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A technique to truncate IIR filter impulse response and its application to real-time implementation of linear-phase IIR filters
Abstract
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</sup>)·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.
... Dans certains domaines d'application tels que : la parole, l'imagerie biomédicale, la géophysique, la cristallographie à rayon X,…etc., [Harasty88], [Raita94], [Rosten03], [Farsi06], [Willson94], [Djokic98], [Kurosu03]. ...
... Cette dernière catégorie présente l'avantage d'engendrer de meilleures performances en retard pur et quantité de mémoire. Ce qui l'a favorisé pour la mise en oeuvre des filtres PN [Powell91], [Willson94], [Djokic98] et [Kurosu03]. Les méthodes de la première catégorie [Czarnach82a], [Arias04], présentent des performances moins bonnes en retard pur et quantité de mémoire requise. ...
... Cependant, leur majeur avantage est la réduction de l'erreur de sectionnement à l'aide de l'état initial du filtrage non causal. Nous montrons que cette démarche s'avère difficile à intégrer pour les schémas en cascade II [Powell91], [Willson94], [Djokic98], [Kurosu03], [Rader06]. L'étude de l'impact du sectionnement sur la qualité de la réponse du filtre PN peut être effectuée dans le domaine temporel. ...
Le travail effectué dans le cadre de cette thèse consiste à étudier et élaborer de nouvelles techniques d’implémentation en temps réel de certaine classe particulière de systèmes, appelés filtres récursifs non causaux à phase nulle (PN). Ces filtres sont souvent utilisés en temps différé (off line). Ils s’implémentent avec des cellules élémentaires récursives, et nécessitent deux inversions temporelles. Ainsi, leur mise en œuvre se réalise par deux filtrages : l’un est causal et s’exécute dans le sens direct (forward filtering), l’autre est non causal, s’effectuant avec une entrée inversée (backward filtering). Ils sont appréciés pour leurs performances intéressantes et non conflictuelles en amplitude et en phase. Ils peuvent avoir à une phase exactement nulle avec une caractéristique d’amplitude très sélective et similaire à celle d’un filtre optimal elliptique. Ils présentent une bonne alternative par rapport aux filtres classiques RII à phase quasi-linéaire et RIF à phase linéaire.
Cependant, leur mise en œuvre en temps réel est délicate ; et rend leur utilisation peu intéressante pour le traitement des signaux de longueur importante. Il existe dans la littérature scientifique, quelques techniques de leur implémentation en temps réel. Celles-ci se basent, pour la plupart, sur des approches utilisant un sectionnement du signal traité. Ceci engendre des erreurs de calcul souvent importantes, avec des retards purs conséquents et qui exigent plus de ressources en mémoire. Notre contribution dans cette thèse portera sur une synthèse critique et une étude de performances des principales techniques de mise en œuvre des filtres PN en temps réel, à savoir : le sectionnement avec chevauchement sauvegardé (SCSauve), et chevauchement additionné (SCAdd), ou sans chevauchement (SSC). Les performances sont évaluées selon quatre critères : l’erreur de calcul, le retard pur engendré, la quantité de mémoire requise et la complexité algorithmique exigée. Une définition des différentes erreurs de calcul dues à la réalisation des filtres PN en temps réel est abordée. Une approche analytique est aussi élaborée dans le but de caractériser l’origine de ces erreurs de calcul. L’impact de sectionnement sur la qualité de la réponse d’un filtre PN est évalué dans les deux domaines temporel et fréquentiel, respectivement par la notion de l’erreur de sectionnement relative et le taux de distorsion harmonique (THD).
Une nouvelle technique s’appuie sur le sectionnement sans chevauchement est proposée. Elle se base sur un calcul récursif de l’état initial du filtrage non causal. Une pile de type FIFO (First In, First Out) est intégrée afin d’organiser les échantillons d’entrées. Les inversions temporelles sont implémentées par des piles de type LIFO (First Out, First In). La technique proposée offre une diminution de la complexité algorithmique et de la quantité de mémoire requise, avec une réduction du retard pur pour des erreurs de calcul acceptables.
Mots Clés :
phase nulle - phase linéaire - filtre RII - filtres RIF - filtre PN - sectionnement par chevauchement sauvegardé - sectionnement par chevauchement additionné - sectionnement sans chevauchement - complexité algorithmique – retard pur- FIFO- LIFO
... Noncausal filters are usually realized using a combination of 2-pass filtering and time reversal [1][2][3][4][5][6][7]. The first pass can be performed in a forward direction using a stable recursive digital filter, and the second pass can be performed in a backward direction using a noncausal subsystem implemented by 2 time reversal operations and a stable recursive digital filter, as shown in Figure 1. ...
