Tomlinson-Harashima Precoding Research Articles

Fractionally spaced nonlinear Tomlinson-Harashima precoding with weight clustering for C-band 100 Gbaud/λ PAM-4 transmission.

A nonlinear Tomlinson-Harashima precoding (NTHP) scheme has been verified for its capability to effectively address both the linear and nonlinear inter-symbol interferences (ISIs) arising in the intensity-modulation direct-detection (IM-DD) fiber optics transmission. Nevertheless, the application of the NTHP scheme may significantly increase the number of levels for the intensity modulated signals, resulting in the reduction of both eye width and receiver sensitivity. Here, we propose a fractionally spaced NTHP with a weight clustering (FS-NTHP-WC) scheme. Consequently, an accurate ISI feedback can be obtained to enlarge the eye width; meanwhile a hardware-efficient implementation without the equalization penalty can be achieved by weight clustering and pruning. When the C-band 100 Gbaud/λ PAM-4 signals are transmitted, our proposed FS-NTHP-WC scheme not only can achieve 0.25 dB and 0.5 dB gains of receiver sensitivity under back-to-back (B2B) and 2-km standard single-mode fiber (SSMF) transmission conditions, respectively, but can also cut down the computational complexity by 90% and 76% in terms of the number of multiplications and additions, respectively, in comparison with the NTHP scheme.

Simplified THP and M-Log-MAP Decoder Based Faster Than Nyquist Signaling for Intra-Datacenter Interconnect

In this paper, a joint application of simplified Tomlinson-Harashima precoding (THP) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -log-maximum a posteriori probability ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -log-MAP) decoder at transceivers is proposed for low complexity faster than Nyquist (FTN) 4-ary pulse amplitude modulation (PAM4) signaling. A THP with only two taps is applied at the transmitter for FTN-induced inter-symbol interference elimination and equalization-induced noise amplification issue mitigation, which effectively relieves the burden of the receiver. At the receiver, a low complexity M-log-MAP decoder is equipped for residual impairments cancellation, thus a feedback channel is not required to accurately obtain the channel state information for the transmitter. Taking advantage of this collaborative approach, the proposed scheme performs much better than applying the two schemes separately. We experimentally demonstrate the proposed scheme in 70, 75, and 80GBaud PAM4 systems with a 32-GHz brick-wall filter, corresponding to 0.914, 0.853, and 0.8 FTN compression factor, respectively. The results show that the 80GBaud PAM4 signal is successfully transmitted over 2-km single mode fiber with the bit-error-rate (BER) under the 7% hard-decision forward error correction (HD-FEC) threshold of 3.8×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> . It is also observed that the proposed scheme shows about 0.7dB receiver sensitivity improvement compared to the error propagation-free decision feedback equalizer, namely the performance upper bound of the THP, at BER of 7% HD-FEC threshold in a 75GBaud FTN PAM4 system. According to the experimental results, the proposed scheme is a promising solution for bandwidth-starved low-cost intra-datacenter interconnect.

Partial response THP for high-speed PAM-4 transmission

Intensity-modulation direct-detection (IM/DD) pulse-amplitude modulation (PAM-4) signals are promising for high-speed data center networks. However, high-speed PAM-4 signals always are subject to critical impairments due to the bandwidth constraint. In the conventional PAM-4 system, a direct detection-faster than Nyquist (DD-FTN) with high complexity is usually appended to the feed-forward equalizer (FFE) for colored noise suppression to maximize transmission performance. In order to yield a simple PAM-4 scheme with similar performance as that of the DD-FTN, we proposed a nonlinear differential coding-generalized Tomlinson-Harashima Precoding (NLDC-GTHP) PAM-4 system. The proposed system consists of a NLDC-GTHP at the transmitter and a simplified two-dimensional constellation distortion (S2DCD) module at the receiver. We experimentally demonstrated the proposed NLDC-GTHP PAM-4 system for 62.5-Gb/s PAM-4 signal transmission over 2-km standard single mode fiber (SSMF) with the 10-dB bandwidth of about 12 GHz. By introducing the NLDC-GTHP and S2DCD, we achieved the similar receiver sensitivity as that of the DD-FTN at the hard-decision forward-error-correction (HD-FEC) bit error rate (BER) threshold, with the decrease of 86% computational complexity.

