Time-varying Digital Filters Research Articles

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

Elimination of temporal dispersion from the finite-difference solutions of wave equations in elastic and anelastic models

Time integration of wave equations can be carried out with explicit time stepping using a finite-difference (FD) approximation. The wave equation is the partial differential equation that governs the wavefield that is solved for. The FD approximation gives another partial differential equation — the one solved in the computer for the FD wavefield. This approximation to time integration in numerical modeling produces a wavefield contaminated with temporal dispersion, particularly at high frequencies. We find how the Fourier transform can be used to relate the two partial differential equations and their solutions. Each of the two wavefields is then a time-frequency transformation of the other. First, this transformation allows temporal dispersion to be eliminated from the FD wavefield, and second, it allows temporal dispersion to be added to the exact wavefield. The two transforms are band-limited inverse operations. The transforms can be implemented by using time-step independent, noncausal time-varying digital filters that can be precomputed exactly from sums over Bessel functions. Their product becomes the symmetric Toeplitz matrix with the elements defined through the cardinal sine (sinc) function. For anelastic materials, the effect of numerical time dispersion in a wavefield propagating in a medium needs special treatment. Dispersion can be removed by using the time-frequency transform when the FD wavefield is modeled in a medium with the frequency-modified modulus relative to the physical modulus of interest. In the rheological model of the generalized Maxwell body, the frequency-modified modulus is written as a power series, which allows a term-by-term Fourier transform to the time domain. In a low-frequency approximation, the modified modulus obtains the same form as the physical modulus, and it can be implemented as changes in the unrelaxed modulus and shifts of the relaxation frequencies and their strengths of the physical modulus.

On elimination of overflow oscillations in linear time-varying 2-D digital filters represented by a Roesser model

This paper proposes a novel linear matrix inequality (LMI)-based stability criterion to avoid overflow oscillations in 2-D digital filters described by a Roesser state-space model with saturation-type overflow characteristics. In contrast to the conventional approaches, the proposed overflow oscillation elimination methodology can be applied to linear parameter varying time-variant digital filters under overflow subject to both state and output disturbances. For such filters, the proposed criterion ensures the zero-input overflow stability as well as robustness against interferences with a guaranteed H∞ performance bound. A numerical example is presented to demonstrate the effectiveness of the proposed approach.

Auralization of Accelerating Passenger Cars Using Spectral Modeling Synthesis

While the technique of auralization has been in use for quite some time in architectural acoustics, the application to environmental noise has been discovered only recently. With road traffic noise being the dominant noise source in most countries, particular interest lies in the synthesis of realistic pass-by sounds. This article describes an auralizator for pass-bys of accelerating passenger cars. The key element is a synthesizer that simulates the acoustical emission of different vehicles, driving on different surfaces, under different operating conditions. Audio signals for the emitted tire noise, as well as the propulsion noise are generated using spectral modeling synthesis, which gives complete control of the signal characteristics. The sound of propulsion is synthesized as a function of instantaneous engine speed, engine load and emission angle, whereas the sound of tires is created in dependence of vehicle speed and emission angle. The sound propagation is simulated by applying a series of time-variant digital filters. To obtain the corresponding steering parameters of the synthesizer, controlled experiments were carried out. The tire noise parameters were determined from coast-by measurements of passenger cars with idling engines. To obtain the propulsion noise parameters, measurements at different engine speeds, engine loads and emission angles were performed using a chassis dynamometer. The article shows how, from the measured data, the synthesizer parameters are calculated using audio signal processing.

Feedback Amplitude Modulation Synthesis

A recently rediscovered sound synthesis method, which is based on feedback amplitude modulation (FBAM), is investigated. The FBAM system is interpreted as a periodically linear time-varying digital filter, and its stability, aliasing, and scaling properties are considered. Several novel variations of the basic system are derived and analyzed. Separation of the input and the modulation signals in FBAM structures is proposed which helps to create modular sound synthesis and digital audio effects applications. The FBAM is shown to be a powerful and versatile sound synthesis principle, which has similarities to the established distortion synthesis methods, but which is also essentially different from them.

Adaptive Filtering [Best of the Web

This article focuses on adaptive filtering or, more generally, adaptive signal processing, the design of time-varying (adaptive) digital filters that would tune themselves to optimally process nonstationary signals in nonstationary environments. Least mean square (LMS) algorithm is widely used in adaptive signal processing, and is the most well-understood approach to training a linear system to minimize the mean square error. The second article described a recursive solution to the discrete-data linear filtering problem. Since that time, the Kalman filter has been the subject of extensive research and application. The area of adaptive signal processing has had a significant impact on a wide variety of signal processing applications.

A Gradient Flow Approach to Optimal Model Reduction of Discrete-Time Periodic Systems

This paper is concerned with the optimal model reduction for linear discrete periodic time-varying systems and digital filters. Specifically, for a given stable periodic time-varying model, we shall seek a lower order periodic time-varying model to approximate the original model in an optimal H2 norm sense. By orthogonal projections of the original model, we convert the optimal periodic model reduction problem into an unconstrained optimization problem. Two effective algorithms are then developed to solve the optimization problem. The algorithms ensure that the H2 cost decreases monotonically and converges to an optimal (local) solution. Numerical examples are given to demonstrate the computational efficiency of the proposed method. The present paper extends the optimal model reduction for linear time invariant systems to linear periodic discrete time-varying systems.

Synthesis of a periodically time-varying digital filter

The paper presents an approximate method of periodically linear time-varying (PLTV) digital filter (DF) synthesis through mean values of their coefficients in time. The recursive and nonrecursive PLTV DFs of the first, second and arbitrary orders are considered. It is shown that PLTV DFs have distinguishing features which give them advantages in some applications in comparison with linear time-invariant (LTI) DFs.

