Delay-free Loops Research Articles

Delay-free phase-locked loop ring: Equilibrium and phase perturbation

Clock signal distribution networks have been studied since the 1960s in the search of topologies and parameters that can optimize performance. The problem can be viewed as the mutual synchronization of coupled oscillators, a field being extensively investigated. However, the majority of these investigations consider a linear coupling factor, unlike in the case of time signal distribution systems, wherein nonlinear coupling terms appear. The latter complicates the analysis, mainly because the nodes are phase-locked loops. In the literature pertaining to telecommunications, this problem is studied using linear analysis and is supported by corrections provided by the experimental work. In this paper, a different approach to this problem is presented: here, it is considered that the basic unit of a fully connected time distribution network is a triangular ring of phase-locked loops. The stability of equilibrium states is derived, starting with a delay-free analytical model for the spatial phase and frequency errors. Perturbations of the equilibrium, step, ramp, and accidental phase modulation are included in the model, producing results reflecting the performance of the entire loop.

Newton–Raphson Solution of Nonlinear Delay-Free Loop Filter Networks

For their numerical properties and speed of convergence, Newton–Raphson methods are frequently used to compute nonlinear audio electronic circuit models in the digital domain. These methods are traditionally employed regardless of preliminary considerations about their applicability, primarily because of a lack of flexible mathematical tools making the convergence analysis an easy task. We define the basin delimiter, a tool that can be applied to the case when the nonlinear circuit is modeled by a delay-free loop network. This tool is derived from a known convergence theorem providing a sufficient condition for quadratic speed of convergence of the method. After substituting the nonlinear characteristics with equivalent linear filters that compute Newton–Raphson on the existing network, through the basin delimiter, we figure out constraints guaranteeing quadratic convergence speed in the diode clipper. Further application to a ring modulator circuit does not lead to comparably useful constraints for quadratic convergence; however, also in this circuit, the basin delimiter has a magnitude roughly proportional to the number of iterations needed by the solver to find a solution. Together, such case studies foster refinement and generalization of this tool as a speed predictor, with potential application to the design of virtual analogue systems for real-time digital audio effects.

Explicit Fixed-Point Computation of Nonlinear Delay-Free Loop Filter Networks

An iterative method is proposed for the explicit computation of discrete-time nonlinear filter networks containing delay-free loops. The method relies on a fixed-point search of the signal values at every temporal step. The formal as well as numerical properties of fixed-point solvers delimit its applicability: On the one hand, the method allows for a reliable prediction of the frequency rates where the simulation is stable, while, on the other hand, its straightforward applicability is counterbalanced by low speed of convergence. Especially in presence of specific nonlinear characteristics, the use of a fixed-point search is limited if the real-time constraint holds. For this reason, the method becomes useful especially during the digital model prototyping stage, as exemplified while revisiting a previous discrete-time realization of the voltage-controlled filter aboard the EMS VCS3 analog synthesizer. Further tests conducted on a digital ring modulator model support the above considerations.

A generalized smith predictor for unstable time-delay SISO systems

In this work, a generalization of the Smith Predictor (SP) is proposed to control linear time-invariant (LTI) time-delay single-input single-output (SISO) systems. Similarly to the SP, the combination of any stabilizing output-feedback controller for the delay-free system with the proposed predictor leads to a stabilizing controller for the delayed system. Furthermore, the tracking performance and the steady-state disturbance rejection capabilities of the equivalent delay-free loop are preserved. In order to place this contribution in context, some modifications of the SP are revisited and recast under the same structure. The features of the proposed scheme are illustrated through simulations, showing a comparison with respect to the corresponding delay-free loop, which is here considered to be the ideal scenario. In order to emphasize the feasibility of this approach, a successful experimental implementation in a laboratory platform is also reported.

Dynamical Systems for Audio Synthesis: Embracing Nonlinearities and Delay-Free Loops

Many systems featuring nonlinearities and delay-free loops are of interest in digital audio, particularly in virtual analog and physical modeling applications. Many of these systems can be posed as systems of implicitly related ordinary differential equations. Provided each equation in the network is itself an explicit one, straightforward numerical solvers may be employed to compute the output of such systems without resorting to linearization or matrix inversions for every parameter change. This is a cheap and effective means for synthesizing delay-free, nonlinear systems without resorting to large lookup tables, iterative methods, or the insertion of fictitious delay and is therefor suitable for real-time applications. Several examples are shown to illustrate the efficacy of this approach.

Generalized Moog Ladder Filter: Part II–Explicit Nonlinear Model through a Novel Delay-Free Loop Implementation Method

One of the most critical aspects of virtual analog simulation of circuits for music production consists in accurate reproduction of their nonlinear behavior, yet this goal is in many cases difficult to achieve due to the presence of implicit differential equations in circuit models, since they naturally map to delay-free loops in the digital domain. This paper presents a novel and general method for non-iteratively implementing these loops in such a way that the linear response around a chosen operating point is preserved, the topology is minimally affected, and transformation of nonlinearities is not required. This technique is then applied to a generalized model of the Moog ladder filter, resulting in an implementation that outperforms its predecessors with only a modest computational load penalty. This digital version of the filter is shown to offer strong stability guarantees w.r.t. parameter variation, allows the extraction of different frequency response modes by simple mixing of individual ladder stage outputs, and is suitable for real-time sound synthesis and audio effects processing.

