Wave Digital Filters Research Articles

An open-source project for wind instrument modeling using digital waveguides

Digital waveguide methods for time-domain modeling of acoustic structures comprised of interconnected piece-wise one-dimensional cylinders, cones and toneholes have been well reported in the past. In this work, an open-source Matlab class is presented that can be used to model arbitrary wind instrument air column structures, defined by length, radii and hole geometric parameters, using digital waveguide methods. Fractional-delay filtering can be enabled or disabled, with support for both Thiran and Lagrange filter types of specified order. Thermoviscous losses can be modeling using either a discrete-time least squares or a shelf-filter fit, both of arbitrary order. Three different tonehole model options are provided (two-port, three-port or wave digital filter). End boundary conditions include closed, anechoic, ideally open, open flanged and open unflanged. The class object can be used either in single-sample iterative contexts (for example, with an attached excitation model) or with arbitrarily-sized input signals (for example, to compute a reflection function or impedance). The modeled geometry can be constructed manually by iteratively adding segments or holes, or a complete geometry can be specified in a file. This project is bundled together with a parallel and previously reported open-source project for frequency-domain transfer matrix modeling of air columns (https://github.com/garyscavone/acmt).

A better SPQR‐tree decomposition of electrical circuits containing multiports and its application to wave digital emulation

SummaryWe report on a systematic method for the port‐wise decomposition of electrical networks. Our method exploits the SPQR‐tree decomposition as an algorithmic way for decomposing a given circuit into a maximal set of one‐port and multiports. We provide a solution for representing multiports by suitable replacement graphs, such that they are encapsulated not decomposed or processed by the SPQR‐algorithm. Contrary to current methods, our replacement graphs are not sophisticated but rather simple, which, in general, enhances the performance of the algorithm, when it comes to decomposing large electrical networks containing multiports. Some electrical devices, such as transistors or op‐amps, are commonly described in terms of their terminals rather than their ports. Thus, we cover the application of the template‐based approach to three‐terminal devices. Lastly, we make use of the new decomposition method for the emulation of a wave digital filter.

Deep learning-based wave digital modeling of rate-dependent hysteretic nonlinearities for virtual analog applications

Electromagnetic components greatly contribute to the peculiar timbre of analog audio gear. Indeed, distortion effects due to the nonlinear behavior of magnetic materials are known to play an important role in enriching the harmonic content of an audio signal. However, despite the abundant research that has been devoted to the characterization of nonlinearities in the context of virtual analog modeling over the years, the discrete-time simulation of circuits exhibiting rate-dependent hysteretic phenomena remains an open challenge. In this article, we present a novel data-driven approach for the wave digital modeling of rate-dependent hysteresis using recurrent neural networks (RNNs). Thanks to the modularity of wave digital filters, we are able to locally characterize the wave scattering relations of a hysteretic reluctance by encapsulating an RNN-based model into a single one-port wave digital block. Hence, we successfully apply the proposed methodology to the emulation of the output stage of a vacuum-tube guitar amplifier featuring a nonlinear transformer.

Wave Digital Models of Piezoelectric Transducers for Audio Applications

The number of consumer electronics devices integrating piezoelectric (PE) transducers as flat-panel loudspeakers has recently experienced a steep increase. Being very thin, in fact, PE transducers well cope with the miniaturization process that characterizes the market. However, at the same time, their reduced dimensions cause the sound pressure level to be poor at low frequencies, impairing the overall acoustic response. In this article, we derive novel efficient discrete-time models of PE transducers in the wave digital (WD) domain such that they can be integrated in the future into signal processing algorithms for audio output enhancement. WD filters are, in fact, making headway among audio digital signal processing techniques based on circuit equivalent models thanks to their efficiency, robustness, and accuracy. Starting from circuital representations of the piezo constitutive equations, we derive both lumped and distributed models. In particular, we show how it is possible to implement the frequency-dependent elements characterizing Mason’s model in the WD domain, contrary to what can be done using mainstream SPICE-like simulators. Such WD implementations come in handy for the simulation and fast prototyping of PE systems, paving the way toward the design of model-based digital signal processing algorithms for enhancing the transducer acoustic performance.

