Wave Digital Filters

2015, Proceedings of the International Conference on Digital Audio Effects (DAFx-15)

Key finding: Introduces a Modified-Nodal-Analysis-derived method that systematically derives WDF adaptors for circuits with complicated topologies and multiport elements (e.g., transformers, controlled sources). This significantly... Read more

RESOLVING GROUPED NONLINEARITIES IN WAVE DIGITAL FILTERS USING ITERATIVE TECHNIQUES

by Michael Jørgen Olsen and

2016, Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16)

Key finding: Proposes iterative zero-finding techniques, specifically Newton's method variants, to solve grouped nonlinearities within WDF structures. This advancement allows for the resolution of multiple and multiport nonlinear elements... Read more

Toward nonlinear wave digital filters

by Augusto Sarti

2016, IEEE Transactions on Signal Processing

Key finding: Develops an extension of classical WDF principles with a novel class of dynamic waves and multiport adaptors capturing intrinsic nonlinear element dynamics. This generalizes the nonlinear mutator concept, allowing nonlinear... Read more

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2. How can digital filters be designed and optimized to ensure stability and performance, particularly for two-dimensional and IIR filters?

This theme investigates methodologies to design stable and efficient digital filters, with emphasis on multi-dimensional (2D) recursive filters and IIR filters, which are known for their stability challenges and nonlinear phase characteristics. Ensuring filter stability while optimizing for narrow transition widths, minimal group delay deviations, and phase linearity is crucial for applications ranging from image processing to biomedical signal enhancement. Constrained optimization frameworks and transformation techniques provide structured solutions to these complex design requirements.

DESIGN TECHNIQUE FOR A CLASS OF STABLE TWO -DIMENSIONAL RECURSIVE DIGITAL FILTERS

by Anthony Constantinides

2018

Key finding: Introduces a design technique for zero-phase, two-dimensional recursive digital filters with guaranteed stability by employing a two-variable reactance function transformation applied to a 1D prototype. Cascading four... Read more

A Nonlinear Optimization Design Algorithm for Nearly Linear-Phase 2D IIR Digital Filters

by abdussalam s omar

2023, Signals

Key finding: Presents a nonlinear constrained optimization framework using interior-point methods to design nearly linear-phase 2D IIR filters with a separable denominator. The method minimizes group delay deviation in the passband under... Read more

Analysis of IIR Filters by Interval Response

by Vinicius Borges

2025, 2020 Moscow Workshop on Electronic and Networking Technologies (MWENT)

Key finding: Proposes an interval size approach to analyze frequency response deviations of IIR filters under finite precision coefficient quantization, particularly for fixed-point implementations. By calculating bounds on frequency... Read more

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3. What advanced computational and architectural techniques can improve digital filter performance in hardware implementations?

This theme centers on optimizing digital filter implementations leveraging mathematical algorithms (e.g., Winograd method), number systems (Residue Number System), and hardware design strategies targeting speed, resource efficiency, and power consumption. These approaches aim to reduce the computational complexity inherent in convolution and filtering operations, crucial for applications such as image processing and convolutional neural networks on FPGA platforms.

Digital Filter Architecture Based on Modified Winograd Method F(2× 2, 5× 5) and Residue Number System

by Andrei Tchernykh

2023, IEEE Access

Key finding: Combines a modified Winograd convolution method with Residue Number System (RNS) arithmetic to construct two-dimensional digital filters with 5×5 masks. This approach reduces multiplication counts while enabling parallel... Read more

A Novel Design and FPGA Implementation of Filters Adapted Using LMS Variants

by Ravi Kumar

2025

Key finding: Proposes an area-efficient adaptive filter architecture that eliminates registers for delayed inputs to save silicon area without power or delay penalty. Implements various LMS algorithm variants on multiple FPGA families,... Read more

Design of digital filters from LC ladder networks

by Anthony Constantinides

2018

Key finding: Develops element-by-element linear voltage-current transformations from analog LC ladder networks to digital domains, ensuring digital filters preserve transfer functions under bilinear transformations and maintain stability.... Read more

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All papers in Wave Digital Filters

Wave Digital Filter Adaptors for Arbitrary Topologies and Multiport Linear Elements

by Jonathan Abel

2024

We present a Modified-Nodal-Analysis-derived method for developing Wave Digital Filter (WDF) adaptors corresponding to complicated (non-series/parallel) topologies that may include multiport linear elements (e.g. controlled sources and... more

Denoting the length-6 vectors of external incident and reflected waves asa = [aa,...,ar]" and b = [ba,.. ., or] and the length-12 vectors of internal incident and reflected waves as a; = [a1,...,@12]' and b; = [b1,...,b12]", we apply the stamp pro- cedures of §4. The matrix resulting from the stamp procedure (Ta- ble 2) is given in Fig. 7. The compatibility matrix (8) relating a; and b; is found according to the next stamp procedure (Table 3): Here light shading indicates one example of an inverse connection and dark shading indicates one example of a direct connection. Plugging Fig. 7 and (12) into (11) yields an identical S to that obtained by the previous method.

Table 3: C stamps for port compatibility. (BCT) [26, 32, 33] always has one parent and two children. How- ever, R-type adaptors cannot be decomposed into smaller adaptors and have N > 6 ports. Hence they have N — 1 > 5 children anda connection tree including them can no longer be assumed binary. To avoid a loss of generality for circuits with 7-type adaptors, we drop the “Binary” from the BCT concept, calling it rather the “Connection Tree” (CT)—this does not require any further alter- ation to standard WDF theory or terminology.

Table 2: S,ings stamps for series/parallel adaptors.

Figure 2: Deriving a WDF adaptor structure for the Fender Bassman tone stack, as in $3. Proc. of the 18" Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015 noncomputable structure is described by the scattering description

Figure 9: Tube Screamer magnitude responses, volume poten- tiometer at 50%, tone potentiometer at 0%, 50%, 95%, and 100%.

Figure 8: Deriving a WDF adaptor structure for the Tube Screamer tone/volume stage, as in §3.

Figure 1: Instantaneous Thévenin port equivalent.