... A continuous infinite-length input is divided into finite-length sections and each section is filtered separately by the 2-pass scheme. The final output can be built using output sections yielded from finite-length section processing and overlap-based approximations such as overlapand-save techniques [2,5], overlap-and-add techniques [3,4,6,7], or the nonoverlapping technique [1]. The final output is never achieved without errors; the magnitude of the systematic errors depends on section length and degrades the linearity of the overall system. ...
... x(4) x (6)]. ...
A novel method for implementing noncausal forward/backward 2-pass recursive digital filters in real time is presented. It is based on a segment-wise block processing scheme without overlapping. Factors that degrade the linearity of the overall system's transfer function are discussed. An analytical condition that corrects the system's linearity is elaborated upon using the state variable approach. A recursive algorithm is developed to compute an implementable condition for real-time filtering. A single first in, first out queue memory is introduced to ensure an organized and continuous data stream into the proposed system. This technique allows real-time, sample-by-sample filtering, and it yields reduced delay and data storage memory compared to previous works. Better performances in total harmonic distortion were also obtained. Experimental results are illustrated.
... Thus, resulting in a IIR filter with approximate linear phase. Later, Kurosu [3] proved the imperfections of the Powell-Chau filter analytically and modifies Powell & Chau's structure. Kurosu's modified Powell-Chau filter is proven to have zero phase disturbances. ...
... . [3] has proven that Kurosu's filter is linear and time invariant. The exact relation to the modified Powell-Chau filter is: Typically, L is chosen where the infinite impulse response (IIR) is near quantization levels. ...
... The exact relation to the modified Powell-Chau filter is: Typically, L is chosen where the infinite impulse response (IIR) is near quantization levels. Exact relation H L (z) is detailed by [3] or estimated by a low order approximation. Also, if L sufficiently long, H L (z) need not be implemented in fixedpoint since coefficient values will be quantized to 0. Thus, H T (z), H T (z −1 ) and H T (z) · H T (z −1 ) are FIR filters but implemented by IIR filters, switches, and L-word Last-In First-Out (LIFO) memory elements. ...
This paper considers high sampling rate digital control of an electromagnetically levitated shaft. Inner loop feedback and outer loop repetitive controls are realized by a Field Pro-grammable Gate Array (FPGA) at 100 kHz rate based on consid-erations made on limited FPGA computing resources, numerical dynamic range, selection of word length for fixed point computa-tion, and distribution of parallel computing. The benefit of high sampling rate feedback control is demonstrated to reduce the sen-sitivity peak when compared to lower sampling rate control. The repetitive control involves inverting non-minimum phase zeros and introducing linear phase low-pass filtering for establishing robust stability. These filtering tasks are realized by a modified Powell-Chau filter, which is an efficient infinite impulse response (IIR) realization of high order finite impulse response (FIR) filter. Experimental results are presented to demonstrate the effective control performance.
... + a M−1 z −(M−1) + a M z −M (6) and h(n) represents the impulse response at time step n. The exact relation for H L (z) has been presented by Kurosu [2] ,which is: ...
... The realization of Kurosu's modified filter [2] is shown inFigure 2. L is chosen such that the remaining portion of the length of the impulse response is near quantization level. Also, we assume that H(z) = H top (z) = H bot (z) for simplicity. ...
... The two differ by a gain γ. Since [2] has proven linearity and time invariance, it follows that there is an exact relation to the modified Powell- Chau filter. Exact input-output relation can be retrieved fromFigure 2 ...
This paper considers a fixed-point realization of approximate stable inversion of systems with non-minimum phase zeros. It is based on the efficient linear phase IIR filter implementation first introduced by Powell and Chau. The proposed filter is used in inversion based feedforward tracking and repetitive controllers which is realized by a Field Programmable Gate Array to control a piezoelectric actuator and generate precise dynamic high bandwidth motion at a 100 kHZ sampling rate.
... As is well known, Noncausal Recursive (NR) digital filters can be realized by a cascade connection of a causal IIR filter with an arbitrary transfer function Hz (forward pass) and an anticausal time reversed version of the same function 1 Hz (backward pass) [1][2][3][4][5][6][7]. The issue of designing this special class of digital filters has been discussed only slightly [1][2], and the most contributions have been focused on its real-time implementation [1][2][3][4], [6][7][8][9][10]. Kormylo and Jain [1] have proposed a classical design in the frequency domain based on bilinear transform and a well-known optimal design of analog elliptic filters, such that the magnitude characteristic of the causal recursive filter Hz has to agree with the square root of the desired ; while, its phase characteristic is inconsequential. ...
