Linear Time-varying Filter Research Articles

Truncated orthogonal expansions of recurrent signals: equivalence to a linear time-variant periodic filter

We show that orthogonal expansions of recurrent signals like electrocardiograms (ECGs) with a reduced number of coefficients is equivalent to a linear time-variant periodic filter. Instantaneous impulse and frequency responses are analyzed for two classical ways of estimating the expansion coefficients: inner product and adaptive estimation with the LMS algorithm. The obtained description as a linear time-variant periodic filter is a useful tool in order to quantify the distortion produced by the effect of using a reduced number of coefficients in the expansion, and to give frequency criteria to select the appropriate number of functions. Moreover, the misadjustment of the LMS algorithm can be explained as a distortion of the instantaneous frequency response. Experimental results are illustrated with the Karhunen-Loeve transform of ECG signals, but this approach can also be applied to any orthogonal transform.

Minimum order input-output equation for linear time-varying digital filters

The objective of this paper is to obtain minimum order input-output equations for a class of linear time-varying digital filters in state equation with constant dimension state vector. It is shown that the minimum order of the input-output equation may not be identical to the dimension of the state vector and there exists a nonunique solution for the minimum order input-output equation.

Reconstruction of signals with known instantaneous frequency using linear time-varying filter

An efficient method for reconstructing signals from an AM-FM noisy signal with known instantaneous frequency (IF) is derived and verified by simulation. Analysis shows that the proposed linear time-varying filter efficiently rejects noise from the received wideband signal in real-time.

System identification using chirp signals and time-variant filters in the joint time-frequency domain

We propose a novel method to identify an unknown linear time invariant (LTI) system in low signal-to-noise ratio (SNR) environment. The method is based on transmitting chirp signals for the transmitter and using linear time-variant filters in the joint time-frequency (TF) domain for the receiver to reduce noise before identification. Due to the TF localization property of chirp signals, a large amount of additive white noise can be reduced, and therefore, the SNR before identification can be significantly increased. This, however, cannot be achieved in the conventional methods, where pseudo-random signals are used, and therefore, noise reduction techniques do not apply. Our simulation results indicate that the method proposed outperforms the conventional methods significantly in a low SNR environment. This paper provides a good application of time-frequency analysis and synthesis.

Necessary and sufficient conditions for stability of LMS

Guo and Ljung (1995) established some general results on exponential stability of random linear equations, which can be applied directly to the performance analysis of a wide class of adaptive algorithms, including the basic LMS ones, without requiring stationarity, independency, and boundedness assumptions of the system signals. The current paper attempts to give a complete characterization of the exponential stability of the LMS algorithms by providing a necessary and sufficient condition for such a stability in the case of possibly unbounded, nonstationary, and non-/spl phi/-mixing signals. The results of this paper can be applied to a very large class of signals, including those generated from, e.g., a Gaussian process via a time-varying linear filter. As an application, several novel and extended results on convergence and the tracking performance of LMS are derived under various assumptions. Neither stationarity nor Markov-chain assumptions are necessarily required in the paper.

Factorability of lossless time-varying filters and filter banks

We study the factorability of linear time-varying (LTV) lossless filters and filter banks. We give a complete characterization of all, degree-one lossless LTV systems and show that all degree-one lossless systems can be decomposed into a time-dependent unitary matrix followed by a lossless dyadic-based LTV system. The lossless dyadic-based system has several properties that make it useful in the factorization of lossless LTV systems. The traditional lapped orthogonal transform (LOT) is also generalized to the LTV case. We identify two classes of TVLOTs, namely, the invertible inverse lossless (IIL) and noninvertible inverse lossless (NIL) TVLOTs. The minimum number of delays required to implement a TVLOT is shown to be a nondecreasing function of time, and it is a constant if and only if the TVLOT is IIL. We also show that all IIL TVLOTs can be factorized uniquely into the proposed degree-one lossless building block. The factorization is minimal in terms of the delay elements. For NIL TVLOTs, there are factorable and unfactorable examples. Both necessary and sufficient conditions for the factorability of lossless LTV systems are given. We also introduce the concept of strong eternal reachability (SER) and strong eternal observability (SEO) of LTV systems. The SER and SEO of an implementation of LTV systems imply the minimality of the structure. Using these concepts, we are able to show that the cascade structure for a factorable IIL LTV system is minimal. That implies that if a IIL LTV system is factorable in terms of the lossless dyadic-based building blocks, the factorization is minimal in terms of delays as well as the number of building blocks. We also prove the BIBO stability of the LTV normalized IIR lattice.

Time-varying matched filters

We examine a generalized matched-filter problem in which the interference is a nonstationary process generated by passing white noise through a general linear time-varying filter. First a matched filter is constructed by transforming the problem into an equivalent formulation involving stationary interference and a time-varying propagation channel. Whereas the response of a time-invariant matched filter is sampled at its peak, the response of this time-varying matched filter is normalized before sampling to account for variations in the signal power. Next a matched filter is constructed using a spectral characterization of the nonstationary interference. This construction is then used to formulate a simplified solution for the case where the rate of variation in the nonstationary interference is sufficiently small. The different solutions are illustrated by a numerical example.

