Truncated Impulse Response Research Articles

Dynamic multi-objective matrix control for a class of switched systems

This article focuses on the tracking and stabilizing issues of a class of discrete switched systems. These systems are characterized by unknown switching sequences, a non-minimum phase, and time-varying or dead modes. In particular, for those governed by an indeterminate switching signal, it is very complicated to synthesize a control law able to systematically approach general reference-tracking difficulties. Taking into account the difficulty to express the dynamic of this class of systems, the present paper presents a new Dynamic matrix control method based on the multi-objective optimization and the truncated impulse response model. The formulation of the optimization problem aims to approach the general step-tracking issues under persistent and indeterminate mode changes and to overcome the stability problem along with retaining as many desirable features of the standard dynamic matrix control (DMC) method as possible. In addition, the formulated optimization problem integrates estimator variables able to manipulate the optimization procedure in favor of the active mode with an appropriate adjustment. It also provides a progressive and smooth multi-objective control law even in the presence of problems whether in subsystems or switching sequences. Finally, simulation examples and comparison tests are conducted to illustrate the potentiality and effectiveness of the developed method.

Identifying concert halls from source presence vs room presence

Identification of concert halls was studied to uncover whether the early or late part of the acoustic response is more salient in a hall's fingerprint. A listening test was conducted with auralizations of measured halls using full, hybrid, and truncated impulse responses convolved with anechoic symphonic music. Subjects identified halls more reliably based on differences in early responses rather than late responses, although varying the late response had more effect on acoustic parameters. The results suggest that in a typical situation with running symphonic music, the early response determines the perceptual fingerprint of a hall more than the late response.

Temporal-moment matching for truncated breakthrough curves for step or step-pulse injection

The method of temporal moments is an efficient approach for analyzing breakthrough curves (BTCs). By matching the moments of the BTCs computed through parametric transfer-function models or one-dimensional transport models to those of the data, one can estimate the parameters characterizing the transfer function or apparent transport parameters. The classical method of moments presumes infinite duration. However, the measurement of BTCs is usually terminated prematurely, before the concentration has reached zero. Unless this truncation of the BTCs has been taken into account, the estimates of the parameters may be in error. Truncated measured BTCs are sometimes extrapolated assuming exponential decay. In this study, we use the concept of moments of the truncated impulse–response function [Jawitz JW. Moments of truncated continuous univariate distributions. Adv Water Res 2004;27:269–81] in the analysis of truncated BTCs corresponding to the commonly encountered step and step-pulse injection modes. The method is straightforward, based on the relation, which we derive, between truncated moments of the impulse–response function and the measured BTC. It is practical to apply and does not require the extrapolation of the measured BTC. The method is also accurate. In a numerical study we discuss how short a step-pulse injection may be so that we can approximate it as instantaneous. Finally, we apply the method to the analysis of a field-scale tracer test.

A technique to truncate IIR filter impulse response and its application to real-time implementation of linear-phase IIR filters

A technique for realizing linear-phase infinite impulse response (IIR) filters has been proposed by Powell and Chau (1991) and gives a real-time implementation of H(z/sup -1/)/spl middot/H(z), where H(z) is a causal IIR filter function. In their system, the input signal is divided into L-sample sections, time-reversed, section convolved with H(z), and time-reversed again. The signal is then filtered by H(z) to give the system output with a processing delay of 3L+1 samples. However, the group delay response of the system exhibits a minor sinusoidal variation superimposed on some constant value. This variation will degrade image quality in image processing and signal quality in signal transmission applications. Furthermore, the output of the system contains harmonic distortion for a sinusoidal input. The main drawbacks of Powell and Chau's technique are the large processing delay of 3L+1 samples and the accompanying phase and harmonic distortions. A smaller processing delay increases the phase and harmonic distortions, yet the amplitude response remains acceptable. Previously, the present authors presented a method of reducing the processing delay by shortening the section length by an integer factor N using a structure with increased number of paths for the time-reversed signal. The authors consider how to reduce the phase and harmonic distortions. We examined the operation of the sectioned convolution and analyzed it based on a state-space representation. Then, we found that the cause of the distortions is a periodic variation of the impulse response length in the sectioned convolution. To overcome this problem, a technique is devised to realize a recursive circuit having a truncated impulse response with a fixed-length L. A system applying this technique to the Powell-Chau system is demonstrated to exhibit perfect linear-phase characteristic and produce virtually no harmonic distortion. Therefore, the section length L can be reduced without limitation due to phase and harmonic distortions. Two methods for reducing the increased computational complexity of this technique assuming fixed L are developed, and simulations are performed for the proposed system to confirm the expected improvements.

