Causal Filters Research Articles

On Causal Estimation From Bandlimited Stationary Sequences

This paper considers the problem of estimating a stationary sequence y from the observation of a stationary correlated sequence x by means of a causal linear filter. Thereby, it is assumed that the spectral density Φx of x vanishes on a subset of the unit circle of positive Lebesgue measure such that the classical derivation of the estimation filter, based on the spectral factorization of Φx, can not be applied. The paper derives the transfer function of such an estimation filter, discusses its stability behavior, and applies the result to the causal reconstruction of deterministic signals from its samples.

On sub-ideal causal smoothing filters

Smoothing causal linear time-invariant filters are studied for continuous time processes. The paper suggests a family of causal filters with almost exponential damping of the energy on the higher frequencies. These filters are sub-ideal meaning that a faster decay of the frequency response would lead to the loss of causality.

EMG Prediction From Motor Cortical Recordings via a Nonnegative Point-Process Filter

A constrained point-process filtering mechanism for prediction of electromyogram (EMG) signals from multichannel neural spike recordings is proposed here. Filters from the Kalman family are inherently suboptimal in dealing with non-Gaussian observations, or a state evolution that deviates from the Gaussianity assumption. To address these limitations, we modeled the non-Gaussian neural spike train observations by using a generalized linear model that encapsulates covariates of neural activity, including the neurons' own spiking history, concurrent ensemble activity, and extrinsic covariates (EMG signals). In order to predict the envelopes of EMGs, we reformulated the Kalman filter in an optimization framework and utilized a nonnegativity constraint. This structure characterizes the nonlinear correspondence between neural activity and EMG signals reasonably. The EMGs were recorded from 12 forearm and hand muscles of a behaving monkey during a grip-force task. In the case of limited training data, the constrained point-process filter improved the prediction accuracy when compared to a conventional Wiener cascade filter (a linear causal filter followed by a static nonlinearity) for different bin sizes and delays between input spikes and EMG output. For longer training datasets, results of the proposed filter and that of the Wiener cascade filter were comparable.

Minimizing the Effect of Sampling Jitters in Wireless Sensor Networks

A wireless sensor network (WSN) consists of low-cost and energy-limited sensors to measure a distributed phenomenon. The finite energy constraint limits the synchronization of sensors at every sampling instant which introduces sampling jitters. In this letter, we model sampling jitters using fractional delay transfer functions. The WSN is modeled using a hybrid multirate filter bank where the objective is to design discrete-time, causal and stable synthesis filters to minimize the effect of sampling jitters. Using a norm-invariant discretization, the hybrid and multirate problem is reduced to a model-matching H2 optimization problem involving linear time-invariant and discrete-time systems. A numerical example is also presented to show the effectiveness of the proposed approach.

Explicit Expression for a Linear Filter in Terms of Its Lattice Representation

A causal digital autoregressive filter is completely defined by a polynomial P. In the general case, and particularly for stable filters, it is well-known that P can be expressed recursively in terms of the so-called reflection coefficients kj appearing in the lattice representation of the associated filter. This letter proposes an explicit expression for polynomial P in terms of the coefficients kj .

Multiple subtraction method based on a non-causal matching filter

The filter operator used in normal multichannel matching filter is physically realizable. This filter operator only delays seismic data in the filtering process. A noncausal multichannel matching filter based on a least squares criterion is proposed to resolve the problem in which predicted multiple model data is later than real data. The differences between causal and non-causal multichannel matching filters are compared using a synthetic shot gather, which demonstrates the validity of the non-causal matching filter. In addition, a variable length sliding window which changes with offset and layer velocity is proposed to solve the count of events increasing with increasing offset in a fixed length sliding window. This variable length sliding window is also introduced into the modified and expanded multichannel matching filter. This method is applied to the Pluto1.5 synthetic data set. The benefits of the non-causal filter operator and variable length sliding window are demonstrated by the good multiple attenuation result.