Digital models of analog circuits for musical audio production: A review of techniques and library for automated circuit solving

In music production, many recording and mixing engineers prefer to use analog equipment as a matter of perceptual preference. Digital models of analog circuits have the potential to achieve similar perceptual qualities as hardware without the drawbacks of cost, maintenance, and availability. Many different techniques of system modeling are used in music production software, ranging on a spectrum from “black-box modeling” to “white-box modeling.” In black-box modeling, the analog system is modeled as a processing block which maps an input signal to an output signal. Examples include the linear impulse response, adaptive filters, Volterra series, and Weiner-Hammerstein models. In white-box modeling, each individual component of the analog circuit is modeled as part of the overall system. Examples include wave digital filters, state-space modeling, and modified nodal analysis. Various other techniques exist on the spectrum between these two types, using some combination of each. One example is Virtual Analog Filtering based on the Topology Preserving Transform. Machine Learning techniques have also had an important role in advancing the accuracy of digital modeling. Lastly, the “Point to Point Library,” developed by the author, will be demonstrated. This MATLAB and C + + library performs automated circuit solving for modeling audio effects.

Loudspeaker virtualization–Part I: Digital modeling and implementation of the nonlinear transducer equivalent circuit

This manuscript, the first of a two-part series, presents a methodology for efficiently implementing an equivalent circuit of nonlinear loudspeakers in the discrete-time domain. This is a crucial step that will allow us to design new algorithms for loudspeaker virtualization in part II of this series. The presented implementation, in fact, is based on Wave Digital Filter (WDF) principles, which lead to a fully explicit model, in the sense that no iterative solvers are needed to compute the output signal. The proposed WDF is highly efficient and modular, since the reference circuit is modeled as a computable interconnection of input-output processing blocks. The accuracy of the implementation is confirmed by comparing it to a ground-truth SPICE simulation of the reference circuit and to measurements on real loudspeakers. This confirms that the proposed Wave Digital modeling approach can be reliably used for the rapid simulation of the transduction behavior of a loudspeaker (within the frequency limits of validity of the equivalent circuit), and can be employed to develop the digital preprocessing method for loudspeaker virtualization described in the second manuscript of this two-part series.

Multidomain modeling of nonlinear electromagnetic circuits using wave digital filters

SummaryAccurate models of electromagnetic systems can be derived by coupling electric and magnetic equivalent circuits together. The different nature of such physical domains constitutes a big challenge that puts a continuous strain on software simulators. To face this problem, a simulation approach based on Wave Digital Filters (WDFs) is proposed in this manuscript. The method is employed to solve nonlinear electromagnetic systems containing complex magnetic equivalent circuits while maintaining the modularity of the electric and magnetic subsystems. The nonlinearities can be locally handled, enabling the possibility to choose a dedicated model for each one of them. The proposed algorithm is a hierarchical generalization of the Scattering Iterative Method, which has shown, over the past few years, promising performance for the simulation of large nonlinear circuits. In addition, the method constitutes a further step towards the development of novel general‐purpose circuit simulators based on WDF principles. In a comparison with mainstream circuit simulation software, the proposed approach turns out to be considerably faster and thus particularly promising for parametric analyses of electromagnetic systems.

Wave-digital filter circuits for single-chip 4-D light field depth-based enhancement

In four-dimensional (4-D) light field (LF) processing, 4-D linear shift-invariant filters having hyperplanar passbands are used for depth-based scene enhancement. In this paper, two low-sensitivity and low-complexity field programmable gate array (FPGA)-based digital hardware architectures for 4-D hyperplanar filters are proposed for on-chip real-time processing of LFs. Both 4-D filters are designed exploiting resonant properties of multi-dimensional passive prototype networks, and are realized as wave digital filters (WDFs). The two 4-D WDF realizations are implemented as raster-scanned processing architectures on a Xilinx Virtex 6 Sx35 FPGA with a real-time clock speed of up to 33 MHz. This corresponds to a real-time throughout of 16.8 LFs/s for an LF of size $$11 \times 11 \times 128 \times 128$$ .