Scattering function characterization of time and frequency spreading in shallow water propagation

Under certain conditions the multipath time delay and frequency spread of signals propagating in shallow water can be described usefully with the ocean modeled as a randomly time-varying linear filter. Starting with an explicit physical model for the time-varying transfer function of a shallow water channel, an alternative system function, the delay-Doppler spread function, can be defined and its autocorrelation determined. A particularly simple form of the spread function autocorrelation, the scattering function, results when the propagation multipaths are uncorrelated and locally stationary in the wide sense. When the scattering function description is valid, signal propagation input–output relations can be expressed in terms of the medium scattering function and the input signal ambiguity function. The scattering function formulation will be reviewed and shown to be most useful for channels supporting many boundary-interacting paths (e.g., shallow water) at mid or high frequency. Using the scattering function the influence of the medium on different sonar signal types will be demonstrated and methods for measuring channel characteristics indicated. Channel characteristics determined from data obtained during ACT II on the New Jersey Shelf under downward refracting propagation conditions will be presented and discussed in terms of the scattering function description. [Work supported in part by ARPA.]

Recursive time-varying filter banks for subband image coding

Filter banks, subband/wavelets, and multiresolution decompositions that employ recursive filters have been considered previously and are recognized for their efficiency in partitioning the frequency spectrum. This paper presents an analysis of a new infinite impulse response (IIR) filter bank in which these computationally efficient filters may be changed adaptively in response to the input. The new filter bank framework is presented and discussed in the context of subband image coding. In the absence of quantization errors, exact reconstruction can be achieved. By the proper choice of an adaptation scheme, it is shown that recursive linear time-varying (LTV) filter banks can yield improvement over conventional ones.

Finite-time-domain synthesis of linear time-variant digital filters by difference equations

This paper presents a method for synthesizing a recursive linear time-variant digital filter in finite time domain. The desired filter can be nonrecursive and is given by its generalized frequency function. The synthesized filter is described by a linear difference equation with time-variant coefficients. These coefficients at a given time instant are derived recursively from those at the previous instants. In addition, for the determined coefficients at the previous instants, the coefficients at the given instant minimize the error between the generalized frequency function of the desired filter and that of the synthesized one at the given instant. The performance of this method is illustrated and compared with another available method through numerical examples. The results show an improvement of nearly two order of magnitude in the normalized mean squared error between the amplitudes of the generalized frequency function of the desired filter and that of the synthesized.

Legal Wigner space filtering and its interpretation

Filtering in Wigner space generates a time—frequency function that does not generally constitute a legal Wigner distribution (WD). The notion ‘legal WD’ implies a time—frequency function, specified in Wigner space, for which a time-domain signal exists. This illegality prevents an exact synthesis of the corresponding time signal, and resorting to an approximate synthesis procedure is unavoidable. Presented herein are necessary and sufficient conditions, on a filter specified in Wigner space, that guarantee the legality of the output signal. We prove that legal Wigner space filters belong either to the class of linear time-variant filters or to the class of nonlinear (quadratic) filters. These two classes possess vastly different characteristics.

Digital video standards conversion in the presence of accelerated motion

We address the standards conversion of digital video containing accelerated motion. The spectral characterization of video containing accelerated motion is performed by means of short time spectral analysis, where the support of the short time spectrum (STS) of video with accelerated motion is derived. Linear time-varying filters that operate along an accelerated motion trajectory are then analyzed, and the time-varying frequency response of these filters is computed. Finally, we show that the support of this time-varying frequency response matches that of the video signal STS, and thus filtering along the accelerated motion trajectory potentially yields the smallest amount of aliasing in standards conversion of video with accelerated motion. Experimental results comparing the performance of motion-compensated filtering along the accelerated motion trajectory with filtering along the constant velocity motion trajectory that is tangent to the accelerated trajectory at the processed frame are provided.

Digital video standards conversion in the presence of accelerated motion

Abstract We address the standards conversion of digital video containing accelerated motion. The spectral characterization of video containing accelerated motion is performed by means of short time spectral analysis, where the support of the short time spectrum (STS) of video with accelerated motion is derived. Linear time-varying filters that operate along an accelerated motion trajectory are then analyzed, and the time-varying frequency response of these filters is computed. Finally, we show that the support of this time-varying frequency response matches that of the video signal STS, and thus filtering along the accelerated motion trajectory potentially yields the smallest amount of aliasing in standards conversion of video with accelerated motion. Experimental results comparing the performance of motion-compensated filtering along the accelerated motion trajectory with filtering along the constant velocity motion trajectory that is tangent to the accelerated trajectory at the processed frame are provided.