... Kormylo and Jain [1] have proposed a classical design in the frequency domain based on bilinear transform and a well-known optimal design of analog elliptic filters, such that the magnitude characteristic of the causal recursive filter Hz has to agree with the square root of the desired ; while, its phase characteristic is inconsequential. Over these four decades, this classical design was the most commonly used in several papers [3][4][5][6][7][8][9][10], where the design procedure is fast and simple. However, despite of using an optimal elliptic digital filter Hz, we are not sure that the squared magnitude response 2 j He is the optimal frequency response of the NR digital filters. ...
In this paper, an optimal design for the special class of Non-Causal Recursive (NR) digital filters with zero phase shift is presented. The approximation problem is formulated as a minimax optimization. It is transformed into a linear programming, which can be solved iteratively by a well-known binary search technique. We demonstrate that the optimal frequency response of the NR filter can be better than that of the classical squared magnitude response of an optimal elliptic filter; where, we get additional attenuation in the stopband. Two examples of filter design are illustrated in agreement with our contribution. Some details of the linear programming implementation and its limits are also discussed.
... This special class of NR digital filters can offer highly magnitude characteristics with theoretically zero phase shift and optimal computational burdens per output sample [4][5][6]. Throughout the last four decades, Kormylo and Jain's design [2] was the most widespread used in designing NR filters [4][5][6][7][8][9][10][11][12][13]. This classical design requires specifications in the frequency domain. ...
... Finally, (9) can be expressed as a linear constraint over the set of the coefficients ...
This paper presents an optimal design for the special class of Non-Causal Recursive (NR) digital filters with zero phase shift. The design
is based on the Chebyshev approximation problem. It can be transformed to an equivalent linear program under linear constraints of the zero phase. The given design yields more interesting pole-zero patterns that are not necessarily restricted to the classical design of Kormylo and Jain. The proposed optimal design allows an accurate zero phase shift and better magnitude characteristics in passband and stopband.
... Willson and Orchard described a variation on this approach that yields higher performance (more stopband loss, less passband ripple and/or narrower transition band) [116]. Kurosu et al. described performance issues in Powell and Chau's original design; a sinusoidal variation in the group delay, and a harmonic distortion with sinusoidal input [117]. To alleviate these issues, they reduce the filter's overall processing delay by using shorter sections with truncated impulse response [117]. ...
... Kurosu et al. described performance issues in Powell and Chau's original design; a sinusoidal variation in the group delay, and a harmonic distortion with sinusoidal input [117]. To alleviate these issues, they reduce the filter's overall processing delay by using shorter sections with truncated impulse response [117]. Azizi proposed an efficient arbitrary sample rate converter using zero phase IIR filtering, which later led to a patented signal interpolator [118]. ...
Audio equalization is a vast and active research area. The extent of research means that one often cannot identify the preferred technique for a particular problem. This review paper bridges those gaps, systemically providing a deep understanding of the problems and approaches in audio equalization, their relative merits and applications. Digital signal processing techniques for modifying the spectral balance in audio signals and applications of these techniques are reviewed, ranging from classic equalizers to emerging designs based on new advances in signal processing and machine learning. Emphasis is placed on putting the range of approaches within a common mathematical and conceptual framework. The application areas discussed herein are diverse, and include well-defined, solvable problems of filter design subject to constraints, as well as newly emerging challenges that touch on problems in semantics, perception and human computer interaction. Case studies are given in order to illustrate key concepts and how they are applied in practice. We also recommend preferred signal processing approaches for important audio equalization problems. Finally, we discuss current challenges and the uncharted frontiers in this field. The source code for methods discussed in this paper is made available at https://code.soundsoftware.ac.uk/projects/allaboutaudioeq.
... This paper introduces another notation which was proposed in our prior works presented in Japanese. In order to show the advantage of using this notation, examples for (1) Chinese remainder theorem used in number theoretic digital filter [1] (2) Hayashi's theorem for modular exponentiation-based cryptography [2] (3) Cyclic code encoder [3] (4) Key circuit to realize a linear phase recursive filter [4] are shown. ...
... The following example is concerned with the method by Kurosu et al. to implement a linear phase recursive filter [4]. Let ...
This paper describes nontraditional notation, which seems to be helpful to learn about a kind of signal processing. Suppose the operator, which extracts the fractional part of a real number. Then a remainder k mod n is expressed using this operator, which relates remainders to real numbers, and hence it becomes easy to explain the properties of residue classes, especially when several moduli must be considered simultaneously. Since a residue class of polynomials has similar properties, analogous calculation is available as shown in examples for a cyclic code encoder and a perfectly linear phase recursive filter.
... Another famous approach is the Powell and Chau approach, which implements a linear phase IIR filter as a tandem connection of an arbitrary transfer function and a time-reversal version of the same function [14]- [16]. Maeng and Lee [15], state in their concluding remarks that the fundamental limitation of the approach's "LIFO based implementation is in the group delay…the application of this method could be limited in real-time interactive signal processing." ...