On the distribution of zeros of mobile channels with application to GSM/EDGE

The distribution of zeros of mobile channels is investigated and the results obtained are applied to the GSM/EDGE system. The taps of the discrete-time overall impulse response can be modeled as correlated complex Gaussian random variables, where the correlations depend on the transmit filter, the power delay profile of the channel, and the receiver input filter. For calculation of the density of zeros of the overall transfer function, a result from the mathematical literature is used. From this density, two cumulative distributions which are relevant for the design of suboptimum receivers are derived. Our investigations show that for the power delay profiles specified for GSM/EDGE, an allpass prefilter which transforms the impulse response in its minimum phase equivalent should be employed if decision-feedback equalization (DFE) or reduced-state sequence estimation (RSSE) are used at the receiver. On the other hand, if impulse response truncation using a linear prefilter is applied, the truncated impulse response should have a length of three as shown.

A parametric error analysis of the backward integration method for reverberation time estimation

Backward integration of a room impulse response has long been used in room acoustics for the estimation of reverberation time. However, the inevitable noise floor limits the minimum level of the measured impulse response, thereby leading to errors. This paper shows how the error is influenced by the selection of truncation time and evaluation range. A general guideline that emerges from this study is to truncate the measured impulse response at the knee where the main decay slope intersects the noise floor, then measure the slope of the backward integrated truncated impulse response down to a level about 5 dB above the noise floor.

System identification via singular value decomposition

The problem of identifying linear time invariant system parameters from their truncated impulse response data is studied. The authors analyse the algorithm originally proposed by Kung, and later modified by others, which is based on the singular value decomposition of the Hankel matrix. They propose a modified algorithm which generates more accurate results.

Some analyses of 2-D median f-k filters

Two‐dimensional median filters can be designed so that they have properties similar to f-k fan filters. This is done by using the coefficients of a truncated impulse response of an f-k filter as the weight coefficients for the weighted median process. The filter is called a median f-k filter and can be used to discriminate between events on the basis of apparent velocity. The filter appears suitable as a poststack coherency filter because it produces less distortion at wavefield terminations than conventional f-k fan filters. One‐dimensional weighted median filters that include negative coefficients are a logical starting point for the analysis of median f-k filters since simple numerical techniques may be used to analyze the behavior of these filters. We show that median filters with negative coefficients do not provide an unbiased estimate of the mean and can misplace the position of steps. Faults on a stacked section may be modeled by steps, and therefore applying a median f-k filter to stacked seismic data could change the position of faults. However, the distortion of steps introduced by median f-k filters is shown to be less than the distortion produced by the corresponding linear f-k filter, and the error in step placement is small. We present simple model examples and a seismic field data example to illustrate differences between linear f-k filters and median f-k filters.

Transfer function phase and truncated impulse response

Transfer function (TF) phase can be analyzed from the impulse response in a vibrating system. This letter analyzes the truncated impulse response of such a system and its phase characteristic by using a simple two-pole model. The estimated phase turns out to be highly sensitive to the window length of the truncated impulse response used for phase analysis. The minimum-phase behavior of a TF for example, is changed into a nonminimum-phase characteristic due to the truncation. An exponential window is recommended for the data analysis of vibrating systems in order to reduce the effect of truncation on the phase.

Parametric modeling of somatosensory evoked potentials.

