Volterra Filter Research Articles

Optical single side-band Nyquist PAM-4 transmission using dual-drive MZM modulation and direct detection.

We present the design and optimization of the optical single side-band (SSB) Nyquist four-level pulse amplitude modulation (PAM-4) transmission using dual-drive Mach-Zehnder modulator (DDMZM)modulation and direct detection (DD), aiming at the C-band cost-effective, high-speed and long-distance transmission. At the transmitter, the laser line width should be small to avoid the phase noise to amplitude noise conversion and equalization-enhanced phase noise due to the large chromatic dispersion (CD). The optical SSB signal is generated after optimizing the optical modulation index (OMI) and hence the minimum phase condition which is required by the Kramers-Kronig (KK) receiver can also be satisfied. At the receiver, a simple AC-coupled photodiode (PD) is used and a virtual carrier is added for the KK operation to alleviate the signal-to-signal beating interference (SSBI).A Volterra filter (VF) is cascaded for remaining nonlinearities mitigation. When the fiber nonlinearity becomes significant, we elect to use an optical band-pass filter with offset filtering. It can suppress the simulated Brillouin scattering and the conjugated distortion by filtering out the imaging frequency components. With our design and optimization, we achieve single-channel, single polarization 102.4-Gb/s Nyquist PAM-4 over 800-km standard single-mode fiber (SSMF).

Nonlinear Structural Damage Detection Based on Adaptive Volterra Filter Model

The majority of the existing damage diagnostic techniques are based on linear models. Changes in the state of the dynamics of these models, before and after damage in the structure based on the vibration measurements, are popularly used as damage indicators. However, the system may initially behave linearly and subsequently exhibit nonlinearity due to the incipience of damage. Breathing cracks that exhibit bilinear behavior are one such example of the damage induced due to nonlinearity. Further many real world structures even in their undamaged state are nonlinear. Hence, in this paper, we present a nonlinear damage detection technique based on the adaptive Volterra filter using the nonlinear time history response. Three damage indices based on the adaptive Volterra filter are proposed and their sensitiveness to damage and noise is assessed through two numerically simulated examples. Numerical investigations demonstrate the effectiveness of the adaptive Volterra filter model to detect damage in nonlinear structures even with measurement noise.

Data-Selective Volterra Adaptive Filters

This work proposes the use of data-selective schemes in Volterra adaptive filters. By working only with input data that brings novelty to the system, thus avoiding unnecessary parameter updates, one can reduce drastically the high computational burden associated with the use of Volterra series. Considering a nonlinear system identification setup, results show that the proposed data-selective Volterra filters update in only 7.5% of the iterations, whereas in a nonlinear channel equalization setup the update rate is around 27%, without compromising the performance in terms of mean squared error or bit-error rate.

Detection of Low‐Flying Target under the Sea Clutter Background Based on Volterra Filter

In order to detect low‐flying small targets in complex sea condition effectively, we study the chaotic characteristic of sea clutter, use joint algorithm combined complete ensemble empirical mode decomposition (CEEMD) with wavelet transform to de‐noise, and put forward a detection method for low‐flying target under the sea clutter background based on Volterra filter. By CEEMD method, sea clutter signal which contains small target can be decomposed into a series of intrinsic mode function (IMF) components, pick out high‐frequency components which contain more noise by autocorrelation function, and perform wavelet transform on them. The de‐noised components and remaining components are used to reconstruct clear signal. In view of the chaotic characteristics of sea clutter, we use Volterra filter to establish adaptive prediction model, detect low‐flying small target hiding in sea clutter background from the prediction error, and compare the root mean square error (RMSE) before and after de‐noising to evaluate de‐noising effect. Experimental results show that the joint algorithm can effectively remove noise and reduce the RMSE by 40% at least. Volterra prediction model can directly detect low‐flying small target under sea clutter background from the prediction error in the cases of high signal‐to‐noise ratio (SNR). In the cases of low SNR, after de‐noised by joint algorithm, Volterra prediction model can also detect the low‐flying small target clearly.

