Volterra Filter Research Articles

Chaotic characterization and chaotic prediction modelling of HFSWR ionospheric echoes excited by typhoon

ABSTRACT This research explores the intricate-coupling relationship between typhoons and the ionosphere by the high-frequency surface wave radar’s (HFSWR) over-the-horizon capabilities. This letter describes the gravity wave features in travelling ionospheric disturbances (TIDs) and ionospheric drift information. Further discussion is raised on the chaotic properties of gravity wave features using the Lyapunov exponents and chaotic attractors, particularly as they are influenced by typhoon. To predict these gravity wave features more accurately, a new model is developed by synergizing phase space reconstruction theory and statistical learning theory, a chaotic prediction model for gravity wave features constructed on a third-order Volterra filter. The experimental findings confirm that the gravity waves features detected via HFSWR present the hyperchaotic behaviour. More importantly, the innovative gravity wave prediction model based on chaotic attractor can capture the dynamics of the original chaotic system. Notably, the model’s prediction accuracy has been improved by 98.6% after the introduction of phase space reconstruction modelling.

A robust adaptive decomposable Volterra filter based on the hyperbolic tangent Leclerc function and its performance analysis

SummaryMost of the existing adaptive filter algorithms pay more attention to improving performance while ignoring the computational complexity and the impact of the impulsive environment. When encountering the impulsive noise environments, the performance of traditional nonlinear adaptive filter may be significantly reduced and usually needs high computational cost. Therefore, this article proposes a hyperbolic tangent Leclerc robust nonlinear adaptive filter based on the low complexity decomposable Volterra model (HTLNAF‐DVM). The filter is implemented by imposing a rank‐one structure on the full Volterra model to get a product of linear filters, and employs a hyperbolic tangent Leclerc function as a robust norm to effectively improve the robustness against the impulsive noise. In addition, we give the theoretical analyses of the steady‐state mean‐square performance of the proposed HTLNAF‐DVM. Finally, the simulation results prove that the proposed HTLNAF‐DVM algorithm has better performance than the existing algorithms and fit well with the theory.

Nonlinear imaging-based bubble cloud size estimates compared with histotripsy treatment zone: An in vitro study

Volterra filtering combined with subharmonic imaging was previously reported to enhance the visualization of the histotripsy bubble cloud. In this study, we compared the bubble cloud size obtained by imaging to the histotripsy ablation region using receiver operating characteristic (ROC) analysis. Histotripsy was performed in a red blood cell (RBC)-doped agarose phantom using a 1 MHz transducer. Ultrasound imaging data was acquired using a C5-2V probe and chirp-coded excitation (bandwidth: 2–5 MHz). These ultrasound acquisitions were then processed by subharmonic (SH) matched filtering alone, SH with quadratic Volterra filtering, and SH with cubic Volterra filtering. The therapy pulse and imaging sequence were interleaved (N = 100 frames), and a video camera was used to visualize the damaged region. A mean bubble cloud image was generated across all treatment sequences and co-registered with the camera image. A ROC curve was formed using binary classification of the mean bubble cloud versus the ablation zone, and the area under the ROC curve was determined. Filtering by quadratic and cubic Volterra filters reduced artifacts significantly [over 20 and 30 dB contrast-to-tissue enhancement (p < 0.01), respectively], along with achieving a high area under the ROC curve (0.97). These findings highlight the potential of Volterra filtering for histotripsy guidance.

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays ( since we are giving more degrees of freedom via the fractional delays ) and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform ( TFT ) of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm ( CSA ) are observed to be superior when compared with other techniques such as real-coded genetic algorithm ( RGA ), particle swarm optimization ( PSO ) and gradient-search ( GS ) methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays U+0028 since we are giving more degrees of freedom via the fractional delays U+0029 and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform U+0028 DTFT U+0029 of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm U+0028 CSA U+0029 are observed to be superior when compared with other techniques such as real-coded genetic algorithm U+0028 RGA U+0029, particle swarm optimization U+0028 PSO U+0029 and gradient-search U+0028 GS U+0029 methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.