Invertibility and inverses of linear time-varying digital filters

This brief studies invertibility and inverses of linear time-varying digital filters in the form of ARMA equation with time-varying dynamic order and relative degree. It presents a necessary and sufficient condition for the d-shift invertibility of linear time-varying filters and computational procedure for obtaining the d-shift left and right inverses. A necessary and sufficient condition for stability of inverse filters is given.

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.

Switched multiple sequence excitation model for low bit rate speech compression

An apparatus for compressing a speech signal into a compressed speech signal that is represented by a plurality of parameters. A time-varying digital filter is used to model the vocal tract. A number of LPC coefficients specify the transfer function of the filter updated on frame basis. An excitation signal is input to the filter analyzed on sub frame basis. This excitation signal includes either an adaptive vector quantiser code or a first pulse sequence, followed by a second pulse sequence. Selection logic is used to determine whether the adaptive vector quantiser code or the first pulse sequence better represents the speech signal. Based thereon, a switch selects either the adaptive vector quantiser code or the first pulse sequence. Thus, the parameters which are transmitted through a channel to a destination decoder include the LPC filter coefficients, either the adaptive vector quantiser code or the first pulse sequence, the second pulse sequence, and one bit indicating the state of the switch.

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.

On cross-correlation function evaluation from a periodically time-varying digital filter's output

Concise mathematical models are established for one-dimensional, linear, periodically time-varying digital filters. The three representations considered include linear difference equations, Green's function, and state-space structures. Linear time-invariant equivalent descriptions are then formulated, and from these, for the particular case of a white noise input, various analytical expressions are derived for both the cross-power spectral density functions and cross-correlation functions of the filter's subsampled output. These results are compared and contrasted with similar formulas for a linear time-invariant system.

On cyclically time-varying autoregressive digital filters

A first-order autoregressive filter is altered by changing the constant gain to two or more gains that cyclically alternate in time. The advantages of this system are shown, and the relation to linear autoregressive moving average difference equations of higher order is derived.

Balanced realisations and model reduction of periodically time-varying state-space digital filters

The authors discuss balanced realisations and model reduction of periodically time-varying (PTV) state-space digital filters. Controllability and observability Grammians of PTV state-space digital filters are discussed. It is extremely interesting to notice that although PTV state-space digital filters can be implemented by using a group of time-invariant coefficient sets, controllability and observability Grammians cannot be evaluated independently by using any one set of these time-invariant coefficients. Also, important physical interpretations of controllability and observability Grammians are considered. Based on these analysis results, balanced realisations of PTV state-space digital filters are defined and a synthesis method for balanced realisations is proposed. As one application of balanced realisations, a reduced-order model of a PTV state-space digital filter can be obtained by taking a subsystem of balanced realisation of the PTV state-space digital filter. Finally, a numerical example is given to illustrate the balanced model reduction procedure.

Approximations of IIR periodically time-varying digital filters

The paper discusses three approximations of IIR periodically time-varying (PTV) digital filters. Two conventional approximations, i.e. time-invariant difference equation and PTV difference equation approximations are first reviewed. It turns out that computation and hardware complexity of the two approximations is in proportion to the period. To reduce the computation and hardware complexity of the two approximations, an efficient PTV difference equation approximation based on extended Pade approximation is proposed. With this approximation, frequency-domain specifications are approximated by a difference equation with PTV coefficients. How to design the difference equation and how to determine the order of the difference equation to obtain the desired approximation precision are considered. A design for the difference equation is presented. As an application of the proposed approximation, a four-channel TDM-FDM transmultiplexer is simulated. It shows that the proposed PTV difference-equation approximation obtains satisfactory precision and compares favourably with above two approximations in respect of computation and hardware complexity.

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.

Coefficient sensitivity analysis of periodically time-varying state-space digital filters

This paper considers an analysis of statistical coefficient sensitivity and frequency coefficient sensitivity of Periodically Time-Varying (PTV) state-space digital filters. A statistical coefficient sensitivity is defined by using a virtual PTV state-space digital filter in which the periodically time-varying coefficients are stochastically varying. In order to analyze frequency coefficient sensitivity, a transfer function, which is called Output Sampling Polyphase (OSP) transfer function, is defined and expressed in terms of the coefficients of PTV state-space digital filters in a closed form. A frequency coefficient sensitivity is defined as an integral measure of the OSP transfer function with respect to the coefficients. It turns out that although the definitions of the two kinds of coefficient sensitivities are extremely different, their expressions are the same and closely related to the controllability and observability Grammians of PTV state-space digital filters. Also, some considerations for the minimization of the coefficient sensitivity are discussed. Finally, a numerical example is given to verify the analysis above and to show the dependence of the coefficient sensitivity on structure of PTV state-space digital filters. >

Inversion of linear periodically time-varying digital filters

This paper discusses the inversion of linear periodically time-varying (LPTV) digital filters using the idea of converting the LPTV filter to the block time-invariant filter. Explicit expressions are given to determine the inversion of LPTV filters. Controllability and observability of the inversion of LPTV filters are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A dedicated processor for time-varying digital audio filters

A dedicated processor for realizing variable digital filters suitable for audio applications is presented. Its architecture is based on the approximation of the time-varying system by finite number of invariant states. A technique which reduces digital filter transfer function distortions caused when filter parameters are changed over time is employed. The proposed transition model is realized by a dedicated unit having a pipeline architecture and allowing a very large number of variable filters to be implemented in real-time. >