Wave Digital Modeling and Implementation of Nonlinear Audio Circuits With Nullors

The nullor is a theoretical two-port element suitable to model several multi-port devices common in audio circuitry, such as ideal operational amplifiers, operational transconductance amplifiers, and transistors operating in linear regime. In this manuscript, we present an approach for the Wave Digital (WD) modeling and implementation of circuits with multiple nullors. In particular, we propose an approach to compute scattering matrices of WD topological junctions absorbing nullors that is less computationally demanding than the techniques available in the literature on WD Filters. We show that the proposed approach turns out to be particularly useful when simulating nonlinear circuits through the Scattering Iterative Method (SIM), a WD fixed-point method recently developed for the solution of circuits with multiple nonlinearities, because it requires a frequent update of the scattering matrices. We also provide a novel convergence analysis of SIM applied to WD structures composed of multiple one-port nonlinear elements and a topological junction absorbing nullors. In order to verify the effectiveness of the proposed methodology, we discuss some WD implementations of analog audio circuits with multiple diodes and opamps, including a precision half-wave rectifier and a wave folder circuit.

Vector Wave Digital Filters and Their Application to Circuits With Two-Port Elements

Wave Digital Filters (WDFs) turn circuits into networks of input-output relationships that can be computed in an explicit fashion. This is done through a linear port-wise mapping of Kirchhoff variables into pairs of incident-reflected waves introducing one scalar free parameter per port, called reference port resistance. Parameters are then used to eliminate the implicit equations relating wave variables, referred to as delay-free-loops. Unfortunately, this methodology can only be applied under strong linearity and topological conditions. This manuscript presents an extension of the WDF formalism involving a novel “cross-port” vector definition of waves, whose reference resistance is a matrix of free parameters. This generalization greatly simplifies the WDF implementation of circuits with two-port elements, such as operational amplifiers. It allows us to derive wave-based descriptions of elements such as nullors, for which no scattering relation is available in the literature. Moreover, it enables a full adaptation of a wide class of two-port elements, thus avoiding the delay-free-loops that would otherwise form in traditional WDFs. This new formalism allows us to implement a wider range of circuits with two-port elements in a modular fashion, since the topology and the elements can be modeled independently.

Optimal design of lattice wave digital fractional‐order Butterworth filter

SummaryThis letter presents the design of digital fractional‐order Butterworth filter (DFOBF) of order (n+α), where n is an integer, and α ∈ (0,1), from the perspective of optimal realization. The magnitude–frequency characteristic of the DFOBF is optimally modeled using the computationally efficient lattice wave digital filters (LWDFs). Design examples for the third‐ and fifth‐order LWDF‐DFOBFs with various values of n, α, and cut‐off frequencies are presented.

Hardware Emulation of a Biorthogonal Wavelet Transform-Based Heart Rate Monitoring Device

In this work, a low-power, area-efficient, and high-performance electrocardiogram (ECG) detector that can be used in modern implantable cardiac pacemaker systems is proposed. Qualitative and quantitative analysis indicates that the Biorthogonal-3.1 wavelet transform combined with adaptive slope prediction is best suited to detect various ECG peaks by effectively denoising various artifacts present in an ECG signal. A demand-based wavelet filter bank (WFB) architecture consisting of a series combination of three lowpass filters is the primary reason for the low-power requirements of the circuit. Power consumption and area requirements of the circuit are further reduced by realizing the lowpass filters using lattice wave digital filter realization that reduces the requirement of delay elements, and multipliers count by 80% and 75%, respectively. The proposed ECG denoising scheme using the biorthogonal 3.1 wavelet transform generates a smoother denoised ECG wave by retaining the important morphology of an ECG signal. Adaptive slope prediction criterion-based ECG processing and detection schemes achieve a high detection accuracy of 99.90%, with the lowest error of 0.002%. The proposed scheme is capable of distinguishing between a normal ECG, paced ECG, arrhythmic ECG, and low and high-resolution ECG. The proposed algorithm is then emulated on the Xilinx Ⓡ-Virtex Ⓡ-7 FPGA hardware platform. Power consumption, area, delay, and switching energy of the proposed ECG processing scheme are reduced when compared to existing ECG detection schemes and are found to be of $0.493~\mu \text{W}$ , 1.1 mm2, 10 ns, and $4.93~\mu \text{J}$ , respectively.