A new hybrid active/passive sound absorber with variable surface impedance

The context of the present paper is the wall treatment of flow ducts, notably aero-engine nacelle intakes and outlets. For this purpose, hybrid active/passive absorbers have been developed at the LMFA for about 15 years. A hybrid cell combines passive absorbent properties of a porous layer and active control at its rear face. Active control is mainly used to increase absorption at low frequencies by cancelling the imaginary part of the surface impedance presented by the absorber. However, the optimal impedance (i.e. the one that produces the highest noise reduction) of an absorber for flow duct applications is generally complex and frequency dependent. A new hybrid absorber intended to realise any of impedance has therefore been developed. The new cell uses one microphone on each side of a resistive cloth. Normal velocity can then be deduced by a simple pressure difference, which allows an estimation of the surface impedance of the absorber. In order to obtain an error signal related to a target impedance, the target impedance has to be reproduced in time domain. The design of a stable and causal filter is a difficult task, considering the kind of frequency response we seek. An alternative way of representing the impedance in time domain is therefore given. The new error signal is integrated into a feedback control structure. Fast convergence and good stability are observed for a wide range of target impedances. Typical optimal impedances with a positive increasing real part and a negative decreasing imaginary part have been successfully realised. Measurements in a grazing-incidence tube show that the new complex impedance absorber clearly outperforms the former active absorber.

A Systemized View of Superluminal Wave Propagation

This paper reviews earlier studies on superluminal wave propagation in anomalously dispersive media that have been carried out in the electronic, microwave, and optical regimes. Those studies are relevant to observation of modulated Gaussian pulses transmitted through various media at speeds apparently faster than c without distortion. This paper also presents the condition for superluminal propagation that is established based on the magnitude-phase relation of a causal and minimum-phase filter. Since the condition is modeled on the basis of filter theory, it is applicable to all types of media. A terahertz experiment with a periodic bandgap structure is also included to illustrate superluminal propagation.

The Relationship Between Causal and Noncausal Mismatched Estimation in Continuous-Time AWGN Channels

A continuous-time finite-power process with distribution P is observed through an AWGN channel, at a given signal-to-noise ratio (SNR), and is estimated by an estimator that would have minimized the mean-square error if the process had distribution Q. We show that the causal filtering mean-square error (MSE) achieved at SNR level snr is equal to the average value of the noncausal (smoothing) MSE achieved with a channel whose SNR is chosen uniformly distributed between 0 and snr. Emerging as the bridge for equating these two quantities are mutual information and relative entropy. Our result generalizes that of Guo, Shamai, and Verdu (2005) from the nonmismatched case, where P = Q, to general P and Q. Among our intermediate results is an extension of Duncan's theorem, that relates mutual information and causal MMSE, to the case of mismatched estimation. Some further extensions and implications are discussed. Key to our findings is the recent result of Verdu on mismatched estimation and relative entropy.

From Neurons to Circuits: Linear Estimation of Local Field Potentials

Extracellular physiological recordings are typically separated into two frequency bands: local field potentials (LFPs) (a circuit property) and spiking multiunit activity (MUA). Recently, there has been increased interest in LFPs because of their correlation with functional magnetic resonance imaging blood oxygenation level-dependent measurements and the possibility of studying local processing and neuronal synchrony. To further understand the biophysical origin of LFPs, we asked whether it is possible to estimate their time course based on the spiking activity from the same electrode or nearby electrodes. We used "signal estimation theory" to show that a linear filter operation on the activity of one or a few neurons can explain a significant fraction of the LFP time course in the macaque monkey primary visual cortex. The linear filter used to estimate the LFPs had a stereotypical shape characterized by a sharp downstroke at negative time lags and a slower positive upstroke for positive time lags. The filter was similar across different neocortical regions and behavioral conditions, including spontaneous activity and visual stimulation. The estimations had a spatial resolution of approximately 1 mm and a temporal resolution of approximately 200 ms. By considering a causal filter, we observed a temporal asymmetry such that the positive time lags in the filter contributed more to the LFP estimation than the negative time lags. Additionally, we showed that spikes occurring within approximately 10 ms of spikes from nearby neurons yielded better estimation accuracies than nonsynchronous spikes. In summary, our results suggest that at least some circuit-level local properties of the field potentials can be predicted from the activity of one or a few neurons.