Hybrid TH precoding with subarray connection for multi‐user mmWave MIMO systems

AbstractExisting linear algorithms based on subarray connection (SC) structures may cause noise enhancement and performance degradation. To solve this problem, this paper presents a nonlinear hybrid Tomlinson‐Harashima precoding (THP) algorithm based on SC structure, named as THP‐SC algorithm. In the proposed algorithm, the sum‐rate optimization problem can be transformed into two equivalent optimization subproblems including the transmitter optimization subproblem and the receiver optimization subproblem. These two subproblems are easy to solve. Furthermore, the analog precoding and combining matrices of the optimization subproblem are solved by a new improved successive interference cancelation (SIC) algorithm based on the singular value threshold shrinkage (SVS), which can effectively reduce the computational complexity while eliminating the interference. In the end, the digital precoding and combining matrices are obtained by TH method. Through the joint optimization of the transmitter and receiver matrices in the system, the optimal solution is obtained by alternate iteration of the matrices. In this paper, a more realistic line‐of‐sight (LOS) and non‐line‐of‐sight (NLOS) channel system is also considered. Simulations verify that the proposed algorithm can improve the spectral efficiency and energy efficiency to some extent with a comparable complexity compared to the previous works.

A low‐complexity transmit combining method for downlink of massive multiuser multiple input multiple output systems

AbstractMultiuser multiple input multiple output (MU‐MIMO) systems can provide high spectral efficiency (SE) that is achieved with transmit/receive combining schemes at the base station (BS) for downlink (DL) and uplink (UL). So far, various combining schemes for DL were proposed, with the regularized zero forcing (RZF) and Tomlinson–Harashima precoding (THP) being usually considered. In a case of massive MU‐MIMO systems, with a large number of antennas, the matrix inversion operation used in the RZF and THP schemes causes high computational complexity that significantly aggravates implementation and raises its costs. The authors propose a low‐complexity transmit combining scheme for the massive MU‐MIMO systems with time division duplex (TDD), devised in a similar manner as the THP scheme, based on the gram‐Schmidt orthogonalization, and show that it achieves slightly better SE than the THP scheme for the estimated channel state information (CSI), but with a much lower complexity than the THP and RZF schemes.

Delay-Doppler Domain Tomlinson-Harashima Precoding for OTFS-Based Downlink MU-MIMO Transmissions: Linear Complexity Implementation and Scaling Law Analysis

Orthogonal time frequency space (OTFS) modulation is a recently proposed delay-Doppler (DD) domain communication scheme, which has shown promising performance in general wireless communications, especially over high-mobility channels. In this paper, we investigate DD domain Tomlinson-Harashima precoding (THP) for downlink multiuser multiple-input and multiple-output OTFS (MU-MIMO-OTFS) transmissions. Instead of directly applying THP based on the huge equivalent channel matrix, we propose a simple implementation of THP that does not require any matrix decomposition or inversion. Such a simple implementation is enabled by the DD domain channel property, i.e., different resolvable paths do not share the same delay and Doppler shifts, which makes it possible to pre-cancel all the DD domain interference in a symbol-by-symbol manner. We also study the achievable rate performance for the proposed scheme by leveraging the information-theoretical equivalent models. In particular, we show that the proposed scheme can achieve a near optimal performance in the high signal-to-noise ratio (SNR) regime. More importantly, scaling laws for achievable rates with respect to number of antennas and users are derived, which indicate that the achievable rate increases logarithmically with the number of antennas and linearly with the number of users. Our numerical results align well with our findings and also demonstrate a significant improvement compared to existing MU-MIMO schemes on OTFS and orthogonal frequency-division multiplexing (OFDM).