Design and FPGA implementation of lattice wave digital notch filter with minimal transient duration

In this study, the design and field-programmable gate array (FPGA) implementation of the digital notch filter with the lattice wave digital filter (LWDF) structure is presented. For reducing the initial signal transient, the variable notch bandwidth filter is designed. During the initial samples, the notch filter has a wide bandwidth in order to diminish signal transient. As time moves forward, the notch bandwidth reduces to attain the possible minimum width. This results in minimised transient duration notch filter with a sufficiently high-quality factor. Previously, the IIR structure has been used for implementing the time varying bandwidth notch filter. Such a filter requires two variable coefficients for varying the notch width with time. The advantage of using a LWDF structure is that only one coefficient has variable values to vary the notch width with time. Therefore, the number of memory locations required to implement the proposed design is reduced by half. Moreover, the LWDF is less sensitive to the word-length effects. Thus, the proposed lattice wave digital notch filter (LWDNF) produces better results compared to the existing literature in terms of error analysis. The suggested LWDNF is then implemented on a field-programmable gate array using a Xilinx system generator for the DSP design suite.

Design of efficient fractional operator for ECG signal detection in implantable cardiac pacemaker systems

SummaryA low power and high‐performance digital electrocardiogram (ECG) detector has become a basic requirement in modern implantable cardiac pacemakers. A fractional operator‐based digital ECG detector for modern pacemaker systems is proposed in this work. Instead of conventional thresholding, an adaptive slope prediction threshold is utilized for the detection of ECG peaks. A stochastic search‐based algorithm, namely, cuckoo search algorithm, is used to design an optimal fractional operator that is used for ECG denoising. It has been found that the proposed adaptive slope prediction threshold increases the QRS complex detection performance. A low detection error rate (DER) ranges from 0.01% to 0.56%, positive predictivity (P+) ranges from 99.32% to 99.98%, sensitivity (Se) ranges from 99.45% to 99.98%, and a detection accuracy (Acc) ranges from 99.43% to 99.96% for different databases are achieved for the proposed ECG detector, which is better compared with the existing ECG detectors. The proposed design of fractional order operator based on the lattice wave digital filter (LWDF) requires a minimum number of the multipliers for its structural realization.

Analytical modeling of laminated composite plates using Kirchhoff circuit and wave digital filters

A physical-based numerical algorithm using Kirchhoff circuit is detailed for modelling the free vibration of moderate thick symmetrically laminated plates based on the first order shear deformation theory (FSDT). With the help of multidimensional passivity of analog circuit and nonlinear optimization solvers, the philosophy gives rise to a nonlinear programming (NLP) model that can apply further to explore stability characteristics and optimum performance of the resultant multidimensional wave digital filtering network representing the FSDT plate. Various optimization methods exploiting gradient-based and direct search methods are adopted with efficient broad search power to tackle the NLP model. As a result, the necessary Courant-Friedrichs-Levy stability criterion can be fully satisfied at all time with least restriction on the spatially discretized geometry of the scattering problem. With full stability guaranteed, the waveform is analyzed by the power cepstrum for spectra peaks detection, which has led to more accurate estimate of various vibration effects in predicting nature frequencies with different fiber orientations, stacking sequences, stiffness ratios and boundary conditions. These results have shown in excellent agreement with early published works based on the finite element solutions of the high-order shear deformation theory and other well known numerical techniques.

