Adaptive Blind Equalization Research Articles

Convolutional Noise PDF at the Convergence State of a Blind Adaptive Equalizer

In the literature, the convolutional noise obtained at the output of a blind adaptive equalizer, is often modeled as a Gaussian process during the latter stages of the deconvolution process where the process is close to optimality. However, up to now, no strong mathematical basis was given supporting this phenomenon. Furthermore, no closed-form or closed-form approximated expression is given that shows what are the constraints on the system’s parameters (equalizer’s tap-length, input signal statistics, channel power, chosen equalization method and step-size parameter) for which the assumption of a Gaussian model for the convolutional noise holds. In this paper, we consider the two independent quadrature carrier input case and type of blind adaptive equalizers where the error that is fed into the adaptive mechanism which updates the equalizer’s taps can be expressed as a polynomial function of the equalized output up to order three. We show based on strong mathematical basis that the convolutional noise pdf at the latter stages of the deconvolution process where the process is close to optimality, is approximately Gaussian if complying on some constraints depending on the step-size parameter, input constellation statistics, channel power, chosen equalization method and equalizer’s tap-length. Simulation results confirm our findings.

Information Entropy and Fuzzy Logic Based Equalizer for PolMux QAM Coherent Optical Communication Systems

In polarization-multiplexed (PolMux) coherent optical communication systems, adaptive blind equalizer is efficient in demultiplexing and mitigating intersymbol interference (ISI). A novel blind algorithm based on Information Entropy and fuzzy logic is proposed, in which the Renyi's $\alpha$ entropy is adopted to measure the uncertainty of error between the desired and estimated probability density function (PDF). The nonparametric PDF estimator of Parzen window method is employed to estimate the PDF of symbols. Meantime, a fuzzy-logic tuning unit is designed to adjust the kernel size of Parzen window, which leads to fast convergence rate and small steady mean-square error. By simulation in PolMux-16 quadrature amplitude modulation (QAM) coherent systems, the correctness and effectiveness of the proposed algorithm are verified.

Blind adaptive correlation-based decision feedback equalizer (DFE) for underwater acoustic communications

Passive time reversal uses a set of matched filters corresponding to individual channel impulse response to reduce intersymbol interferences (ISIs) in underwater acoustic communications. Because of residual ISIs after time reversal, a coupled single decision feedback equalizer (DFE) is necessary. The coupled DFE uses a fixed tap number applicable to most shallow oceans with less user supervision. However, this correlation-based DFE needs the training sequence to estimate the individual channel response and initialize DFE, which decreases the effective bit rate and increases user supervision. We therefore in this paper propose a new blind structure of time reversal coupling with DFE, and it can complete multichannel identification and equalization at the same time without the training sequence. This new receiver has a reversible structure. In the first mode, it is linear and adapted by blind algorithms, e.g. constant modulus algorithm (CMA), to estimate the multichannel impulse responses and initialize the equalizer. In the second mode, the receiver becomes correlation-based DFE. From the viewpoint of user supervision and spectral efficiency, the new structure with no training sequence is more attractive. The blind correlation-based DFE exhibits the same steady-state mean square error (MSE) with the trained structure, and has been validated on real underwater communication data.

A robust and low complexity clustering-based blind equalizer for PolMux QAM optical coherent systems

In polarization-multiplexed (PolMux) coherent systems, adaptive blind equalizer is efficient in demultiplexing and mitigating inter-symbol interference. A novel clustering-based blind algorithm is proposed, which has simpler computational requirement than the traditional clustering-based blind algorithms. And furthermore, the variance of Gaussian clusters and step size in the novel algorithm can adjust adaptively in the process of convergence, which means it has excellent robust ability. By simulation in a PolMux-16/64QAM coherent systems, we show that the proposed algorithm outperforms existing blind algorithms.

A New Equalization Performance Analyzing Method for Blind Adaptive Equalizers Inspired by Maximum Time Interval Error

Up to now, the Mean Square Error (MSE) criteria, the residual Inter-Symbol Interference (ISI) and the Bit-Error-Rate (BER) were used to analyze the equalization performance of a blind adaptive equalizer in its convergence state. In this paper, we propose an additional tool (additional to the ISI, MSE and BER) for analyzing the equalization performance in the convergence region based on the Maximum Time Interval Error (MTIE) criterion that is used for the specification of clock stability requirements in telecommunications standards. This new tool preserves the short term statistical information unlike the already known tools (BER, ISI, MSE) that lack this information. Simulation results will show that the equalization performance of a blind adaptive equalizer obtained in the convergence region for two different channels is seen to be approximately the same from the residual ISI and MSE point of view while this is not the case with our new proposed tool. Thus, our new proposed tool might be considered as a more sensitive tool compared to the ISI and MSE method.