... The error in magnitude at a frequency is , where is the desired magnitude response, and the delay error is , where is the filter delay at some nominal center frequency in the passband, and is the desired delay response of the filter relative to . The gain that minimizes is given by (16) The Fletcher and Powell method used in the iteration is described as follows. ...
A novel approach to designing approximately linear phase infinite-impulse-response (IIR) digital filters in the passband region is introduced. The proposed approach yields digital IIR filters whose numerators represent linear phase finite-impulse-response (FIR) filters. As an example, low-pass IIR differentiators are introduced. The range and high-frequency suppression of the proposed low-pass differentiators are comparable to those obtained by higher order FIR low-pass differentiators. In addition, the differentiators exhibit almost linear phases in the passband regions
... The impulse response function is a cross-correlation function that describes the characteristics of the time-domain system and is widely used in radar, sonar, digital communication, and geology [43]. The advantage of the impulse response function is that the algorithm's parameters and convolution operation are suitable for signals mixed with additive noise or delayed samples [44]. For a given time series, this function can solve for the distance of a real target submerged in noise. ...
Landslide deformation is the most intuitive and effective characterization of the evolution of landslides and reveals the inherent risk of landslides. Considering the inadequacy of existing deformation monitoring data for early warnings regarding landslide hazards, resulting in insufficient disaster response times, this paper proposes a time-domain correlation model. Based on the process of rainfall-induced landslide deformation, the time-domain correlation between regional rainfall and landslide deformation is proposed, which can reflect the temporal characteristics of landslide responses to rainfall, and the calculation method of the impulse response function is designed to quantitatively model and calculate the correlation. Furthermore, rainfall monitoring data are used to optimize the landslide deformation monitoring indicator system for early warnings regarding landslide instability. The feasibility of the method proposed in this paper is verified by analyzing the historical monitoring data of rainfall and landslide deformation at nine typical locations in five landslide hazard areas in Fengjie County, Chongqing city. (1) The correlation models for the XP landslide involve a delayed rainfall response time of 5 for deformation, respectively, as well as the existence of a cycle of 55–56 days, which means that the above area can advance the landslide warning by one lag time based on the cycle; (2) The correlation models for the OT landslide show consistent correlations under a 48–50-day cycle, which means that the deformation in the above areas can be predicted based on rainfall accumulation. (3) The HJWC landslide presents a turbulence correlation, which means that other monitoring data need to be supplemented and analyzed.
... Furthermore, IIR filters are not well-suited to high dynamic range environments (> 60 dB) where large clutter returns can produce transient ringing in the filters [23]. This issue can be resolved using tail cancellation techniques, such as truncated IIR filtering [24], [25]. These techniques have not been tested in this work due to the low-dynamic range environment encountered in these measurements. ...
In this paper, we describe three digital moving target indication (MTI) and moving target segmentation techniques (based on target speed) and apply them to short-range FMCW radar data. The described approaches are applicable to many short-range radar sensors. In particular, we focus on FMCW radar, which are ubiquitous in numerous applications including gesture recognition radar, automotive radar and imaging radar. The three digital MTI filtering methods explored are background subtraction, FIR filtering, and IIR filtering. Each of the methods is implemented in the time domain for simpler logic implementation. We apply the MTI methods on datasets gathered using a Cband FMCW radar in both a short-range, direct line-of-sight scenario and a complex cluttered through wall radar scenario. Based on the analyses, it is shown that each of the MTI techniques are extremely effective when deployed in the right scenario. Background subtraction is found to be well suited for slow-moving targets. FIR and IIR filtering techniques provide the simplest, one-step processes for moving target segmentation. IEEE
... Although exactly linear phase IIR filters are either unstable or noncausal, low complexity techniques for handling noncausality in block-based applications are well documented (see, e.g. [30,65,81]). These techniques make IIR filters [25,76,111]. ...
... Kurosu et. al reduce LF IIR filter delay [4]. Also, Azizi has patented a signal interpolator using a zero-phase filter [2], [1]. ...
In the paper infinite impulse response (IIR) digital filters with linear-phase (LF IIR) for audio signal interpolator are discussed. This article presents the causal realization of LF IIR and especially the realization of LF IIR filter with overlap is considered. As the LF IIR filters are an alternative for ordinary finite impulse response (FIR) filters, a comparison of a particular design of FIR and LF IIR filter is made. There are also compared realizations of signal interpolators using FIR filter and LF IIR filter.
... There is a number of approaches for designing IIR HB filter [5]- [7] but they invariably lead to nonlinear phase HB filters. Although exactly linear phase IIR filters are either unstable or noncausal, low complexity techniques for handling noncausality in block-based applications are well documented (see, e.g., [8]- [10]). They make IIR filters with exactly linear phase practicable in various applications, especially in image processing [11]- [13]. ...