Rational b-spline adaptive filter for active noise control of machinary noises

Spline adaptive filter has a different architecture than linear-in-parameter nonlinear filters like Volterra filter. There are various kinds of splines like Bezier spline, B-spline and rational B-spline. Rational B spline has weights associated with control points and hence, provides more flexibility than B-spline to create a curve. This paper presents development of rational b-spline adaptive filter for active noise control. Updating equations to update control points and weights of control points are derived. A simulation study is presented to study performance of proposed rational B-spline adaptive filter to reduce machinary noises. Performance of proposed rational B-spline filter are compared with B-spline filter and Bezier filter for active noise control of experimentally recorded noises of band-saw, CNC machine and compressor

Nonlinear active noise control with tap-decomposed robust volterra filter

The Volterra filter is proven to be a structurally simple method for nonlinear active noise control (NLANC). However, its wide application is deeply constrained by computational complexity. This paper proposes a tap-decomposed Volterra filter, in which a long-tap Volterra filter is approximated as multiple short-tap filters by the nearest Kronecker decomposition. Since the number of taps of decomposed filters is much less than that of the original counterpart, the proposed method greatly reduces the implementation cost of the Volterra filter. Moreover, for impulsive noise sources in NLANC, decomposed filters are updated using the maximum correntropy criterion derived from information theory. Simulation experiments with different noise source types highlight the noise control performance of the proposed approach.

Identification of the parametric array loudspeaker system using differential Volterra filter

The self-demodulation property of a parametric array loudspeaker (PAL) system can peel off audible sound modulated on ultrasonic wave and deliver it with high directivity, but also introduces nonlinear distortion. The PAL system can be modeled by the Volterra series expansion in any acoustic field region. However, measurements of Volterra kernel using simple frequency sweep excitation signals may not provide adequate information about the nonlinear system, and the use of wideband signals such as speech signals makes convergence difficult. Based on the Westervelt model, we propose a PAL system identification method utilizing the differential Volterra kernel. The results demonstrate superior fitting performance for real swept frequency output data and speech signal output data compared to the existing one-dimensional Volterra filter (ODVF). Additionally, the proposed method exhibits fewer parameters and faster convergence speed than the traditional volterra kernel.

Low-Complexity Conditional Generative Adversarial Network (c-GAN) Based Nonlinear Equalizer for Coherent Data-Center Interconnections

Nonlinear impairments emerging from both fiber optical links and optoelectrical devices are the major bottleneck for enhancing the throughput of photonic data-center interconnection (DCI). Here, we propose a data-driven nonlinear equalizer based on a conditional generative adversarial network (c-GAN) for high-speed and large-capacity photonic DCI. Its performance is experimentally evaluated by transmitting C-band 40-channel 60-GBaud dual polarization-16 quadrature amplitude modulation (DP-16QAM) dense wavelength division multiplexing (DWDM) signals over 20-km standard single-mode fiber (SSMF). Our experimental results verify that, the proposed c-GAN equalizer outperforms both traditional nonlinear equalizers and other data-driven counterparts, in terms of both the bit-error ratio (BER) and computational complexity. Only the proposed c-GAN equalizer can guarantee the BER performance of all 40 channels to reach the threshold of 20% soft decision forward error correction (SD-FEC) at BER=0.027, leading to a net transmission capacity of 16 Tb/s. Meanwhile, the computational complexity of the proposed c-GAN equalizer can be significantly reduced by 31.8%, 80.7%, and 98.8%, respectively, in comparison with Volterra filter equalizer (VFE), multilayer perceptron (MLP), and long short-term memory neural network (LSTM-NN) schemes.

Contrast-Enhanced Imaging of Histotripsy Bubble Clouds Using Chirp-Coded Excitation and Volterra Filtering.

Histotripsy is a focused ultrasound therapy that ablates tissue via bubble cloud activity. Real-time ultrasound image guidance is used to ensure safe and effective treatment. Plane-wave imaging enables tracking of histotripsy bubble clouds at a high frame rate but lacks adequate contrast. Furthermore, bubble cloud hyperechogenicity is reduced in abdominal targets, making the development of contrast-specific sequences for deep-seated targets an active area of research. Chirp-coded subharmonic imaging was reported previously to enhance histotripsy bubble cloud detection by a modest 4-6 dB compared to the conventional sequence. Incorporating additional steps into the signal processing pipeline could enhance bubble cloud detection and tracking. In this study, we evaluated the feasibility of combining chirp-coded subharmonic imaging with Volterra filtering for enhancing bubble cloud detection in vitro. Chirped imaging pulses were used to track bubble clouds generated in scattering phantoms at a 1-kHz frame rate. Fundamental and subharmonic matched filters were applied to the received radio frequency signals, followed by a tuned Volterra filter to extract bubble-specific signatures. For subharmonic imaging, the application of the quadratic Volterra filter improved the contrast-to-tissue ratio from 5.18 ± 1.29 to 10.90 ± 3.76 dB, relative to the application of the subharmonic matched filter. These findings demonstrate the utility of the Volterra filter for histotripsy image guidance.

