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IIR Cascaded-Resonator-Based Filter Design for Recursive Frequency Analysis
Abstract
The cascaded-resonator (CR)-based filter structure has been proposed in previous works as a convenient approach to the dynamic harmonic analysis. This CR filter can be with multiple resonator (MR) poles or ones dispersed properly. In all previous papers the resulting filters of the CR structure are finite-impulse-response (FIR)-type. In this letter, the aim is to use the infinite-impulse-response (IIR) CR-based filters for a full spectrum estimation. Thanks to the linearized model, including stability conditions, filter coefficients of the IIR filter transfer function are derived through the minimax optimization using a linear programming. This way, the frequency responses of the harmonic analyzer can be improved in accordance with actual requirements, sometimes even without any increase of computational burden.
... Resonator-based filter banks, based on the structure of parallel resonators with common feedback, are an example of complex filter banks [9][10][11]. Multiple-resonator (MR)based filters [12,13] and their more general version, cascaded-resonator (CR)-based filters [14][15][16], have been proposed for usage in the spectrum analysis of dynamic signals. In comparison to single-resonator-based analyzers, analyzers with a higher multiplicity of resonators provide lower side lobes. ...
... In comparison to single-resonator-based analyzers, analyzers with a higher multiplicity of resonators provide lower side lobes. In [16], infinite-impulse-response (IIR)-type CRbased filter banks were used as a computationally more efficient solution, rather than the finite-impulse-response (FIR)-type. Even more, through simultaneous optimization of the frequency responses of the whole harmonic bank, the same shapes of all harmonic frequency responses were assured, thanks to symmetrical pole placement. ...
... In this paper, the approach used in [16], with certain modifications, is used to design online adaptive filter banks. Bandpass filters of arbitrary width can be obtained by connecting filters of the appropriate number of adjacent harmonics of the primary filter bank. ...
The use of a filter bank of IIR filters for the spectral decomposition and analysis of signals has been popular for many years. As such, a new filter-bank resonator-based structure, representing an extremely hardware-efficient structure, has received a good deal of attention. Recently, multiple-resonator (MR)-based and general cascaded-resonator (CR)-based filters have been proposed. In comparison to single-resonator-based analyzers, analyzers with a higher multiplicity of resonators in the cascade provide lower side lobes and a higher attenuation in stopbands. In previous works, it was shown that the CR-based filter bank with infinite impulse response (IIR) filters, which is numerically more efficient than one with finite impulse response (FIR) filters, is suitable for dynamic harmonic analysis. This paper uses the same approach to design complex digital filter banks. In the previous case, the optimization task referred to the frequency responses of harmonic filters. In this work, the harmonic filters of the mother filter bank are reshaped so that the frequency response of the sum (or difference, depending on the parity of the number of resonators in the cascade) of two adjacent harmonic filters is optimized. This way, an online adaptive filter base can be obtained. The bandwidth of the filters in the designed filter bank can be simply changed online by adding or omitting the output signals of the corresponding harmonics of the mother filter.
... This results in poor performance under a low signal-to-noise ratio for detection and estimation. Recently, for the purpose to improve the performance of LFM signal detection and estimation, several researchers have associated WVD with the FrFT, LCT, and OLCT, respectively [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Results show that such transforms exploit the advantages of both transforms, which is why we call them hybrid transforms. ...
... Urynbassarova et al. presented the WVD associated with the instantaneous autocorrelation function in the LCT domain, named WL, which has elegance and simplicity in marginal properties and affine transformation relationships compared to the WVD [17]. Similar to this in [27] Xin and Li proposed a new definition of WVD associated with LCT, and its integration form, which estimates two phase coefficients of LFM signal simultaneously and effectively suppresses cross terms for multi-component LFM signal. In [19] introduced the WVD association with the OLCT (WVD-OLCT), which is a generalization of the WVD-LCT and its special cases. ...
