Linear Time-invariant Filter Research Articles

Generalization of the Synthetic Aperture Radar Azimuth Multi-Aperture Processing Scheme—MAPS

This paper analyzes the advantages and the drawbacks of using the Synthetic Aperture Radar (SAR) azimuth multichannel technique known as Multi-Aperture Processing Scheme (MAPS), in a set of relevant application cases that are far from the canonical ones. In the scientific literature on this topic, equally distributed azimuth channels with the quasi-monostatic deployment are assumed. With this research, we aim at extending the models from the current literature to (i) a generic bistatic acquisition geometry, (ii) a set of cases where the number of receiving tiles is not the same for each channel, or (iii) the tiles are shared between adjacent channels thus creating an overlapping configuration. The paper introduces the mathematical models for the listed non-conventional MAPS cases. Dealing with the bistatic MAPS, we first solve the problem by interpreting multichannel acquisition as a bank of Linear Time Invariant (LTI) filters. Then, a more physical approach, based on discrimination of the direction of arrivals (DoAs) is pursued. The effectiveness of the two methods and the advantages of the second approach on the first are proved by using a simplified 1D end-to-end simulation. Even limiting to the monostatic configuration, the azimuth antenna tiles have always been supposed equally partitioned among the RX channels. Overcoming this limit has two advantages: (i) more MAPS possible solutions in case few azimuth tiles are available, as in the ROSE-L mission; (ii) the number of channels can be designed independently of the number of tiles, also allowing asymmetric solutions, useful for a phase array antenna with an odd number of tiles such as in the SAOCOM-1 mission. Conversely, sharing one or more receiving tiles in different receiving channels makes the input noise partially correlated. The drawback is an increase in the noise level. A trade-off is determined for the different solutions obtained using simulations with real mission parameters. The theoretical performance and the end-to-end simulations are compared.

Combining optimal SASS (Sparsity Assisted Signal Smoothing) and notch filters for transient measurement signals denoising of large power generating unit

Denoising and harmonic interference removal from the measurement signal is crucial for accurate parameters calculation of a generating unit model. Determining the actual parameters values requires various unit transient signals data recorded during a well-defined testing procedure. However, the recorded signals usually contain noise and harmonic interferences which are difficult to remove due to a presence of different kinds of discontinuities.The paper presents a proposal of optimal denoising and harmonic interference removal method from transient measurement signals of 200 MW power generating unit. The method combines notch filters for harmonic rejection followed by SASS (Sparsity Assisted Signal Smoothing) algorithm for wideband noise rejection. Parameters of the notch filters as well as the SASS are determined using popular optimization algorithms (simulated annealing, Nelder-Mead simplex and gradient algorithm).Performance of the proposed method is compared to its direct Linear Time-Invariant (LTI) competitors such as low-pass and notch filters alone, notch and low-pass filters combined and to the optimal SASS method alone. The performance evaluation is based on a set of specially generated test signals which are using the developed mathematical model of the signals (to set the level of noise and interferences) as well as the mathematical model of the power generating unit (to provide the reference clean signals). The obtained results confirm that the optimal notch-SASS method might be a valuable noise removal tool for considered class of signals and leads to superior results than the LTI filters or the SASS method alone.

Comparisons between Fourier and STFT multipliers: The smoothing effect of the short-time Fourier transform

We study the connection between STFT multipliers A1⊗mg1,g2 having windows g1,g2, symbols a(x,ω)=(1⊗m)(x,ω)=m(ω), (x,ω)∈R2d, and the Fourier multipliers Tm2 with symbol m2 on Rd. We find sufficient and necessary conditions on symbols m,m2 and windows g1,g2 for the equality Tm2=A1⊗mg1,g2. For m=m2 the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier A1⊗mg1,g2, also called localization operator, presents a smoothing effect due to the so-called two-window short-time Fourier transform which enters in the definition of A1⊗mg1,g2. As a by-product we prove necessary conditions for the continuity of anti-Wick operators A1⊗mg,g:Lp→Lq having multiplier m in weak Lr spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multipliers are better known as linear time invariant (LTI) filters.