Adaptive Function Expansion 3-D Diagonal-Structure Bilinear Filter for Active Noise Control of Saturation Nonlinearity

In this paper, a general function expansion bilinear (FEB) filter with a 3-D diagonal structure is proposed to deal with the problem of saturation nonlinearity in active noise control (ANC) systems. In an ANC system, the reference microphone and/or loudspeaker may be saturated when the acoustic noise and/or the controller output exceeds the dynamic limits of electronic devices. Such saturation nonlinearity degrades the control performance of the linear and some nonlinear control filters equipped with a filtered-x least mean square (FXLMS) algorithm. In order to tackle the problem of signal saturation, we use a new nonlinear saturation ANC (NSANC) model and derive a function expansion diagonal-structure bilinear FXLMS (FEDBFXLMS) algorithm. The performance of the proposed filters equipped with the associated FEDBFXLMS algorithm is validated through the analysis of computational complexity and simulations of various nonlinearities for NSANC systems. Computer simulation results demonstrate that the proposed FEB filter can achieve significant performance improvement in reducing saturation effects in comparison with the diagonal-channel bilinear filter and the recursive second-order Volterra filter based on the FXLMS algorithm, which often outperform the conventional nonlinear Volterra and functional link artificial neural network (FLANN) filters.

Partially Decoupled Adaptive Filter Based Multifunctional Three-Phase GPV System

This paper deals with partially decoupled adaptive Volterra filter (PDAVF) based control for a three-phase two-stage grid-interfaced photovoltaic (GPV) system. Besides maximum power extraction, the proposed control is having the potential of grid currents balancing, harmonics currents mitigation, reactive power compensation, and adaptive adjustment of DC bus voltage. The control technique is efficient in extraction of weight component of reference grid currents and an adaptation routine of filter weights uses the principle of fifth-order PDAVF with single-element observation vector implemented by using the method of least mean squares (LMS). An increased order of partial filter assures accurate estimation of filter weights by adaptive weight update at each filter step. The convergence is ensured by providing bounds on algorithm's step size. The algorithm overcomes drawbacks of Adaline-based LMS and LMF (least mean fourth) based weight estimations without imposing high computational burden. The switching losses in voltage source converter are minimized using adaptive DC bus voltage. Effectiveness of PDAVF is presented through simulation and test results. These results are found satisfactory with improved steady state and dynamic performances and total harmonic distortion of grid currents meet an IEEE-519 standard.

Blurred fingerprint image enhancement: algorithm analysis and performance evaluation

A conventional automatic fingerprint matching process uses similarity score to quantify similarity between fingerprint images to be matched, and the similarity score can be determined with a minutiae extraction algorithm (MEA) which extracts minutiae from fingerprint images. The performance of MEA relies on the quality of fingerprint images. In case of blurred fingerprint images, it becomes difficult to obtain a reliable similarity score. As the result, an image enhancement algorithm should be incorporated with MEA when the fingerprint image is blurred. In this study, Volterra filter is proposed to enhance blurred fingerprints and compared against different enhancement algorithms. Experimental results show that Volterra filter outperforms other techniques such as Laplacian, Wiener, and Gabor filters for enhancing blurred images and its calculation complexity is moderate among techniques considered in this study.

Sparse Volterra model based on optical single side-band NPAM-4 direct-detection system

Abstract Signal-to-signal beating interference (SSBI) is one of the main drawbacks in direct-detection based optical transmission systems. Volterra filter is a common equalization method to mitigate the nonlinear distortion. However, the computational complexity may be unacceptable as the transmission capacity increases. In this paper, we propose a sparse Volterra model combining the feed forward equalization (FFE) and higher order terms of a modified Volterra filter with Schmidt orthogonal searching to mitigate the linear and nonlinear interference and reduce the complexity significantly in an optical single-side band (SSB) Nyquist pulse-shaped four-level pulse amplitude (NPAM-4) system. The experimental results show that the sparse Volterra filter and full Volterra filter have comparable performance, but the former only needs half kernels of the latter.

