For faster-than-Nyquist Research Articles

Joint Optimization of Probabilistic Shaping and Geometric Shaping for Faster-than-Nyquist Signaling

Abstract We propose a joint optimization algorithm of probabilistic shaping and geometric shaping for faster-than-Nyquist (FTN) signaling. The theoretical analysis and bit error performance both represent that the FTN probabilistic-geometric shaping method is superior to the conventional Nyquist regular QAM system. When spectral efficiency (SE) is 3.077 bits/s/Hz, the proposed 16QAM FTN probabilistic-geometric shaping scheme can obtain 1.65 dB theoretical gain compared to the conventional Nyquist regular 16QAM system, while the simulation gain is 2.90 dB with spectral efficiency at 3.077 bits/s/Hz. The proposed 16QAM FTN probabilistic-geometric shaping scheme can obtain 1.63 dB theoretical gain compared to the conventional FTN regular 16QAM system, while the simulation gain is 3.28 dB with spectral efficiency at 3.282 bits/s/Hz. Therefore, the proposed method is reliable, which can be regarded as a potential scheme for the 6G communications.

Signal detection for FTN system under unknown noise model with imperfect CSI

The problem of signal detection for faster-than-Nyquist (FTN) signaling system with imperfect channel state information (CSI) is studied. Also, the non-Gaussian noise in real applications is considered. A general model, Gaussian mixture noise (GMN) model, is applied to describe the unknown probability density function (PDF) of the non-Gaussian noise. After that, a framework over variational inference is constructed to simplify the problem through finding an auxiliary PDF with factorization form. Based on the factor graph over the signal, joint response and the parameters of GMN, an iterative solution is obtained. Simulations demonstrate the effectiveness of the proposed approach.

SNR-Optimal Spreading Sequences for Chip-Wise Faster-Than-Nyquist Signaling

This letter proposes signal-to-noise ratio (SNR)-optimal spreading sequences for faster-than-Nyquist (FTN) transmission. The peculiar association of direct sequence spread spectrum with FTN proves relevant to mitigate noticeable cyclostationary features, leading up to covert communications schemes. Considering a matched-filter-based receiver, the proposed SNR-optimal spreading sequences and their closed-form approximations outperform traditional pseudo-random noise sequences by achieving near-interference-free bit-error-probability. Our results pave the way for the design of FTN-specific multiple access techniques while ensuring a low probability of detection/intercept.

Low Complexity Classification Approach for Faster-Than-Nyquist (FTN) Signaling Detection

In this letter, we investigate the use of machine learning (ML) to reduce the detection complexity of faster-than-Nyquist (FTN) signaling. In particular, we view the FTN signaling detection problem as a classification task, where the received signal is considered as an unlabeled class sample that belongs to the set of all possible classes samples. We observe that by jointly considering <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{p}$ </tex-math></inline-formula> samples, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{p} \ll N$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is the transmission block length, for the FTN signaling detection, the distance between the classes samples of any distance-based classifier increases, and hence, the detection performance improves. That said, we propose a low-complexity classifier (LCC) that exploits the ISI structure of FTN signaling to perform the classification task in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{p}$ </tex-math></inline-formula> -dimension space. The proposed LCC consists of two stages: 1) offline pre-classification that constructs the labeled classes samples in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{p}$ </tex-math></inline-formula> -dimensional space and 2) online classification where the detection of the received samples occurs. The proposed LCC is extended to produce soft-outputs as well. Simulation results show the effectiveness of the proposed LCC in balancing performance and complexity.

MetNet: A Novel Low-Complexity Neural Network-Aided Detection for Faster-Than-Nyquist (FTN) Signaling in ISI Channels

This paper studies the application of neural networks to Viterbi detection of FTN signals in an intersymbol interference (ISI) channel. The main contribution of this paper is to propose a receiver structure for detecting FTN signals in unknown static ISI channel. In particular, we propose a novel low-complexity neural network structure for calculating the branch metrics, and we explore its suitability for FTN signalling with channel uncertainty. We compare the proposed network, which we call the Metric Net (MetNet), to a benchmark neural network-based technique for metric calculation, the ViterbiNet, which was originally designed for ISI channels. The simulation results confirm that the MetNet outperforms the ViterbiNet, with two orders of magnitude lower complexity, and is much more resilient to channel uncertainty than the traditional Viterbi detector, which uses Euclidean distance for metric calculations. We further show that the MetNet exhibits robustness to being trained at mismatched SNR values and FTN pulse acceleration factors, meaning that the number of trained models required can be significantly reduced. Additionally, the results show that the proposed MetNet remains a favorable alternative at much higher levels of channel uncertainties. The results also reflect that we can generalize the MetNet to work with different channel models defined by different decaying factors. Finally, we show that we succeed in achieving a bandwidth efficiency gain of 33% due to FTN by using the MetNet in the presence of channel uncertainty.