Table 1: Modified Nodal Analysis element stamps. the equivalent circuit is assigned an index, and then the contribu- tion of each element is added into X one by one according to the element stamps. A fine point of this process is that one node in the equivalent circuit is chosen as the “datum” node, and neither its row nor its column appear in X. Since the number of independent KCL equations in a circuit is always one less than the number of nodes [31], X would always be singular without this step.

Figure 3: Instantaneous Thévenin port equivalent to Fig. 2f. C can also be populated by inspection; every internal incident wave is equal to the reflected wave at the port it is connected to. sometimes with a sign inversion if the polarities at that port don’t match. C = C' on account of the reciprocity of port connections. We abstract this process into another stamp procedure. Stamps fot direct connections and inverse connections” are shown in Table 3.

Figure 4: Bassman MNA system matrix—example resistor stamp in light shading, example voltage source stamp in dark shading. Proc. of the 18" Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015

Figure 5: Bassman magnitude responses, low and high poten- tiometers at 50%, mid potentiometer at 20% increments.

tions. (b) Detail on noncomputable system of series/parallel adaptors.

Figure 7: Bassman loop resolution matrix—example parallel stamp in light shading, example series stamp in dark shading Proc. of the 18" Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015

Resolving Wave Digital Filters with Multiple/Multiport Nonlinearities

by Jonathan Abel

2024

We present a novel framework for developing Wave Digital Filter (WDF) models from reference circuits with multiple/multiport nonlinearities. Collecting all nonlinearities into a vector at the root of a WDF tree bypasses the traditional... more

Emulation of Operational Amplifiers and Diodes in Audio Distortion Circuits

by Vesa Välimäki

2023, IEEE Transactions on Circuits and Systems II: Express Briefs

This brief presents a generic model to emulate distortion circuits using operational amplifiers and diodes. Distortion circuits are widely used for enhancing the sound of guitars and other musical instruments. This brief introduces a new... more

Fig. 5. WDF models for the circuits in Fig. 4: (a) noninverting and (b) inverting distortion. Fig. 6. Swept-sine results for the model in SPICE [26] and using the proposed WDF model. (a) Results for the noninverting configuration in Fig. 4(a). (b) Results for the inverting configuration in Fig. 4(b).

Fig. 1. (a) Typical operational amplifier connections and models based on input-output the behavior: (b) Traditional model [23]. (c) Simplified circuit model. (d) WDF model for operational amplifiers. behavior of different circuits using operational amplifiers. How- ever, the voltage source used in this model has instantaneous dependence on other voltages in the circuit, causing the solution for the circuit using this model to involve implicit equations. They are typically avoided when implementing real-time audio effects since they require computationally heavy operations, such as a matrix inversion or iterative methods.

Fig. 2. WDF diode model including a p-n junction and series resistance.

Fig. 3. Evaluation of the diode model for k = 0, 1, and 2 iterations for the Lambert W function approximation with a sinusoidal input. (a) Voltage ap- proximation error as a function of the diode current. (b) Error for approximating individual harmonics.

Fig. 4. Distortion circuits used in the simulations. (a) Noninverting distortion [5]. (b) Inverting distortion [10].

COMPUTATIONAL COMPLEXITY PER OUTPUT SAMPLE the proposed WDF model and the SPICE reference simulation. In both configurations, the energy of the even harmonics was negligible when compared with the odd harmonics and, hence, were omitted from the results.

On the derivation of the nonlinear discrete equations numerically integrating the Euler PDEs

by Fotis N. Koumboulis

2023, Discrete Dynamics in Nature and Society

The Euler equations, namely a set of nonlinear partial differential equations (PDEs), mathematically describing the dynamics of inviscid fluids are numerically integrated by directly modeling the original continuous-domain physical system... more

Generalized Wave Digital Filter Realizations of Arbitrary Reciprocal Connection Networks

by Julius Smith

2023, IEEE Transactions on Circuits and Systems I: Regular Papers

In this paper, an existing approach for modeling and efficiently implementing arbitrary reciprocal connection networks using Wave Digital scattering junctions based on voltage waves is extended to be used in a broader class of Wave... more

Structural equivalences in wave digital systems based on dynamic scattering cells

by Giovanni De Sanctis

2023

In this article we prove that a computable tree-like interconnection of parallel/series Wave Digital adaptors with memory (which are characterized by reflection filters instead of reflection coefficients) is equivalent to a standard... more

D–T adaptor: Applications to the design of non-linearN-ports

by Jose Huertas

2023, International Journal of Circuit Theory and Applications

A new network element is presented which can be of interest for synthesizing non-linear n-ports. It is shown that the use of n D-T adaptors reduces the problem to the synthesis of a multidimensionai transfer-characteristic (TC) plot. Some... more

Simulation of a “Giannini True Reverber” vacuum-tube guitar amplifier by using wave digital filters

by Thomaz Chaves A. Oliveira

2023

The objective of this work is to simulate a tube guitar amplifier, the Giannini True Reverber designed by Carlos Alberto Lopes in the nineteen sixties. The nonlinear “overdrive” characteristics of these devices make them attractive for... more

Figure 3: Asymmetric waveshaper proposed by [Gallo, 2011] where ky = a’, ky = 1+ 2a, kg = 6? and ky = 1 — 2b. The values of a and b can be freely chosen between —1.0 and +1.0 in order to control the characteristics of the nonlinear function. Since these two parameters are independent of each other, the positive and negative values of the input signal can be treated separately, which helps mimic the behavior of real vacuum-tube amplifiers. Small signals in the range a < x < b remain undistorted.

Figure 7: Giannini True Reverber pre-amplifier WDF representation.

Figure 6: Wave Digital Filters Signal Propagation of Kirchoff Variables. The main characteristic of the WDF models is that the bidirectional interactions of circuit elements are considered. This is accomplished by the concept of wave scattering propagation. The formalism of WDF theory is based on ‘Voltage’ wave notations, in Equation 4, a corresponds to incoming wave and b corresponds to the reflected wave, V is the Voltage, / stands for current in the same ways as Kirchoff variables and R, is the port or reference resistance, where all these variables are represented in Figure 6.