Suppressed Noise Enhancement Equalization Combined With Nonlinear-MLSE Techniques for Bandwidth-Limitation IM/DD System

Bandwidth limitation introduced inter-symbol interference (ISI) is the major impediment for high baud rate intensity modulation and direct detection (IM/DD) system with commercial devices. Least mean square (LMS) criterion based feedforward equalizer (FFE) is the most common method to eliminate ISI in single-carrier system. However, there is a tradeoff between the residual ISI and enhanced noise when the system bandwidth is severely limited. Although a post filter (PF) followed by maximum likelihood sequence estimation (MLSE) can suppress the enhanced noise effectively, the residual ISI left by post-FFE will degrade the system performance as the subsequent MLSE only handles ISI introduced by PF. In this paper, an adaptive partial response equalization realized by a channel-shortening filter (CSF) is investigated, which is combined with MLSE. As the target of CSF is a partial response signal with desired low-pass feature, the noise will not be enhanced in this scheme. And the residual ISI with determined response can be fully solved by MLSE. The performance of CSF based MLSE (CSF-MLSE) is evaluated and compared to PF based MLSE (PF-MLSE) in 60/70/75Gbaud four-level pulse amplitude modulation (PAM-4) system with 10-dB bandwidth of 16.2GHz. The experimental results show that CSF-MLSE has a better capability of ISI elimination compared to PF-MLSE. The bit error rate (BER) can be reduced from 7.4×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> to 4×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> and from 3×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> to 1×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> for 70Gbaud and 75Gbaud in optical back-to-back (OBTB) transmission. To further compensate the system nonlinear impairments, several nonlinear MLSE (NL-MLSE) techniques assisted by look-up table (LUT) or Volterra filter are proposed and investigated in this paper. The BER of 75Gbuad PAM-4 can be below 3×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> after using NL-MLSE schemes.

A Short-Term Forecast Method of Maximum Usable Frequency for HF Communication

To further improve the quantitative forecasting ability of high-frequency (HF) communication frequency, we propose a maximum usable frequency (MUF) adaptive forecasting method. This method has four new features: 1) The Volterra filter is introduced to this method, which is easy to implement due to the simple structure. 2) This method can forecast M(3000)F2 based on past measurements without using any previous or forecast data of geomagnetic and solar activity parameters. 3) The spatial characteristics of the M(3000)F2 can be obtained by the improved reconstruction method based on geomagnetic coordinates. 4)The high prediction accuracy can be obtained by the training data set on the 27-day (one solar rotation period). Through the statistical analysis of the measurements and predictions, the results show that the forecast values are in good agreement consistent with the measurement during the geomagnetic quiet or the disturbance periods. This method is proven to have good adaptability and usability and has a further performance improvement compared with ITU and long-term forecast models. And the root-mean-square error of MUF forecasts decreased by 1.9MHz and 0.9MHz on average, and the relative root-mean-square error decreased by 12.6% and 2.9% on average.

Recurrent Neural Network Based Joint Equalization and Decoding Method for Trellis Coded Modulated Optical Communication System

In this paper, a joint equalization and decoding method based on recurrent neural networks (RNNs) is proposed for trellis coded modulation (TCM) systems. For traditional methods based on concatenated equalizers and Viterbi decoder, the information may be lost between the equalization and decoding. However, our proposed joint method can obtain the information bits directly from the received symbols and the error also can be corrected in iterations. In two-dimensional eight-level pulse amplitude modulation (2D-PAM8) links, we experimentally demonstrate that, compared with the traditional method based on Volterra filter and Viterbi decoder, the proposed joint method based on RNNs with similar complexity can achieve bit-error rate (BER) performance improvements of 1 dB, 1 dB, and 2 dB in 87.5 Gb/s, 100 Gb/s, and 112.5 Gb/s transmissions at 7% hard-decision forward-error-correction (HD-FEC) limit, respectively. For the 10 km standard single-mode fiber (SSMF) link at 87.5 Gb/s, the traditional method cannot get the BER below the 7% HD-FEC limit whereas the joint RNNs method with similar complexity can get the BER below the 7% HD-FEC limit. As the nonlinearity increases, the proposed joint method can achieve greater improvement of BER performance.