Design and FPGA implementation of lattice wave fractional order digital differentiator

This paper deals with the design and FPGA implementation of a fractional order digital differentiator. It is realized using computationally efficient lattice wave digital filter (LWDF) which requires minimum multipliers in comparison to the direct form structure. A nature inspired ant lion optimization (ALO) algorithm is utilized to compute optimal coefficients of the LWDF based digital differentiator. The proposed LWDF based digital differentiator is compared with the existing literature in terms of magnitude response, percentage magnitude error and root mean square magnitude error. LWDF structure comprises of constant coefficient multipliers, adders and delay units which are implemented on FPGA with the help of Xilinx system generator for DSP design tool. For efficient implementation of multipliers, multiplier-less logic through digit recoding techniques such as canonic signed digit (CSD) representation and radix-2r encoding are described through Verilog HDL and incorporated in the implementation model as black box. Post-implementation results show that implementation based on the radix-2r encoding is found to be more efficient than that of the CSD encoding. The design and implementation results are also reported to highlight the improvements.

Nonlinear Circuit Simulation by Means of Alfred Fettweis' Wave Digital Principles

A dependable circuit simulation plays an elementary part in the process of commercial circuit design. However, when Alfred Fettweis invented the concept of Wave Digital Filters (WDFs) back in the early 1970s, that was clearly not the case as tools like SPICE hadn't been published yet. But even though Fettweis' new WDFs were able to provide an accurate digital model of analogue reference circuits, they were mostly perceived as an elaborated technique for digital filter design. In this article we provide an insight into Wave Digital Filters with a focus on nonlinear circuit simulation, highlighting assets and drawbacks of the method. In particular, recent results that extend the Wave Digital concept towards a general circuit simulation strategy with interesting properties are presented.

Robust Digital Filter Structures: A Direct Approach

One of the many contributions of Prof. Fettweis was the invention of wave digital filters. These filters are obtained from classical RLC filters, in particular doubly terminated lossless two-ports, by using some transformations. Namely, the voltages and currents in the circuit elements are transformed into wave-variables and then a bilinear transformation is performed. If the wave transformation is performed appropriately it results in a realizable digital filter structure which furthermore enjoys a number of robustness properties such as low passband sensitivity, low roundoff noise, and freedom from limit cycle oscillations. Prof. Fettweis and his colleagues also showed that these properties are due to the inheritence of passivity properties from the continuous-time domain into the digital filter domain. Subsequent to this landmark work, a number of researchers worked on the problem of obtaining robust digital filter structures without starting from continuoustime circuits. One of these is the structurally bounded or structurally passive class of digital filters. These structures are based on inducing structural passivity directly into the implementation and are therefore simpler, both conceptually and from a practical viewpoint. They are also more general and lead to new structures which have no natural connection to electrical circuits. This paper gives an overview of some of these developments.

Design and FPGA implementation of 11th order Efficient IIR Wavelet Filter Banks with Approximate Linear-phase

In this paper. Bireciprocal Lattice Wave Digital Filters (BLWDFs) are utilized in an approximate linear-phase in pass-band design of order IIR wavelet filter banks (FBs). These filter banks are efficiently designed by replacement one of branches for (BLWDFs) by only a unit delay. The coefficients of the designed filter are achieved by simulating the IIR response suggested in [1]. The design is first simulated using Matlab programming in order to investigate the resulting wavelet filter properties and to find the suitable wordlength for the BLWDFs coefficients. FPGA implemtation of the proposed IIR wavelet filter bank is also achived for three levels with less complexity and high operating frequency.

Generalized Wave Digital Filter Realizations of Arbitrary Reciprocal Connection Networks

In this paper, an existing approach for modeling and efficiently implementing arbitrary reciprocal connection networks using wave digital (WD) scattering junctions based on voltage waves is extended to be used in a broader class of WD filters (WDFs) based on different kinds of waves. A generalized wave definition which includes traditional voltage waves, current waves, and power-normalized waves as particular cases is employed. Closed-form formulas for computing the scattering matrices of the junctions are provided. Moreover, the approach is also extended to the family of Biparametric WDFs, which have been recently introduced in the literature.