Figure 4: A 1 kHz sine wave filtered by Eq. 1 in the time and frequency domains with parameters a = 0.3 and b = 0.7. Figure 4 exhibit the distortion introduced by Eq 1 into a | kHz test tone. It is worth noting that this asymmetric distortion introduces odd harmonics that have have larger magnitude than the even harmonics. Values a and b can be altered creating many different curves and distortion characteristics for Eq. 1. The patent by [Gallo, 2011] also describes other parameters that can be added to this function to increase the system versatility, which are not presented in this paper for the sake of clarity. Benchmarks works for vacuum-tube amplifier simulations using waveshaping functions are present in the patents by Yamaha [Araya and Suyama, 1996] and Line Six [Doidic et al., 1998].

Figure 1: Vacuum Tube Electric Guitar Amplifiers - (a) Giannini True Reverber, (b) Giannini Tremendao, source [Giannini, 1962].

Figure 2: Giannini True Reverber pre-amplifier schematic

The most straightforward way to generate nonlinear distortion in digital audio signals i: by applying a nonlinear function into each sample of the signal, as illustrated in Fig ure 5. These functions can be of many types and are known as static waveshapers where the term “static” is due to the fact that such waveshapers are memoryless sys: tems. Waveshaping functions for audio applications were introduced by [Arfib, 1979. and [Le Brun, 1979] in the late 1970’s. Many simple DSP devices for tube amplifier sim- ulation use waveshaping functions to generate the nonlinearites. Normally the waveshap- ing functions are run oversampled up to eight times the original sampling rate to avoic aliasing in the output signal[Pakarinen and Yeh, 2009]. A example of these functions was implemented by [Gallo, 2011] proposed in a patent that comprises a complex waveshape! to emulate the effects of vacuum-tube amplifiers:

where LE, is the plate voltage, E’, is the grid voltage (where the electric guitar signal is applied), J, is the plate current, ky, kp, H,X and kg, are fitting parameters that are altered according to the type of electron tube to be modelled. For that reason, Koren’s triode equations are able to model many different triodes. In this model, the plate current is always J, > 0 for all positive plate voltages (EL, > 0). Another important aspect of these equations is that grid current is absent, as the grid circuit’s impedance is considered to be infinite. Novel equations developed by [Cohen and Helie, 2012] include the grid current, but have not yet shown their potential since they were developed very recently. Vacuum tube phenomenological equations, that is, equations that are not derived from fundamental physics but model the behaviour of a physical phenomena using fitted pa- rameters were developed by [Koren, 1996]. These models were successfully used in most of the physically informed vacuum tube guitar amplifiers digital emulators devel- oped over the last few years, and in SPICE (Simulation Program with Integrated Circuit Emphasis) simulations. Most physically informed vacuum tube amplifier simulations still use Koren’s models [Yeh et al., 2010] [Macak and Schimmel, 2011] to model the nonlin- ear vacuum tube transfer. The triode equations are presented by Eq. 2 and Ea. 3:

Exposure Bias and State Matching in Recurrent Neural Network Virtual Analog Models

2023, 2021 24th International Conference on Digital Audio Effects (DAFx)

Virtual analog (VA) modeling using neural networks (NNs) has great potential for rapidly producing high-fidelity models. Recurrent neural networks (RNNs) are especially appealing for VA due to their connection with discrete nodal... more

Rounding corners with BLAMP

2023

The use of the bandlimited ramp (BLAMP) function as an antialiasing tool for audio signals with sharp corners is presented. Discontinuities in the waveform of a signal or its derivatives require infinite bandwidth and are major sources of... more

Figure 1: Time-domain waveform of the (a) impulse, (b) bandlim- ited impulse, (c) Heaviside unit step, (d) BLEP, (e) ramp, and (f) BLAMP functions. Parameter T is the sampling period.

Figure 3: (a) Cubic B-spline basis function, (b) its first integral, (c) its second integral or polyBLAMP approximation and (d) the polyBLAMP residual, i.e., the difference between the polyBLAMP and the trivial ramp functions.

Figure 2: BLAMP residual function is the difference of the BLAMP and trivial ramp functions. Cf. Figs. 1(f) and I(e).

Figure 4: (a) Continuous-time triangular waveform with arbitrary fundamental frequency fo, (b) its first and (c) second derivatives.

Figure 5: Waveform and magnitude spectra of a (a)-(b) 1661.2-Hz trivial triangle wave, and (c)-(d) the same signal after four-point polyBLAMP correction. Circles indicate non-aliased components.

Figure 6: (a) Continuous-time fo-Hz sinusoid hard-clipped with clipping threshold L = 0.8, (b) its first and (c) second derivatives.

Figure 7: Waveform and magnitude spectra of a (a)-(b) 1660-Hz trivial hard-clipped sine wave (L = 0.3), and (c)-(d) the same signal after four-point polyBLAMP correction.

Figure 9: Waveform and magnitude spectrum of a (a)-(b) 1660-Hz trivial half-wave rectified sine wave, and (c)-(d) the same signal after four-point polyBLAMP correction. Figure 8: (a) Half-wave rectified continuous-time sine wave, (b) its first and (c) second derivatives.

Proceedings of the 19"" International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5-9, 2016

Figure 10: (a) Full-wave rectified continuous-time sine wave, (b) its first and (c) second derivatives.

Figure 11: Waveform and spectrum for a (a)-(b) synthetic string recording and the same signal after full-wave rectification (c)-(d) before and (e)-(f) after four-point polyBLAMP correction.

loosely resembles the central lobe of the bandlimited impulse [see Fig. 1(b)], we can observe the characteristic bell-shaped curve of B-spline interpolators. Integrating this basis function once yields the B-spline polynomial form of the BLEP function (known as the polyBLEP [12, 11]), and integrating once more results in the four- point polyBLAMP function [20]. The polynomials for these two functions and their corresponding waveforms are shown in Table 1, and Figs. 3(b) and 3(c), respectively. Finally, the bottom four rows of Table 1 show the piecewise polynomial coefficients for the polyBLAMP residual evaluated by substituting D = d+ 1 and computing difference between the polyBLAMP and the ramp function. A two-point version of the polyBLAMP function can be found in [21]. However, due to its superior performance, this work focuses solely on the four-point method. Table 1: Third-order B-spline basis functions, its first integral (polyBLEP), its second integral (polyBLAMP), and polyBLAMP residual (1 < D < 2and0 <d < 1) [20].