In this paper, we examine methods of characterizing somatosensory evoked potentials (SEP's) in both the time and frequency domains. We have found that the truncated impulse response (TIR) method produced an accurate time domain model of the SEP signals at model orders greatly reduced from the original state space matrix. The TIR method was valuable for smoothing signals that were slightly corrupted by noise. In this case, the simulated data sequence was close to the original data sequence in the mean squared error sense. For signals that were greatly corrupted by noise, the TIR method was not able to perform as well. Therefore, the TIR method was not a feature extraction method but was valuable for data simulation. In the frequency domain, we have used the autoregressive moving average model (ARMA) to parameterize the SEP signal. An overdetermined set of Yule-Walker equations was created to determine the autoregressive (AR) parameters of the original data with the model order established by the singular value decomposition. From these AR parameters, a residual time series was generated which was used to find the moving average parameters. The resulting ARMA model was used to produce a simulated data sequence. The frequency domain characteristics of the simulated sequence and the corresponding power spectral density of the ARMA filter were very close to the periodogram of the original data sequence. Accurate parameterization was achieved for the SEP waveforms at low filter lengths.

Approximate realization algorithms for truncated impulse response data

The objectives of this paper are: 1) to present a novel derivation of an approximate realization algorithm using singular value decomposition proposed by Kung; 2) to explain the difference between two versions of this algorithm which have been used interchangeably in the literature; and 3) to introduce a new version of this algorithm which exhibits the accuracy of the original algorithm and the theoretical guarantee of stability.

Realization of Truncated Impulse Response Sequences with Prescribed Uncertainty

Many algebraic algorithms have been described in the literature to compute a discrete time state space realization of a system with a known impulse response . Six years ago De Jong (1975) focused attention on the numerical stability of these algorithms when using finite precision arithmetic. Only recently however full attention has been paid to the problem of measurement noise, where the uncertainty level on the available data is fixed by the measurement configuration. Accepting this as a fact of life we consider the problem of finding an approximate state space model that takes into account all modes of the signal which are distinguishable from the noise i.e. all modes which have a signal-to-noise-ratio better than a required minimum. General properties and an algorithm for the solution of this problem are given. We obtain conclusions on the extractability of modes and useful schemes for the automatic setting of decision parameters necessary for practical implementation. From a practical point of view, the numerical qualities and the behaviour of the particular class of one-time-step-matching algorithms on noisy data are shown in a feasability study.

A General Class of Numerically Reliable Algorithms for the Realization of Truncated Impulse Responses

The basic goal of this paper is to show that the problem of impulse response-realization allows a very simple formalism . This formalism generalizes the MBH-statement of the realization problem considerably. In this way it unifies many known algorithms, but it also introduces a wealth of possible new procedures. Moreover the geometric interpretations of the various algorithms and numerical techniques can be easily visualized, allowing to discern their strong and weak points. These ideas motivate appropriate decisions for handling noisy data, which is a major concern when developing programs for real life applications. Algorithms have been developed, using the formulation of this paper and numerically reliable techniques, such as Singular Value Decomposition and Generalized Inverses. They are presented together with a feasability study in a companion paper

Limit-cycle predictions in sampled data systems

For sampled data systems, it is possible to express discrete time convolution in terms of appropriate matrix multiplication. A limiting process then yields the steady-state response to a periodic input. The matrix involved in this operation is a circulant matrix based on the impulse response of the system. Circulant matrices are known to have useful structural properties and permit the association of the frequency domain with the time domain. The use of circulants in the analysis of nonlinear systems provides the means for converting the unwieldy nonlinear equations of continuous systems to simple matrix multiplications. It is then possible to apply numerical range techniques and rederive the circle criterion. The direct application of this approach yields a criterion which is less conservative than that obtained by the simple application of the small-gains theorem. Use of the approach in conjunction with a loop transformation, on the other hand, provides an alternative derivation of the circle criterion for discrete systems. The method can be extended to multivariable systems, and, because of its association with the time domain, it permits the assessment of system stability in the face of imperfect system descriptions, namely truncated impulse responses.

Truncation error in 2-D block mode filtering

An investigation is made of the error due to truncated impulse response for the block mode or sectioning technique. The upper bound on the error in terms of the sectioning parameters is obtained and shown to be independent of block size; this results in a simpler approach to obtaining an optimal sectioning procedure.

On the zeros of polynomials used in digital filters

The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transform of the truncated impulse response of a digital filter with poles, is a polynomial that is important in the stability analysis of certain types of block recursion and inverse digital filters. This paper presents new results on the zeros of these polynomials, extending the earlier work of Read and Burrus and also studies the effect of coefficient quantization on the zeros.