A Simple Ladder Realization of Maximally Flat Allpass Fractional Delay Filters

This brief proposes a new ladder structure for the Thiran fractional delay filter (i.e., maximally flat allpass fractional delay filter given by the Thiran approximation). The proposed ladder structure is based on a continued fraction representation. Although there exists a similar approach that was proposed by Tassart and Depalle, their structure is not realizable because of delay-free loops. On the other hand, we show that the proposed method avoids generating delay-free loops and thus successfully yields a realizable ladder structure for the Thiran fractional delay filter in a very simple form.

Modeling of the EMS VCS3 Voltage-Controlled Filter as a Nonlinear Filter Network

A previously developed nonlinear differential equation system, representative of the diode-based voltage-controlled filter on board of the EMS VCS3 analog synthesizer, can be modeled in terms of a filter network. The properties of this network can be converted block-by-block into a discrete-time version of the same model, in which the relationships existing between the system variables take the form of delay-free loops. In the presence of such loops, the discrete-time filter network is solved by iterating over a fixed-point scheme. Simulations running at 176.4 kHz show reasonable fidelity of the model in the regimes of interest, as well as shorter computation time compared to previous implementations of the same system. While reducing aliasing, such a sampling rate optimizes also the computation time as it requires fewer fixed-point iterations. The new model represents an improved alternative to previously developed voltage-controlled filter designs in the field of virtual analog.

Computation of Delay-Free Nonlinear Digital Filter Networks: Application to Chaotic Circuits and Intracellular Signal Transduction

A method for the computation of nonlinear digital filter networks containing delay-free loops is proposed. By preserving the topology of the network this method permits the inspection of all signals flowing across the loops. Furthermore, it enables to discretize analog filter networks described by nonlinear ordinary differential equation systems on a block-by-block basis, in such a way that ad hoc analog-to-digital maps can be individually applied to each filter. A nonlinear implicit system must be solved at every computation step using iterative methods, holding certain sufficient conditions for the existence of the solution. This condition is in algebraic relationship with the causality and structure of the network and its filtering blocks. The proposed method can be straightforwardly applied to the computation of several interconnected networks. This property adds modularity to the procedure, and enables to handle changes in the network topology. Examples are presented showing the applicability of the method to the computation of the Chua-Felderhoff RLC circuit and to the dynamic simulation of a known intracellular signal transduction model of circadian cycles in Drosophila melanogaster.

Interconnection of state space structures and wave digital filters

State space structures (SSSs) and wave digital filters (WDFs) are two major paradigms for the realization of digital filters. Both approaches are well established, but there are no proven methods for a mixed design of digital filters consisting of parts which are realized as SSSs and parts realized as WDFs. This contribution shows how to add a wave port to the conventional SS representation. This wave port allows to interconnect SSSs and WDFs without creating delay-free loops. Such interconnections allow to build discrete-time structures by reusing existing designs from both classes of digital filters.

Evolutionary synthesis of digital filter structures using genetic programming

This paper presents a synthesis method for infinite-impulse response (IIR) digital filter structures using genetic programming with automatically defined functions (GP-ADF). In the proposed method, digital filter structures are represented as S-expressions with subroutines, which are written directly from the set of difference equations. This paper also shows the condition for the constructing the S-expressions that represent the filter structures without delay-free loops. Numerical examples synthesize two-filter structures: the low-coefficient sensitivity fourth-order filter structure and the low-output roundoff noise second-order filter structure.

Computation of linear filter networks containing delay-free loops, with an application to the waveguide mesh

A method that computes linear digital filter networks containing delay-free loops is proposed. Compared to existing techniques the proposed method does not require a rearrangement of the network structure, conversely it makes use of matrices describing this structure and specifying the connections between the filter blocks forming the network. For this reason the efficiency of the method becomes interesting when the filter blocks are densely interconnected. The Triangular Waveguide Mesh is an example of dense filter network: Using the proposed method we can compute a transformed, delay-free version of this mesh, obtaining simulations that are significantly more accurate compared to those provided by the traditional, explicitly computable formulation of the triangular mesh.

Implementation of frequency-warped recursive filters

Frequency-warped filters are beneficial in many audio applications because they inherently use nonuniform frequency resolution which may be matched so that it approximates the frequency representation of hearing. Warped transversal filters are immediately obtained by replacing the unit delays of a conventional filter with first-order allpass blocks. Implementation of warped recursive filters is a problem since those filters contain delay-free loops. A technique to implement any such filter without changing the coefficients or the structure of the filter is presented in this paper. In addition, a method to modify such a filter to a new structure where the delay-free loops are eliminated is introduced.