Table 3: Averaged computation time (in ms) for oversampling by factors of 2 and 4, the EPTR, and 4-point polyBLAMP methods.

4.2. Alias-Free Hard Clipping Table 2: SNR measurements in dB for test signals of 1661 Hz (A6) and 4186 Hz (A8). The best SNR on each row is bolded. Hard clipping is another example of an audio application where discontinuities in the first derivative of a signal are introduced [20]. Signal clipping is a form of distortion that limits the values of a sig- nal that lie above or below a predetermined threshold. Symmetric hard clipping can be expressed as

Aliasing reduction in soft-clipping algorithms

2023, 2015 23rd European Signal Processing Conference (EUSIPCO)

Soft-clipping algorithms used to implement musical distortion effects are major sources of aliasing due to their nonlinear behavior. It is a research challenge to design computationally efficient methods for alias-free distortion without... more

Fig. 4. Waveform for the continuous (a) bandlimited unit im- pulse, (b) BLEP, (c) BLAMP and (d) BLAMP residual. T is the sampling interval 1/ f..

Fig. 2. Waveforms for a 1245 Hz sinewave with L = 0.6 (a) hard clipping, (c) soft clipping and their respective magnitude spectra (b) and (d). The circles indicate non-aliased compo- nents, which correspond to harmonic distortion.

Fig. 6. Waveforms for a 1245 Hz sinewave with L = 0.6 (a) hard clipping with polyBLAMP correction, (c) soft clipping after polyBLAMP correction, and their respective magnitude spectra (b) and (d). Circles indicate non-aliased components. Fig. 7 shows the spectrograms for two 5-second linear chirps going from 20 Hz to 5 kHz, soft-clipped trivially and with the polyBLAMP method, respectively. For the poly- BLAMP implementation, fractional clipping points and their respective slopes were estimated following the aforemen-

Fig. 1. Input—output relationship for the hard clipper and the soft clipper designed by Araya and Suyama [11].

Fig. 5. (a) Linear interpolation basis function: triangular pulse, (b) its first integral, (c) second integral (solid line) and trivial ramp (dashed line), and (d) polyBLAMP residual.

Fig. 7. Spectrograms for a linear sine sweep with L = 0.6 (a) trivial soft clipping and (b) soft clipping with polyBLAMP aliasing reduction, respectively.

Emulation of Operational Amplifiers and Diodes in Audio Distortion Circuits

2023, IEEE Transactions on Circuits and Systems II: Express Briefs

New Family of Wave-Digital Triode Models

2023, IEEE Transactions on Audio, Speech, and Language Processing

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail.

Resolving Wave Digital Filters with Multiple/Multiport Nonlinearities

by Vaibhav Nangia

2023

Comparison of Germanium Bipolar Junction Transistor Models for Real-Time Circuit Simulation

2022

The Ebers-Moll model has been widely used to represent Bipolar Junction Transistors (BJTs) in Virtual Analogue (VA) circuits. An investigation into the validity of this model is presented in which the Ebers-Moll model is compared to BJT... more

On the Evaluation of the MD Passive WDF Discretization Method for a Flow through Cylinder

by Fotis N. Koumboulis

2022

In this paper a complete algorithm is developed to evaluate the multidimensional (MD) wave digital filter (WDF) method for the numerical integration of nonlinear (NL) partial differential equations (PDEs), and particularly those referring... more

Modeling nonlinear wave digital elements using the Lambert function

by Augusto Sarti

2022, IEEE Transactions on Circuits and Systems I: Regular Papers

A large class of transcendental equations involving exponentials can be made explicit using the Lambert W function. In the last fifteen years, this powerful mathematical tool has been extensively used to find closed-form expressions for... more

Structural equivalences in wave digital systems based on dynamic scattering cells

by Augusto Sarti

2022

Simulation of a “Giannini True Reverber” vacuum-tube guitar amplifier by using wave digital filters

by Gilmar Barreto

2022

Combined switched-capacitor FIR N-path filter using only grounded capacitors

by George Moschytz

2022, Electronics Letters

In contrast to known SC ‘N-path filters with lowpass cells, the presented FIR N-path filter does not need a bandpass prefilter to filter out the bandpass range at f.. All that is neces- sary is a lowpass prefilter that can also be built with grounded capacitors. The SC FIR 4-path filter of Fig. 9 has been breadboarded using discrete IC components (C,/C, = 1/1463, f. = 1 kHz). The measurements confirmed the theory presented as shown in Fig. 11. The number of clock phases can be halved by introduc- ing more switched capacitors. The tradeoff between clock phases and the number of switched capacitors depends on the application in question.

the amplitude response is obtained as shown in Fig. 2. At a given commutation rate f. the amplitude response is indepen. dent of the number N of paths and depends only on the lowpass cell. If the frequency of the input signal is equal to f., N determines the number of samples per period. This is why A has to be greater than 2 (sampling theorem). Introduction: Switched-capacitor N-path filters allow the reali- sation of very narrow bandpass filters.'-* An N-path filter consists of N identical cells which are cyclically connected be- tween the input and the output at a commutation rate of f, as shown in the 4-path example of Fig. 1.1 Using SC lowpass cells,

period T, at the output, which are held for the time T,/4, a more exact formula can be given by

Designing an N-path filter, N SC networks have to be linked together according to Fig. 1. Using 4 LP cells, the circuit shown in Fig. 4 is obtained. The numbers at the switches indi- cate the number of the phase at which the particular switch is closed. Using an 8-phase nonoverlapping clock scheme, the number of input capacitors can be reduced to 1 (see Fig. 5). The amplitude response of such a configuration is shown in Fig. 2.

