Multi-modulus Algorithm Research Articles

Robust multimodulus blind equalization algorithm for multilevel QAM signals in impulsive noise

As a blind equalization algorithm, the fractional lower-order statistics based multimodulus algorithm (FLOS-MMA) is suitable for equalizing multilevel quadrature amplitude modulation (QAM) signals in impulsive noise. However, the equalization performance of FLOS-MMA is severely degraded in strong impulsive noise because the first-order gradient of the cost function constrains the ability of FLOS to suppress large outliers in the equalizer output signal and is sensitive to large outliers in the equalizer input signal. To enhance the robustness of multimodulus blind equalization algorithm against impulsive noise, a FLOS based momentum fractional-order multimodulus algorithm (FLOS-MFOMMA) is proposed in this paper. The proposed algorithm mitigates the adverse effects of impulsive noise on the updating of equalizer coefficients by utilizing the fractional-order gradient and accelerates the convergence speed by incorporating a momentum term. Then, the steady-state excess mean square error and the computational complexity of FLOS-MFOMMA are analyzed theoretically. Finally, numerical simulations, for equalizing 16-QAM signals under impulsive noise environments, show that the proposed algorithm is superior to FLOS-MMA in terms of robustness and convergence speed.

Secure WDM communication based on second order differential feedback electro-optical chaotic mutual injection synchronization

An optical phase chaos and mutual injection chaos synchronization scheme based on second order differential feedback electro-optical loop is proposed for chaotic communication. The filtered chaotic waves for mutual injection synchronization and multi-channel signals are transmitted through wavelength division multiplexing (WDM) to realize large-capacity confidential communication. The results show that the bandwidth of the optical chaos is 41.38 GHz, the normalized permutation entropy is 0.999, and the chaos has the performance of time delay suppressing (TDS). This scheme achieves up to 7-channel 25 GBaud 16QAM with the multi-modulus algorithm (MMA) algorithm at OSNR of 25 dB after 100 km transmission. The bit-rate distance of 280,000 Gb/s km is successfully realized by 500 km transmission of 4-channel 35 GBaud 16QAM signals at 25 dB OSNR. The analysis proves that the performance of the second order digital differential electro-optical phase feedback (2nd order DEOP) chaos is better than that of the 1st order DEOP chaos and 3rd order DEOP chaos. The 2nd order DEOP chaos can be generated at a smaller feedback coefficient, the bandwidth of radio frequency chaos is wider, and the TDS suppression is better. This scheme realizes high-speed and large-capacity secure optical chaos communication, which can be widely used in WDM networks and has practical feasibility and application value.

A New Dual Mode Blind Equalization Method Based on Neighborhood Symbol‐Based Decision and Decision‐Directed Algorithm

AbstractIn this paper, a new dual mode blind equalization method using neighborhood symbol‐based decision and decision‐directed (NSBD‐DD) algorithm is proposed to equalize the satellite communication signals. The proposed algorithm adopts the combined cost function to employ the advantages of symbol decision algorithm and the decision‐directed algorithm, which can further improve the universality of the proposed algorithm. Compared with the conventional blind equalization algorithm, the compensation ability of intersymbol interference and convergence speed of the proposed algorithm can be improved simultaneously. To verify the performance of the proposed algorithm, the computer simulation is performed, and the simulation results show that the residual intersymbol interference of the 64QAM signals can be decreased to −50 dB under the satellite channel, which has 30 dB lower than multi‐modulus algorithm (MMA), 26 dB lower than NSBD algorithm, and 10 dB lower than SBD algorithm. And compared with SBD algorithm, the number of iterations required for the NSBD‐DD algorithm can be decreased from 8000 iterations to 6000 iterations, which convergence speed is increased by 25%. © 2023 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Fast Multimodulus Blind Deconvolution Algorithms

A novel class of fast Multi-Modulus algorithms (fastMMA) for Blind Source Separation (BSS) and deconvolution are presented in this work. These are obtained through a fast fixed-point optimization rule used to minimize the Multi-Modulus (MM) criterion. Here, two BSS versions are provided to separate the sources either by finding the separation matrix at once or by separating a single source each time using a fast deflation technique. Further, the latter method is extended to cover systems of convolutive nature. Interestingly, these algorithms are implicitly shown to belong to the fixed step-size gradient descent family, henceforth, an algebraic variable step-size is proposed to make these algorithms converge even much faster. Apart from being computationally and performance-wise attractive, the new algorithms are free of any user-defined parameters.