This paper proposes a novel method to design exactly linear phase infinite impulse response half-band filters with arbitrary regularity. Broadly speaking, the design problem is formulated as a semi-infinite program, which is then turned into a semidefinite program of minimal order via a new linear matrix inequality characterization of convex hulls of trigonometric polynomials. In contrast to maximally flat approach, the proposed method allows direct control of various design parameters, which in turn enables the synthesis of filters with better transition response. The viability of the proposed method is demonstrated through several numerical examples.
... For the filtering of the EEG data a forward-backward scheme using Butterworth filters was employed that is being applied in one step to the integrality of the ingoing signal. This choice was made for reasons of simplicity, however in order to enable realtime classification in online BCI systems, the filter should be realized either by using a windowed implementation of the forward-backward scheme (Djokic et al., 1998; Kurosu et al., 2003), or by using a traditional causal filtering scheme, for instance based on finite impulse response (FIR) filters. ...
A brain-computer interface (BCI) is a system that enables control of devices or communication with other persons, only through cerebral activity, without using muscles. The main application for BCIs is assistive technology for disabled persons. Examples for devices that can be controlled by BCIs are artificial limbs, spelling devices, or environment control systems. BCI research has seen renewed interest in recent years, and it has been convincingly shown that communication via a BCI is in principle feasible. However, present day systems still have shortcomings that prevent their widespread application. In part, these shortcomings are caused by limitations in the functionality of the pattern recognition algorithms used for discriminating brain signals in BCIs. Moreover, BCIs are often tested exclusively with able-bodied persons instead of conducting tests with the target user group, namely disabled persons. The goal of this thesis is to extend the functionality of pattern recognition algorithms for BCI systems and to move towards systems that are helpful for disabled users. We discuss extensions of linear discriminant analysis (LDA), which is a simple but efficient method for pattern recognition. In particular, a framework from Bayesian machine learning, the so-called evidence framework, is applied to LDA. An algorithm is obtained that learns classifiers quickly, robustly, and fully automatically. An extension of this algorithm allows to automatically reduce the number of sensors needed for acquisition of brain signals. More specifically, the algorithm allows to perform electrode selection. The algorithm for electrode selection is based on a concept known as automatic relevance determination (ARD) in Bayesian machine learning. The last part of the algorithmic development in this thesis concerns methods for computing accurate estimates of class probabilities in LDA-like classifiers. These probabilities are used to build a BCI that dynamically adapts the amount of acquired data, so that a preset, approximate bound on the probability of misclassifications is not exceeded. To test the algorithms described in this thesis, a BCI specifically tailored for disabled persons is introduced. The system uses electroencephalogram (EEG) signals and is based on the P300 evoked potential. Datasets recorded from five disabled and four able-bodied subjects are used to show that the Bayesian version of LDA outperforms plain LDA in terms of classification accuracy. Also, the impact of different static electrode configurations on classification accuracy is tested. In addition, experiments with the same datasets demonstrate that the algorithm for electrode selection is computationally efficient, yields physiologically plausible results, and improves classification accuracy over static electrode configurations. The classification accuracy is further improved by dynamically adapting the amount of acquired data. Besides the datasets recorded from disabled and able-bodied subjects, benchmark datasets from BCI competitions are used to show that the algorithms discussed in this thesis are competitive with state-of-the-art electroencephalogram (EEG) classification algorithms. While the experiments in this thesis are uniquely performed with P300 datasets, the presented algorithms might also be useful for other types of BCI systems based on the EEG. This is the case because functionalities such as robust and automatic computation of classifiers, electrode selection, and estimation of class probabilities are useful in many BCI systems. Seen from a more general point of view, many applications that rely on the classification of cerebral activity could possibly benefit from the methods developed in this thesis. Among the potential applications are interrogative polygraphy ("lie detection") and clinical applications, for example coma outcome prognosis and depth of anesthesia monitoring.
Heave measurement methods are beneficial to the advancement of ocean engineering and modern navigation technology. Unfortunately, these methods have several drawbacks in practice. Firstly, the high-pass filter used in these methods introduces positive phase and then decreases the real-time performance of the heave measurement. Secondly, the peak-to-peak error existing in traditional methods is typical 2%. In order to address the above issues, a novel heave measurement method was proposed in this study. It employs a band-pass filter instead of the high-pass filter existing in traditional methods. The band-pass filter possesses a zero-phase and zero-magnitude-attenuation at center frequency. Weighted-frequency Fourier linear combiner (WFLC) method, as an adaptive frequency estimator, is designed to ensure algorithm accuracy under unknown sea conditions. The proposed method eliminates the peak-to-peak error existing in traditional methods, and it avoids phase correction by adding an addition filter. The simulation and trial results demonstrated efficiency of the proposed heave measurement method and illustrated that it has a higher accuracy than conventional methods. This method plays an important role in ocean engineering.