... However these cross terms become troublesome if the frequency rate of one component approaches other. This drawback of AF gave rise to a series of different classes of time-frequency representation tools (see [20][21][22][23][24][25][26][27]). In Ref. [28], authors used fractional instantaneous auto-correlation ω t þ k τ 2 À ...
... In accordance with the prevailing trends in works dealing with this issue, in the initial works [13,16,22,23,25,26] the resulting filters of CR structures were of the FIR type. Later, in [27], IIR filters were used, which represent a computationally more efficient solution [28,29]. The unit characteristic polynomial of the transfer function is replaced by the optimized one. ...
... The task of optimization is to design a filter Bz ðÞ =Az ðÞwhere the order of the characteristic polynomial Az ðÞis ÀÁ . In order to make the notation as simple and short as possible, let us form a virtual transfer function so that in each bandwidth centered in mf 1 with width of f 1 , i.e., for Similarly, a virtual unique desired transfer function in an angular frequency ω has the following form [27]: ...
It is well known that recursive algorithms for harmonic analysis have better characteristics in terms of monitoring the change of the spectrum in comparison to methods based on the processing of blocks of consecutive samples, such as, for example, discrete Fourier transform (DFT). This property is particularly important when applying spectral estimation in real-time systems. One of the recursive algorithms is the resonator-based one. The approach of the parallel cascades of multiple resonators (MR) with the common feedback has been generalized as the cascaded-resonator (CR)-based structure for recursive harmonic analysis. The resulting filters of the CR structure can be finite impulse response (FIR) type or the infinite impulse response (IIR) ones as a computationally more efficient solution, optimizing the frequency responses of all harmonics simultaneously. In the case of the IIR filter, the unit characteristic polynomial present in the FIR filter is replaced with an optimized characteristic polynomial of the transfer function. Such a change does not lead to an increase in computing requirements and changes only the resonator gain values. By using a conveniently linearized iterative algorithm for stability control purpose, based on the Rouche’s theorem, the iterative linear-programming-based or the constrained linear least-squares (CLLS) optimization techniques can be used.
... The ABC method is utilised as an optimizer with the numerous design procedures related to filter design and yields decent results [23]. Few author, proposed work for digital IIR filter design [1]. Based on the above literature review, it has been observed that most existing works treat digital filter design as an objective problem. ...
This paper presents a new method for designing matched digital filters with discrete valued coefficients. The fuzzy particle swarm optimization vector quantization (FPSOVQ) has been applied to obtain the optimum codebook in design of matched wavelet function. Abelian group has been used to extract the similarity present in the input voiced signal. Fuzzy particle swarm optimization (FPSO) process is used to find approximate ideal vector quantization (VQ) codebook to be carried out for compression of data. FPSOVQ scheme utilises features of fuzzy inference method (FIM) and expert particle swarm optimization (PSO). The generated codebook consists of set of highly ideal features, which are considered as the filter coefficients. These coefficients are used in the designing of the filter. All of the phonemes in the American English language were included in the 30 sentences that were chosen from the IEEE database. The sentences were originally down-sampled from 25 kHz to 8 kHz. The magnitude responses of each filter have been drawn, which indicates the characteristics of the filter. A comparison has been provided using energy compaction ratio as a parameter to judge the performance of matched designed filter with db4 filter. Experimental results show the advantage of the developed algorithm as the average value of energy compaction ratio for sampled voice signals is 2.8046 times lower for matched designed filter.
Although voiced speech signals are physical signals which are approximately harmonic and electric power signals are true harmonic, the algorithms used for harmonic analysis in electric power systems can be successfully used in speech processing, including in speech enhancement, noise reduction, speaker recognition, and hearing aids. The discrete Fourier transform (DFT), which has been widely used as a phasor estimator due to its simplicity, has led to the development of new DFT-based algorithms because of its poor performance under dynamic conditions. The multiple-resonator (MR) filter structure proposed in previous papers has proven to be a suitable approach to dynamic harmonic analysis. In this article, optimized postprocessing compensation filters are applied to obtain frequency responses of the transfer functions convenient for fast measurements in dynamic conditions. An optimization design method based on the constrained linear least-squares (CLLS) is applied. This way, both the flatness in the passband and the equiripple attenuation in the stopband are satisfied simultaneously, and the latency is reduced.