Robust Wiener filtering with non-parametric spectral uncertainty

In this article, robust least-squares filtering problems are considered for non-parametric multivariate spectral uncertainty defined by the so-called spectral band and generalised-moment constraints. Its major aim is to provide a basis for computing approximate solutions to worst-case, Wiener-filtering optimisation problems involving causal filters and multivariate signals. It hinges upon associating upper and lower bounds on the minimum worst-case performance achievable with causal filters with linear-cost/linear matrix inequality (LC/LMI)-constraint optimisation problems. On the basis of a Lagrangean duality formulation for the worst-case, least-squares performance of a given filter, upper bounds on it are obtained as the optimal values of LC/LMI problems. Then, for linearly parameterised classes of filter transfer functions, a causal filter which optimises such an upper bound on worst-case performance can also be obtained from an LC/LMI optimisation problem. To estimate the amount of conservatism incurred when relying on such upper bounds, optimal, nominal, least-squares performance for a given pair of power spectral densities (for the information and noise signal) is maximised over finite-dimensional, linearly parameterised classes of the latter. Again, such problems are shown to be equivalent to LC/LMI problems and the corresponding optimal values are lower bounds on the minimum worst-case, least-squares error achievable in the original robust filtering problem (say, μ*). Finally, two simple numerical examples are presented to illustrate how causal filters can be obtained whose worst-case, least-squares performance is quite close to the optimal one (i.e. μ*).

Robust Structure Transformation for Causal Lagrange-Type Variable Fractional-Delay Filters

We have rigorously proved that an odd-order causal Lagrange-type variable fractional-delay (VFD) digital filter can be implemented as the Farrow structure with symmetric or antisymmetric subfilters through utilizing matrix transformation. This paper reveals a numerical problem that occurs in the matrix transformation due to the so-called catastrophic cancellation in numerical computation. Our computer simulations have verified that utilizing the matrix transformation can get numerically stable solutions for the VFD filter whose order N is below about 20, but fails for higher order cases. To solve this problem, we present a robust approach to the structure transformation for both even- and odd-order causal Lagrange-type VFD filters, which does not rely on matrix operations and, thus, can yield numerically stable solutions even for high-order cases. Moreover, the symmetry and antisymmetry of the subfilter coefficients after the robust structure transformation are also rigorously proved. Numerical examples are included to illustrate the robustness of the proposed structure transformation and show the coefficient symmetries.

Two Classes of Cosine-Modulated IIR/IIR and IIR/FIR NPR Filter Banks

This paper introduces two classes of cosine-modulated causal and stable filter banks (FBs) with near perfect reconstruction (NPR) and low implementation complexity. Both classes have the same infinite-length impulse response (IIR) analysis FB but different synthesis FBs utilizing IIR and finite-length impulse response (FIR) filters, respectively. The two classes are preferable for different types of specifications. The IIR/FIR FBs are preferred if small phase errors relative to the magnitude error are desired, and vice versa. The paper provides systematic design procedures so that PR can be approximated as closely as desired. It is demonstrated through several examples that the proposed FB classes, depending on the specification, can have a lower implementation complexity compared to existing FIR and IIR cosine-modulated FBs (CMFBs). The price to pay for the reduced complexity is generally an increased delay. Furthermore, two additional attractive features of the proposed FBs are that they are asymmetric in the sense that one of the analysis and synthesis banks has a lower computational complexity compared to the other, which can be beneficial in some applications, and that the number of distinct coefficients is small, which facilitates the design of FBs with large numbers of channels.

Optimal unbiased reduced order filtering for discrete-time descriptor systems via LMI

In this paper, the reduced order unbiased optimal H ∞ filtering problem for discrete-time descriptor linear systems is considered. Necessary and sufficient conditions for the existence of unbiased causal time invariant filters are obtained. Solutions of the filtering problems are given in terms of linear matrix inequalities (LMIs). Parametrization of all optimal causal unbiased filters is provided. The proposed method generalizes the existing results in the standard case.

Novel Approach to Signal Matched Two Channel Multirate Filter Bank Using Nyquist Filter

In this paper, we propose the novel approach to estimate the filters of 2-channel biorthogonal filter bank. In this approach, first FIR causal half-band filter from a given signal or its second statistics is estimated. By estimating the present sample, from the linear combination of samples which are at odd samples away from it, the coefficients of half-band filters are obtained. From the estimated half-band filter, by using spectral factorization, analysis and synthesis filters of the first channel are obtained. To obtain the filters of the second channel, of the 2-channel filter bank, theory of biorthogonal filter bank is used and the filters of second channel are obtained. It is observed that analysis and synthesis filters are found to be FIR. It is also observed that, using proposed approach, estimation of filter bank takes fewer computations as compared to other approaches reported in the literature. When this filter bank is used, in the context signal/image compression, it is found to give better results as compared to 2-channel standard biorthogonal filter bank. It is also seen that the average subband coding gain obtained in this case is higher than the 2-channel standard biorthogonal filter bank.

