed MB-MMSE-THP algorithms retain better performance than conventional counterparts under spatial correlation, but imperfect CSI degrades their advantage at higher error variance. The comparison also identifies structural trade-offs in BER, sum rate, and feedback overhead.

B. Correlated Channel State Information Scenario

  • With spatial correlation |r| = 0.5, MB-MMSE-THP algorithms still outperform conventional counterparts, while MMSE-cTHP is more sensitive to correlation.The study generates correlated channels using an exponential correlation model and assumes an uncorrelated receive side.Source: B. Correlated Channel State… · p. 10

C. The impact of imperfect channels

  • With imperfect CSI at Eb/N0 = 20 dB, BER worsens for all precoders as σ2_e increases, and MB-MMSE-THP's advantage degrades faster at higher error variance.At low σ2_e, the proposed algorithms' performance advantage remains unchanged; the authors attribute faster degradation at high error variance to nonlinearity.Source: C. The impact of imperfect channels · p. 11
  • For channel-error variance above σ2_e = 0.14, MMSE-cTHP loses its BER advantage over MMSE-dTHP, indicating greater cTHP sensitivity to imperfect channels.The authors state that cTHP therefore needs more feedback bits than dTHP in realistic systems.Source: C. The impact of imperfect channels · p. 11
  • dTHP requires its scaling matrix to be sent to each distributed receiver, adding control overhead or feedforward information despite requiring less feedback information in practice.The paper leaves the feedback issue for further research.Source: C. The impact of imperfect channels · p. 11
  • MB-MMSE-cTHP provides better BER, while MB-MMSE-dTHP can provide higher system sum rates and more flexible sub-channel management.The cTHP sum-rate performance can be improved by increasing the branch count.Source: C. The impact of imperfect channels · p. 11

VI. CONCLUSIONS

The paper concludes that MB-MMSE-cTHP and MB-MMSE-dTHP exploit transmission degrees of freedom through branch lists, adding transmit diversity while keeping computational complexity reasonable. Simulations indicate that substantial improvement is achieved with only 2 or 4 branches.

VI. CONCLUSIONS

  • The proposed MB-MMSE-cTHP and MB-MMSE-dTHP algorithms target MU-MIMO systems with multiple receive antennas.Source: §VI · p. 11
  • Branch-list construction exploits transmission degrees of freedom and provides extra transmit diversity, while LQ-derived filters keep computational complexity reasonable.Source: §VI · p. 11
  • Simulations show considerable improvement with only 2 or 4 branches, which the authors identify as valuable for practical applications.The set of parallel channels also permits power-loading schemes to optimize power across channels.Source: §VI · pp. 11–12
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