AFC-CE Hybrid Precoding for Energy-Efficient mmWave MIMO Systems

In this letter, an energy-efficient hybrid precoding method based on cross-entropy (CE) is proposed for the adaptive fully-connected (AFC) structure in millimeter wave massive multiple-input multiple-output systems, named the AFC-CE algorithm. By the nonlinear Tomlinson-Harashima precoding method, the non-convex spectrum efficiency maximization problem is transformed into a matrix residual minimization problem between the hybrid precoding matrix and the optimal precoding matrix. Then, using CE theory, the matrix residuals minimization problem is transformed into a probability distribution optimization problem of switching precoding matrix elements. Finally, the probability distribution parameters are updated adaptively by minimizing the CE between the estimated value and the expected value, until the residual converges and the switching precoding matrix is obtained. On this basis, the generalized power method and the least square method are used to give the phase shifts precoding matrix and the digital precoding matrix, respectively. Theoretical analysis and simulation results verify that the proposed method not only has lower computational complexity but also can provide better spectral efficiency and energy efficiency.

Low-complexity Tomlinson-Harashima Precoding Update Algorithm for Massive MIMO System

Low-complexity Tomlinson-Harashima Precoding Update Algorithm for Massive MIMO System

Demonstration of 200 Gbps PAM-4 transmission in a limited-bandwidth system using a two-tap THP with nonlinearity compensator

In this paper, we have proposed a digital signal processing (DSP) method for a 200 Gbps pulse-amplitude modulation level-4 (PAM-4) transmission in a bandwidth-limited system. To mitigate inter-symbol interferences (ISI) in the system and reduce the propagation of errors in the receiver, the Tomlinson-Harashima precoding (THP) technique was applied to the transmitter. For simplicity, the number of THP taps was limited to two (i.e., one coefficient to be optimized); the residual ISI resulting from the short tap was compensated for with a decision feedback equalizer (DFE). The performance of the proposed DSP was analyzed via experiments and simulations. Compared to conventional DSP techniques (such as DFE and the maximum likelihood sequence estimation (MLSE) without THP), the proposed DSP performs better within the limited-bandwidth region. As a result, the soft-decision forward error correction (SD-FEC) threshold (of 2 × 10−2) was satisfied even when the bandwidth fell to below 35% of the baud rate. To enhance this performance, post-filter and MLSE were additionally applied, after which its performance was investigated. In addition, the nonlinear characteristics of the components were alleviated by applying a pre-distortion technique to the THP using look-up tables; this further enhanced performance, lowering the SD-FEC threshold’s bandwidth to 32% of the baud rate.

Hybrid Nonlinear Transceiver Optimization for the RIS-Aided MIMO Downlink

The hybrid nonlinear transceiver optimization problem of reconfigurable intelligent surface (RIS)-aided multi-user multiple-input multiple-output (MU-MIMO) downlink is investigated. Specifically, the Tomlinson-Harashima precoder (THP) and the hybrid transmit precoder (TPC) of the base station are jointly optimized with the linear digital receivers of mobile users. The triangular feedback matrix of the THP is optimized and the optimal solution is derived in closed form based on a matrix inequality. Moreover, in order to tackle the nonconvexity of the constant-modulus constraints imposed on the analog TPC, the Majorization-Minimization (MM) based reconfigurable optimization framework is proposed, which strikes a trade-off between the implementation complexity and system performance in a reconfigurable manner. Explicitly, our MM-based reconfigurable optimization framework is capable of optimizing the analog TPC in a dynamically reconfigurable manner on an element-by-element, column-by-column, row-by-row or block-by-block basis. Moreover, an MM-based reconfigurable algorithm is proposed for the optimization of the phase shifting matrix at RIS, which also suffers from constant-modulus constraints. In the proposed MM-based reconfigurable algorithm, the RIS can be partitioned into a series of subarrays for striking different performance vs. complexity tradeoffs. Finally, our numerical results demonstrate the performance advantages of the proposed nonlinear hybrid transceiver optimization techniques.