Fig. 1 Schematic cross-section of groove laser if The present work describes a new groove laser diode, which is illustrated in Fig. 1, fabricated on SI substrate utilising the SI material to provide electrical current confinement. This forces the injection current to flow through the active region. The leakage through regions of surrounding SI InP, which have very high specific resistivity, is reduced to negligible proportions.

Resolving Wave Digital Filters with Multiple/Multiport Nonlinearities

by vaibhav nangia

2022

On the Evaluation of the MD Passive WDF Discretization Method for a Flow through Cylinder

by Fotis N . Koumboulis

2022

Macromodeling and demonstration of the LT6600 amplifier and lowpass filter

by Cheng-Wei Pei

2021

The goal of this thesis is to demonstrate the abilities of the Sevastopoulos-LaPorte active low-pass filter topology in Linear Technology Corporation’s LT6600 integrated circuit (IC). The thesis is split into two parts, representing two... more

Antiderivative Antialiasing for Stateful Systems

2021, Applied Sciences

Nonlinear systems, such as guitar distortion effects, play an important role in musical signal processing. One major problem encountered in digital nonlinear systems is aliasing distortion. Consequently, various aliasing reduction methods... more

Nonlinear-Phase Basis Functions in Quasi-Bandlimited Oscillator Algorithms

2021

Virtual analog synthesis requires bandlimited source signal algorithms. An efficient methodology for the task expresses the traditionally used source waveforms or their time-derivatives as a sequence of bandlimited impulses or step... more

Modeling Nonlinear Wave Digital Elements using the Lambert Function

by Kurt J A M E S Werner and

2018, IEEE Transactions on Circuits and Systems I: Regular Papers

Wave Digital Filter Modeling of Circuits with Operational Amplifiers

2018, Proceedings of the 24th European Signal Processing Conference (EUSIPCO)

We extend the Wave Digital Filter (WDF) approach to simulate reference circuits that involve operational amplifiers (op-amps). We handle both nullor-based ideal op-amp models and controlled-source-based linear op-amp macromodels in... more

A Computational Model of the Hammond Organ Vibrato/Chorus Using Wave Digital Filters

2018, Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16)

We present a computational model of the Hammond tonewheel organ vibrato/chorus, a musical audio effect comprising an LC ladder circuit and an electromechanical scanner. We model the LC ladder using the Wave Digital Filter (WDF) formalism,... more

Figure 11: Studying a single 1760 Hz sinusoid.

Figure 6: MNA matrix. Red and blue cells respectively show examples of resistor and voltage source “stamps” [27]

Figure 3: Gain applied to each scanner terminal ty - - - to.

Figure 10: Magnitude response at various interpolations. Recall Fig. 3 and Table 2.

In [8], Werner et al. propose a method for handling multiple nonlinearities that does not resort to these tactics. All of the non- linearities are grouped as sub-elements of a WDF structure at the root of the WDF tree. Inside that structure, and after proper modifi- cation of the circuit graph, those elements end up being connected to each other through a complex R-type adaptor that also inter- faces those elements to the rest of the circuit. The method of [27] is used to solve for the scattering behavior of this R-type adap- tor. Because of the non-adaptable nature of the root elements, the response of the root adaptor structure from the perspective of the rest of the tree forms an implicit loop that we can resolve using ei- ther a tabulated solution [8] or an iterative solution [28,29]. These approaches extends readily to nonadaptable linear elements, but is unnecessarily complex. Here we propose a novel more efficient approach for the case of multiple nonadaptable linear elements.

Table 2: Taps for different depth settings. 2.1. LC Ladder Circuit dard digital linear interpolation has a well-known lowpass charac- teristic [18]* that digital audio effect designers often try to avoid by using, e.g., allpass interpolation [18]*. Ironically, the scanner of the Hammond Organ vibrato/chorus essentially implements linear interpolation—meaning it does not have an allpass characteristic. The input signal is represented as an ideal voltage source vin. 19 LC ladder stages are composed of inductors L 1 --- L1i9, capac- itors C1 ---Ci9, and voltage divider pairs Ry and Ry_, k € [1---6]. A termination resistor R; ends the ladder. A switch con- trols whether R. is shorted or not. Electrical component values for the circuit are given in Table 1 [1].

Proceedings of the 19" International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5-9, 2016 where ® and Y embody the wave-domain behavior of the linear elements. Qroot and Byoot are related to the a; and b; by

Proceedings of the 19" International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, September 5-9, 2016 Figure 2: Vibrato/Chorus Schematic Partitioned.

Figure 8: Responses of the LC ladder, using unwarped BLT:

Figure 4: LC ladder Schematic, rearranged towards WDF.

Figure 7: WDF 2-port series adaptor / inverter. Interestingly, the discretized transfer functions of the capacitor and inductor do not depend on C, L, or T’. However, all of these do affect their adapted port resistance.

Figure 9: Responses of the LC ladder, using warped BLT:

RT-WDF—A Modular Wave Digital Filter Library With Support For Arbitrary Topologies And Multiple Nonlinearities

2018, Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16)

Wave Digital Filters (WDF) [1] are a popular approach for virtual analog modeling [2]. They provide a computationally efficient way to simulate lumped physical systems with well-studied numerical properties. Recent work by Werner et al.... more

The model is described by Equations (1a)—(1c) with perveances G, Gz, adaption factors C’, C's and positive exponents y, €. The nonlinear two port model is defined in terms of the port voltages Upk = Up — Uk, Ugk = Vg — Vr and port currents 2,, 7g. To be used with the Newton solver in this library, it is desirable that the modeling equations are continuously differentiable with respect to their port voltages in a region around the solution and the Jacobian must be invertible [12]. solver (Listing 6). In its current state the library supports a multi- dimensional Newton Solver as described in [12] in detail. For this type of root, the setRootMat rixData() method must config- ure the root’s system matrices correctly [4]. These matrices im- plicitly contain a w—K converter to transform the wave variables into the Kirchhoff domain and back. All iterative nonlinear models are currently evaluated in the iv domain. For this circuit the ac- tual nonlinearity is specified from a user expandable list of models as 12AX7_DW, a triode model after Dempwolf et al. [30].

