Linear Time-invariant Filter Research Articles

Time-varying coherent AM–FM demodulation and denoising of acoustic signals

Noise removal via linear time-invariant (LTI) filtering is most effective when the signal and noise spectra have minimal overlap in frequency. In particular, it can be difficult to extract, via LTI filtering, broadband signals from broadband noise, because often their spectra overlap. However, many broadband signals are locally narrow band (e.g., AM–FM signals with large FM and moderate to small AM), and this characteristic can be exploited to improve noise suppression for such signals. We present a method for extracting locally narrow band signals from broadband noise, based on an AM–FM decomposition of the signal and time-varying filtering. The center frequency and passband of a linear time-varying filter are determined from estimates of the instantaneous frequency and instantaneous bandwidth of the signal. Results on both synthetic signals and recorded whale sounds in ambient noise demonstrate a significant improvement in SNR compared to LTI-based filtering. [Supported by ONR Grant N00014-98-1-0680.]

Minimum order input–output equation for LTV digital filters with time-varying state dimension

This paper develops a connection between linear time-varying state equations and minimum order input–output equations for linear time-varying digital filters. A procedure is presented for computing a minimum order input–output equation from a given linear time-varying state equation. The linear time-varying input–output equations and state equations considered in this paper have time-varying orders and time-varying state dimensions, respectively. These models are appropriate for representing practical signal processing systems and are efficient in the computation and implementation of digital filters. It is shown that, in contrast to linear time-invariant filters, the solution for the minimum time-varying order input–output equation can be non-unique.

A Linear Time-Invariant Filtering Approach for Ultrasonic Transducer Normalization

The increased use of automatic defect detection and characterization systems of the self-learning type has created a demand for means capable of normalizing signals from ultrasonic transducers. Measurements obtained using different measurement setups should be normalized with reference to a standard transducer. It is usually an unfeasible task to optimize characterization procedures for all combinations of measurement parameters that are usually available in a modern complex measurement system. For instance, a change of transducer or only a change in cable length may result in substantial differences in measured data. We propose a linear filtering approach for normalizing ultrasonic pulse-echo measurements as a preprocessing step before presenting the data to a characterization system. The approach requires two data sets: one for the reference transducer and one for the transducer to normalize. We formulate the normalization problem as a general linear approximation problem and derive an optimal linear transformation for an ideal situation with known transducer and noise characteristics. Due to the properties of the optimal linear transformation, a close approximation of this transformation can be implemented using a linear time-invariant filter. We verify by simulations that the filter approximation is valid, and we also examine some properties concerning the accuracy of the estimates obtained using the filter approximation. The filter is obtained using the output error method, one of the standard system identification methods. The proposed method is tested on real ultrasonic data obtained from carbon-fiber—reinforced epoxy composites. The results of experiments with real data, illustrating one of the possible applications, are used to point out some practical considerations that have to be taken into account when implementing the proposed method.

A Linear Time-Invariant Filtering Approach for Ultrasonic Transducer Normalization

The increased use of automatic defect detection and characterization systems of the self-learning type has created a demand for means capable of normalizing signals from ultrasonic transducers. Measurements obtained using different measurement setups should be normalized with reference to a standard transducer. It is usually an unfeasible task to optimize characterization procedures for all combinations of measurement parameters that are usually available in a modern complex measurement system. For instance, a change of transducer or only a change in cable length may result in substantial differences in measured data. We propose a linear filtering approach for normalizing ultrasonic pulse-echo measurements as a preprocessing step before presenting the data to a characterization system. The approach requires two data sets: one for the reference transducer and one for the transducer to normalize. We formulate the normalization problem as a general linear approximation problem and derive an optimal linear transformation for an ideal situation with known transducer and noise characteristics. Due to the properties of the optimal linear transformation, a close approximation of this transformation can be implemented using a linear time-invariant filter. We verify by simulations that the filter approximation is valid, and we also examine some properties concerning the accuracy of the estimates obtained using the filter approximation. The filter is obtained using the output error method, one of the standard system identification methods. The proposed method is tested on real ultrasonic data obtained from carbon-fiber-reinforced epoxy composites. The results of experiments with real data, illustrating one of the possible applications, are used to point out some practical considerations that have to be taken into account when implementing the proposed method.