A momentum fractional order multimodulus blind equalization algorithm

The multimodulus algorithm (MMA) is widely used to suppress inter-symbols interference (ISI) in communication systems because of its advantages that it can complete blind equalization and carrier phase recovery simultaneously. However, the slow convergence speed of MMA causes the equalizer to be unable to eliminate ISI effectively, which deteriorate the communication quality of the system. For that, we first present a momentum fractional order MMA (MFOMMA) to improve the performance of MMA for blind equalization in this paper. By optimizing MMA with the momentum term and the fractional order gradient information of the cost function, the proposed algorithm can get better ISI suppression performance. Then, the convergence performance of MFOMMA is analyzed to reveal the relationship between algorithm parameters and equalization ability. Moreover, in order to further enhance the ISI elimination capability of MFOMMA, we develop a variable fractional order scheme, which guarantee that the algorithm not only obtains a faster convergence speed but also a lower residual ISI. Finally, the effectiveness and superiorities of the proposed algorithm are demonstrated by some simulation experiments.

Adaptive blind equalization for multi‐level QAM signals in impulsive noise environment

A maximum correntropy criterion-blind clustering multi-modulus algorithm (MCC-BCMMA) is proposed to equalize multi-level quadrature amplitude modulation (QAM) channels in impulsive noise environment. The novel cost function is based on the maximum correntropy criterion (MCC), theoretical analysis shows that this criterion is useful when the noise is impulsive. Moreover, this cost function is decomposed into in-phase term and orthogonal term to eliminate the phase ambiguity, treats high-order QAM signal to classical QAM4 signal to reduce the error, and calculates the weight coefficients by a simple way. Simulation results show that the newly proposed algorithm MCC-BCMMA can be used to equalize multi-level QAM signals in both Gaussian and impulsive noise environments, and it shows better equalization performance under impulsive noise. The relationship between the kernel size of MCC, the convergence speed and accuracy is also analysed.

4×112 Gb/s/λ MCF Transmission Using Field PDM-PAM4 and Coherent Detection for Datacenter Applications

The inter-data center links are meeting the challenge of increasing transmission capacity continuously. Here, the tradeoff between transmission capacity and cost and power consumption especially needs to be considered. The field modulation polarization-division-multiplexed (PDM) four-level pulse amplitude modulation (PAM4) with coherent detection is a potential candidate for 400 Gb/s and beyond inter-data center transmissions, which has several advantages including low-cost, low optical signal to noise ratio (OSNR) requirement and high receiver sensitivity. In this paper, the feasibility and effectiveness of the field PDM-PAM4 coherent system are investigated over multi core fiber (MCF). In order to adapt to the field PAM signals, the signal-phase aid least-mean square (SP-LMS) and training multi-modulus algorithm (TMMA) are modified and analyzed for equalization and polarization demultiplexing. Finally, it is demonstrated that 4×112 Gb/s field PDM-PAM4 is successful transmitted over 16 km MCF and 384 km SSMF in C-band with a BER below 3.8e-3.

Comparison of Real- and Complex-Valued NN Equalizers for Photonics-Aided 90-Gbps D-band PAM-4 Coherent Detection

5G defines below 100 GHz as the millimeter-wave bands, whereas 100 GHz - 3 THz is categorized as THz band in 6G. Deep leraning (DL) is expected to enable a significant paradigm shift in 6G wireless networks. In this paper, D-band 90-Gbps single channel PAM-4 signal generation and transmission over 10-km SMF and 3-m wireless link at 140-GHz can be achieved. A novel complex-valued neural network (CVNN) equalizer using &#x2018;&Copf;ReLU&#x2019; activation function to directly recover PAM-4 signals from received noised signals is demonstrated. There are three DSP options in the experiment. In <i>Opt</i>.1, firstly conducted cascaded multi-modulus algorithm (CMMA) pre-equalization (pre-EQ) at transmitter, then processed via frequency offset estimation (FOE), carrier phase recovery (CPR) and finally real-valued neural network (RVNN) equalization at receiver. Here, the RVNN equalizers include DNN with a softmax output layer, two-step joint-DNN equalizer and LSTM. The experimental results show that LSTM-based equalizer outperforms the other real NN-based equalizers by average 0.5 to 1.5 dB at BER of 10^&#x2212;3 magnitude. Differently from <i>Opt</i>. 1, real-valued CMMA pre-EQ at transmitter is unexpected for CVNN in the other two options. In <i>Opt</i>. 2, we only combine down-conversion and CVNN regarded as &#x2018;pure data-driven&#x2019; training at receiver. This pure data-driven CVNN equalizer improves BER a lot and also has a larger computation burden, especially BER is as low as 1 &#x00D7; 10^&#x2212;4 with <i>n</i>₀ &#x003D; 571 and <i>n</i>₁ &#x003D; 200 training cells and the time complexity reaches 350000 in one iteration. Thanks to the aid of traditional mathematical-oriented models including FOE and CPR, the computation burden of CVNN in <i>Opt</i>. 3 is released significantly. Furthermore, we compare the performance of CVNN and RVNN in terms of BER decision accuracy, time complexity and receiver sensitivity. Followed by the same DSP with the same complexity, the comparison result between DNN and CVNN in the same structure of [371-260-1] with 11000 samples and 300 epochs shows that CVNN performs better due to its reservation of phase information. Therefore, we believe that the joint use of model-based, e.g., FOE, CPR steps and complex DL-based techniques has a potential for the future 6G wireless physical layer algorithms.