In this paper, a new design approach of approximately linear phase infinite impulse response (IIR) low pass digital differentiator (LPDD) is proposed and studied. The proposed design is based on a transfer function that has a numerator with anti-symmetric coefficients. To better control the magnitude response of the designed LPDD, the differential evolution (DE) optimization algorithm is used to find the coefficients of the transfer function that meets an appropriate pass band and stop band edge frequencies. The use of appropriate pass band and stop band edge frequencies gives the designer of the LPDD direct control on the width of the transition band. The designed LPDD using the proposed approach has approximately linear phase and much better magnitude response than that of the IIR LPDDs designed using other techniques reported in literature. In addition, the designed LPDD using the proposed approach has steeper roll-off magnitude response and narrower transition band than that of designed IIR and high order FIR LPDDs available in the literature.
Chapter 2 is devoted to problems of analog signal acquisition for the digital control circuits in power electronics devices. In this chapter the problems of the measurements in power electronics circuits are highlighted. Typical systems for current and voltage measurements are discussed. Particular attention is paid to galvanic isolation and the impact of a high slew rate in common mode voltage. A discussion is presented on the selection of the sample rate and number of bits. Consideration is given to the methods and circuit design to reduce quantization noise. Included is a discussion of noise shaping circuits useful for power electronics output circuits. Also included is a section on the impact of the phenomenon of jitter on the signal to noise ratio. In this chapter a method for calculating the resultant signal to noise ratio is also shown. Finally, at the end of this chapter a presentation of selected A/D converters suitable for use in power electronics circuits is made. Special attention is paid to simultaneously sampling A/D converters.
Selected methods of filtration and separation of signals and their implementation using digital signal processors are presented in this Chapter. At the beginning of the chapter there is discussion of classical finite impulse response (FIR) digital filters and infinite impulse response (IIR) digital filters. However, special attention is paid to lattice wave digital filters, which are excellent in implementation. Modified lattice wave digital filters for modern digital signal processors are considered. The next section presents infinite impulse response digital filters with linear-phase, and a discussion of the their causal realizations. Following this section multirate circuits are introduced; both circuits for reducing the sample rate—decimators, and circuits for increasing the sample rate—interpolators are examined. Consideration is given to interpolator circuits based on lattice wave digital filters. After this a section dedicated to digital filter banks follows. Particular attention has been paid to digital filter banks useful for power electronics applications: wave digital filters, sliding DFT, sliding Goertzel, moving DFT and strictly complementary filter banks. Presented are some implementations of digital filters using digital signal processors. The last part of this chapter is devoted to selected digital signal processors from Texas Instruments and Analog Devices.
In this article a class of digital correction filters is presented. This class of filters has been synthesized in an optimization procedure where two criteria are considered. The first criterion is an approximation of the series interconnection of a distorting system and a compensating filter. The second is the stability of the compensating filter. Their characteristic advantage is full correction of phase distortions independent from the values of established criteria. These filters can be useful for compensation of parts of measurement systems when the discrete transfer function has the zeros on the unit circle or in its proximity. Theoretical results have been verified in numerical simulations.
A novel method for designing IIR digital filters satisfying both the phase and magnitude specifications is presented. The proposed method is based on the global search method of an evolutionary algorithm (EA). Minimum phase filters have the lowest group delay for an IIR filter with zeros lying inside of the unit circle. The EA method can solve both the nonlinear phase problem of IIR filters and the selection of the optimum implementation coefficients. Designed examples demonstrate the efficiency of the proposed method for the specified phase and magnitude of an IIR filter.
Selected methods of filtration and separation of signals and their implementation using digital signal processors are presented in this section. At the beginning of the chapter, there is discussion of classical finite impulse response digital filters and infinite impulse response digital filters. However, special attention is paid to lattice wave digital filters, which are excellent in implementation. Considered as modern digital signal processors are the lattice modified wave digital filters. Presented in the next section are infinite impulse response digital filters with linear phase, where there is discussion of the their causal realizations. In the following section, these are presented multirate circuits, circuits for reducing the sample rate–decimators, and circuits for increasing the sample rate–interpolators. Consideration is given to interpolator circuits based on lattice wave digital filters. After this a section dedicated to digital filter banks follows. Particular attention has been paid to digital filter banks useful for power electronics applications: wave digital filters, sliding DFT, moving DFT, and strictly complementary. There are presented some implementations of digital filters using digital signal processors. The last part of this chapter is devoted to selected digital signal processors.