A new class of invertible FIR filters for spectral shaping

Spectral shaping filters find a rich set of applications in the statistical signal processing problems such as signal modeling, spectral estimation, system identification, and channel equalization. In this paper, we present a new class of LTI causal FIR filters for converting a set of autocorrelation samples representing the input to a prescribed set of output autocorrelation samples. The input autocorrelation samples may represent a MA, AR, or ARMA process, with or without additive white noise, whose stable model is minimum or non-minimum phase. In contrast to the existing methods, derivation of a FIR filter of this class is direct requiring no intermediate step and no direct application of orthogonality principle or minimization of a cost function. The FIR filter is minimum-phase and its output autocorrelation lags, except the lag zero, match the set of prescribed values precisely. We characterize the filters solution space and present a novel time-domain algorithm for finding a unique invertible FIR filter of minimum-order in the solution space. We refer to the solution filter as an “optimal invertible FIR (OIFIR)” filter, in general, and as a “semi-whitening filter (SWF)”, in particular, when the prescribed output autocorrelation lags, except the lag zero, are all equal to zero. It is shown that the whitening filter obtained by Yule–Walker equations (or by the orthogonality principle) belongs to the solution space of SWF only if the set of input autocorrelation samples accurately represents an AR process of finite order. Examples show performance of the proposed class of filters in spectral shaping applications including the AR modeling of a stationary process.

Causal Cubic Splines: Formulations, Interpolation Properties and Implementations

The paper presents two formulations of causal cubic splines with equidistant knots. Both are based on a causal direct B-spline filter with parallel or cascade implementation. In either implementation, the causal part of the impulse response is realized with an efficient infinite-impulse-response (IIR) structure, while only the anticausal part is approximated with a finite-order finite-impulse-response (FIR) filter. Resulting cubic coefficients are computed from the causal B-spline coefficients by using a third-order output FIR filter with either single-input multiple-output (SIMO) or multiple-input multiple-output (MIMO) structure, depending on the chosen formulation of the cubic spline. The paper demonstrates and proves that the properties of the resulting causal splines are quite different, whether they are based on a more popular B-spline formulation, or a bit neglected tridiagonal matrix formulation. It is shown that the proposed low-complexity but accurate causal interpolators can be realized for many practical applications with the delay of only a few samples.

What is the real shape of extracellular spikes?

We show that the standard filters used for on-line spike detection in most hardware acquisition systems introduce distortions in the recorded spike shapes. This is because on-line spike detection is done after band pass filtering the data with causal filters. As illustrated with three clusters of spike shapes from a real single cell recording in a human subject, causal filtering can create a spurious negative rebound and a smooth looking appearance of the spikes. We also show that these filtering distortions can make artifacts look similar to real spikes.

Time Scales in Video Surveillance

Events in surveillance video occur over many time scales, but common approaches to background subtraction and video representation are implicitly based on a single temporal scale. In this work, we derive a set of causal filters which define a temporal scale-space representation for the activity at each pixel. This scale-space can be maintained and continuously updated in real time and, for static cameras viewing dynamic scenes, has several interesting properties. In particular, it directly characterizes interesting temporal features and supports approximate reconstruction of the video history under challenging noise conditions. The temporal scale-space grounds novel approaches to several applications, including a natural visualization tool to summarize recent video behavior in a single image, and a tool to directly report how long the object has been present in a scene without reexamining any video data.

Neville–Lagrange wavelet family for lossless image compression

This paper presents a new wavelet family by combining the Neville filter theory and Lagrange interpolation. The filter banks of the new wavelet family are built and named as Neville–Lagrange lifting wavelet filter banks (N–LLWFBs for short). The prediction filters of N–LLWFBs are obtained by considering the signal sampling and Lagrange interpolation, and the corresponding update filters are given by using Neville filter theory. Examples are given by using this approach. The Neville–Lagrange prediction filters are obtained; causal lifting wavelet filter banks are also constructed by using this approach. Several N–LLWFBs for image compression are designed, and they are normalized in terms of the normalization conditions of the first generation wavelet filter bank. As a special example, the lifting scheme of 5/3 wavelet of JPEG2000 is obtained; it is the two-channel N–LLWFB of order 2 both dual and primal vanishing moments. Experiment results show that the performance of N–LLWFBs for image compression becomes better with the increase of their vanishing moments.