Performance Analysis of Random Fourier Features-Based Unsupervised Multistage-Clustering for VLC

Visible light communication (VLC) has emerged as a secure, cost-effective, and green supplement to the existing radio frequency (RF) communication systems. However, the performance of a typical VLC link is degraded by the following factors: (a) inter-symbol interference (ISI), which arises due to limited modulation bandwidth of light-emitting diode (LED), (b) the dispersion due to multi-path reflections, and (c) nonlinear transfer characteristics of LED. To mitigate ISI and LED nonlinearity, existing blind post-distorters, like the Volterra modified cascaded multimodulus algorithm (MCMMA) and the Hammerstein improved multistage clustering (HIMSC) approaches, experience degradations from modeling errors due to abrupt polynomial-series truncation. To mitigate these degradations, we propose a novel random Fourier features (RFF) based improved multi-stage clustering (RFF-IMSC) algorithm for blind post-distortion. Simulations performed over standardized IEEE 802.15 PAN VLC channels indicate that the proposed RFF-RIMSC post-distorter delivers improved performance as compared to the existing blind post-distorters. Also, analytical guarantees are presented and error-rate performance are analyzed for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -optimality of this approach with respect to supervised loss-functions.

Time Reverse Equalization Algorithm for 16 QAM Coherent Optical Communication Systems

The high speed rotation of state of polarization (RSOP) is a key issue for 16 quadrature amplitude modulation (QAM) coherent optical polarization division multiplexing (PDM) communication systems in strong lightning weather. To reduce the bit error rate (BER) caused by the high speed of RSOP, the historical information is first proposed to correct the demodulation process in coherent optical communication systems. By adding the time reverse error term into the cost function, a new equalization algorithm is proposed for coherent optical communication systems. Compared with the conventional Multi-modulus Algorithm (MMA), simulation results show that the average BER and steady-state error (SSE) of the new algorithm is reduced by 33.47% and 4.08%, respectively. Moreover, the average tolerance of RSOP of the new algorithm is increased by 28.57% compared with the MMA in 16 QAM coherent optical PDM communication systems.

Extensive evaluation of the MIMO along with sliced multi modulus algorithm and estimate strategies in the direction of MIMO OFDM solutions

A MIMO usage multiple antennas at both sending and receiving ends. These antennas communicate with each various other on the exact same frequency band and also help in linearly raising the channel ability. Because of the multi paths cordless networks face the issue of network fading which cause Inter Sign Interference. Amongst several formulas to blind deconvolution of a multiple-input multiple-output systems, the one that utilizes higher-order cumulants has benefits in pertains to of convergence price. Influenced on this algorithm and on a stochastic gradient technique, we recommend a formula with capability of recovering simultaneously all resources. The crucial feature of this particular DFE is actually a linear-complexity slim stations estimator, which possesses the capacity to see as well as refuse instincts located upon pair of audio variations: infected Gaussian as well as symmetrical alpha constant.

Blind nonlinearity mitigation of 10G DMLs using sparse Volterra equalizer in IM/DD PAM-4 transmission systems

Abstract The nonlinear characteristics of the electro-optic components are the major limiting factor to realize a multi-level signal transmission. Cascaded multi-modulus algorithm (CMMA) is a popular blind equalization scheme that can compensate for only linear distortions without using any training sequence. In this paper, a modified CMMA is proposed to blindly adapt the tap weights of a receiver-side nonlinear Volterra equalizer (NLVE) to mitigate both linear and nonlinear distortions. Furthermore, a nonlinear sparse Volterra equalizer (NLSVE) is suggested in this paper that can achieve similar performance with a substantial reduction of complexity, in comparison with an NLVE. Experimental results for a 25 Gb/s PAM-4 transmission in the 1310-nm band over a 20-km single-mode fiber (SMF-28e) using a commercial 10 GHz-class directly modulated laser (DML) for an intensity modulation direct-detection (IM/DD) optical access network verify that the proposed NLSVE can achieve a gain of ~ 2-dB in comparison with a traditional CMMA.