Modeling microgrid power flow dynamics with covariance based realization

Reducing transient effects of power quality in the distribution network of a microgrid requires a better understanding of the dynamics of power flow and distributed energy resources (DERs) present within the microgrid. Instead of labor intensive and detailed physics-based distribution network and DER modeling, this paper shows how power flow dynamics can also be modeled in a dynamic equivalent with a data-driven approach that exploits the time dependent covariance of power flow at different locations within a microgrid. The proposed approach results in multi-input multi-output (MIMO) linear time invariant (LTI) discrete-time filter concatenated by a static nonlinearity to model both dynamics and power losses. Estimation of the MIMO LTI dynamics is done by the Covariance Based Realization Algorithm (CoBRA) that can handle noisy perturbed measurements and provide a low order MIMO LTI filter with a normalized steady-state gain. The additional static nonlinearity is found by the estimation of the admittance matrix of the distribution network within the microgrid. The main contribution of this paper is proposing a physics-informed data-driven approach to modeling microgrid power flow dynamics, where the CoBRA method is implemented to learn parameters from data while observing a priori physical constraints. The approach is illustrated on experimental data from a microgrid with multiple DERs and it is shown that an excellent agreement in power flow dynamics is obtained over a large operating range, which demonstrates its advantage over the N4SID method.

Linear time-invariant (LTI) modeling for aerial and underwater acoustics

Most newcomers to acoustic signal processing understand that linear time-invariant (LTI) filters can remove out-of-band noise from time series signals. What many acoustics researchers may not realize is that LTI models can be applied much more broadly, including to non-linear and time-variant systems. This presentation covers an overview of the autoregressive (AR), moving-average (MA), and autoregressive moving-average (ARMA) family of LTI models and their many useful applications in acoustics. Examples include analytic time-frequency processing of multi-component echolocation signals, fractional-delay filtering for acoustic time series simulations, broadband acoustic array beamforming, adaptive filtering for noise cancelation, and system identification for acoustic equalizers (i.e., flattening the frequency response of a source-receiver pair). This talk serves as a brief tutorial and inspiration for researchers who want to expand their use of signal processing, especially those in the fields of animal bioacoustics, aerial acoustics, and underwater acoustics.

The continuous fractional Bessel wavelet transform and its applications

In this paper, the fractional Bessel wavelet transform is introduced by exploiting the theory of the fractional Hankel transform and the boundedness of the fractional Bessel wavelet transform obtained. Time invariant linear filter associated with the fractional Hankel transform is investigated and its various properties are obtained. In the present paper, authors also expressed time-invariant linear filters in the form of the fractional Bessel wavelet transform and applications of the aforesaid filters in the integral equation are given.

Gaussian process repetitive control: Beyond periodic internal models through kernels

Repetitive control enables the exact compensation of periodic disturbances if the internal model is appropriately selected. The aim of this paper is to develop a novel synthesis technique for repetitive control (RC) based on a new more general internal model. By employing a Gaussian process internal model, asymptotic rejection is obtained for a wide range of disturbances through an appropriate selection of a kernel. The implementation is a simple linear time-invariant (LTI) filter that is automatically synthesized through this kernel. The result is a user-friendly design approach based on a limited number of intuitive design variables, such as smoothness and periodicity. The approach naturally extends to reject multi-period and non-periodic disturbances, exiting approaches are recovered as special cases, and a case study shows that it outperforms traditional RC in both convergence speed and steady-state error.

Tensor Wiener Filter

In signal processing and data analytics, Wiener filter is a classical powerful tool to transform an input signal to match a desired or target signal by a linear time-invariant (LTI) filter. The input signal of a Wiener filter is one-dimensional while its associated least-squares solution, namely Wiener-Hopf equation, involves a two-dimensional data-array, or correlation matrix. However, the actual match should often be carried out between a multi-dimensional filtered signal-sequence, which is the output of a multi-channel filter characterized as a linear-time-invariant MIMO (multi-input and multi-output) system, and a multi-dimensional desired signal-sequence simultaneously. In the presence of such a multi-channel filter, the solution to the corresponding Wiener filter, which we call MIMO Wiener-Hopf equation now, involves a correlation tensor. Therefore, we call this optimal multi-channel filter Tensor Wiener Filter (TWF). Due to lack of the pertinent mathematical framework of needed tensor operations, TWF has never been investigated so far. Now we would like to make the first-ever attempt to establish a new mathematical framework for TWF, which relies on the inverse of the correlation tensor. We propose the new parallel block-Jacobi tensor-inversion algorithm for this tensor inversion. A typical application of the new TWF approach is illustrated as a multi-channel linear predictor (MCLP) built upon a multi-channel autoregressive (MCAR) filter with multi-dimensional input data. Numerical experiments pertaining to seismic data, optical images, and macroeconomic time-series are conducted in comparison with other existing methods. The memory- and computational-complexities corresponding to our proposed parallel block-Jacobi tensor-inversion algorithm are also studied in this paper.