Neural-Network-Based Nonlinear Tomlinson-Harashima Precoding for Bandwidth-Limited Underwater Visible Light Communication

Underwater visible light communication (UVLC) based on light-emitting diodes (LEDs) is considered a potential candidate for underwater wireless data transmission. However, the implementation of high-speed UVLC remains a challenge due to bandwidth limitation and nonlinear effects. In this paper, we investigate the performance of Tomlinson-Harashima precoding (THP) in the bandwidth-limited UVLC system for the first time. Apart from research on linear and Volterra series-based nonlinear THP, neural network (NN)-based THP is first proposed and conducted. At the transmitter, the intersymbol interference (ISI) and nonlinearity can be partially mitigated through a feedback neural network (FBN) without error propagation. A modulo operator is employed to eliminate instability. In addition, for the precoded signals, the power spectral density (PSD) is near-white. Therefore, it is possible to avoid the spectrum-shaping-induced signal-to-noise ratio (SNR) loss problem that is common in pre-emphasis methods. At the receiver, another modulo operator is used for decoding. An adaptive feedforward neural network (FFN) is applied to compensate for the channel state information (CSI) mismatch of the FBN and mitigate the remaining ISI and nonlinear impairments. In the demonstrated UVLC system, the experimental results prove the feasibility of the proposed method. The Q factor is increased by 3.39 dB at a data rate of 2.2 Gbps. 630 MBd CAP-16 transmission under a 7&#x0025; HD-FEC threshold is realized, which is 90 MBd higher than that when employing linear postequalization only.

Joint Precoding and Pre-Equalization for Faster-Than-Nyquist Transmission Over Multipath Fading Channels

Faster-than-Nyquist signaling (FTNS) has emerged as a promising technique to increase communication capacity in bandwidth-limited channels. However, the presence of FTN-induced inter-symbol interference (FTN-ISI) in the received observations, is detrimental to channel estimation (CE) and data detection in terms of computational complexity and performance. This paper copes with these problems by incorporating linear pre-equalization (LPE) and composite precoding formed by linear spectral precoding and Tomlinson-Harashima precoding (THP), into the FTNS. Specifically, LPE completely pre-equalizes the FTN-ISI, while spectral precoding resolves the LPE-caused signal spectral broadening by introducing proper artificial ISI, which is pre-equalized by THP. Channel-induced ISI, as the only ISI component in the observations, is estimated and equalized using the classical frequency-domain low-complexity schemes. We show that there are four advantages of the LPE-aided CE over the CE designed for the FTN transmissions without FTN-ISI pre-equalization, namely lower pilot overhead, simpler yet optimal pilot sequence design, lower mean-squared error of CE, and more robust against the FTN-ISI. Simulation results show that our scheme improves the performance of CE and detection, compared to existing FTN frequency-domain CE and equalization schemes.

Advanced DSP Enabled C-Band 112 Gbit/s/λ PAM-4 Transmissions With Severe Bandwidth-Constraint

The performance of high-speed intensity modulation direct detection (IM-DD) transmissions is severely degraded by the linear inter-symbol interference (ISI), due to both bandwidth-constrained optoelectrical components and the chromatic dispersion (CD). Meanwhile, nonlinear ISI due to intensity modulation and square-law detection can further deteriorate the IM-DD system performance significantly. Here, we first propose three metrics to quantify those impairments in IM-DD transmissions. Then, we compare both the BER performance and computational complexity of different DSP schemes to enable short-reach high-speed IM-DD transmissions with severe bandwidth constraint, using a C-band PAM-4 experimental test-bed with an end-to-end 10-dB bandwidth of 17 GHz. Those DSP schemes include receiver-side feedforward equalization (FFE), receiver-side Volterra filter equalization (VFE), transmitter-side Tomlinson-Harashima precoding (THP) together with receiver-side FFE, transmitter-side THP together with receiver-side VFE, channel-shortening filter (CSF) together with maximum likelihood sequence estimation implemented with linear branch metrics (MLSE-LI-BM), and CSF together with MLSE implemented with nonlinear branch metrics (MLSE-NL-BM). Our experimental results indicate that, only the combination of CSF and MLSE-NL-BM enables a 112 Gbit&#x002F;s transmission over 5 km fiber at the KP4-FEC threshold, but with an increased computational complexity.