Adaptive Blind Postdistortion and Equalization of System Impairments for a Single-Channel Concurrent Dual-Band Receiver

Concurrent dual-band receivers use fewer RF components to flexibly receive signals in the dual bands. To further reduce the number of components, a compact concurrent dual-band receiver architecture has been demonstrated in this paper. It employs only a single down-conversion channel to down-convert the dual-band signals simultaneously. The down-converted signal is a relatively narrowband signal without the wide gap between the two bands. Hence, slower analog-to-digital converters can be adopted in the subsequent circuits to digitize the received signal. Due to the nonlinearity of the low-noise amplifier, there might be noticeable nonlinearity distortion and memory effects in the received dual-band signals. To ensure high signal quality, a blind postdistortion algorithm is proposed to cancel out the nonlinearity distortion in the two received signals adaptively. Subsequently, the blind finite-impulse response equalizers for each band are built based on the modified constant modulus algorithm and the decision-directed equalization method. In addition, the nonideal wireless channel response is also able to be compensated using the proposed algorithm. Experimental results have shown perfect postcompensation performances of the overall algorithms in the dual bands.

An Adaptive Momentum CMA Blind Equalization Based on Error Energy

To solve the problem of large steady state residual error of momentum constant modulus algorithm (CMA) blind equalization, a momentum CMA blind equalization controlled by energy steady state was proposed. The energy of the equalizer weights is estimated during the updating process. According to the adaptive filtering theory, the energy of the equalizer weights reaches to the steady state after the algorithm is converged, and then the momentum can be set to 0 when the energy change rate is less than the threshold, which can avoid the additional gradient noise caused by momentum and further improve the convergence precision of the algorithm. The proposed algorithm takes advantage of momentum to quicken the convergence rate and to avoid the local minimum in the cost function to some extent; meanwhile, it has the same convergence precision with CMA. Computer simulation results show that, compared with CMA, momentum CMA (MCMA) and adaptive momentum CMA (AMCMA) blind equalization, the proposed algorithm has the fastest convergence rate and the same steady state residual error with CMA.

Distortion Mitigation in Multiband OFDM RoF Transmission Employing Blind Post Equalizer

We propose and experimentally demonstrate a blind adaptive equalizer for a multiband (MB) orthogonal frequency-division multiplexing (OFDM) radio-over-fiber (RoF) system. The proposed equalizer adaptively decreases nonlinear and linear distortion, which might lead to serious signal degradation in MB OFDM transmission, without the use of a training signal. Instead of a training signal, error vector magnitude (EVM) and the power difference between the signal and the third-order intermodulation distortion components were observed in the equalizer. Therefore, the performance enhancement of MB OFDM transmission was experimentally verified in terms of EVM from 5.52% to 2.75% by mitigation of distortion. Consequently, the dynamic range of radio frequency input power was improved when the proposed equalizer was employed in an MB OFDM RoF system without increasing system overhead.

Low-Complexity Hybrid Adaptive Blind Equalization Algorithm for High-Order QAM Signals

It is well known that the constant modulus algorithm (CMA) presents a large steady-state mean-square error (MSE) for high-order quadrature amplitude modulation (QAM) signals. In this paper, we propose a low-complexity hybrid adaptive blind equalization algorithm, which augments the CMA error function with a novel constellation matched error (CME) term. The most attractive advantage of the proposed algorithm is that it is computationally simpler than concurrent CMA and soft decision-directed (SDD) scheme (CMA+SDD), and modified CMA (MCMA), while the approximation of steady-state MSE of the proposed algorithm is same with CMA+SDD, and lower than MCMA. Extensive simulations demonstrate the performance of the proposed algorithm.

A double-DD blind equalizer for PolMux QAM optical coherent systems

In polarization-multiplexed (PolMux) coherent system, adaptive blind equalization is efficient in demultiplexing and mitigating inter-symbol interference (ISI). We propose a double decision-directed (double-DD) blind algorithm which combines the output of soft decision-directed (SDD) algorithm with that of the equalizer in variable step size decision-directed (VSS-DD) mode. In our scheme, two equalizers can softly switch when the mean-square error (MSE) is sufficiently low and it can achieve fast convergence rate and small steady MSE in steady state. By simulation in PolMux-16QAM/64QAM coherent systems, we show that the double-DD algorithm outperforms existing blind equalizers.