The paper presents a fixed-point realization of non-minimum phase system inversion for inversion based control algorithms such as feedforward, repetitive controller, and iterative learning control. Based on the author's previous work on efficient dynamic inversion using stable pole-zero cancelation and non-minimum phase zero inversion using Kurosu filter, this paper employs the use of the Delta Operator filter realization to mitigate quantization effects in fixed-point computations. The Kurosu filter based inversion, where the difference of two IIR filters are used to approximate a high order FIR filter and switches are used to perform time reversals for phase compensation, is computationally efficient and well suited for low level fixed-point realization by digital signal processors or field programmable gate arrays. However, substantial quantization noise is evident in high sampling rate implementation. The Delta Operator realization increases computational complexity only slightly while providing substantial reduction of quantization noise. The method is employed in the feedforward and repetitive control of a piezoelectric actuator and the experimental results are presented to demonstrate its effectiveness.
This paper presents a stable inversion of nonminimum phase systems with highly efficient computation for high-sampling rate applications. The stable filter that inverts the dynamics of a nonminimum system is based on cascading a stable pole-zero cancellation infinite impulse response (IIR) filter with a high-order finite impulse response (FIR) filter which inverts the unstable zero dynamics. The high-order FIR filter is realized based on efficient IIR filter implementation first introduced by Powell and Chau then later modified by Kurosu. As a demonstrative example, the inversion filters are applied to feedforward tracking and repetitive control algorithms and realized by a field programmable gate array. The controllers are implemented at 100-kHz sampling rate to control the motion of a 4 degrees-of-freedom magnetically levitated shaft in experiment.
▶ Discusses problems concerning the design and realization of digital
control algorithms for power electronics circuits using digital signal
processing (DSP) methods
▶ Covers the general problems of analog signal acquisition from power
electronics digital control circuitry, such as: sampling rate, number
of bits, galvanic isolation, signal-to-noise ratio $SNR$, anti-aliasing
filter, AC and DC current sensors, bandwidth of signal, signal range
▶ Includes Matlab examples for illustration of considered problems
Many digital control circuits in current literature are described using analog transmittance.
This may not always be acceptable, especially if the sampling frequency and power
transistor switching frequencies are close to the band of interest. Therefore, a digital
circuit is considered as a digital controller rather than an analog circuit. This helps to avoid
errors and instability in high frequency components.
Digital Signal Processing in Power Electronics Control Circuits covers problems concerning
the design and realization of digital control algorithms for power electronics circuits
using digital signal processing (DSP) methods. This book bridges the gap between power
electronics and DSP. The following realizations of digital control circuits are considered:
digital signal processors, microprocessors, microcontrollers, programmable digital circuits.
Discussed in this book is signal processing, starting from analog signal acquisition,
through its conversion to digital form, methods of its filtration and separation, and ending
with pulse control of output power transistors. The book is focused on two applications
for the considered methods of digital signal processing: an active power filter and a digital
class D power amplifier.
The major benefit to readers is the acquisition of specific knowledge concerning
discussions on the processing of signals from voltage or current sensors using a digital
signal processor and to the signals controlling the output inverter transistors. Included are
some Matlab examples for illustration of the considered problems.
A model describing the time constant of a wide-band microwave differentiator is presented. The time constant combined with the amplitude response dictates the circuit behaviour of a microwave differentiator. By representing the formulations of differentiators in the discrete-time (or Z ) domain, the authors implement the differentiators with equal-length transmission lines. The approach method consists of discrete signal processing techniques, transfer functions in the Z domain and optimisation algorithms. Three differentiators with different time constants and frequency bands are built and tested. The experimental results are in good agreement with theoretical values.
An improvement to the realization of the linear-phase IIR filters
is described. It is based on the rearrangement of the numerator
polynomials of the IIR filter functions that are used in the real-time
realizations proposed in literature. The new realization has better
total harmonic distortion when a sine input is used, and it has smaller
phase and group delay errors due to finite section length
A new implementation of an IIR digital filter transfer function is presented that is structurally passive and, hence, has extremely low pass-band sensitivity. The structure is based on a simple parallel interconnection of two all-pass sections, with each section implemented in a structurally lossless manner. The structure shares a number of properties in common with wave lattice digital filters. Computer simulation results verifying the low-sensitivity feature are included, along with results on roundoff noise/dynamic range interaction. A large number of alternatives is available for the implementation of the all-pass sections, giving rise to the well-known wave lattice digital filters as a specific instance of the implementation.
