The Fast Griffin-Lim Algorithm (FGLA) is one of the most widely used iterative methods for phase reconstruction and signal estimation from modified short-time Fourier transforms. Unlike the classic Griffin-Lim Algorithm (GLA), however, the convergence of FGLA has not yet been fully proved, with existing theoretical guarantees holding only for momenta significantly smaller than those found to perform best in practice. In this letter, we build upon the appendix of the original paper by Griffin and Lim and formulate GLA as a gradient descent algorithm. From this, we show that FGLA corresponds to an accelerated gradient descent method with constant momentum. We then derive a sufficient condition ensuring convergence over a broad range of momenta and formulate a criterion to assess its validity in numerical experiments, which we find is typically satisfied within the first few iterations.

Sound field reconstruction aims to estimate pressure fields in areas lacking direct measurements. Existing techniques often rely on strong assumptions or face challenges related to data availability or the explicit modeling of physical properties. To bridge these gaps, this study introduces a zero-shot, physics-informed dictionary learning approach to perform sound field reconstruction. Our method relies only on a few sparse measurements to learn a dictionary, without the need for additional training data. Moreover, by enforcing the Helmholtz equation during the optimization process, the proposed approach ensures that the reconstructed sound field is represented as a linear combination of a few physically meaningful atoms. Evaluations on real-world data show that our approach achieves comparable performance to state-of-the-art dictionary learning techniques, with the advantage of requiring only a few observations of the sound field and no training on a dataset.

Virtual Analog effects refer to the class of digital audio effects that recreate the sound of analog equipment. Vacuum tubes, although old-fashioned, still play a crucial role in defining the distinctive tone of certain audio gear, e.g., guitar amplifiers. Among tubes, variable-µ pentodes stand out for their peculiar nonlinear behavior but are notoriously difficult to simulate using conventional methods. In this article, we extend a method previously proposed for the simulation of circuits containing a single nonlinear two-port element to the case of circuits containing a single variable-µ pentode. We show that by combining Wave Digital Filters based on vector waves and neural networks, we are able to solve said class of circuits in a fully explicit fashion, contrary to what can be done in standard simulators. Our approach is validated through the simulation of the Dogzilla preamplifier stage, offering a promising pathway for the efficient and accurate simulation of circuits containing nonlinear elements characterized by an arbitrary number of ports.

Circuits containing multiple nonlinear elements are typically solved by means of iterative techniques. Recently, Wave Digital Filters (WDFs) have demonstrated good performance for the simulation of circuits containing a high number of nonlinearities. In particular, the vast majority of the approaches available in the literature consider WDFs characterized by reciprocal lossless connection networks. In this article, we provide, instead, a comparison of iterative methods for the solution of WDFs characterized by non-lossless junctions with involutory scattering matrices, e.g., connection networks absorbing nullors. We run different tests at different amplitudes and frequencies, unveiling and giving insights into the trade-off between convergence speed and simulation time.

Time-varying circuits, either due to inherent physical phenomena or due to external user interactions, are ubiquitous, especially in audio processing gear. Examples of such circuits include amplification stages based on a single Bipolar Junction Transistor (BJT) with time-varying controls that adjust, for example, the transistor biasing point. In this manuscript, we extend a previously proposed Wave Digital (WD) methodology for the fully explicit simulation of circuits incorporating a single nonlinear two-port element to accommodate time-varying circuits. To this end, we introduce a novel definition for the explicit BJT WD model, using a nonlinear vector wave scattering equation implemented through a neural network architecture. The proposed approach is validated through the discrete-time simulation of a fuzz guitar pedal with two potentiometers, achieving accuracy levels comparable to standard circuit simulation software.

Wave Digital Filter (WDF) theory has been widely used to design nonlinear digital filters that behave like reference analog circuits, especially in the field of Virtual Analog modeling, i.e., the digital emulation of audio circuits. WDF principles allow us to implement circuits with one nonlinearity in an explicit fashion, by properly setting the free parameters that are introduced in the Wave Digital (WD) domain. Although this property does not extend to the WDF realization of circuits with multiple nonlinearities, which instead requires the use of iterative solvers, a proper setting of the free parameters is beneficial also in these cases, in terms of both robustness and efficiency of the implementation. In particular, recent research shows that a common optimal policy for the setting of free parameters can be adopted when either a WD fixed-point method or a WD Newton-Raphson (NR) method is used to solve the same nonlinear circuit. In this work, we present a class of WD extended fixed-point solvers with order from zero to infinity that generalizes the aforementioned iterative methods; the zero-order case corresponds to the existent WD fixed-point method, while the infinite-order case to WD NR. The proposed solvers do not require to compute inverse Jacobian matrices and are characterized by superlinear speed of convergence. Moreover, we show that the very same policy of free parameter setting can be applied independently of the order, causing, in any case, an increase of robustness and convergence speed.