Waveform Design for High-Order QAM Faster-Than-Nyquist Transmission in the Presence of Phase Noise

The state-of-the-art radio-frequency (RF) devices limit the deployment of extremely high-order quadrature amplitude modulation (QAM) formats (e.g., 16384-QAM) to meet the high-capacity demand on microwave backhaul links. This paper turns to faster-than-Nyquist (FTN) transmission using lower-order constellations and lower-cost RF devices as a solution to the demand. To realize low-complexity interference cancellation, we pre-equalize the FTN-induced inter-symbol interference at the transmitter by using Tomlinson-Harashima precoding (THP), while at the receiver suppressing the phase noise (PHN) generated by the RF local oscillators with pilot symbol assisted approaches. However, the THP may distort the pilots, which degrades the performance of PHN compensation. To resolve this problem, we propose two pilot designs that are distortion-free to precisely estimate the PHN samples. Moreover, we derive a closed-form expression of the symbol detection signal-to-noise ratio (SNR), in terms of the THP-FTN waveform parameters. With the SNR expression, a waveform optimization procedure is developed to maximize the SNR and enhance the achievable FTN capacity. The proposed scheme is validated in the simulated platform of 4096-QAM microwave link. The results demonstrate that the FTN signaling achieves the system capacity equivalent to that of the 16384-QAM Nyquist signaling with an SNR gain of 5.8 dB.

A Low-Complexity Hybrid Linear and Nonlinear Precoder for Line-Of-Sight Massive MIMO With Max-Min Power Control

In line-of-sight (LOS) massive MIMO, there is a nonnegligible probability that the channel vectors of some users become correlated. In these correlated scenarios, nonlinear precoders can be used instead of linear precoders at the cost of high computational complexity. To reduce the complexity of nonlinear precoders, hybrid linear and nonlinear precoders have been suggested in 5G New Radio (NR). In this paper, we find the probability that there is at least one pair of correlated users and we find the average number of correlated users. We propose a hybrid linear and nonlinear precoder (HLNP) with max-min power control for which the served users are divided into two groups. By employing a proposed modified Tomlinson-Harashima Precoding (THP), we design and combine the transmit vectors of the two groups such that inter-group interference is removed. Simulation results show that by employing HLNP instead of zero-forcing, the required transmit power to assure a given average block error rate (BLER) with 95% probability is reduced. For a 64-antennas BS, when modified THP is used for 3 out of 10 users in HLNP, the transmit power is reduced by up to 4.70 dB to assure an average BLER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−2</sup> using 16QAM and 64QAM constellations with NR low-density parity-check codes.

Mitigating pilot contamination in Rician faded massive MIMO 5G systems using enhanced zero forcing precoding and ring partitioning

In this paper, pilot contamination, in massive multiple-input multiple-output (MIMO) system is reduced to a significant extent by using base station (BS) rotation, user (UE) scheduling and enhanced Zero Forcing (e-ZF) precoding. Cellular network is divided, based on the angle, and partitioned into rings for grouping mobile UEs. Grouping is done by creating specifically designed policies using three significant network parameters. Neural network is used for scheduling grouped UEs and then UEs are assigned with optimal pilots. Hybrid particle swarm optimization with fuzzy (PSOF) logic detects optimal pilot in accordance to channel quality and additive white gaussian noise (AWGN). Simulations are performed in MATLAB-R2017b and performance is experimentally evaluated in terms of achievable rate, noise ratio and spectral efficiency in Rician Fading environment. Comparative results are evaluated with respect to conventional precoding techniques as dirty paper coding (DPC), block diagonalization (BD), matched filter (MF) and Tomlinson Harashima (TH) precoding.

PDM-16QAM transmission of 135 GBaud enabled by 120 GSa/s DAC and Tomlinson-Harashima pre-coding.