ℓ 2 and ℓ 1 Trend Filtering: A Kalman Filter Approach [Lecture Notes

Two of the most popular denoising algorithms are ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> trend filtering, which are used in science, engineering, and statistical signal and image processing. They are typically treated as separate entities, with the former as a linear time-invariant (LTI) filter, which is commonly used for smoothing the noisy data and detrending the time-series signals, while the latter is a nonlinear filtering method suited for the estimation of piecewise-polynomial signals (e.g., piecewise constant, piecewise linear, piecewise quadratic, and so on) observed in additive white Gaussian noise.

Sampling Theorems with Nonlinear Signal Reconstruction Scheme

In this paper, sampling Theorems for perfect signal reconstruction using samples of a signal taken below the Nyquist rate involving nonlinear signal reconstruction technique are presented. The presented results are in contrast with the linear time-invariant filtering based signal reconstruction in the celebrated Shannon sampling theory. It is shown that using kth order nonlinearity in the signal reconstruction system, the required sampling rate for perfect signal reconstruction can be reduced by the same factor k for positive odd values of it. The extensions of the proposed sampling expansion to the fractional Fourier transform and linear canonical transform domains are also derived. Simulation results of the proposed technique are given to validate the proposed approach.

The Entropy Gain of Linear Systems and Some of Its Implications.

We study the increase in per-sample differential entropy rate of random sequences and processes after being passed through a non minimum-phase (NMP) discrete-time, linear time-invariant (LTI) filter G. For LTI discrete-time filters and random processes, it has long been established by Theorem 14 in Shannon’s seminal paper that this entropy gain, , equals the integral of . In this note, we first show that Shannon’s Theorem 14 does not hold in general. Then, we prove that, when comparing the input differential entropy to that of the entire (longer) output of G, the entropy gain equals . We show that the entropy gain between equal-length input and output sequences is upper bounded by and arises if and only if there exists an output additive disturbance with finite differential entropy (no matter how small) or a random initial state. Unlike what happens with linear maps, the entropy gain in this case depends on the distribution of all the signals involved. We illustrate some of the consequences of these results by presenting their implications in three different problems. Specifically: conditions for equality in an information inequality of importance in networked control problems; extending to a much broader class of sources the existing results on the rate-distortion function for non-stationary Gaussian sources, and an observation on the capacity of auto-regressive Gaussian channels with feedback.

Sparsity-Assisted Signal Denoising and Pattern Recognition in Time-Series Data

We address the problem of signal denoising and pattern recognition in processing batch-mode time-series data by combining linear time-invariant filters, orthogonal multiresolution representations, and sparsity-based methods. We propose a novel approach to designing higher-order zero-phase low-pass, high-pass, and band-pass infinite impulse response filters as matrices, using spectral transformation of the state-space representation of digital filters. We also propose a proximal gradient-based technique to factorize a special class of zero-phase high-pass and band-pass digital filters so that the factorization product preserves the zero-phase property of the filter and incorporates a sparse-derivative component of the input in the signal model. To demonstrate applications of our novel filter designs, we validate and propose new signal models to simultaneously denoise and identify patterns of interest. To denoise or detect a pattern of interest in the signal, our proposed signal models simultaneously combine linear time-invariant (LTI) filters and sparsity-based methods with orthogonal multiresolution representations, such as wavelets and short-time Fourier transform. We illustrate the capabilities of our proposed signal models using sleep-electroencephalography (EEG) data to detect K-complexes and sleep spindles. Reproducible research is available at https://github.com/prateekgv/sasdpr.

Unmixing deep-sea paleoclimate records: A study on bioturbation effects through convolution and deconvolution