135-GHz D-Band 60-Gbps PAM-8 Wireless Transmission Employing a Joint DNN Equalizer With BP and CMMA

In this article, a novel scheme to effectively mitigate the nonlinear impairments in a PAM-8 radio-over-fiber (ROF) delivery is proposed by a joint deep neuron network (J-DNN) equalizer, which has more superiority in terms of good training accuracy, satisfactory tracking speed, and over-fitting suppression compared with a typical deep neuron network (DNN) equalizer. Our proposed J-DNN equalization scheme is mainly based upon back-propagation (BP) algorithm and blind cascaded multi-modulus algorithm (CMMA), which can be trained via two steps including DNN initialization and DNN optimization. By using the proposed J-DNN equalizer, 60-Gbps PAM-8 signal generation and transmission over 10-km SMF and 3-m wireless link at 135-GHz can be achieved. For the digital signal processing (DSP) at receiver, comparisons between CMMA equalizer, DNN equalizer, and J-DNN equalizer are demonstrated. The results indicate that J-DNN equalizer has a much better BER performance in receiver sensitivity than the traditional CMMA, and an improvement of receiver sensitivity can be achieved as much as 1 dB compared with a DNN equalizer at the BER of 3.8 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> . To the best of our knowledge, this is the first time to propose a novel joint DNN equalizer, which is promising for the development in integrated microwave photonics and microwave/millimeter-wave photonics for 5G applications and beyond.

Stabilised blind interference suppression of constant modulus‐based adaptive array over Rayleigh fading channels by initial antenna selection

This article proposes an initial antenna selection for constant modulus-based blind interference suppression algorithms to stabilise their prominent interference suppression performance. Constant modulus algorithm (CMA) is a well-known blind adaptive array scheme, but it cannot suppress the interference when signal-to-interference power ratio (SIR) is <0 dB. It faces strong limits on their applicable region over Rayleigh fading channels where instantaneous channel gain fluctuates over 10 dB range. Even if the expected desired signal strength is larger than interference, CMA may still miscapture the interference and incorrectly suppress the desired signal. The authors’ proposal is simplified approach to select the antenna element whose reception power is maximal. Certain antenna element can be expected to capture the desired signal precisely under the condition where SIR is statistically positive, so that the CMA processor can utilise it as an initial input. This condition will be ensured by increasing antenna element number, supported by the trends that base station antenna elements are going massive. Computer simulations verifies improved interference suppression performance provided by the proposed scheme. This article also verified applicability for multi-modulus algorithm to accommodate M-ary quadrature amplitude modulation (QAM) use.

100 Gbit/s VSB-PAM-n IM/DD transmission system based on 10 GHz DML with optical filtering and joint nonlinear equalization.

We experimentally demonstrate the transmission of 100 Gbit/s and beyond vestigial sideband (VSB) n-level pulse amplitude modulation (PAM-n) signals, with a commercial 10 GHz-class directly modulated laser (DML) in the C-band, using optical filtering. To mitigate transmission impairments at the transmitter side, Nyquist pulse shaping and Kaiser Window filtering techniques are used to overcome the limited bandwidth of optoelectronic devices. At the receiver side, the joint nonlinear equalization based on cascaded multi-modulus algorithm (CMMA) and Volterra Filter (VF) is used to reduce the strong nonlinear impairments from chirp and chromatic dispersion (CD). 100 Gb/s PAM-4, 107.5 Gb/s PAM-4, and 101.25 Gb/s PAM-8 signals can be successfully transmitted over 45 km, 10 km and 10 km standard single-mode fiber (SSMF) under the bit-error-ratio (BER) of 3.8 × 10-3, respectively.

SVR Based Blind Signal Recovery for Convolutive MIMO Systems With High-Order QAM Signals

This paper addresses the blind signal recovery for convolutive multiple-input multiple-output systems with high-order quadrature amplitude modulation (QAM) signals. First, a family of batch blind recovery algorithms is proposed. Concretely, they introduce the error function of multimodulus algorithm and the cross-correlation among different equalizer output vectors into the penalty term of the support vector regression (SVR) framework to recover all sources simultaneously. Then, the corresponding dual-mode blind recovery schemes are constructed to further decrease the interference. The new blind formulation through iterative re-weighted least square achieves low complexity optimization. The SVR framework, in essence, determines that the proposals perform better than the conventional methods in terms of data block size, total interference, and symbol error rate. Moreover, the excellent initialization provided by the first mode and the accurate error expression in the second mode ensure that the SVR-based dual-mode schemes work well with the high-order QAM signals. Finally, the efficiency of the proposals over the classical approaches is evaluated by simulations.