Phasor measurement units (PMU) are increasingly used in electrical power transmission networks, to maintain stability and protect the network. PMUs accurately measure voltage, phase, frequency, and rate of change of frequency (ROCOF). For reliability, it is desirable to implement a PMU using an FPGA. This paper describes a novel algorithm, suited to implementation in an FPGA and based on a simple PMU block diagram. A description of its realization using low hardware complexity infinite impulse response (IIR) filters is given. The IEC/IEEE standard 60255-118-1:2018 Part 118-1: Synchrophasor measurements for power systems, describes “reference” Finite Impulse Response (FIR) filters for implementing PMU hardware. At the 10 kHz sampling frequency used for our implementation, each “reference” FIR filter requires 100 multipliers, while an 8th order IIR filter only requires 12 multipliers. This paper compares the performance of different order IIR filter-based PMUs with the performance of the same PMU algorithm using the IEC/IEEE FIR reference filter. The IIR-based PMU easily satisfies all the requirements of IEC/IEEE standard and has a much better out of band signal rejection performance than a FIR-based PMU. Steady state errors for a rated voltage ± 10% and a rated frequency ± 5 Hz are < 0.000001% for total vector error (TVE) and < 1 µHz for frequency, with a latency of two mains cycles.
Digital filters for recursively computing the discrete Fourier transform (DFT) and estimating the frequency spectrum of sampled signals are examined, with an emphasis on magnitude-response and numerical stability. In this tutorial-style treatment, existing recursive techniques are reviewed, explained and compared within a coherent framework; some fresh insights are provided and new enhancements/modifications are proposed. It is shown that the replacement of resonators by (non-recursive) modulators in sliding DFT (SDFT) analyzers with either a finite impulse response (FIR), or an infinite impulse response (IIR), does improve performance somewhat; however, stability is not guaranteed as the cancellation of marginally stable poles by zeros is still involved. The FIR deadbeat observer is shown to be more reliable than the SDFT methods, an IIR variant is presented, and ways of fine-tuning its response are discussed. A novel technique for stabilizing IIR SDFT analyzers with a fading memory, so that all poles are inside the unit circle, is also derived. Slepian and sum-of-cosine windows are adapted to improve the frequency responses for the various FIR and IIR DFT methods.
The classical “frequency sampling method” (1975) based
on the direct utilization of the Lagrange interpolation technique is
extended in a natural way to a rather efficient Hermite interpolation
scheme. This structure is a new extension related to the resonator-based
digital filter family introduced by M. Padmanabhan et al. (1996). The
development resulted in a system having good properties if measurement
of signals consisting of sinusoidal components is to be performed
The multiple-resonator (MR) filter structure has been proposed in previous works as a convenient approach to the dynamic harmonic analysis. Later, the cascaded-resonator (CR)-based filters for the off-nominal frequency harmonics analysis, MR-based like, has been proposed. In this article, the aim is to use the infinite-impulse-response (IIR) CR-based filters for a single phasor estimation, instead of a dead-beat observer (i.e. the FIR filter) for a full spectrum estimation. Filter coefficients of the IIR filter transfer function are derived through the minimax optimization. This way, both the flatness in the passband and the attenuation in the stopband are significantly improved simultaneously.