Efficient Computation of Bilinear Approximations and Volterra Models of Nonlinear Systems

Bilinear and Volterra models are important when dealing with nonlinear systems which arise in several signal processing applications. The former can approximate a large class of systems affine in the input with relatively low parametric complexity. Such an approximate bilinear model can be derived by means of Carleman bilinearization (CB). Then, a Volterra model can be computed from it, having the advantage of being linear in the parameters, but often involving a large number of them. In this paper, we develop efficient routines for CB and for computing the Volterra kernels of a bilinear system. We argue that they are useful for studying a class of systems for which a reference physical model is known. In particular, the so-derived kernels allow assessing the suitability of a Volterra filter and of other alternatives for modeling the system of interest. Techniques exploiting sparsity and low rank of involved matrices are proposed for alleviating computing cost. Several examples are given along the paper to illustrate their use, based on existing physical models of loudspeakers.

Single-Lane 112-Gbit/s SSB-PAM4 Transmission With Dual-Drive MZM and Kramers–Kronig Detection Over 80-km SSMF

We have experimentally demonstrated the signal-to-signal beating interference (SSBI) mitigation capability of Kramers–Kronig (KK) detection in a single sideband (SSB) four-ary pulse-amplitude-modulation (PAM4) direct detection (DD) system. A 112-Gbit/s (56-GBaud) SSB-PAM4 optical signal suitable for KK detection is generated by using a single commercial, low-cost dual-drive Mach–Zehnder modulator. The system bandwidth constraint is mitigated by Nyquist pulse shaping and bandwidth precompensation. The SSBI mitigation capability of KK algorithm is identified by comparing with two order Volterra filter. After 80-km dispersion-uncompensated standard single-mode fiber transmission, linear equalizer with KK algorithm and two order Volterra filter based nonlinearity equalizer show similar bit error rate (BER) performance, which are 3.2 × 10–3 and 2.8 × 10–3 , respectively. Both BERs are below 7% hard-decision forward error correction threshold of 3.8 × 10 –3. The experiment results present the excellent SSBI mitigation capability of KK detection in SSB-PAM4 DD system. Above all, our experiments reveal the potential of KK detection in DD system for data center interconnects.

Improved EEG Segmentation Using Non-linear Volterra Model in Bayesian Method

ABSTRACTIn order to analyze non-stationary signals, like Electroencephalogram (EEG), it is sometimes easier to segment signals into pseudo-stationary segments. In this paper, the cascade of linear predictive coding (LPC) and non-linear Volterra filter is employed for modeling of noise in EEG signal and this methodology is applied to the procedure of change-point detection, for estimating the number of change-points and their exact location which is a powerful way to detect the change-points as precisely as possible. The earlier results are completed by constructing algorithms that use the cascade of LPC and non-linear Volterra filter for modeling the relation between noisy signal and noise in practical situations. In a Bayesian configuration, the posterior distribution of the change-point sequence is constructed and then Markov Chain Monte Carlo procedure is used for sampling this posterior distribution. The simulation results for segmentation of synthetic and real EEG data show that by applying our newly proposed methodology, the specificity and sensitivity of the segmentation are highly improved. In the case of synthetic data, the change-points are estimated completely precise (100% correct) in 70% of times and they are estimated with at least 98% accuracy in other times.

PAM4 based symmetrical 112-Gbps long-reach TWDM-PON

We experimentally demonstrate cost effective symmetrical 112-Gbps long-reach passive optical network (LR-PON) over 70-km standard signal mode fiber (SSMF), based on pulse amplitude modulation (PAM)-4. Four 10G-class directly modulated lasers (DMLs) at C-band are used for achieving 4×28-Gbps downstream transmission, while two 18G-class DMLs at O-band are used to realize 2×56-Gbps upstream transmission, without any optical amplification in optical distributed network (ODN). Both dispersion compensation fiber (DCF) for downstream signal and praseodymium-doped fiber amplifier (PDFA) for upstream signal are equipped at optical line terminal (OLT). Meanwhile, sparse Volterra filter (SVF) equalizer is proposed to mitigate the transmission impairments with substantial reduction of computation complexity. Finally, we can successfully provide a loss budget of 33 dB per downstream wavelength channel, indicating of 64 optical network units (ONUs) with more than 1.25 Gbps per ONU.