Marine sediments are typically subject to bioturbation from benthic macrofauna, which disrupt sediments and can have a substantial impact on the amplitude and timing of climate proxy signals recorded in marine sediment cores. This study explores a computational approach to quantify and remove these post-depositional effects from marine climate proxy records. The bioturbation process is modeled as a linear time-invariant filter building upon prior work (Guinasso and Schinck, 1975). Using forward modeling we find that, given modeled mixing layer depths ≥ 5 cm, simulated centennial scale climate variability is generally not preserved even when the sedimentation rate in our model is above 15 cm/kyr. On the scale of ice ages (104-105 yr), the observed effects of bioturbation have the smallest impact, considering the event scale and typical sedimentation rates and mixed layer depth in the deep sea. For millennial scale events, the signal attenuation strongly depends on the event scale and specific bioturbation and sedimentation parameters. To account for the bioturbation effect on a given climate proxy series, a deconvolution method is proposed. We apply the approach to three individual benthic foraminifera oxygen isotope records to reconstruct δ18O across the last interglacial (Marine Isotope Stage 5e; MIS 5e), and to assess the magnitude of the deglacial shift between MIS 6 and MIS 5e. In the highest resolution records (Site MD952042, Site 1012), the deconvolution results reveal an increase in the amplitude of the recovered deglacial δ18O shift by approximately 0.1‰, with implications for reconstructed global average sea level variations and temperature estimates from benthic foraminifera δ18O. The recovered signal at Site 677 suggests high amplitude δ18O variability during MIS 5e, but exhibits larger uncertainty due to a lower sampling frequency, lower sedimentation rate, and an inferred higher noise level. While uncertainties on the deconvolution results can be large, the results provide a new view of the potential impact of bioturbation on paleoclimate reconstruction. Future work can reduce the uncertainty in the deconvolution estimates, through quantitative integration of additional constraints, provided by our knowledge of the climate and depositional systems.

Measure for degree of time variance and measure for degree of non-stationarity: application to discrete LPTV systems

Processing and analysis of time-varying systems and associated non-stationary signals are of great practical importance. The issue of quantifying the amount of time variance and amount of non-stationarity has gained importance recently. This work is also an effort in that direction. In particular, in this paper, we define two measures, one for the degree of time variance (MDTV) in the context of linear periodically time-varying systems (LPTV) and second for the degree of non-stationarity (MDNOST) of cyclostationary processes as they naturally arise when a wide-sense stationary (WSS) process passes through an LPTV system. While MDTV is a system property, on the other hand, MDNOST is a signal property. This paper also tries to find a relationship between the amount of time variation of a system with the amount of non-stationarity it generates. In this light, two concepts are analyzed for two different scenarios. First, we consider a system consisting of an upsampler followed by a linear time-invariant (LTI) filter and second a dual rate system (DRS). Both these systems are LPTV. We have shown that the MDTV generated by both LPTV systems is the same as the MDNOST of their output process through simulations and rigorous mathematical proofs.

On a class of filters in the Watson Fourier wavelet setting

In this paper, using the theory of harmonic analysis related to the Watson-Fourier transform, we study a linear time invariant filter. Also, we show that this linear time invariant filter can be expressed in the form of Watson Fourier wavelet transform. Finally, the Fredholm integral equation is defined and we give a solution of this integral equation. Next, an application of the linear time invariant filter is given in the theory of the aforesaid integral equation. <br><br> У даній роботі, використовуючи пов'язаний з перетворення Уотсона-Фур’є гармонічний аналіз, вивчено лінійний інваріантний за часом фільтр. Показано, що цей лінійний інваріантний за часом фільтр може бути представлений у вигляді вейвлет-перетворення Уотсона Фур’є. Також визначено інтегральне рівняння Фредгольма і наведено розв'язок цього інтегрального рівняння. Надано застосування лінійного інваріантного за часом фільтра до теорії вищезгаданого інтегрального рівняння.

A New Physically Motivated Clutter Model With Applications to Nondestructive Ultrasonic Testing.

In this article, we construct a new physically motivated model for the grain scattering noise (clutter). We assume the clutter is the output of a random linear time-invariant (LTI) filter, the impulse response of which is considered to be a realization of a Gaussian wide sense stationary (WSS) random process, when the transmitted ultrasonic pulse is at the input. In an ultrasonic non-destructive testing (NDT) session, the clutter noise, which is signal-dependent and caused by the microstructure of the material, presents a substantial challenge in identifying defects in the material under testing. The model is used to aid in the detection of a defect in the material. The model incorporates, explicitly, many important physical characteristics of the generated clutter, such as the average grain size, the random shape, and orientation of the grains, and emphasizes the single scatterer assumption (Rayleigh region). The statistical properties of the model are formulated and derived. A comparison with the usual matched filter to indicate the model advantage is performed. An application to real ultrasonic data is conducted with excellent results. Furthermore, we explored the effect of the choice of the model parameters, and the model shows robustness toward parameter misspecification. We then tested the performance under a deviation from the single scatterer assumption, for a more complex target, using simulated noise and obtained promising results.