Listing 6: (partial) Common Cathode Triode Amplifier tree class with nonlinear root element and appropriate root matrix data update function. The final example high handle multiple/multiport Kirchhoff-nonlinearities in circuits via the wdfRootNL and nlModel classes. Here we model the com- mon cat. been studied for exam WDF re extend t Cex, Cop and Cox as we rent Ip. Deriving the W as in the previous two Figure 4b. The extension of the wdfTree class for this circuit again implements all e tree nodes and registers all subtree entry nodes (not shown). lights the ability of the RT-WDF library to hode triode tube amplifier shown in Figure 4a which has ple in [29] as well. The results of recent search [3, 4] to support arbitrary topologies enable us to he model from [29] to include the parasitic capacitances 1 as continuously evaluated triode grid cur- DF adaptor structure is again accomplished examples, the result of which is shown in ements and their topology in the form of

WDF element, which has a closed-form reflection coefficient. It is thus treated as a non-adaptable root Node in this framework and can be implemented using a wdfRoot Simple type root. A pointer for the switch is declared as a private member of the class. The constructor of the class begins with the creation of the tree and root nodes. Adapted elements are created and initialized according to their physical parameters and the unadapted switch is initialized to be ‘open’. The next step is conceptually impor-

In general, these nonlinear model objects are managed by the nlNewtonSolver as shown in Figure la/ 1d. They always con- sist of acalculate () function that reflects the physical behav- ior and a getNumPorts() method in the base class for house- keeping.

Figure 1: General RT-WDF framework overview: (a) involved classes and their dependencies, (b) high level functions involved to initialize and run a WDF structure, (c) call graph of a simple WDF root implementation, (d) call graph of a nonlinear root with iterative solver.

Table 1: Comparison of processing times of WDF and SPICE cir- cuit simulations for all case studies.

The Newton Solver iteratively evaluates its specified models for each sample, converging towards a solution of the nonlinear sys- tem within a certain tolerance. This solution is then transformed back into the wave domain and returned as descending waves from the root down into the subtrees.

Figure 3: Deriving a WDF adaptor structure for the Fender Bass- man tone stack: a) circuit, b) graph, c) SPOR tree, d) WDF adap- tor structure. Modified from [3]

Figure 2: Deriving a WDF adaptor structure for the switchable attenuator: (a) circuit, (b) graph, (c) SPQR tree, (d) WDF adaptor structure.

Figure 4: Common Cathode Triode Amplifier: a) circuit, b) WDF adaptor structure with Kirchhoff nonlinearity. Grey: wdfRoOotNL; Dark Grey: 12AX7DwModel.

Modeling a Class of Multi-Port NonLinearities in Wave Digital Structures

2018, Proceedings of the European Signal Processing Conference (EUSIPCO)

Wave Digital Structures (WDS) are particularly interesting for applications of interactive modeling of nonlinear (NL) elements in the context of Virtual Analog modeling. NL circuits, however, often include multiple nonlinearities or... more

ALTERNATIVE APPROACH TO DESIGN OF WAVE DIGITAL FILTERS

by Anthony Constantinides

2018

The purpose of the letter is to outline an approach for designing wave digital filters that is based on a suitable interpretation of the scattering parameters of a network. The approach can be used for simulating analogue, doubly... more

Thus we have the incident- and reflected-wave formulation in the form

NETWORK VARIABLE PRESERVING STEP-SIZE CONTROL IN WAVE DIGITAL FILTERS

by Michael Jørgen Olsen

2017, Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17)

In this paper a new technique is introduced that allows for the variable step-size simulation of wave digital filters. The technique is based on the preservation of the underlying network variables which prevents fluctuation in the stored... more

Figure 9: Spectrums of the output voltages of a simulation run without polyBLEP (top) and with (bottom). The theoretical ampli- tude of the harmonics are indicated by asterisks. Third-order Lagrange polynomial polyBLEP residual functions were used to reduce the aliasing for a simulation run with a fun- damental frequency of 1007 at a sampling rate of 44.1 kHz with a higher rate of 16 x F’; for a frequency error of 9.7e—4 cents. The result of applying the polyBLEP residual function to the out- put voltage is shown in Figure|9|where the upper image shows the spectrum of the original output voltage and the lower image shows the spectrum of the output voltage with the polyBLEP residual ap- plied. Clearly, the aliasing in the spectrum of the output voltage with polyBLEP residuals applied is falling off at a much steeper rate than the original waveform’s spectrum. It is also evident that the amplitude of the higher harmonics in the spectrum of the alias suppressed waveform are falling off more quickly than the theoret- ical amplitudes. This can be fixed using a high shelf equalization filter. Example filter coefficients are given in (24).

Figure 5: Relaxation Oscillator Rearranged Schematic and WDF-

(a) Capacitor voltage error when simulation frequency is less than the (b) C desired frequency. the de he (b) Capacitor voltage error when simulation frequency is greater than the desired frequency.

Figure 1: RLC Circuit Schematic. Proceedings of the 20" International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5-9, 2017

Figure 4: Traces of the op amp input voltage, op amp output volt- age and the capacitor voltage. Figure 3: Relaxation Oscillator Schematic. The basic operation of the circuit is to behave as a comparator whose output swings high or low based on the whether the input voltage to the op amp is above or below a certain threshold value.

Algorithm 2: Extrapolation/Step-Size Algorithm It should be noted that anti-aliasing is not performed at the decimation stage. This is because the anti-alias filtering of Vou will change the dynamics of the system, through modification of the output voltage, affecting the energy stored in the capacitor there- fore affecting the frequency of oscillation. In order for the simula- tion frequency to be as accurate as possible, which corresponds to the simulation error being as small as possible, we allow the alias- ing to occur within the simulation. How to suppress the aliasing inherent in the output voltage is the subject of Section3.6]

Figure 8: Incident waves and the stored energy in the capacitor for the relaxation oscillator circuit run with a fixed step-size and different variable step-size methods. Proceedings of the 20" International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5-9, 2017

One ee The techniques of [3] are used to determine the scattering be- havior of the 7-type adaptor which is represented by the matrix S. The following system of equations fully describes the relationship between the R-type adaptor and the two-port nonlinearity: and ag = [az a4 5 ag] is the vector of incident waves from the R-type adaptor.