Volterra equalization of complex modulation utilizing frequency chirp in directly modulated lasers

We apply Volterra-based equalization for complex modulated optical signals utilizing the frequency chirp in DMLs. We experimentally demonstrate that the higher order Volterra filter is necessary in the higher speed transmissions. For further study, we isolate the adiabatic chirp by injection locking and realize the optical PM transmission. We make a comparison among IM, FM and PM with Volterra equalization, finding that PM and FM are more power insensitive and suitable for high speed, power limited fiber transmission. The performance can be further improved by exploiting the diversity gain.

Volterra and Wiener Equalizers for Short-Reach 100G PAM-4 Applications

Unlike ultralong coherent optical systems that seriously suffer from fiber nonlinearities, short-reach noncoherent systems such as data center interconnections, which utilize small, cheap, and low-bandwidth components, are sensitive to nonlinearities that are mainly produced by devices responsible for electrical signal amplification, modulation, and demodulation. One of the most promising schemes for these applications is the four-level pulse amplitude modulation format combined with intensity modulation and direct detection; however, it can be significantly degraded by linear and nonlinear intersymbol interference. Linear and nonlinear signal degradation can efficiently be handled by different types of equalizers. In many cases, the straightforward linear equalizer cannot lower the error rate at the acceptable level. Therefore, much stronger equalizers based on nonlinear models such as the Volterra series are proposed. Volterra filter that can also be orthogonalized by the Wiener model is well described in the existing literature, and, in this paper, we investigate the most critical points related to high-speed Volterra filter design and implementation. Several experiments are carried out in order to indicate filter requirements/complexity, acquisition, and stability. We also provide a simple guidance for filter complexity reduction and useful hints for channel acquisition.

Experimental demonstration of an efficient hybrid equalizer for short-reach optical SSB systems

We propose an efficient enhanced hybrid equalizer combining the feed forward equalization (FFE) with a modified Volterra filter to mitigate the linear and nonlinear interference for the short-reach optical single side-band (SSB) system. The optical SSB signal is generated by a relatively low-cost dual-drive Mach–Zehnder modulator (DDMZM). The two driving signals are a pair of Hilbert signals with Nyquist pulse-shaped four-level pulse amplitude modulation (NPAM-4). After the fiber transmission, the neighboring received symbols are strongly correlated due to the pulse spreading in time domain caused by the chromatic dispersion (CD). At the receiver equalization stage, the FFE followed by higher order terms of modified Volterra filter, which utilizes the forward and backward neighboring symbols to construct the kernels with strong correlation, are used as an enhanced hybrid equalizer to mitigate the inter symbol interference (ISI) and nonlinear distortion due to the interaction of the CD and the square-law detection. We experimentally demonstrate that the optical SSB NPAM-4 signal of 40 Gb/s transmitting over 80 km standard single mode fiber (SSMF) with a bit-error-rate (BER) of 7.59×10−4.

Characterization of Quadratic Nonlinearity between Motion Artifact and Acceleration Data and its Application to Heartbeat Rate Estimation.

Accelerometers are applied to various applications to collect information about movements of other sensors deployed at diverse fields ranging from underwater area to human body. In this study, we try to characterize the nonlinear relationship between motion artifact and acceleration data. The cross bicoherence test and the Volterra filter are used as the approaches to detection and modeling. We use the cross bicoherence test to directly detect in the frequency domain and we indirectly identify the nonlinear relationship by improving the performance of eliminating motion artifact in heartbeat rate estimation using a nonlinear filter, the second-order Volterra filter. In the experiments, significant bicoherence values are observed through the cross bicoherence test between the photoplethysmogram (PPG) signal contaminated with motion artifact and the acceleration sensor data. It is observed that for each dataset, the heartbeat rate estimation based on the Volterra filter is superior to that of the linear filter in terms of average absolute error. Furthermore, the leave one out cross-validation (LOOCV) is employed to develop an optimal structure of the Volterra filter for the total datasets. Due to lack of data, the developed Volterra filter does not demonstrate significant difference from the optimal linear filter in terms of t-test. Through this study, it can be concluded that motion artifact may have a quadaratical relationship with acceleration data in terms of bicoherence and more experimental data are required for developing a robust and efficient model for the relationship.