Automatic quantitative diagnosis for rolling bearing compound faults via adapted dictionary free orthogonal matching pursuit

Vibration signals of rolling bearings can reflect not only the concrete faulty component, but also the faulty status, e.g., the severity. Motivated by the fault status assessment, a sparse step-impact characteristic-oriented quantitative diagnosis method is developed for the automatic size estimations of compound faults. First, based on the excitation mechanism of bearing step-impact signals, a right-angle quadrilateral model is established for size estimation. Then the linear time-invariant (LTI) filter and autoregressive model (AR model) are used to pre-process the original fault signal. Technically supported by the adapted dictionary free orthogonal matching pursuit (ADOMP), through correlating the Asymmetric Gaussian Chirplet Model (AGCM) atoms and bearing signals, the location of step-impact points are parameterized and the selection strategy for the most suitable AGCM atoms are also investigated, guaranteeing the successful separation of severe and slight fault step-impact components. Finally, the estimations for compound fault sizes are automatically realized via combining the time shift factors of AGCM atoms and proposed right-angle quadrilateral model. The simulated result shows that the proposed right-angle quadrilateral model and quantitative diagnosis method can be reliably applied to the automatic estimations of compound fault sizes. The experimental results show that, the estimation deviations for compound sizes (2.5 mm and 1.0 mm) come only to 1.61% and 6.49% at 300r/min, respectively, superior to Symlet5 wavelet decomposition. Furthermore, the proposed method is applied to the quantitative analysis at different speeds and compound fault sizes, and the satisfactory diagnosis results are obtained.

STATIONARY LINEAR MEAN SQUARE FILTER FOR THE OPERATION MODE OF CONTINUOUS-TIME MARKOVIAN JUMP LINEAR SYSTEMS

This paper makes a further foray on the study of the filtering problem for the class of Markov jump linear systems (MJLSs) with partial observations of the Markov parameter (the operation mode). We derive a stationary filter for the best linear mean square filter (BLMSF) devised in a recent paper by the authors. It amounts here to obtain the convergence of the error covariance matrix of the best linear mean square filter to a stationary value under some suitable assumptions which includes ergodicity of the Markov chain. The advantage of this scheme is that it is easier to implement since the filter gain computation can be performed offline, leading to a linear time-invariant filter.

Arithmetic Approaches for Rigorous Design of Reliable Fixed-Point LTI Filters

In this paper we target the Fixed-Point (FxP) implementation of Linear Time-Invariant (LTI) filters evaluated with statespace equations. We assume that wordlengths are fixed and that our goal is to determine binary point positions that guarantee the absence of overflows while maximizing accuracy. We provide a model for the worst-case error analysis of FxP filters that gives tight bounds on the output error. Then we develop an algorithm for the determination of binary point positions that takes rounding errors and their amplification fully into account. The proposed techniques are rigorous, i.e., based on proofs, and no simulations are ever used. In practice, Floating-Point (FP) errors that occur in the implementation of FxP design routines can lead to overestimation/underestimation of resulting parameters. Thus, along with FxP analysis of digital filters, we provide FP analysis of our filter design algorithms. In particular, the core measure in our approach, Worst-Case Peak Gain, is defined as an infinite sum and has matrix powers in it. We provide fine-grained FP error analysis of its evaluation and develop multiple precision algorithms that dynamically adapt their internal precision to satisfy an a priori absolute error bound. Our techniques on multiple precision matrix algorithms, such as eigendecomposition, are of independent interest as a contribution to Computer Arithmetic. All algorithms are implemented as C libraries, integrated into an open-source filter code generator and tested on numerical examples.

Gait Cycle Validation and Segmentation Using Inertial Sensors.

In this paper, we develop an algorithm to automatically validate and segment a gait cycle in real time into three gait events, namely midstance, toe-off, and heel-strike, using inertial sensors. We first use the physical models of sensor data obtained from a foot-mounted inertial system to differentiate stationary and moving segments of the sensor data. Next, we develop an optimization routine called sparsity-assisted wavelet denoising (SAWD), which simultaneously combines linear time invariant filters, orthogonal multiresolution representations such as wavelets, and sparsity-based methods, to generate a sparse template of the moving segments of the gyroscope measurements in the sagittal plane for valid gait cycles. Thereafter, to validate any moving segment as a gait cycle, we compute the root-mean-square error between the generated sparse template and the sparse representation of the moving segment of the gyroscope data in the sagittal plane obtained using SAWD. Finally, we find the local minima for the stationary and moving segments of a valid gait cycle to detect the gait events. We compare our proposed method with existing methods, for a fixed threshold, using real data obtained from three groups, namely controls, participants with Parkinson disease, and geriatric participants. Our proposed method demonstrates an average F1 score of 87.78% across all groups for a fixed sampling rate, and an average F1 score of 92.44% across all Parkinson disease participants for a variable sampling rate.