Harmonics filtering for nominal frequencies is well known and plain task. Discrete Fourier transform (DFT), Taylor Fourier transformation (TFT) and cascaded integrator comb (CIC) filters are only some of widely used digital signal processing techniques. However, in case of frequency deviation, this task becomes much more complex. The reason is that harmonic frequency deviation from the nominal one is proportional to the harmonic order. This paper proposes the cascaded-dispersed-resonators (CDR) based filter technique for off-nominal frequency harmonics analysis, similar to the multiple-resonator (MR) based filters. The technique previously used for synthesis of the so called quasi MRbased filters is applied. Unlike the previous intention to design filters with frequency responses as much close to true MR filters as possible, that caused putting poles in the cascade as much close to each other as possible, here is necessary to place poles with a displacement width necessary to cover whole band of the fundamental frequency deviation proportional to the orders of the harmonics.
This Letter proposes a modified method for the dynamic harmonic analysis of signals with non-stationary harmonics, which is based on the previously proposed multiple-resonator observer structure. An optimisation technique is applied to reshape frequency responses of the basic transfer functions. This method, by using a parallel structure with common feedback, is very robust and, in addition, allows a reduction of the computational burden. The proposed method has been investigated for up to 64 harmonics, using LabView software package, under different conditions, and confirmed to be valuable and efficient tool for signal components estimation.
This paper presents an improved approach to the recently proposed multiple-resonator-based method for the harmonic analysis that has been provided in the previous papers. Previously, two inherent particular cases have been considered. In these cases, reference points in which estimation is performed are located either in the middle or at the end of the observation interval. The first case exhibits a good noises and unwanted harmonics attenuation but possesses a large delay time. In the second case, the filters are able to form a zero-flat phase response about the operation frequency and hence able to provide instantaneous estimates, but with large overshoots caused by resonant frequencies at the edges of the passband, and the high level of the sidelobes, that also makes it susceptible to interharmonics and noise interference. The aim of this paper is to propose a compromised solution provided by the tradeoff between those opposite requirements by shifting this point along the observation interval. This way the frequency responses of the estimator are reshaped. A maximally flatness of the frequency response in the operation harmonic frequency is kept in all cases, but only locating the reference point in a fraction around the center of the observation interval provides flat-top frequency responses. The effectiveness of the proposed estimation technique is shown through simulations.
It has been recently shown that the multiple-resonator-based observer is convenient for so called Taylor-Fourier transform (TFT) estimating not only Fourier components but also their derivatives. In order to provide lower side lobes and to avoid leakage and picket fence effects, the estimator with higher multiplicity of resonators should be used. Although the development of the estimators with higher-multiplicity resonators is simple in general, the design process could be time consuming since the higher order derivatives of the estimator characteristic polynomial are to be calculated. This paper presents a filter design technique based on quasi multiple resonators that approximates the exact multiple resonators technique. By using the quasi multiple-resonator-based harmonic analysis nearly the same frequency responses of the transfer functions can be obtained but with a much simpler design. Even more, in this paper closed form solutions for resonator gain coefficients are given in the case of a dead-beat observer.
In this paper, a new approach to the harmonic estimation based on the linear transformation named Taylor-Fourier transform has been proposed. A multiple-resonator-based observer structure has been used. The output taps of the multiple resonators may fix not only the complex harmonic values but also, according to the actual resonator multiplicity, their first, second, third, fourth, and so on, derivatives at the corresponding frequency. The algorithm is recursive, which allows its implementation in embedded environment with limited memory. The estimation technique is suitable for application in a wide range of frequency changes, transient conditions, and interharmonics presence, with benefits in a reduced complexity and computational effort. To demonstrate the performance of the developed algorithm, computer simulated data records are processed.
A new dynamic harmonic estimator is presented as an extension of the fast Fourier transform (FFT), which assumes a fluctuating complex envelope at each harmonic. This estimator is able to estimate harmonics that are time varying inside the observation window. The extension receives the name “Taylor-Fourier transform (TFT)” since it is based on the McLaurin series expansion of each complex envelope. Better estimates of the dynamic harmonics are obtained due to the fact that the Fourier subspace is contained in the subspace generated by the Taylor-Fourier basis. The coefficients of the TFT have a physical meaning: they represent instantaneous samples of the first derivatives of the complex envelope, with all of them calculated at once through a linear transform. The Taylor-Fourier estimator can be seen as a bank of maximally flat finite-impulse-response filters, with the frequency response of ideal differentiators about each harmonic frequency. In addition to cleaner harmonic phasor estimates under dynamic conditions, among the new estimates are the instantaneous frequency and first derivatives of each harmonic. Two examples are presented to evaluate the performance of the proposed estimator.