IMPROVED NONLINEAR NOISE ESTIMATION IN EEG SIGNAL BASED ON VOLTERRA FILTERS

Recognition and compensation of undesired nonlinearity is one of the important subjects in the field of digital signal processing. The Volterra model is widely used for nonlinearity identification in practical applications. The current tendency in the digital systems design is the identification and compensation of unwanted nonlinearities. In this paper, we employed a nonlinear noise estimation approach for electroencephalogram (EEG) signal based on a combination of linear predictive coding (LPC) and Volterra filter that is a new and good way to estimate noise in EEG signal. We initially used LPC filter to estimate the noise present in EEG signal (correlated and uncorrelated noise) plus the uncorrelated portion of the signal (the part of the signal that has no linear relation to its past samples). After that, we employed nonlinear Volterra model to estimate the existing noise in EEG signal (correlated and uncorrelated noise). We show that by employing the cascade of LPC and Volterra filter, we can considerably improve the signal-to-noise ratio (SNR) in EEG signal by the ratio of at least 1.94. Also, we compared the simulation results to the case where we used just Volterra filter. In comparison with just Volterra filter, we have a significant increase in the SNR.

Compensation for Nonlinear Distortion of the Frequency Modulation-Based Parametric Array Loudspeaker

The parametric array loudspeaker (PAL) modulates the audio signal on an ultrasonic carrier. When the modulated signal is transmitted in air, an audio beam is created based on the nonlinear acoustic principle. Each modulation method has advantages and disadvantages. The frequency modulation (FM) is favorable for its low cost and high volume, but the tradeoff is its complicated nonlinear distortion, which is difficult to be reduced. In this paper, the Volterra filter is adopted to model the nonlinearity of the FM-based PAL. A novel complex inverse system is devised to effectively reduce the nonlinear distortion. Three practical aspects are addressed. First, the computational complexity of the Volterra filter is reduced by the parallel cascade structure with almost no compromise to the model accuracy. Second, Volterra filters are identified at discrete input levels to treat the nonlinearity that keeps changing with the time-varying audio input. Third, when the input level is high, a separation approach is proposed to refine the identified Volterra filters, which eventually improves the performance of the proposed inverse system.

2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs).

We experimentally demonstrate 2 × 64 Gb/s PAM-4 transmission over a 70 km standard single-mode fiber (SSMF) using two O-band 18G-class directly modulated lasers (DMLs). Only one praseodymium-doped fiber amplifier (PDFA) at the receiver side is used to compensate the transmission loss. Meanwhile, transmission impairments are compensated by a sparse Volterra filter (SVF) equalizer, which can achieve similar system performance but with half the computational complexity (CC), in comparison with a traditional VF equalizer. Finally, we optimize the insignificant factor (IF) of SVF to identify the trade-off between the transmission performance and the CC. Thus, the redundancy of individual SVF kernels can be reasonably removed.

Nonlinear Adaptive Volterra Filter for Control of Distribution Static Compensator

This paper presents nonlinear adaptive Volterra filter for control of three-phase four wires Distribution Static Compensator under mixed type linear and nonlinear load. The proposed adaptive filter identifies and measures active and reactive weights of the fundamental components of load currents. These weights are used to generate the reference signals such that the currents drawn from a supply are close to balanced and sinusoidal. The performance of the proposed control algorithm is observed under nonlinear load in the laboratory environment. The test results are found satisfactory as total harmonic distortion in supply currents are within the 5% levels specified by the IEEE Standard 519-1992.