Contrast-specific imaging of histotripsy: Chirp-coded subharmonic imaging combined with Volterra Filtering

Histotripsy is a non-thermal focused ultrasound therapy under development to ablate tissue mechanically via bubble cloud activity. Real-time ultrasound imaging is used for treatment guidance. Bubble cloud hyperechogenicity is reduced in deep abdominal targets, making contrast-specific imaging an active area of research. Subharmonic imaging with chirp-coded excitation can improve bubble cloud detection, though the contrast-to-tissue ratio (CTR) was still limited to nearly 6 dB. Nonlinear components of a signal can be delineated with Volterra filtering, including those associated with bubble oscillations. In this study, we tested Volterra filtering as a means to enhance bubble cloud detection. Histotripsy bubble clouds were generated in scattering tissue-mimicking phantoms. Imaging of the bubble clouds was performed with a chirped pulse (1.9-μs duration, 7–12 MHz bandwidth). The scattered signals were processed with a subharmonic matched filter, followed by a tuned second-order Volterra filter. Volterra filtering improved the CTR for bubble cloud detection two-fold relative to matched filtering alone (12.5 dB vs 5.6 dB). Further improvement in bubble contrast was observed for third-order Volterra filtering (CTR of 20.3 dB) but at the cost of underestimating the bubble cloud area. Overall, these findings indicate the utility of Volterra filtering as a means to improve histotripsy image guidance.

Design of robust adaptive Volterra noise mitigation architecture for sEMG signals using metaheuristic approach

Surface Electromyogram (sEMG) signals, like other electrophysiological measurements, get corrupted by several artefacts; much critical helpful information regarding a person’s clinical conditions may alter or be lost entirely. Therefore, the sEMG signal must be filtered to minimise such artefacts. The drive of this paper is to propose an efficient adaptive Volterra noise mitigation architecture (AVNMA) for the sEMG signal. Further, the optimal coefficients for designing the Volterra filter architecture are achieved by applying the recently proposed metaheuristic algorithm, gannet optimisation algorithm and compared to the performance of other benchmark optimisation algorithms, namely harmony search optimisation algorithm and teaching learning-based optimisation algorithm to the proposed architecture. The quantitative analysis based on the proposed architecture is measured in terms of several metrics such as mean squared error, normalised root mean square error, peak reconstruction error, mean difference, maximum error and signal-to-noise ratio at various noisy environments in the presence of additive white Gaussian noise, including artefacts such as muscle noise, baseline wandering, electrode misplacements, and electrical interference. The outcomes of experimental research (SNR = 102.236 dB andMSE = 6.71E-09 for healthy-sEMG) ensure the superiority of the gannet optimisation algorithm based-AVNMA over the other metaheuristic algorithm based-AVNMAs. The proposed approach is verified parametrically and non-parametrically in a statistical sense with the help of two sample t-tests and the Mann-Whitney U test, respectively.

Interpolated Individual Weighting Subband Volterra Filter for Nonlinear Active Noise Control

Nonlinear active noise control (NLANC) is always a challenge in algorithm design. Most of the NLANC algorithms are based on the linear-in-the-parameters (LIP) filters, and no work is conducted on subband adaptive filter (SAF). In this brief, a novel nonlinear SAF, termed the Volterra filtered-x normalized SAF (VFxNSAF), is proposed for NLANC, whose input vector is obtained according to the size of the analysis filter bank. Moreover, the virtual microphone strategy is applied to VFxNSAF to suppress the noise at a remote location. To mitigate the computational burden, the interpolated IWVFxNSAF with individual weighting (IW) (I-IWVFxNSAF) is proposed, in which the quadratic kernel is obtained by the interpolated filter. Simulation results corroborate the superiority of the VFxNSAF and I-IWVFxNSAF algorithms for NLANC.