This paper presents a method for the frequency domain design of
infinite impulse response (IIR) digital filters. The proposed method
designs filters approximating prescribed magnitude and phase responses.
IIR filters of this kind can have approximately linear-phase responses
in their passbands, or they can equalize magnitude and phase responses
of given systems. In many cases, these filters can be implemented with
less memory and with fewer computations per output sample than
equivalent finite impulse response (FIR) digital filters. An important
feature of the proposed method is the possibility to specify a maximum
radius for the poles of the designed rational transfer function.
Consequently, stability can be guaranteed, and undesired effects of
implementations using fixed-point arithmetic can be alleviated by
restricting the poles to keep a prescribed distance from the unit
circle. This is achieved by applying Rouche's theorem in the proposed
design algorithm. We motivate the use of IIR filters with an unequal
number of poles and zeros outside the origin of the complex plane. In
order to satisfy simultaneous specifications on magnitude and phase
responses, it is advantageous to use IIR filters with only a few poles
outside the origin of the z-plane and an arbitrary number of zeros.
Filters of this type are a compromise between IIR filters with optimum
magnitude responses and phase-approximating FIR filters. We use design
examples to compare filters designed by the proposed method to those
obtained by other methods. In addition, we compare the proposed general
IIR filters with other popular more specialized structures such as FIR
filters and cascaded systems consisting of frequency-selective IIR
filters and phase-equalizing allpass filters
This paper presents a new approach to the frequency domain design
of infinite impulse response (IIR) digital filters with arbitrary
magnitude and phase responses and a constraint on the maximum pole
radius. The proposed method uses the polynomial coefficients of the
filter's transfer function as design parameters which provides better
solutions in most cases than other parameterizations of the transfer
function. An arbitrarily specified constraint on the pole radii can be
satisfied using the implications of Rouche's theorem. Using these
results we modify a standard method for the solution of nonlinear least
squares problems to incorporate this maximum pole radius constraint
which we consider to be important for implementing the filter with
finite wordlength. A design example demonstrates the superiority of the
proposed method over two other methods considering the same design
problem
In this brief, new composite polyphase filter banks are presented
for the implementation of the recursive Fourier transformation and some
of its generalizations. These structures can be operated both in sliding
window and block recursive modes with a computational complexity of the
order of the fast algorithms. The parallelization applied enables very
high speed and also a considerably higher sampling rate. Based on this
structure almost all issues of the so-called transform-domain digital
signal processing can be reconsidered
In this work, an iterative linear programming approach is
presented to design stable IIR digital filters with prescribed magnitude
and phase responses. At each iteration, the complex error of the
frequency response is transformed into a linear form by treating the
denominator polynomial obtained from the preceding iteration as a part
of the weighting function, and the poles restricted inside the unit
circle by using a set of linear constraints. After solving the standard
linear programming problem at each iteration, the design algorithm
converges to the minimax solution. Design examples demonstrate that our
method provides better design results than the conventional linear
programming method
A technique is presented whereby a class of recursive digital filters can be designed to approximate simultaneously given magnitude and linear phase characteristics. The underlying approach is to linearize the inherently nonlinear approximation problem, and thereby use linear programming to carry out the approximation. The linear phase is specified in terms of a desired constant group delay. Therefore, the constraints of the linear program become a function of desired group delay. Thus the design algorithm consists of carrying out a univariant search within a range of group delay values to obtain a minimum error of approximation. Several examples of design are presented to illustrate the usefulness of the method.