Non-Wiener behavior of the filtered LMS algorithm

This brief presents some analytical results for the non-Wiener solutions of the filtered least mean square algorithm. Secondary paths are modeled as linear time-invariant filters. Results are presented for the steady-state adaptive weight behavior and algorithm stability. Stability regions are implicitly determined as a function of the algorithm step size /spl mu/, the number of filter taps N, and the reference signal power.

Oversampling PCM techniques and optimum noise shapers for quantizing a class of nonbandlimited signals

We consider the efficient quantization of a class of nonbandlimited signals, namely, the class of discrete-time signals that can be recovered from their decimated version. The signals are modeled as the output of a single FIR interpolation filter (single band model) or, more generally, as the sum of the outputs of L FIR interpolation filters (multiband model). These nonbandlimited signals are oversampled, and it is therefore reasonable to expect that we can reap the same benefits of well-known efficient A/D techniques that apply only to bandlimited signals. We first show that we can obtain a great reduction in the quantization noise variance due to the oversampled nature of the signals. We can achieve a substantial decrease in bit rate by appropriately decimating the signals and then quantizing them. To further increase the effective quantizer resolution, noise shaping is introduced by optimizing prefilters and postfilters around the quantizer. We start with a scalar time-invariant quantizer and study two important cases of linear time invariant (LTI) filters, namely, the case where the postfilter is the inverse of the prefilter and the more general case where the postfilter is independent from the prefilter. Closed form expressions for the optimum filters and average minimum mean square error are derived in each case for both the single band and multiband models. The class of noise shaping filters and quantizers is then enlarged to include linear periodically time varying (LPTV)/sub M/ filters and periodically time-varying quantizers of period M. We study two special cases in great detail.

A Linear Time-Invariant Filtering Approach for Ultrasonic Transducer Normalization

A Linear Time-Invariant Filtering Approach for Ultrasonic Transducer Normalization

On the eigenstructure of linear quasi-time-invariant systems

In this paper, the eigenstructure of a class of linear time-varying systems, termed linear quasi-time-invariant (LQTI) systems, is investigated. A system composed of dynamic devices such as linear time-varying capacitors and resistors can be an example of the class. To describe and analyze the LQTI systems effectively, a differential operator G composed of the derivative operator D and some time functions is adopted. Then, the dynamic systems described by the operator G are studied in terms of eigenvalue, frequency characterisics, stability and an extended convolution. Some basic attributes of the operator G are compared with those of the differential operator D. The corresponding generalized Laplace transform pair is defined and relevant properties are derived for frequency-domain analysis and design of the filters. It is also noted that the stability is determined by the position of poles in the G frequency domain, where the stable region in the complex plane is different from the classical left half s plane. A point in the extended Laplace transform space represents a complex exponential function modulated by some functions. As an application example, an LQTI filter is examined and designed by using the concept of eigenstructure of LQTI system. Also, a new type of modulation or frequency shift property is examined as in the linear time-invariant filter theory.

Optimal construction of subband coders using Lloyd-Max quantizers

A new method is presented for the analysis of the effects of Lloyd-Max quantization in subband filterbanks and for the optimal design of such filterbanks. A rigorous statistical model of a vector Lloyd-Max quantizer is established first, consisting of a linear time-invariant filter followed by additive noise uncorrelated/with the input. On the basis of this model, an expression for this variance of the error of a subband coder using Lloyd-Max quantizers is explicitly determined. Given analysis filters that statistically separate the subbands, it is shown that this variance is minimized if the synthesis filters are chosen, which mould achieve perfect reconstruction in lossless coding. The globally optimum of such a filterbank, minimizing the coder error variance, is further obtained by proper choice of its analysis filters. An alternative design method is also evaluated and optimized. In this, the errors correlated with the signal are set to zero, leaving a random error residue uncorrelated with the signal. This design method is optimized by choosing the analysis filters so as to minimize the random error variance. The results are evaluated experimentally in the realistic setting of a logarithmically split subband image coding scheme.