Sign algorithm for blind channel equalisation

A new admissible hardware-efficient adaptive blind sign equalisation algorithm for quadrature amplitude modulation systems is presented. The main idea is to amplify the equalised sequence while constraining it to not to exceed the largest signal amplitude. Simulation results demonstrate improved performance in terms of mean squared error and inter-symbol interference measures.

Inspection of the Output of a Convolution and Deconvolution Process from the Leading Digit Point of View—Benford’s Law

In the communication field, during transmission, a source signal undergoes a convolutive distortion between its symbols and the channel impulse response. This distortion is referred to as Intersymbol Interference (ISI) and can be reduced significantly by applying a blind adaptive deconvolution process (blind adaptive equalizer) on the distorted received symbols. But, since the entire blind deconvolution process is carried out with no training symbols and the channel’s coefficients are obviously unknown to the receiver, no actual indication can be given (via the mean square error (MSE) or ISI expression) during the deconvolution process whether the blind adaptive equalizer succeeded to remove the heavy ISI from the transmitted symbols or not. Up to now, the output of a convolution and deconvolution process was mainly investigated from the ISI point of view. In this paper, the output of a convolution and deconvolution process is inspected from the leading digit point of view. Simulation results indicate that for the 4PAM (Pulse Amplitude Modulation) and 16QAM (Quadrature Amplitude Modulation) input case, the number “1” is the leading digit at the output of a convolution and deconvolution process respectively as long as heavy ISI exists. However, this leading digit does not follow exactly Benford’s Law but follows approximately the leading digit (digit 1) of a Gaussian process for independent identically distributed input symbols and a channel with many coefficients.

Recursive Blind Adaptive Equalization for 16-QAM applications

Abstract: An innovative blind adaptive identification algorithm based on least-squares type arguments is offered in this project. Devoid of any matrix inversion operation, parameter estimates are recursively updated with each output measurement. It is confirmed that the parameter estimates converge almost surely (a.s.) toward a scalar multiple of the true parameters. The viable utilization of this algorithm to the channel equalization problem is explained in this paper [3].

Design and Implementation of a Blind Adaptive Equalizer Using Frequency Domain Improved Square Contour Algorithm

Improved Square Contour Algorithm (ISCA) is a blind adaptive equalization technique which employs estimated outputs from the decision device along with the dispersion constant in its cost function, thereby, forcing the equalizer outputs to lie on multiple zero error square moduli instead of a single contour in Square Contour Algorithm (SCA). This leads to greater ISI suppression and an improved convergence rate. In this paper, frequency domain implementation of block based ISCA is developed. Simulation results show that the FDISCA exhibits similar performance as time domain ISCA but with a significantly lower computational complexity. However, by employing normalization in the frequency bins, a noteworthy improvement in the convergence rate for FD-ISCA is observed. Complete hardware architecture of FD-ISCA along with its simulation results is also discussed.

Performance analysis of a family of adaptive blind equalization algorithms for square-QAM

Multimodulus algorithms (MMA) based adaptive blind equalizers mitigate inter-symbol interference and recover carrier-phase in communication systems by minimizing dispersion in the in-phase and quadrature components of the received signal using the respective components of the equalized sequence in a decoupled manner. These equalizers are mostly incorporated in bandwidth-efficient digital receivers which rely on quadrature amplitude modulation (QAM) signaling. The nonlinearities in the update equations of these equalizers tend to lead to difficulties in the study of their steady-state performance. This paper presents originally the steady-state excess mean-square-error (EMSE) analysis of different members of multimodulus equalizers MMAp–q in a non-stationary environment using energy conservation arguments, and thus bypassing the need for working directly with the weight error covariance matrix. In doing so, the exact and approximate expressions for the steady-state mean-square-error of several MMA based blind equalization algorithms are derived, including MMA2–2, MMA2–1, MMA1–2, and MMA1–1. The accuracy of the derived analytical results is validated using Monte–Carlo experiments and found to be in close agreement.