Efficient Realization for Third-Order Volterra Filter Based on Singular Value Decomposition

Nonlinear distortion in loudspeaker systems degrades sound quality and must be properly compensated for by linearization techniques. One technique to reduce nonlinear distortion is to use a Volterra Filter, which approximates the nonlinearity of the target loudspeaker using the Volterra series expansion. In general, the Volterra Filter is computationally very expensive, and the amount of computation needs to be reduced for real-time processing. In this paper, we propose an efficient implementation of the third-order Volterra filter based on singular value decomposition. The proposed method determines the necessary coefficients based on the symmetry of the third-order Volterra filter and applies singular value decomposition to them. In the filter structure consisting of singular values and their corresponding singular vector, the computational complexity of the third-order Volterra filter can be reduced by eliminating the part of the filter with small singular values. By focusing on the magnitude of the singular values, the proposed method can improve the computational efficiency of the third-order Volterra filter without decreasing its approximation accuracy. Simulation results show that the proposed method can improve the computational efficiency by 60% while maintaining the nonlinear distortion compensation performance of the micro-speaker for smartphones by about 8 dB.

Widely nonlinear quaternion-valued second-order Volterra recursive least squares filter

Recently, a family of the quaternion-valued second-order Volterra LMS (QSOV-LMS) algorithms were proposed to deal with 3d and 4d signals. However, it should be noted that the QSOV-LMS algorithms suffer from low convergence speed and high steady-state properties. To this end, a widely nonlinear quaternion recursive least square algorithm is proposed to train the second-order Volterra filter (WNL-QSOVRLS) for enhancing the performance of QSOV-LMS algorithms. Furthermore, to avoid the immense computation complexity, we also present a novel widely nonlinear quaternion Volterra recursive least square dichotomous coordinate descent filtering model (WNL-QSOVRLS-DCD) which introduce the quaternion dichotomous coordinate descent (QDCD) algorithm for the first time. Moreover, based on the augmented model of quaternion second-order Volterra filter, the two proposed algorithms can process quaternion-valued circular and non-circular signals well. Finally, experiments verify the performance of the proposed algorithms.

3×3 MIMO 60-GHz OFDM-RoF System With Long-Distance Wireless Transmission Enabled by MIMO Volterra Filtering

A 3×3 60-GHz multiple-input multiple-output (MIMO) radio-over-fiber system with an MIMO Volterra filter was demonstrated. The main objective of the proposed scheme is to compensate for the nonlinear distortion caused by radio frequency (RF) power amplifiers (PAs) before the signals reaches the transmitter antennas—particularly when the PAs operate with a high input power to achieve a high output RF power for long-distance wireless transmission. When the required wireless transmission distance is only 7 m, the 3×3 MIMO system with careful control of the PA input power can achieve a data rate of 126 Gbps. However, the data rate decreases rapidly over a longer wireless distance because of the larger loss. By eliminating nonlinear distortion and interchannel interference simultaneously, MIMO Volterra filtering can significantly increase the achievable data rate of a long-distance system. When the required data rate is 90 Gbps (or 100 Gbps), the maximum wireless transmission distance is 118 m (or 70 m) with the aid of MIMO Volterra filtering, but it is only 74 m (or 40 m) without MIMO Volterra filtering.

ROBUSTNESS ANALYSIS OF THE DATA-SELECTIVE VOLTERRA NLMS ALGORITHM

Recently, the data-selective adaptive Volterra filters have been proposed;however, up to now, there are not any theoretical analyses on its behavior rather than numerical simulations. Therefore, in this paper, we analyze the robustness (in the sense of l_2-stability) of the data-selective Volterra normalized least-mean-square (DSVNLMS) algorithm. First, we study the local robustness of this algorithm at any iteration, then we propose a global bound for the error/discrepancy in the coefficient vector. Also, we demonstrate that the DS-VNLMS algorithm improves the parameter estimation for the majority of the iterations that an update is implemented. Moreover, we also prove that if the noise bound is known, then we can set the DS-VNLMS so that it never degrades the estimate. The simulation results corroborate the validity of the executed analysis and demonstrate that the DS-VNLMS algorithm is robust against noise, no matter how its parameters are adopted.