Optimal pyramidal decomposition for progressive multidimensional signal coding using optimal quantizers

Optimal hierarchical coding is sought, for progressive or scalable multidimensional signal transmission, by minimizing the variance of the error difference between the original image and its lower resolution renditions. The optimal, according to the above criterion, pyramidal coders are determined for images quantized using the optimal vector Lloyd-Max quantizers. A rigorous general statistical model of a vector Lloyd-Max quantizer is used, consisting of a linear time-invariant filter followed by additive noise uncorrelated with the input. Given arbitrary analysis filters, the optimal synthesis filters are found. The optimal analysis filters are subsequently determined, leading to formulas for globally optimal structures for pyramidal multidimensional signal decompositions. These structures produce replicas of the original image, which at lower resolutions retain as much similarity to the original as possible. This is highly useful for the progressive coding of two- or three-dimensional (2-D or 3-D) images needed in applications such as fast browsing through image databases. Furthermore, the minimization of the variance of the error image leads to minimization of the variance of the quantization noise for this image and, hence, to its optimally efficient compression. Experimental results illustrate the implementation and performance of the optimal pyramids in application for the coding of still 2-D images.

Impact of non-white noises in pulse amplitude measurements: a time-domain approach

The contribution of the 1 f - noise to the spectral line broadening in pulse amplitude measurements is derived with a time-domain analysis. The known time-domain relationships which provide the contributions of the series and parallel white noises are generalised for the case of 1 f and other typical non-white noises, by using the fractional derivative of either the system impulse response (time-invariant linear filters) or its weight function folded (time-variant linear filters). It is shown that a time-domain approach is also effective to determine the contribution of Lorentzian noises. A simple rule suitable to derive numerically the fractional derivative is given, which permits to calculate the effect of non-white noises even when the filter impulse response is not known analytically but only in sampled form.

Non-stationary time series, linear interpolators and outliers

We define the linear interpolator for a class of discrete-parameter real valued non-stationary time series. We assume that a linear time-invariant filter exists so that the filtered time series is second-order stationary. Methods for estimating the time-invariant weights of the linear interpolator are examined and consistent estimates are derived. The usefulness of the linear interpolator for missing values estimation and multiple outliers detection is discussed.

Optimized estimation of moments for nonstationary signals

The behavior of nonstationary moment estimators that are based on linear time-invariant filtering is analyzed. The performance of such estimators is evaluated in terms of their time-averaged variance and time-averaged squared bias. Optimal estimators that minimize a convex combination of bias and variance are derived. The superiority of such optimally weighted filtering over the conventional (exponentially windowed) moment estimation technique is demonstrated by means of a simple example. The same example also serves to illustrate the difficulties encountered when the construction of optimal estimators relies on uncertain prior information, as well as to demonstrate the feasibility of overcoming such difficulties by using appropriately designed (robust) optimal estimators. We also include a brief discussion of the relevance of the proposed moment estimation technique to identification of linear time-variant systems and to estimation of time-variant spectra.

HOS based algorithm for autofocusing of spotlightSAR images

The authors propose a novel SAR (synthetic aperture radar) image autofocusing technique based on higher order statistics (HOS) analysis. The SAR imaging system has been modelled as a bandlimited complex linear time invariant (LTI) filter, so the autofocusing problem is equivalent to the problem of reconstructing the phase of a bandlimited complex LTI system. The complex transfer function of the imaging system is recovered from the estimated bispectrum of the received data by means of an ad-hoc algorithm.

Blind identification of linear subsystems of LTI-ZMNL-LTI models with cyclostationary inputs

Discrete-time nonlinear models consisting of two linear time invariant (LTI) filters separated by a finite-order zero memory nonlinearity (ZMNL) of the polynomial type (the LTI-ZMNL-LTI model) are appropriate in a large number of practical applications. We discuss some approaches to the problem of blind identification of such nonlinear models, It is shown that for an Nth-order nonlinearity, the (possibly non-minimum phase) finite-memory linear subsystems of LTI-ZMNL and LTI-ZMNL-LTI models can be identified using the N+1th-order (cyclic) statistics of the output sequence alone, provided the input is cyclostationary and satisfies certain conditions. The coefficients of the ZMNL are not needed for identification of the linear subsystems and are not estimated. It is shown that the theory presented leads to analytically simple identification algorithms that possess several noise and interference suppression characteristics.