Robust Estimators for Faster-Than-Nyquist Signaling

Novel, robust, and computationally attractive non-data-aided (NDA) and data-aided (DA) single and joint parameter estimators for faster-than-Nyquist (FTN) signaling are presented. By using the sixth moment alongside the second and fourth, it is possible to formulate an estimator allowing the joint blind estimation of the FTN symbol-packing ratio (SPR) (speed-up parameter) as well as the signal-to-noise ratio (SNR). The proposed estimators are robust in that they are based on uniformly spaced samples of the FTN signal taken at a fairly arbitrary FTN rate, insensitive to carrier and timing phase errors. While avoided in Nyquist signaling, the inter-symbol interference (ISI) term deliberately introduced in FTN signals is advantageously utilized for the estimation of SNR and SPR. The derivations are performed on a general complex base-signaling pulse and are illustrated for the commonly used root-raised cosine (RRC). Novel FTN Cramer-Rao lower bounds (CRLB) are derived to benchmark the efficiency of the proposed estimators. Extensive simulations are provided to illustrate the performance of the proposed estimators with respect to variations in all the relevant parameters and indicate that the estimators work best for the practical range of lower-intermediate values of SPR and SNR. The computational features of the proposed estimators, in addition to their robustness against carrier and sampling errors, make them valuable in many practical applications that incorporate cognitive radio (CR) alongside FTN.

A Novel Sum-Product Detection Algorithm for Faster-Than-Nyquist Signaling: A Deep Learning Approach

A deep learning assisted sum-product detection algorithm (DL-SPDA) for faster-than-Nyquist (FTN) signaling is proposed in this paper. The proposed detection algorithm works on a modified factor graph which concatenates a neural network function node to the variable nodes of the conventional FTN factor graph to approach the maximum a posterior probabilities (MAP) error performance. In specific, the neural network performs as a function node in the modified factor graph to deal with the residual intersymbol interference (ISI) that is not considered by the conventional detector with a limited complexity. We modify the updating rule in the conventional sum-product algorithm so that the neural network assisted detector can be complemented to a turbo equalization receiver. Furthermore, we propose a compatible training technique to improve the detection performance of the proposed DL-SPDA with turbo equalization. In particular, the neural network is optimized in terms of the mutual information between the transmitted sequence and the extrinsic information. We also investigate the maximum-likelihood bit error rate (BER) performance of a finite length coded FTN system. Simulation results show that the error performance of the proposed algorithm approaches the MAP performance, which is consistent with the analytical BER.

M-BCJR algorithm with channel shortening based on ungerboeck observation model for faster-than-Nyquist signaling

The M-BCJR algorithm based on the Ungerboeck observation model is a recent study to reduce the computational complexity for faster-than-Nyquist (FTN) signaling [1]. In this paper, we propose a method that can further reduce the complexity with the approximately same or better bit error rate (BER) performance compared to [1]. The information rate (IR) loss for the proposed method is less than 1% compared to the true achievable IR (AIR). The proposed improvement is mainly by introducing channel shortening (CS) before the M-BCJR equalizer. In our proposal, the Ungerboeck M-BCJR algorithm and CS can work together to defeat severe inter-symbol interference (ISI) introduced by FTN signaling. The ISI length for the M-BCJR algorithm with CS is optimized based on the criterion of the IR maximization. For the two cases τ = 0.5 and τ = 0.35, compared to Ungerboeck M-BCJR without CS benchmark [1], the computational complexities of Ungerboeck M-BCJR with CS are reduced by 75%. Moreover, for the case τ = 0.35, the BER performance of Ungerboeck M-BCJR with CS outperforms that of the conventional M-BCJR in [1] at the low signal to noise ratio region.