A technique for realizing linear phase IIR filters has been proposed by Powell and Chau. This technique contains a real-time realization of a noncausal transfer function using the following three procedures. 1) Approximate the IIR filter's impulse response to a finite length L. 2) Divide the input and output signal sequences of the IIR filter by this length L and perform a time reversal operation at each section in real time. 3) Realize a time-adjusted overlap-add for the divided section with length L by means of a two-path IIR filter. However, in 2), an L-sample delay is generated at each signal section, and in 3), an L + 1-sample delay is generated. Hence, in the entire system, a 3L + 1–sample delay is generated. In this article, the delay in 2) is discussed. It is shown that the delay in the system can be reduced by shortening the divided section length L of the signal sequence to 1 over an integer (shortened section length is also an integer). © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 83(3): 1–11, 2000
The paper deals with the design of digital differentiators and Hilbert transformers to be implemented as noncausal recursive systems. For this case suitable design methods will be presented and the resulting filters will be compared to well known equivalent nonrecursive ones.
The authors introduce a modular VLSI architecture for a novel
linear-phase FIR (finite impulse response) filter structure. The
proposed linear-phase FIR filters have a highly reduced number of
general multiplications per sample compared to conventional FIR filters.
The novel filter structure is based on an IIR subfilter whose infinite
impulse response is truncated into a finite one. Although the subfilter
has a nonlinear phase response, it can be made exactly linear by
reversing the data stream in time and using the same filter again.
Another possibility is to use a maximum-phase version of the FIR filter
in cascade. The choice between different realizations depends on the
filter specifications. In the general case the filter coefficients
cannot be represented with the simple shift-and-add procedure and the
time reversal technique should be used. The authors prefer the
maximum-phase FIR alternative
A technique using Jacobian elliptic functions is given which by removing a previous method's (Powell and Chau, 1991) double-zero constraint, yields improved designs of linear phase IIR filters
Introduces efficient linear-phase FIR (finite-impulse response)
filter structures using novel recursive subfilters. The novel filter
structure is based on subfilters whose filter coefficients are related
to each other recursively. A well-known structure that uses this method
is the recursive running sum. The use of recursive subfilters is
extended to cover the simple structures utilizing real and complex poles
of radius r . The proposed linear-phase FIR filter structures
result in computationally efficient implementations which require fewer
general multipliers than conventional filters. This structure is
beneficial for filters with narrow transition bands since the use of a
single pole pair near the transition region lowers the required
numerator order radically. Since the proposed filters have finite
impulse responses, they can be realized efficiently using block
processing
The problem of synthesis of recursive digital filters to give a desired pulse response over a specified interval is studied. Realizability conditions are stated and a linear design method is developed. Several design procedures that require only linear calculations are given for approximate realization of recursive filters. Finally, an error analysis of the techniques is made.
A real-time IIR filter structure is presented that possesses exact
phase linearity with 10~1000 times fewer general multiplies than
conventional FIR filters of similar performance and better magnitude
characteristics than equiripple or maximally flat group delay IIR
filters. This structure is based on a technique using local time
reversal and single pass sectioned convolution methods to realized a
real-time recursive implementation of the noncausal transfer function
H ( z <sup>-1</sup>). The time reversed section technique
used to realize exactly linear phase IIR filters is described. The
effects of finite section length on the sectional convolution are
analyzed. A simulation methodology is developed to address the special
requirements of simulating a time reversed section filter. A design
example is presented, with computer simulation to illustrate
performance, in terms of overall magnitude response and phase linearity,
as a function of finite section length. Nine example filter
specifications are used to compare the performance and complexity of the
time reversed section technique to those of a direct FIR implementation
A systematic method is outlined to realize an m th-order all-pass digital transfer function using only m multipliers as a cascade of first-order and/or second-order all-pass sections. The realization is based on the multiplier extraction approach in which the n th-order filter section is considered as a digital (n + 1) -pair of which n pairs of input and output terminal variables are constrained by n multipliers. The transfer matrix parameters of the digital (n + 1) -pair, containing only delays and adders, are first identified from which the realization is obtained by inspection. Both canonic and noncanonic realizations are derived. All realizations are then compared with regard to the effect of multiplication roundoff and hardware requirements.
Digital filters having exactly linear phase are usually realized by nonrecursive systems. However, the filter degree, and thus the expense, becomes rather large in the case of steep filters with high attenuation in the stopband. This paper deals with an alternative possibility using noncausal recursive systems After known methods are generalized, suitable algorithms are proposed and systematic errors are investigated. Finally, the method is compared with nonrecursive filtering.
A two-pass recursive scheme is proposed for realizing zero phase shift filters with arbitrary magnitude characteristics. The first pass is performed in forward time and the second in reverse time. The effect of initial and reverse time transients is discussed, and a scheme for quasi on-line adaptation is presented.
An analysis of linear phase IIR filters and a condition for perfect linear phase
- A Kurosu
- S Miyase
- S Tomiyama
- T Takebe
A. Kurosu, S. Miyase, S. Tomiyama, and T. Takebe, " An analysis of linear phase IIR filters and a condition for perfect linear phase, " in Proc. School Eng. Tokai Univ. Series J, vol. 39, 1999, pp. 37–42.