Reconstructing the sound field in a room is an important task for several applications, such as sound control and augmented (AR) or virtual reality (VR). In this paper, we propose a data-driven generative model for reconstructing the magnitude of acoustic fields in rooms with a focus on the modal frequency range. We introduce, for the first time, the use of a conditional Denoising Diffusion Probabilistic Model (DDPM) trained in order to reconstruct the sound field (SF-Diff) over an extended domain. The architecture is devised in order to be conditioned on a set of limited available measurements at different frequencies and generate the sound field in target, unknown, locations. The results show that SF-Diff is able to provide accurate reconstructions. We conduct a comparative analysis with two state-of-the-art baseline methods, one relying on kernel interpolation and the other on deep learning.

In the past, the field of drum source separation faced significant challenges due to limited data availability, hindering the adoption of cutting-edge deep learning methods that have found success in other related audio applications. In this letter, we introduce StemGMD, a large- scale audio dataset of isolated single-instrument drum stems. Each audio clip is synthesized from MIDI recordings of expressive drum performances using ten real-sounding acoustic drum kits. Totaling 1224 h, StemGMD is the largest audio dataset of drums to date and the first to comprise isolated audio clips for every instrument in a canonical nine-piece drum kit. We leverage StemGMD to develop LarsNet, a novel deep drum source separation model. Through a bank of dedicated U-Nets, LarsNet can separate five stems from a stereo drum mixture faster than real-time and is shown to considerably outperform state-of-the-art nonnegative spectro-temporal factorization methods.

State-of-the-art soundfield reconstruction methods are computationally expensive and their performance is generally undermined by the presence of strong reverberation and of near-field sources, which are usually modeled using omnidirectional radiation patterns. In this work, we propose a compressive sensing approach for the reconstruction of the soundfield produced by arbitrary directive sources in a reverberant room. Assuming sparsity in the distribution of sources, along with a loose prior knowledge on their position and on the geometry of the environment, we reconstruct both the direct and the reverberant components of the soundfield by modeling early reflections as near-field sources. Moreover, the directivity of sources is explicitly modeled by first expressing the soundfield produced by arbitrarily directive sources as an expansion of multipoles, and then introducing group sparsity constraints. Numerical simulations in two rooms with different reverberation characteristics are conducted to perform the validation of the proposed method.

In this work, we propose a novel approach for distributed 3D sound source localization and tracking based on networks of planar microphone arrays, each of which estimates a 2D Direction Of Arrival (DOA). The proposed method is computationally distributed and eliminates the need for a specialized node to collect and process all information. Sound source localization is achieved by considering the task as a distributed optimization problem approached using the Adapt Then Combine (ATC) diffusion technique. This approach also allows the development of cooperation strategies between sensor nodes (i.e., microphone arrays). We propose the use of a cooperation strategy that improves the localization accuracy by exploiting the estimated error statistics of each sensor node and penalizing the noisy arrays. We then evaluate the proposed approach in terms of localization accuracy and robustness to noisy sensor measurements.

A family of iterative root-finding methods for nonlinear discrete-time systems of equations is presented, with a formulation that puts it in between the Fixed-Point (FP) and Newton-Raphson (NR) methods. Applicability of this family is allowed, provided that the Jacobian matrix of the nonlinear system has a spectral radius less than one. By varying the order of a matrix geometric sum that approximates the inverse Jacobian matrix, root-finding at any iteration can be steered toward the FP or conversely toward the NR method, becoming identical to either of them if the order is equal to zero or infinitely large, respectively. Since the methods in this family do not need the solution of a linear system at each iteration as required by NR, their computational cost makes them palatable for the online digital implementation of nonlinear models. As an example of application, a Virtual Analog model of the voltage-controlled filter onboard a popular music synthesizer is tested, showing that for some orders of the aforementioned geometric sum the proposed methods perform better than FP and NR in terms of computational cost, while exhibiting the same accuracy.

A method for circuit system inversion has been recently employed to develop digital algorithms for loudspeaker virtualization. In this brief, building on such promising results, we propose an algorithm that flips the paradigm by virtualizing sensors rather than actuators. In particular, we derive direct and inverse nonlinear circuital models of guitar pickup systems by assuming string vertical excitations, and we then present a virtualization algorithm based on circuit inversion. The proposed circuital models are then implemented in the discrete-time domain in a fully explicit fashion (i.e., with no use of iterative solvers) by employing Wave Digital Filter principles. Finally, we validate and test the designed algorithm on the physical output voltage of a guitar pickup system, making it sound as if acquired by two other magnetic pickups characterized by a different nonlinear behavior.