Joint Channel Estimation and Precoding for Faster-Than-Nyquist Signaling

The performance of existing frequency-domain channel estimation and equalization algorithms for faster-than-Nyquist (FTN) signaling is seriously hampered with the noise enhancement phenomenon that results from their inverse or pseudo-inverse operations. Through simulations, we show that the linear precoding can effectively mitigate the above-mentioned noise enhancement phenomenon. At first, a low complexity precoding-based channel estimation (PCE) algorithm is proposed for FTN signaling over frequency-selective fading channels. In contrast with most existing frequency-domain channel estimation algorithms, the proposed PCE algorithm has much better mean square error performance, and the performance improvement enlarges with the increase of signal to noise ratio. Furthermore, a joint channel estimation and precoding (JCEP) algorithm is proposed to perform data detection for FTN signaling over frequency-selective fading channels. On the one hand, compared with the existing frequency-domain channel estimation and equalization algorithms, the JCEP algorithm greatly reduces the complexity of signal processing at receivers since it performs the linear precoding processing at transmitters. On the other hand, even with estimated channel state information (CSI), the proposed JCEP algorithm can approach the bit error rate (BER) performance of the Nyquist signaling for all the modulation types adopted in digital video broadcasting-satellite-second generation extension (DVB-S2X). More precisely, the BER performance degradation with perfect and estimated CSI is 0.06 dB and 0.07 dB respectively when the time acceleration parameter equals to 0.7 and the rolling factor is 0.45.

Code-Based Channel Shortening for Faster-Than-Nyquist Signaling: Reduced-Complexity Detection and Code Design

A novel code based channel shortening (CCS) algorithm for faster-than-Nyquist (FTN) signaling is proposed, where special convolutional codes are used to absorb the channel memory. These convolutional codes have a special type of generator matrix that allows previous code symbols to be determined by the current code trellis state and thus been referred to as output-retainable convolutional codes (ORCCs). Different from conventional schemes, the CCS algorithm performs joint detection and decoding (JDD) based only on the code trellis by exploiting the ORCC structure. Therefore, it provides a new view for channel shortening techniques, i.e., absorbing the channel memory by using channel codes. Properties of ORCCs are discussed. Based on these properties, we derive the bit error rate (BER) bound for the CCS algorithm. According to the bound, a code search algorithm is proposed to facilitate the code design. Furthermore, two concatenated codes based on ORCCs are designed. Simulation results show that with a 16-states BCJR algorithm for JDD, the BER performance of the designed self-concatenated convolutional code incorporated with FTN signaling is only 0.75 dB away from the Shannon limit of the shaping pulse, and the required signal-to-noise ratio is below the BPSK limit of Nyquist signaling.

Blind symbol packing ratio estimation for faster‐than‐Nyquist signalling based on deep learning

This letter proposes a blind symbol packing rartio estimation for faster-than-Nyquist (FTN) signaling based on state-of-the-art deep learning (DL) technology. The symbol packing rartio is a vital parameter to obtain the real symbol rate and recover the origin symbols from the received symbols by calculating the intersymbol interference (ISI). To the best of our knowledge, this is the first effective estimation approach for symbol packing rartio in FTN signaling and has shown its fast convergence and robustness to signal-to-noise ratio (SNR) by numerical simulations. Benefiting from the proposed blind estimation, the packing-ratio-based adaptive FTN transmission without dedicate channel or control frame becomes available. Also, the secure FTN communications based on secret symbol packing rartio can be easily cracked.

Pre-Equalized Interference Cancellation for Faster-Than-Nyquist Signaling

An affine projection (AP)-based equalizer (APE) is introduced to eliminate the inter-symbol interference (ISI) for faster-than-Nyquist (FTN) signaling. Based on the APE, a pre-equalized interference cancellation (PIC) algorithm is proposed to eliminate the ISI for FTN signaling. By utilizing interference factors in the FTN system and low complexity APE, the computational complexity of the proposed PIC algorithm is much lower than the most existing block-based estimation algorithms, which makes it more practical for implementation. Besides, the proposed PIC algorithm has higher estimation accuracy by comparison with most existing algorithms. The simulation results show that the APE has a satisfactory bit error rate (BER) performance in the moderate ISI cases. In uncoded FTN systems, for all the modulation types adopted in digital video broadcasting-satellite-second generation extension (DVB-S2X), the proposed PIC algorithm can approximate the BER performance of the ISI-free Nyquist signaling when the time acceleration parameter and rolling factor equal to 0.8 and 0.3, respectively, which is beyond the performance of the most existing low-complexity algorithms. Even for 256-amplitude phase shift keying (APSK), the BER performance degradation is no more than 0.05 dB when the BER is 10 -5 . Furthermore, compared with the state-of-the-art frequency-domain equalization algorithm, the proposed PIC algorithm performs well in coded FTN systems.