Time-frequency analysis of modulated speech noises

The production of some speech sounds involves the periodic modulation of a noise component (e.g., the voiced fricatives in French, or aspiration noises in breathy vowels). According to the linear acoustic model of speech production, the speech signal is produced by filtering an excitation signal with a linear time-varying filter (which represent the vocal tract transfer function and sound radiation). For instance, in voiced fricatives, the excitation signal is a mixed signal resulting from the modulation of a frication noise source by the glottal flow. Two representations, taking into account the nonstationary nature of mixed-excitation speech sounds, were studied: cyclostationary analysis (using a spectral correlation (SC) estimator of the cyclic frequency-frequency spectrum) and nonstationary analysis [using a smoothed pseudo Wigner–Ville (WV) estimator of the Wigner–Ville spectrum]. The theoretical and experimental results obtained on test signals and actual speech show that some acoustic parameters can be estimated using these analysis methods (frequency of modulation using SC; time structure of the modulation spectrum and vocal tract filter transfer function using WV). Nevertheless, estimation of other acoustic parameters (for instance the spectral density of the excitation noise) appeared difficult, if not impossible.

The indoor radio propagation channel

In this tutorial survey the principles of radio propagation in indoor environments are reviewed. The channel is modeled as a linear time-varying filter at each location in the three-dimensional space, and the properties of the filter's impulse response are described. Theoretical distributions of the sequences of arrival times, amplitudes and phases are presented. Other relevant concepts such as spatial and temporal variations of the channel, large-scale path losses, mean excess delay and RMS delay spread are explored. Propagation characteristics of the indoor and outdoor channels are compared and their major differences are outlined. Previous measurement and modeling efforts are surveyed, and areas for future research are suggested.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis and filtering using the optimally smoothed Wigner distribution

The authors consider the analysis and filtering of a deterministic signal with slowly time-varying spectra using the optimally smoothed Wigner distribution (OSWD). They compare this mixed time-frequency representation (MTFR) to other MTFRs such as the spectrogram, the short-time Fourier transform (STFT), and the Wigner and pseudo-Wigner distributions. The authors propose an approach to designing linear time-varying filters for slowly time-varying signals which is based on the concept of local nonstationarity cancellation and show that it is equivalent to masking the optimal STFT. The performance of the filter in suppressing white noise and in decomposing a slowly time-varying signal into its components is studied and compared to the performance of the techniques based on the STFT. >

Finite-time-domain synthesis of recursive linear time-variant causal digital filters by separable sequences

A linear time-variant (LTV) digital filter having a separable sequence as its impulse response has been proved to be recursive. Such filters have the potential of saving computation time and storage. Two techniques are presented for synthesizing a desired LTV digital filter given in numerical form in a finite time domain by a separable sequence. The first is a realization technique which decomposes the impulse response of a desired filter into a separable sequence and theoretically leads to the solution of a separable sequence with the lowest order. For a numerical solution, the order of the separable sequence is dependent on an error tolerance that reflects the realization accuracy. If the desired filter is recursive, then the exact order can be solved. The solved order, as well as the decomposition error, is independent of the error tolerance in a wide range. The second is an approximation technique which finds a separable sequence of a given order by minimizing the normalized mean squared error between the impulse response of the desired filter and the separable sequence. The technique searches the separable sequence by alternately using two nonlinear restrictions that an optimally approximated filter must satisfy. The performances of the proposed techniques are illustrated and compared with other available techniques through numerical examples. The results of the comparisons show that our approximation technique results in smaller approximation errors than those of the others. >

A Computationally Efficient Realization of Recursive Linear Time-Variant Digital Bandpass Filters by Transforming Lowpass Ones

The paper presents a computationally efficient realization technique for recursive linear time-variant (LTV) digital bandpass filters by transforming lowpass ones. The transformation is based on the principle that an input signal is first demodulated, then filtered by lowpass filters, and at last modulated back again. The presented technique simples this procedure and needs no modulators and demodulators. The performance of this transformation is illustrated

A linear, time-varying system framework for noniterative discrete-time band-limited signal extrapolation

A technique for the analysis and design of noniterative algorithms for discrete-time, band-limited signal extrapolation is described. The approach involves modeling the extrapolation process as a linear, time-varying (LTV) system, or filter. Together with a previously developed Fourier theory for LTV systems, this model provides a frequency-domain transfer function representation for the extrapolation system. This representation serves as a powerful tool for characterizing and comparing the reconstruction properties of several well-known least squares optimal algorithms for band-limited extrapolation. Moreover, the frequency-domain setting provides a conceptually attractive means for understanding the process of extrapolation itself. Additionally, a least squares approximation methodology for designing LTV filters for band-limited extrapolation is developed. The design technique is shown to unify a broad class of algorithms for extrapolating discrete-time data and, further, to provide a means for designing new and improved extrapolation algorithms. >

Adaptive algorithms with filtered regressor and filtered error

This paper presents a unified framework for the analysis of several discrete time adaptive parameter estimation algorithms, including RML with nonvanishing stepsize, several ARMAX identifiers, the Landau-style output error algorithms, and certain others for which no stability proof has yet appeared. A general algorithmic form is defined, incorporating a linear time-varying regressor filter and a linear time-varying error filter. Local convergence of the parameters in nonideal (or noisy) environments is shown via averaging theory under suitable assumptions of persistence of excitation, small stepsize, and passivity. The excitation conditions can often be transferred to conditions on external signals, and a small stepsize is appropriate in a wide range of applications. The required passivity is demonstrated for several special cases of the general algorithm.