Figure 2: Inductor incident waves from RLC circuit [28) 3. RELAXATION OSCILLATOR CIRCUIT CASE STUDY

The circuit to be modeled is called a relaxation oscillator and is also known as an astable multivibrator circuit . It contains the following components: a resistor R1, a capacitor C1, a voltage divider (Rz and R3) and an op amp. Throughout this paper it is assumed that R2 = Rez and that the op amp operates from rail-to- rail. The circuit schematic is shown in Figure]3

Block-Based Physical Modeling for Digital Sound Synthesis

by Cumhur Erkut

2017

lock-based physical modeling is a methodology for modeling physical systems with different subsystems. It is an important concept for the physical modeling of real or virtual musical instruments where different components may be modeled... more

[FIG2] The interconnection of two one-ports described by the physical port variables u, and i‘, and alternatively by the inciden waves a, and the reflected wave b,, n = 1, 2.

[FIG11] Top: A K-node (left) and a wave node (right) forming a part of a hybrid waveguide in a source-free structure. There is a KW- converter (type Ill) between K- and wave nodes, and Y% is its wave admittance. U; and U2 are the junction voltages of the K-node and wave node, respectively. Y; and Y3 provide terminations for the corresponding node, following the principles developed in [39]. Bottom: Abstraction of the structure. In the signal flow diagram of Figure 11, a K-node Ny (a) anda wave node N (b) are aligned in a source-free structure with Multiparadigm modeling is another key design principle in BC. The first category of objects supports conventional DSP and one-directional signal data flow between object terminals. This includes elementary blocks such as adders, multipliers,

[FIG3] Interconnection of two wave ports with a parallel adaptor.

[FIG4] (a) Three-port parallel adaptor and equivalent network with Kirchhoff variables. The direction of the wave variables is given with respect to the connected block elements. (b) Three-port serial adaptor and equivalent network with Kirchhoff variables.

which is also shown in Figure 6(a). Each first-order system is driven by samples of the excitation f,[k] and delivers a contri- bution to the velocity of the vibration y[k]. However for the communication with other blocks, the Kirchhoff variables force f.[k] and velocity y[k] have to be converted to the wave variables a[k] and b[k]. [23]-[25]. In detail, the vibration of the membrane is modeled by the parallel arrangement of first-order systems with complex frequencies By,n = 1,... , N. With the weighting con- stants by, and cy following directly from the application of the functional transformation method [25], the output is calculated by the discrete system

[FIG10] Typial BCT configuration with tree updating after a two- port value change. Root and leaves are represented by rectangles. The nodes consist of serial and parallel adaptors according to Figure 4 and are represented by circles. While the wave and K-nodes could be formulated for multi- paradigm, block-based physical modeling without any reference to a specific software system, we consider their implementation in block compiler (BC) in this section [38].

[FIG8] Chain structure: each adaptor is connected to at least one two-port, and there is n branching. It was recently proved [34], [35] that a computable tree-like interconnec ion of adaptors with memory is completely equiva- lent to a memoryless macro-adaptor (a computable tree-like interconnec adaptors) w tion of standard WDF adaptors, i.e., instantaneous hose outer ports are connected to mutators (two- port adaptors with memory) [32], [33]. It is thus easy to define a process of memory extraction from [35] that preserves the inter- connection opology. This approach simplifies the implementa- tion to WD structures based on one-port WD elements (generally with memory) interconnected by a tree-like arrange- ment of standard WDF adaptors. The nonlinear element is gen- erally connected to the adapted port of the resulting instantaneous MA, possibly through a mutator. A method [27], [36] for automatically implementing a WD structure of this sort through a direct inspection (scanning) of the tree-like topologi-

[FIG7] Example network to simulate the excitation of a membrane with a mallet. The membrane is modelled by a distributed parameter model according to Figure 6 (bottom left). The mass of the mallet is represented by a lumped parameter reactive element (bottom right) and the elasticity of the mallet felt by a memoryless nonlinearity (top left). The dissipation of the excitation mechanism is contained in a lumped parameter resistive element (top right). The parallel and series adaptors are the backbone of the interconnection network.

The definition of the vectors of incident and reflected waves a[k] and b[A] with the matrix of port resistances R and the iden- tity matrix I is given in the first line of (13). This definition can be solved for the quantities that should leave the KW-converter, i.e., for the input v[x ] of the state-space structure and for the reflected wave b[A] [(13) second line. The result defines a first version of the KW-converter called KW-converter I. It is only applicable if the direct path in the state space structure, i.e., the matrix D is zero. Otherwise, a delay-free loop would result [see Figure 5(a)]. The corresponding KW-converter for D 40 follows by inserting the output equation of the state-space representation into the definition of the wave variables (13). Setting the port resistance R = D and solving for the input v[A] and the reflect- ed wave b[/] yields the KW-converter II, which avoids delay free loops [see Figure 5(b)].

[FIG6] (a) Structure of the membrane implementation with N complex harmonics. (b) Structure of the membrane implementation that uses the wave variables a[k ] and b[k] for input and output instead of the Kirchhoff variables f.[k ] and y[k ].

lock-based physical modeling is a methodology for modeling physical systems with different subsys- tems. It is an important concept for the physical modeling of real or virtual musical instruments where different components may be modeled accord- ing to different paradigms. Connecting systems of diverse nature in the discrete-time domain requires a common inter- connection strategy. This contribution presents suitable inter- connection strategies that incorporate a wide range of modeling blocks and considers the automatic implementation of block- based structures. Software environments are presented, which allow to build complex sound synthesis systems without burden- ing the user with problems of block compatibility.