Maximizing SNR in improper complex noise and applications to CDMA

The optimum linear time-invariant (LTI) filter that maximizes the signal-to-noise ratio (SNR) is, in general, the solution to an integral equation. This is the well-known matched filter. With regards to the detection of bandpass signals, we present the full form of the integral equation using the complex baseband representation of signals. The correct form of the complex integral equation includes, in addition to the autocorrelation function, a pseudoautocorrelation function which vanishes for most applications. However, in the presence of improper complex noise, overlooking the pseudoautocorrelation leads to a filter. To illustrate the concept, we apply the theory to a direct-sequence code-division multi-access (DS-CDMA) system in which improper complex noise arises. In the application, we derive the SNR maximizing filter and the nonoptimal filter and compare their near-far resistances.

Multivariable Frequency Response Methods for Coloured Noise Optimal Discrete-Time Kalman Filter

This paper is concerned with the problem of multi-input multi-output, infinite time, linear time-invariant optimal coloured-noise discrete-time Kalman filter in frequency domain. A simple new algorithm is given for the solution to the steady state gain of the coloured noise of the optimum filter based on a transfer function approach. The algorithm is based on spectral factorization of the observed spectral density matrix of the filter which generates directly the return-difference matrix of the optimal filter. This leads to the proof of necessary condition of optimality of filter. The method seems to be more direct than by algebraic Riccati equation solution and can easily be implemented on a digital computer.

On taming chaos using LTI filters

In a recent paper [l] the possibility of suppressing chaos using linear time-invariant (LTI) filters is considered with numerical experiments concerning a specific chaotic system (the logistic map) and low-order filters. In this letter, simple arguments based on the frequency response of linear systems and on the Liapunov exponents are used to point out that the experimental result of [l] (i.e., a low-order LTI filter cannot destroy nor delay chaos in the logistic map) has general validity, namely it can be extended to any continuous or discrete-time chaotic system and to LTI filters of any order.

The adaptive linear combiner with a periodic-impulse reference input as a linear comb filter

In this paper we present a particularization of the adaptive linear combiner (ALC) filter structure, that results in a linear time-invariant comb filter suitable for the estimation of periodic signals and repetitive time-locked signals. The ALC is used in its transversal form, and the reference input is a periodic unitary-impulse train signal. When the LMS algorithm is used we show that the structure results in a simple and efficient linear time-invariant comb filter, taking as output the output of the ALC. This comb filter has lobe widths proportional to the μ gain parameter of the LMS algorithm, and the separation between lobes is controlled by the period L of the periodic impulse train. We have also analyzed this filter when applied to estimate repetitive signals time-locked to a stimulus, and we show that the effect of a temporal misalignment in the determination of the stimulus results in a low-pass filtering effect, with cut-off frequency inversely proportional to the dispersion of the impulse estimation. This effect is specially important when the time occurrence of the stimulus is not directly accessible and needs to be estimated from some ‘noise affected’ procedures, as in Electrocardiographic signals. The filter is also shown to be equivalent to a time-sequenced adaptive filter with one weight. Finally, an application to somatosensory evoked potentials estimation is presented.

A probabilistic approach to multivariable robust filtering and open-loop control

A new approach to robust filtering, prediction, and smoothing of discrete-time signal vectors is presented. Linear time-invariant filters are designed to be insensitive to spectral uncertainty in signal models. The goal is to obtain a simple design method, leading to filters which are not overly conservative. Modeling errors are described by sets of models, parameterized by random variables with known covariances. These covariances could either be estimated from data or be used as robustness "tuning knobs." A robust design is obtained by minimizing the /spl Hscrsub 2/-norm or, equivalently, the mean square estimation error, averaged with respect to the assumed model errors. A polynomial solution, based on an averaged spectral factorization and a unilateral Diophantine equation, is derived. The robust estimator is referred to as a cautious Wiener filter. It turns out to be only slightly more complicated to design than an ordinary Wiener filter. The methodology can be applied to any open-loop filtering or control problem. In particular, we illustrate this for the design of robust multivariable feedforward regulators, decoupling and model matching filters.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>