We experimentally verified that Tomlinson-Harashima (TH) pre-coding for an optical coherent system can enhance the speed of polarization division multiplexing (PDM) 16 quadrature amplitude modulation (QAM) signal to be higher than 135 GBaud with 120 GSa/s digital-to-analog converter (DAC) and electrical/optical/electrical bandwidth of the system narrower than 57 GHz at -30dB. The improvement of achievable capacity based on the received SNR in optical back-to-back conditions could be obviously observed by comparing the PDM-16QAM signal with and without TH pre-coding. The normalized generalized mutual information of the 135 GBaud PDM-16QAM signal for 120 km standard single mode fiber transmission was much higher than the threshold of error-free operation of a 5/6 code rate of digital video broadcasting satellite second generation low-density parity check forward error correction. To the best of our knowledge, the ratio of the symbol rate to the DAC sampling rate (135 GBaud over 120 GSa/s) is the highest ever reported for DAC-based optical coherent systems. The reachable distance of a 140 GBaud PDM-16QAM signal can be predicted to be approximately 100 km, even with the imperfect timing recovery operation.

MIMO-Aided Nonlinear Hybrid Transceiver Design for Multiuser Mmwave Systems Relying on Tomlinson-Harashima Precoding

Hybrid analog-digital (A/D) transceivers designed for millimeter wave (mmWave) systems have received substantial research attention, as a benefit of their lower cost and modest energy consumption compared to their fully-digital counterparts. We further improve their performance by conceiving a Tomlinson-Harashima precoding (THP) based nonlinear joint design for the downlink of multiuser multiple-input multiple-output (MIMO) mmWave systems. Our optimization criterion is that of minimizing the mean square error (MSE) of the system under channel uncertainties subject both to realistic transmit power constraint and to the unit modulus constraint imposed on the elements of the analog beamforming (BF) matrices governing the BF operation in the radio frequency domain. We transform this optimization problem into a more tractable form and develop an efficient block coordinate descent (BCD) based algorithm for solving it. Then, a novel two-timescale nonlinear joint hybrid transceiver design algorithm is developed, which can be viewed as an extension of the BCD-based joint design algorithm for reducing both the channel state information (CSI) signalling overhead and the effects of outdated CSI. Moreover, we determine the near-optimal cancellation order for the THP structure based on the lower bound of the MSE. The proposed algorithms can be guaranteed to converge to a Karush-Kuhn-Tucker (KKT) solution of the original problem. The simulation results demonstrate that our proposed nonlinear joint hybrid transceiver design algorithms significantly outperform the existing linear hybrid transceiver algorithms and approach the performance of the fully-digital transceiver, despite its lower cost and power dissipation.

Improved Tomlinson–Harashima Precoding for Ultra Reliable Communication in Intelligent Transportation Systems

Cyber-physical systems (CPSs) are characterized by integrating computation and physical processes. To cope with the challenges of the application of the CPSs in all kinds of environments, especially the cellular vehicle-to-everything (C-V2X) which needs high quality end-to-end communication, the robustness and reliability for CPSs are very crucial. Aiming at the technical challenges of information transmission caused by the fading effect and the fast time-varying characteristics of the channel for C-V2X communication, an improved Tomlinson–Harashima precoding (THP) algorithm for multiple input multiple output (MIMO) systems is proposed. Channel state information (CSI) and correlation are exploited to compensate instantaneous CSI, which could reflect current real-time channel status exactly. Further, the iterative water filling power allocation algorithm and the multiuser scheduling algorithm based on the greedy algorithm are jointly optimized and applied to the THP, which could improve the system performance. Simulation results show that the proposed algorithm can be efficiently applied to high-speed mobility scenarios and improve bit error ratio (BER) performance as well as spectrum utilization.

THP-UO Algorithm for Multi-User mmWave MIMO Systems

This paper presents an enhanced nonlinear hybrid Tomlinson-Harashima precoding method based on user ordering for millimeter wave (mmWave) massive multi-user mutiple-input multiple-out (MU-MIMO) systems, named as THP-UO algorithm. In the proposed method, making use of the fact that the performance of THP strongly depends on the ordering of precoding symbols, the enhanced THP solution is derived via the minimum mean square error (MMSE) criterion, which can obtain an optimal ordering of precoding symbols by rearranging the user ordering. Then, a digital precoding matrix can be given by THP method directly. It can further eliminate the influence of inter-user interference. Simulation results verify that the proposed hybrid THP-UO algorithm can reduce the bit error rate (BER) effectively, which is closer to the performance of the fully digital precoder than the previous algorithms.