Pulse-Overlapping Super-Nyquist WDM System

In this paper, a pulse-overlapping super-Nyquist (Pol-SN) wavelength-division multiplexing (WDM) scheme is proposed to improve spectral efficiency of coherent optical transmission. In this scheme, two tributaries of polarization-division multiplexing quadrature phase-shift keying (PDM-QPSK) signals at the transmitter, carried by the same wavelength, are grouped together to allow for pulse overlapping in the time domain. Then, several wavelength channels are packed tightly in the frequency domain to form a super-Nyquist WDM system. At the receiver, three partial-response detection schemes are investigated and compared: (1) constant modulus algorithm (CMA) + duobinary shaping + maximum likelihood sequence estimation (MLSE); (2) duobinary shaping + multimodulus algorithm (MMA) + MLSE; and (3) channel shortening filter. Among these schemes, the easy-to-implement duobinary shaping +MMA + MLSE is selected by virtue of its best performance and lowest complexity. A 224-Gb/s Pol-SN PDM-QPSK system is investigated through numerical simulations. The simulation results show that the channel spacing of the 224 Gb/s Pol-SN PDM-QPSK system can be reduced to 20 GHz (11.2 bit/s/Hz SE) with 7% overhead hard-decision forward-error correction (HD-FEC, 3.8 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> ). At 26-GHz channel spacing, Pol-SN PDM-QPSK signals show 5.5-dB OSNR improvement, as compared with the signals of 224-Gb/s super-Nyquist PDM-16QAM. The simulation results also show that 224-Gb/s Pol-SN PDM-QPSK system, with 20-GHz spacing, can transmit up to 1000-km SSMF with 7% overhead HD-FEC.

Space-Time Blind Equalization of Dispersive MIMO Systems Driven by QAM Signals

This paper investigates space-time blind equalization (BE) of dispersive multiple-input multiple-output communication systems driven by high throughput quadrature amplitude modulation signals. Multistage processing is adopted to overcome the severe local convergence problem in BE. First, a good initial value for the equalizer is obtained by forcing one of the system inputs with desired delay to be recovered. Then, the fast multimodulus algorithm (MMA) is used to roughly equalize the communication system. After that, an improved MMA (IMMA) is developed to effectively search for a fine equalizer for the system. Moreover, the novel modified Newton method (MNM) proposed in our previous work is employed to fast optimize the MMA and IMMA cost functions, which significantly reduces the computational load. Furthermore, the proposed algorithm has an additional benefit that once a signal is recovered, its corresponding single-input multiple-output channel impulse response is estimated by the classical least squares methods. Then, the influence of the recovered signal to the original received signals is removed to avoid extracting this recovered signal repeatedly. The quadratic rate of convergence of the MNM is theoretically analyzed, and the good equalization performance of the IMMA is explained. Finally, simulation results are provided to illustrate the effectiveness of the proposed algorithm.

Experimental demonstration of a real-time PAM-4 Q-band RoF system based on CMMA equalization and interleaved RS code

Abstract In this paper, we experimentally demonstrated a complete real-time 4-level pulse amplitude modulation (PAM-4) Q-band radio-over-fiber (RoF) system with optical heterodyning and envelope detector (ED) down-conversion. Meanwhile, a cost-efficient real-time implementation scheme of cascaded multi-modulus algorithm (CMMA) equalization is proposed in this paper. By using the proposed scheme, the CMMA equalization is applied in the system for signal recovery. In addition, to improve the transmission performance of the system, an interleaved Reed-Solomon (RS) code is applied in the real-time system. Although there is serious power impulse noise in the system, the system can still achieve a bit error rate (BER) at below 1 × 10−7 after 25 km standard single mode fiber (SSMF) transmission and 1-m wireless transmission.

Joint compensation scheme of polarization crosstalk, intersymbol interference, frequency offset, and phase noise based on cascaded Kalman filter

A joint compensation scheme based on cascaded Kalman filter is proposed, which can implement polarization tracking, channel equalization, frequency offset, and phase noise compensation simultaneously. The experimental results show that the proposed algorithm can not only compensate multiple channel impairments simultaneously but also improve the polarization tracking capacity and accelerate the convergence speed. The scheme has up to eight times faster convergence speed compared with radius-directed equalizer (RDE) + Max-FFT (maximum fast Fourier transform) + BPS (blind phase search) and can track up polarization rotation 60 times and 15 times faster than that of RDE + Max-FFT + BPS and CMMA (cascaded multimodulus algorithm) + Max-FFT + BPS, respectively.