A Maximum Entropy inspired model for the convolutional noise PDF

In this paper we consider a blind adaptive deconvolution problem in which we observe the output of an unknown linear system (channel) from which we want to recover its input using an adaptive blind equalizer (adaptive linear filter). Since the channel coefficients are unknown, the optimal equalizer's coefficients are also unknown. Thus, the equalizer's coefficients used in the deconvolution process are only approximated values leading to an error signal in addition to the source signal at the output of the deconvolutional process. We define this error signal throughout the paper as the convolutional noise. It is well known that the convolutional noise probability density function (pdf) is not a Gaussian pdf at the early stages of the deconvolutional process and only at the latter stages of the deconvolutional process the convolutional noise pdf tends to be approximately Gaussian. Despite this knowledge, the convolutional noise pdf was modeled up to recently as a Gaussian pdf because it simplifies the Bayesian calculations when carrying out the conditional expectation of the source input given the equalized or deconvolutional output and since no other model was suggested for it. Recently, a new model was suggested by the same author for the convolutional noise pdf based on the Edgeworth expansion series. This new model leads to improved deconvolution performance for the 16 Quadrature Amplitude Modulation (QAM) input and for a signal to noise ratio (SNR) of 30 dB. Thus, the question that arose here was whether we may find another model for the convolutional noise pdf that will also lead the system with improved deconvolutional performance compared to the case when the Gaussian model is applied for the convolutional noise pdf. In this paper, we propose a new model for the convolutional noise pdf inspired by the Maximum Entropy density approximation technique. We derive the relevant Lagrange multipliers and obtain as a by-product new closed-form approximated expressions for the conditional expectation and mean square error (MSE). Simulation results indicate that improved system performance is obtained from the residual ISI point of view for the 16QAM input case with our new proposed model for the convolutional noise pdf compared to the case when the Gaussian model or Edgeworth expansion series are applied for the convolutional noise pdf. For two other chosen input sources, a faster convergence rate is observed with the algorithm using our new proposed model for the convolutional noise pdf compared to the Maximum Entropy and Godard's algorithm.

Under What Condition Do We Get Improved Equalization Performance in the Residual ISI with Non-Biased Input Signals Compared with the Biased Version

Recently, closed-form approximated expressions were obtained for the residual Inter Symbol Interference (ISI) obtained by blind adaptive equalizers for the biased as well as for the non-biased input case in a noisy environment. But, up to now it is unclear under what condition improved equalization performance is obtained in the residual ISI point of view with the non-biased case compared with the biased version. In this paper, we present for the real and two independent quadrature carrier case a closed-form approximated expression for the difference in the residual ISI obtained by blind adaptive equalizers with biased input signals compared with the non-biased case. Based on this expression, we show under what condition improved equalization performance is obtained from the residual ISI point of view for the non-biased case compared with the biased version.

Nyquist-WDM With Low-Complexity Joint Matched Filtering and Adaptive Equalization

In this letter, we investigate the application of jointly processing the matched filtering and blind adaptive equalization functions in a receiver for a Nyquist wavelength division multiplexing (Nyquist-WDM) system employing a root-raised-cosine (RRC) pulse shaping filter. We show that blind equalization is able to also provide the matched filtering function, if the number of filter taps is increased. In addition, if implemented in the frequency domain, this form of blind equalization is more computationally efficient than when a separate RRC is used. It is also verified that this method is not sensitive to frequency offset. We demonstrate a 16-channel 19-GBd Nyquist-WDM system with QPSK and 16-quadrature-amplitude-modulation formats, and find the frequency domain blind equalizer can replace the RRC filter without adding a performance penalty.

Steady-state performance of multimodulus blind equalizers

Multimodulus algorithms (MMA) based adaptive blind equalizers mitigate inter-symbol interference in a digital communication system by minimizing dispersion in the quadrature components of the equalized sequence in a decoupled manner, i.e., the in-phase and quadrature components of the equalized sequence are used to minimize dispersion in the respective components of the received signal. These unsupervised equalizers are mostly incorporated in bandwidth-efficient digital receivers (wired, wireless or optical) which rely on quadrature amplitude modulation based signaling. These equalizers are equipped with nonlinear error-functions in their update expressions which makes it a challenging task to evaluate analytically their steady-state performance. However, exploiting variance relation theorem, researchers have recently been able to report approximate expressions for steady-state excess mean square error (EMSE) of such equalizers for noiseless but interfering environment.In this work, in contrast to existing results, we present exact steady-state tracking analysis of two multimodulus equalizers in a non-stationary environment. Specifically, we evaluate expressions for steady-state EMSE of two equalizers, namely the MMA2-2 and the βMMA. The accuracy of the derived analytical results is validated using different set experiments and found in close agreement.

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