RESOLVING GROUPED NONLINEARITIES IN WAVE DIGITAL FILTERS USING ITERATIVE TECHNIQUES

by Michael Jørgen Olsen and

2016, Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16)

In this paper, iterative zero-finding techniques are proposed to resolve groups of nonlinearities occurring in Wave Digital Filters. Two variants of Newton's method are proposed and their suitability towards solving the grouped... more

Wave Digital Filter Modeling of Circuits with Operational Amplifiers

by Kurt J A M E S Werner and

2016, Proceeding of the European Signal Processing Conference

Fig. 2. Symbols and port definitions for nullator, norator, and nullor.

Fig. 7. Results of four simulations in the frequency domain.

include potentially dozens of transistors [19]. To simplify cir- cuit design, analysis, and simulation, op-amp behavior is often idealized completely or approximated using macromodels.

Fig. 3. Nullor (a) and element stamp (b) for Modified Nodal Analysis. and current embody the restrictions on the op-amp’s input terminals. The port between output terminal and ground is equivalent to a norator—its arbitrary port voltage and current embody the ideal op-amp’s infinite gain and zero output impedance [4].

Fig. 4. (a) Op-amp-based Bridged-T Resonator schematic; (b) op-amp symbol replaced by nullor; and (c) op-amp symbol replaced by linear macromodel. topologies with or without absorbed multiport linear elements is infinitely large. To form a WDF involving one of these topologies one must derive its scattering behavior. [13] gives a general procedure, which we review briefly, emphasizing cases with currents which are not port currents (involving, e.g., VWCVSs and nullors) and how to handle nullor stamps.

Fig. 5. Reference circuits rearranged to highlight WDF adaptor structures corresponding to (a) the nullor-based model (Fig. 4b); and (b) the macromodel-based model (Fig. 4c). WDF adaptors are represnted by shaded boxes, whose darker shaded edges indicate the adapted port as in [2]. 7e-type adaptors in each are shown with Thévenin equivalents (A--- F and A---K, respectively) and node labels necessary for their scattering matrix derivations.

Fig. 6. Forming MNA matrix X, highlighting examples of resistor (red, Rg), voltage source (blue, e,), and nullor (green, Fig. 3) element stamps.

where entries of S are denoted as S8mn, m,n € [A--- F], meaning the contribution of a, to b,,. For example, sr, = 884.00. The scattering b = Sa for the R-type adaptor in Fig. Sa is Fig. 7 shows simulation results in the frequency domain and Table Ic summarizes extracted parameters. Accounting for the warping introduced by the bilinear transform [31], [32], f- and @ for the ideal simulation closely match the values predicted by (5). The macromodel simulations diverge from ideal behavior, including audible differences in frequency and decay time.

The Wave Digital Reed: A Passive Formulation

by Richard Kronland-Martinet

2016

In this short paper, we address the numerical simulation of the single reed excitation mechanism. In particular, we discuss a formalism for approaching the lumped nonlinearity inherent in such a model using a circuit model and the... more

Wave digital modeling of the output chain of a vacuum-tube amplifier

by Miikka Tikander

2016

This article introduces a physics-based real-time model of the output chain of a vacuum-tube amplifier. This output chain consists of a single-ended triode power amplifier stage, output transformer, and a loudspeaker. The simulation... more

A generalized method for the derivation of non-linear state-space models from circuit schematics

2016, 23rd European Signal Processing Conference (EUSIPCO)

Digital emulation of analog circuits for musical audio processing, like synthesizers, guitar effect pedals, or vintage amplifiers, is an ongoing research topic. David Yeh proposed to use the nodal DK method to derive a non-linear... more

NONLINEAR-PHASE BASIS FUNCTIONS IN QUASI-BANDLIMITED OSCILLATOR ALGORITHMS

2016

Proportional plus integral control of ladder circuits modeled in the form of two-dimensional (2D) systems

by Krzysztof Galkowski

2015

In this paper, a 2D systems setting is used to develop new results on control of active electrical ladder circuits. In particular, the proportional plus integral control method has been extended to this case and the problem of how to... more

An Improved and Generalized Diode Clipper Model for Wave Digital Filters

by Kurt J A M E S Werner and

2015, Proceedings of the Audio Engineering Society

We derive a novel explicit wave-domain model for “diode clipper” circuits with an arbitrary number of diodes in each orientation, applicable, e.g., to wave digital filter emulation of guitar distortion pedals. Improving upon and... more

Calculating the node-to-output transfer in ladder wave digital filters (ldwdfd)

by Diego Andina

2015

In this article we will calculate the roundoff noise produced in ldwdf, from the noisy node through to the output, with input zero, via its transfer function g(z). In addition to the error produced through having to limit the exactness of... more

Resolving Wave Digital Filters with Multiple/Multiport Nonlinearities

2015, Proceedings of the International Conference on Digital Audio Effects (DAFx-15)

Wave Digital Filter Adaptors for Arbitrary Topologies and Multiport Linear Elements

2015, Proceedings of the International Conference on Digital Audio Effects (DAFx-15)

We present a Modified-Nodal-Analysis-derived method for developing Wave Digital Filter (WDF) adaptors corresponding to complicated (non-series/parallel) topologies that may include multi-port linear elements (e.g. controlled sources and... more

Multi-Port Nonlinearities in Wave Digital Structures

by Alberto Bernardini and

2015, Proceedings of the IEEE International Symposium on Signals, Circuits, and Systems (ISSCS)

Wave Digital Structures (WDS), with their inherent stability and robustness, would be particularly suitable for nonlinear (NL) circuit modeling in Virtual Analog applications, if it were possible for them to accommodate multi-port... more

A Simple Digital Model of the Diode-Based Ring-Modulator

by Julian Parker

2011, recherche.ircam.fr

The analog diode-based ring modulator has a distinctive sound quality compared to standard digital ring modulation, due to the non-linear behaviour of the diodes. It would be desirable to be able to recreate this sound in a digital... more

On Minimizing the Look-Up Table Size in Quasi-Bandlimited Classical Waveform Oscillators

by Jussi Pekonen

2011

In quasi-bandlimited classical waveform oscillators, the aliasing distortion present in a trivially sampled waveform can be reduced in the digital domain by applying a tabulated correction function. This paper presents an approach that... more