Adaptive Channel Memory Truncation for Maximum Likelihood Sequence Estimation

Maximum likelihood data sequence estimation, implemented by a dynamic programming algorithm known as the Viterbi algorithm (VA), is of considerable interest for data transmission in the presence of severe intersymbol interference and additive Gaussian noise. Unfortunately, the required number of receiver operations per data symbol is an exponential function of the duration of the channel impulse response, resulting in unacceptably large receiver complexity for high-speed PAM data transmission on many channels. We propose a linear prefilter to force the overall impulse response of the channel/prefilter combination to approximate a desired truncated impulse response (DIR) of acceptably short duration. Given the duration of the DIR, the prefilter parameters and the DIR itself can be optimized adaptively to minimize the mean-square error between the output of the prefilter and the desired prefilter output, while constraining the energy in the DIR to be fixed. In this work we show that the minimum mean-square error can be expressed as the minimum eigenvalue of a certain channel-dependent matrix, and that the corresponding eigenvector represents the optimum DIR. An adaptive algorithm is developed and successfully tested. The simulations also show that the prefiltering scheme, used together with the VA for two different channel models, compares favorably in performance with another recently proposed prefiltering scheme. Limiting results for the case where the prefilter is considered to be of infinite length are obtained; it is shown that the optimum DIR of length two must be one of two possible impulse responses related to the duobinary impulse response. Finally we obtain limiting results for the case where the transmitting filter is optimized.

Effects on Speech Perception of Modifying the Impulse Response of a Small Room

A linear model of the ear as a parallel bank of filters and the image solution for room response are outlined. These models are used to predict the perception of speech under certain controlled variations in the impulse response of a small room. Using either a hardware digital filter or a computer simulation, room impulse responses have been measured, modified, and their effect on speech examined. The investigations included linearizing the phase of and truncating the impulse response, examining an all-pass filter with a real room phase characteristic and observing the effect of the time complement of a truncated impulse response.

Use of the geometry of partial sums in digital filter analysis

The z transform of the truncated impulse response of a digital filter with poles is a polynomial that is important in the stability analysis of certain types of block and inverse digital filters. The geometry of partial sums is the theory that describes the behavior of the zeros of these polynomials. This paper presents specific results and also develops guidelines for use in stability analysis and design of block and inverse digital filters.

REALIZATION OF SHARP CUT‐OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS

ABSTRACTIt was found in Part I of this paper that approximating the sharp cut‐off frequency characteristic best in a mean square sense by an impulse response of finite length M produced a characteristic whose slope on a linear frequency scale was proportional to the length of impulse response, but whose maximum overshoot of ±9% was independent of this length (Gibbs' phenomenon). Weighting functions, based on frequency tapering or arbitrarily chosen, were used in Part II to modify the truncated impulse response of the sharp cut‐off frequency characteristic, and thereby obtain a trade‐off between the value of maximum overshoot and the sharpness of the resulting characteristic. These weighting functions, known as apodising functions, were dependent on the time‐bandwidth product Mξ, where 2ξ, corresponded to the tapering range of frequencies.Part III now deals with digital filters where the number 2N–1 of coefficients is directly related to the finite length M of the continuous impulse response. The values of the filter coefficients are taken from the continuous impulse response at the sampling instants, and the resulting characteristic is approximately the same as that derived in Part II for the continuous finite length impulse response. Corresponding to known types of frequency tapering, we now specify a filter characteristic which is undefined in the tapering range, and determine the filter coefficients according to a mean square criterion over the rest of the frequency spectrum. The resulting characteristic is dependent on the time bandwidth product Mξ= (N–1/2)ξ up to a maximum value of 2, beyond which undesirable effects occur. This optimum partially specified characteristic is an improvement on the previous digital filters in terms of the trade‐off ratio for values of maximum overshoot less than 1%. Similar to the previous optimum characteristic is the optimum partially specified weighted digital filter, where greater “emphasis is placed on reducing the value of maximum overshoot than of maximum undershoot”. Such characteristics are capable of providing better trade‐off ratios than the other filters for maximum overshoots greater than 1/2%. However these filters have critical maximum numbers 2.NC–1 of coefficients, beyond which the resulting characteristics have unsuitable shapes. This type of characteristic differs from the others in not being a biassed odd function about its cut‐off frequency.