Interference cancellation and signal detection technique based on QRD‐M algorithm for FTN signalling

This Letter presents interference cancellation and signal detection technique for faster-than-Nyquist (FTN) signalling. The proposed scheme uses QR decomposition (QRD)-M algorithm to execute interference cancellation and signal detection. Based on the Toeplitz structure of the inter-symbol interference coefficient matrix, the QRD-M algorithm is adopted to detect the transmitted symbols from an FTN signal. The computational complexity comparisons and numerical results demonstrate that the proposed technique can achieve the complexity <10% of Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm while maintaining BER performance similar to that of BCJR algorithm.

A Hybrid BP-EP-VMP Approach to Joint Channel Estimation and Decoding for FTN Signaling over Frequency Selective Fading Channels

This paper deals with low-complexity joint channel estimation and decoding for faster-than-Nyquist (FTN) signaling over frequency selective fading channels. The inter-symbol interference (ISI) imposed by FTN signaling and the frequency selective channel are intentionally separated to fully exploit the known structure of the FTN-induced ISI. Colored noise due to the faster sampling rate than that of the Nyquist signaling system is approximated by autoregressive process. A Forney style factor graph representation of the FTN system is developed and Gaussian message passing is performed on the graph. Expectation propagation (EP) is employed to approximate the message from channel decoder to Gaussian distribution. Since the inner product between FTN symbols and channel coefficients is infeasible by belief propagation (BP), we propose to perform variational message passing (VMP) on an equivalent soft node in factor graph to tackle this problem. Simulation results demonstrate that the proposed low-complexity hybrid BP-EP-VMP algorithm outperforms the existing methods in FTN system. Compared with the Nyquist counterpart, FTN signaling with the proposed algorithm is able to increase the transmission rate by over 40%, with only negligible BER performance loss.

Variational Inference-Based Frequency-Domain Equalization for Faster-Than-Nyquist Signaling in Doubly Selective Channels

This work deals with frequency-domain equalization for faster-than-Nyquist (FTN) signaling in doubly selective channels (DSCs). To handle the interference of frequency-domain symbols, the minimum mean square error (MMSE) equalizer involves high complexity in DSCs. To overcome the problem, we propose low-complexity receivers based on two variational methods, i.e., mean field (MF) and Bethe approximations. Compared with the MF method, the Bethe approximation takes into account the conditional dependencies of pairwise symbols. By only considering a small set of the frequency-domain symbols that have strong interference to each other, the complexity of the proposed algorithms increases linearly with the block length. Simulation results demonstrate that the proposed algorithms for FTN signaling are able to perform close to the MMSE equalizer in DSCs while with significantly reduced computational complexity.

Method for carrier frequency‐offset estimation of faster‐than‐Nyquist signalling

A new non-data aided frequency-offset estimation scheme for faster-than-Nyquist (FTN) signalling system is proposed. Considering the intentionally introduced intersymbol interference, the envelope of carrier signals is time varying, which makes it difficult to estimate the carrier frequency-offset at the receiver. In order to improve the estimation performance, a new estimation scheme is proposed which is based on Newton's method for minimising the non-linear least-squares error. Moreover, the proposed estimation scheme can be also applied into M-ary pulse-amplitude modulation. Further, the performance of proposed frequency-offset estimation scheme for FTN is verified and tested.

An 0.8-mm&lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;$^2$&lt;/tex&gt; &lt;/formula&gt; 9.6-mW Iterative Decoder for Faster-Than-Nyquist and Orthogonal Signaling Multicarrier Systems in 65-nm CMOS

This paper presents an iterative decoder for faster-than-Nyquist (FTN) and orthogonal signaling multi-carrier systems. FTN signaling is a method of improving bandwidth efficiency at the expense of higher processing complexity in the transceiver. The decoder can switch between orthogonal and FTN signaling modes and exploits channel properties to improve bandwidth efficiency. The decoder is fabricated in a 65-nm CMOS process and occupies a total area of 0.8 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> with decoder core taking up 0.567 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The power consumption of the chip is 9.6 mW at 1.2 V when clocked at 100 MHz, providing a peak information throughput of 1 Mbps and with an energy efficiency of 0.6 nJ per bit per iteration. To the best of our knowledge, those measurement results are from the first ever silicon implementation of a decoder for FTN signaling.