Elimination of delay-free loops in discrete-time models of nonlinear acoustic systems

Nonlinear acoustic systems are often described by means of nonlinear maps acting as instantaneous constraints on the solutions of a system of linear differential equations. This description leads to discrete-time models exhibiting noncomputable loops. We present a solution to this computability problem by means of geometrical transformation of the nonlinearities and algebraic transformation of the time-dependent equations. The proposed solution leads to stable and accurate simulations even at relatively low sampling rates.

Computable real lattice structures for cochlea-like digital filters

A synthesis algorithm for a pipelined lattice implementation of cochlea-like digital filters is presented, based upon the properties of real, lossless lattice synthesis of ARMA filters. The algorithm operates on a simplified characterization of elementary lattice sections of degree one or degree two. This leads to a structure that is recursively designed and for which each lattice is precisely implemented by a pair of complex conjugate transmission zeros via Richard's function extractions. Except for zeros of transmission on the unit circle, all other types and multiplicities are allowed. Necessary and sufficient conditions are derived for the degree-two lattices to guarantee computability, i.e., realizability with no delay-free loops. In addition to being suitable for VLSI realization, the structure enables a systematic cochlea assessment from the scattering ear parameters.

Complex reference networks to realize transfer functions with arbitrary transmission zeros

This paper proposes a method of realizing arbitrary complex coefficient transfer functions with complex double resistively terminated lossless filters as the reference network for complex wave digital filters (CWDFs). For this purpose, the Darlington synthesis method, which is known for realizing arbitrary real coefficient transfer functions, is expanded into the complex domain. Furthermore, a new construction method for a digital adapter without a delay-free loop is presented. Finally, a complex coefficient transfer function using some realization method is constructed and the validity of the proposed method is confirmed. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(2): 12–23, 1997

Design and analysis of bilinear digital ladder filters

Methods for designing bilinear switched-capacitor (SC) ladder filters are well known. However, they cannot be directly extended to digital filters because of the delay-free loops allowed in SC filters. In the present paper, design methods for low-sensitive bilinearly transformed voltage-current ladder filters are developed. The matrix approach for SC filter design is extended to digital filters, and the delay-free loops are eliminated. Design equations for general order low-pass, high-pass, bandpass, bandstop, and allpass filters are obtained in a simple closed form. The design procedure gives a canonical number of delays and a number of distinct multipliers equal to the number of reactive elements in the prototype. A number of test filters are analyzed with respect to magnitude sensitivity, magnitude response with respect to coefficient quantization, occurrence of limit cycles, stability, quantization noise levels, and structural complexity. The results are compared to wave digital filters (WDF), and cascade-form filters. It is found that bilinear digital ladder filters (BDLF) in most cases perform better than WDFs, while they at the same time have a simpler structure.

An improved technique for realization of switched-capacitor filters from LC networks

A systematic approach for the synthesis of switched-capacitor (SC) filters is presented based on (1) the nodal potential relation of passive doubly terminated LC networks (i.e., LC filters), represented in matrix form using a virtual vector to express the variables; (2) switched-capacitor fundamental blocks constructed with customary SC integrator circuit structures. The main feature of our new technique is its capability to realize SC filters without delay-free loops. This characteristic improves the transfer function estimation of the active components used. This method is well suited to the mass production of integrated SC filters using the programmable capacitor matrix (PCM), which can result in substantial savings in production time and costs. It is also shown that this method can be extended to directly transforming passive all-passes to SC all-passes.

Existence of structurally bounded wave digital filters for arbitrary transfer functions

Structurally bounded circuits such as properly designed wave digital filters (WDFs) are known to have low sensitivity in the passband. Existing structurally bounded WDFs cannot always realize arbitrary transfer functions. This paper shows first the existence of structurally bounded WDF sections which correspond to reactance E sections and Richards sections and then derives their circuit structures. Such a natural assumption is made in this derivation that multiplier coefficients having the same nominal value continue to have the same value even under finite wordlength condition. Interconnections of the proposed sections cause delay-free loops, which can be removed using Kuroda's identity. Thus, it is shown that such WDFs that can realize arbitrary transfer functions while inheriting the structural boundedness of prototype reactance filters really exist.

Synthesis of leapfrog‐type digital filters based on the distributed circuitsimulation

AbstractIt is known that a leapfrog‐type digital filter obtained by simulating an LC ladder filter terminated with resistors on both ends is a digital filter with a low coefficient sensitivity. In this paper, an analog circuit to be simulated is chosen as a distributed circuit in which the real frequency and the discrete variable correspond rigorously. By means of an equivalent transformation, a leapfrog‐type digital filter is constructed in which no delay free loop is generated. In this method, an LC ladder circuit expressed in terms of the Richard variable p is transformed to the circuit expressed in terms of the LDI variable γ. the digital filter is configured by simulation of the relationship between the node voltage and series branch current in each element of LC ladder circuit. However, if the configuration is based on general distributed LC ladder circuits, a delay‐free loop is always generated and configuration becomes unstable. As a solution to this problem, unit elements are introduced in this paper. It is shown that a realizable signal‐flow graph can always be obtained if stub circuits are used as the original circuit. the basic section necessary to circuit configuration is presented. Finally, as a practical example, the present method is used for design of a fourth‐order bandpass filter. the results are described and compared with those by other configurations.