Virtual Bass Enhancement (VBE) refers to a class of digital signal processing algorithms that aim at enhancing the perception of low frequencies in audio applications. Such algorithms typically exploit well-known psychoacoustic effects and are particularly valuable for improving the performance of small-size transducers often found in consumer electronics. Though both time- and frequency-domain techniques have been proposed in the literature, none of them capitalizes on the latest achievements of deep learning as far as music processing is concerned. In this letter, we propose a novel time-domain VBE algorithm that incorporates a deep neural network for music demixing as part of the processing pipeline. This technique is shown to improve the bass perception and reduce inharmonic distortion, i.e., the main issue of existing time-domain VBE algorithms. The results of a perceptual test are then presented, showing that the proposed method is able to outperform state-of-the-art algorithms both in terms of bass enhancement and basic audio quality.

Differential Microphone Arrays (DMAs) are of great interest in the literature on small-sized microphone arrays, due to their good directivity properties and nearly frequency-invariant spatial responses. Recently developed beamforming techniques combine multiple DMA units to form flexible two-stage spatial filtering systems, where the output of each DMA is fed into a higher-level filter, called virtual filter, for further processing. In this manuscript, we analyze and discuss some properties of a broad class of two-stage beamformers with arbitrary planar geometry. In this context, the DMA units are all assumed to have the same directivity pattern of arbitrary order and can be characterized by a variable number of omnidirectional sensors organized in an arbitrary geometry. For any given choice of the virtual array filter, we introduce a closed-form optimization procedure to design DMA filters that maximize the White Noise Gain (WNG) or the Directivity Factor (DF) of the resulting two-stage beamformer at any frequency. Based on this frequency-dependent design, we propose a frequency-invariant design of the two-stage beamformer and we compare the performance of the two approaches. Finally, we propose two possible computational schemes for the proposed generic two-stage spatial filtering system and discuss their efficiency in performing filtering, steering, and changing beampattern.

In consumer electronics, the advent of ultra-thin devices has raised interest in Virtual Bass Enhancement (VBE) algorithms for enhancing the acoustic performance of their small-size loudspeakers. In fact, due to physical limitations, large volume velocities cannot be achieved, impairing thus the reproduction of low frequencies. VBE techniques exploit psychoacoustic effects originated by the sound signal processing happening in the inner ear and brain. In this paper, we propose a nonlinear circuital model of a generic time-domain VBE system, and we implement it in the discrete-time domain. As required by most applications of interest, the proposed VBE algorithm is able to operate in real-time. A MUSHRA-like test is then employed to evaluate the bass enhancement performance of the proposed algorithm using different nonlinear devices and parameter configurations.

The literature is rich with techniques for the design of small-size Differential Microphone Arrays (DMAs), known for their almost frequency-invariant beampatterns and low computational cost. Few works, instead, discuss the properties of beamformers based on multiple DMA units. In this paper, we consider arbitrarily shaped planar arrays of DMA units. In turn, each DMA unit is a first-order continuously-steerable differential microphone characterized by an arbitrary configuration of omnidirectional sensors and a symmetric beampattern. We present a beamforming technique that, assumed all the DMA units to steer identical beams in the same direction, allows us to approach the behavior of a Delay-And-Sum beamformer or Super-Directive beamformer by solely varying a single scalar parameter. Efficient implementations of the proposed beamformers can be developed by taking into account that, for a wide range of frequencies, the values of such a parameter are practically invariant with respect to the geometry of the array.

The digital implementation of a nonlinear audio circuit often employs the Newton-Raphson (NR) method for solving the corresponding system of implicit ordinary differential equations in the discrete-time domain. Although its quadratic convergence speed makes NR attractive for real-time audio applications, quadratic convergence is not always guaranteed, since it depends on initial conditions, and also divergence might occur. For this reason, especially in the context of Virtual Analog modeling, techniques for increasing the robustness of NR are in order. Among the various approaches, the Wave Digital (WD) formalism recently showed potential to rethink traditional circuit simulation methods. In this manuscript, we discuss an original formulation of the NR method in the WD domain for the solution of audio circuits with multiple one-port nonlinearities. We provide an in-depth theoretical analysis of the proposed iterative method and we show how its quadratic convergence strongly depends on the free parameters (called port resistances) introduced when modeling the reference circuit in the WD domain. In particular, we demonstrate that the size of the basin where the WD NR solver can be initialized to converge on a solution with quadratic speed is a function of the free parameters. We also show that by setting each port resistance value as close as possible to the derivative w.r.t. current of the nonlinear element v-i characteristic we keep the basin size large. We finally implement an audio ring modulator circuit with four diodes in order to test the proposed iterative method.

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.

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.

We present a spatial filtering approach to first-order steerable Differential Microphone Arrays (DMAs) with arbitrary planar geometry. In particular, the design of the spatial filter is based on a recently proposed frequency-domain design methodology that approximates, in a least-square sense, a target beampattern using the Jacobi-Anger expansion involving Bessel functions. Despite the generality of that approach, however, its computational cost turns out to be excessive when working with limited processing resources. The beamforming technique proposed in this manuscript overcomes this issue by exploiting the fact that in DMAs the spacing between sensors is typically smaller than the smallest wavelength of audio signals of interest. This allows us to substitute zero- and first-order Bessel functions with their Taylor series approximation truncated to the first order. Moreover, we show that this approximation allows us to derive an efficient discrete-time-domain implementation of first-order steerable differential beamformers based on arrays with arbitrary geometries.

There is a growing interest in Virtual Analog modeling algorithms for musical audio processing designed in the Wave Digital (WD) domain. Such algorithms typically employ a discretization strategy based on the trapezoidal rule with fixed sampling step, though this is not the only option. In fact, alternative discretization strategies (possibly with an adaptive sampling step) can be quite advantageous, particularly when dealing with nonlinear systems characterized by stiff equations. In this paper, we propose a unified approach for modeling capacitors and inductors in the WD domain using generic linear multi-step discretization methods with variable time-step size, and provide generalized adaptation conditions. We also show that the proposed approach for implementing dynamic (energy-storing) elements in the WD domain is particularly suitable to be combined with a recently developed technique for efficiently solving a class of circuits with multiple one-port nonlinearities, called Scattering Iterative Method. Finally, as examples of application, we develop WD models for a Van Der Pol oscillator and a dynamic diode-based ring modulator, which use different discretization methods.

We propose a spatial filtering method for linear arrays of First-Order Steerable Differential Microphones (FOSDMs), which operates in two layers. In the former, signals acquired by individual microphones are locally filtered to produce the outputs of the FOSDMs. In the latter, the outputs of the FOSDMs are processed by another filter. We analyse different design methodologies and study the conditions under which the two filtering layers can be decoupled. The proposed two-layer spatial filter can be flexibly controlled with a single scalar parameter, which can be chosen, for example, to maximize the White Noise Gain (like in a Delay-and-Sum beamformer); or to maximize the Directivity Factor (like in a Super-Directive beamformer); without needing any matrix inversion. The effectiveness of the proposed beamforming method is compared with traditional spatial filtering techniques using different metrics.

This research investigates the effects that electrical mismatches and partial shading can have on the performance of photovoltaic arrays. The analysis adopts a probabilistic point of view where the most relevant parameters of the solar units are seen as random variables. The analysis relies on an efficient and robust simulation technique, based on Wave Digital principles, that is tailored to the modular topology of solar arrays. It is shown how electrical mismatch, solar shading and array topology can interact among them in a quite complex way.

In this letter, a novel time-domain implementation of robust first-order differential microphone arrays (DMAs), based on wave digital filters, is presented. The proposed beamforming method is extremely efficient, as it requires at most two multipliers and one delay for each filter, where the necessary number of filters equals the number of physical microphones of the array, and it avoids the use of fractional delays. The update of the coefficients of the filters, required for reshaping the beampattern, has a significantly lower computational cost with respect to the time-domain methods presented in the literature. This makes the proposed method suitable for real-time DMA applications with time-varying beampatterns.

Sallen-Key filters are widespread in audio circuits. Therefore, accurate and efficient digital models of such filters are highly desirable in audio Virtual Analog applications. In this paper, we will discuss a possible strategy, based on Wave Digital Filters (WDFs), for implementing all the analog filters described in the historical 1955 manuscript by Sallen and Key. In particular, we will group the eighteen filter models presented by Sallen and Key into nine classes, according to their topological properties. For each class we will describe the corresponding WDF structures. Finally, we will compare the output signals of WDFs to the output signals of the same models implemented in LTSpice.

Performing continuous beam steering, from planar arrays of high-order differential microphones, is not trivial. The main problem is that shape-preserving beams can be steered only in a finite set of privileged directions, which depend on the position and the number of physical microphones. In this letter, we propose a simple and computationally inexpensive method for alleviating this problem using planar microphone arrays. Given two identical reference beams pointing in two different directions, we show how to build a beam of nearly constant shape, which can be continuously steered between such two directions. The proposed method, unlike the diffused steering approaches based on linear combinations of eigenbeams (spherical harmonics), is applicable to planar arrays also if we deal with beams characterized by high-order polar patterns. Using the coefficients of the Fourier series of the polar patterns, we also show how to find a tradeoff between shape invariance of the steered beam, and maximum angular displacement between the two reference beams. We show the effectiveness of the proposed method through the analysis of models based on first-, second-, and third-order differential microphones.