Lutz Trautmann

Nonlinear acoustic echo cancellation with 2nd order adaptive Volterra filters

Acoustic echo cancellers in todays speakerphones or video conferencing systems rely on the assumption of a linear echo path. Low-cost audio equipment or constraints of portable communication systems cause nonlinear distortions, which limit the echo return loss enhancement achievable by linear adaptation schemes. These distortions are a super-position of different effects, which can be modelled either as memoryless nonlinearities or as nonlinear systems with memory. Proper adaptation schemes for both cases of nonlinearities are discussed. An echo canceller for nonlinear systems with memory based on an adaptive second order Volterra filter is presented. Its performance is demonstrated by measurements with small loudspeakers. The results show an improvement in the echo return loss enhancement of 7 dB over a conventional linear adaptive filter. The additional computational requirement for the presented Volterra filter is comparable to that of existing acoustic echo cancellers.

Digital sound synthesis by physical modeling using the functional transformation method

1. Introduction.- 2. Sound-Based Synthesis Methods.- 1 Wavetable synthesis.- 1.1 Looping.- 1.2 Pitch shifting.- 1.3 Enveloping.- 1.4 Filtering.- 2 Granular synthesis.- 2.1 Asynchronous granular synthesis.- 2.2 Pitch-synchronous granular synthesis.- 3 Additive synthesis.- 4 Subtractive synthesis.- 5 FM synthesis.- 6 Combinations of sound-based synthesis methods.- 3. Physical Description of Musical instruments.- 1 General notation.- 2 Subdivision of a musical instrument into vibration generators and a resonant body.- 2.1 Division of stringed instruments into single strings and the resonant body.- 2.1.1 Construction of stringed instruments.- 2.1.2 Fixed strings filtered with the resonant body.- 2.1.3 Strings terminated with independent impedances.- 2.1.4 Strings terminated with an impedance network.- 2.2 Division of a kettle drum into a membrane and the kettle.- 2.2.1 Construction of drums.- 2.2.2 Drum body simulation by modifying the physical parameters of the membrane.- 2.2.3 Drum body simulation by room acoustic simulation with the membrane as vibrating boundary.- 3 Physical description of string vibrations.- 3.1 Longitudinal string vibrations.- 3.2 Torsional string vibrations.- 3.3 Transversal string vibrations.- 3.3.1 Basic linear model.- 3.3.2 Nonlinear excitation functions.- 3.3.3 Nonlinear PDE with solution-dependent coefficients.- 4 Physical description of membrane vibrations.- 4.1 Bending membrane vibrations.- 5 Physical description of resonant bodies.- 6 Chapter summary.- 4. Classical Synthesis Methods Based on Physical Models.- 1 Finite difference method.- 1.1 FDM applied to scalar PDEs.- 1.2 FDM applied to vector PDEs.- 2 Digital waveguide method.- 2.1 Digital waveguides simulating string vibrations.- 2.2 Digital waveguide meshes simulating membrane vibrations.- 3 Modal synthesis.- 4 Chapter summary.- 5. Functional Transformation Method.- 1 Fundamental principles of the FTM.- 1.1 FTM applied to scalar PDEs.- 1.1.1 Laplace transformation.- 1.1.2 Sturm-Liouville transformation.- 1.1.3 Transfer function model.- 1.1.4 Discretization of the MD TFM.- 1.1.5 Inverse Sturm-Liouville transformation.- 1.1.6 Inverse z-transformation.- 1.2 FTM applied to vector PDEs.- 1.2.1 Laplace transformation.- 1.2.2 Sturm-Liouville transformation.- 1.2.3 Transfer function model.- 1.2.4 Discretization of the MD TFM.- 1.2.5 Inverse Sturm-Liouville transformation.- 1.2.6 Inverse z-transformation.- 1.3 FTM applied to PDEs with nonlinear excitation functions.- 1.4 FTM applied to PDEs with solution-dependent coefficients.- 1.5 Stability and simulation accuracy of the FTM.- 1.6 Section summary.- 2 Application of the FTM to vibrating strings.- 2.1 Transversal string vibrations described by a scalar PDE.- 2.2 Longitudinal string vibrations described by vector PDEs.- 2.2.1 Boundary conditions of second kind.- 2.2.2 Boundary conditions of third kind.- 2.2.3 Two interconnected strings.- 2.3 Transversal string vibrations with nonlinear excitation functions.- 2.3.1 Piano hammer excitation.- 2.3.2 Slapped bass.- 2.4 Transversal string vibrations with tension-modulated nonlinearities.- 3 Application of the FTM to vibrating membranes.- 3.1 Rectangular reverberation plate.- 3.2 Circular drum heads.- 4 Application of the FTM to resonant bodies.- 5 Chapter summary.- 6. Comparison of the Ftm with the Classical Physical Modeling Methods.- 1 Comparison of the FTM with the FDM.- 2 Comparison and combination of the FTM with the DWG.- 2.1 Comparison of the FTM with the DWG.- 2.2 Combination of the DWG with the FTM.- 2.2.1 Designing the loss filter.- 2.2.2 Designing the dispersion filter.- 2.2.3 Designing the fractional delay filter.- 2.2.4 Adjusting the excitation function.- 2.3 Limits of the combination.- 3 Comparison of the FTM with the MS.- 4 Chapter conclusions.- 7. Summary, Conclusions, and Outlook.

Digital sound synthesis of string instruments with the functional transformation method

The theory of multidimensional continuous and discrete systems is applied to derive a parametric description of musical sounds from a physical model of real or virtual string instruments. The mathematical representation of this model is given by a partial differential equation for a vibrating string. Suitable functional transformations with respect to time and space turn this partial differential equation into a multidimensional transfer function. It is the starting point for the derivation of a discrete-time system by classical analog to discrete transformations. The coefficients of this discrete model depend explicitly on the geometric properties and material constants of the underlying physical model. This ensures a meaningful behaviour of the discrete system under varying conditions and allows for an intuitive control by the user. Furthermore, the performance of real-time implementations is discussed. Finally, several extensions of this synthesis method for computer music applications are presented.

Multidimensional transfer function models

Transfer functions are a standard description of one-dimensional linear and time-invariant systems. They provide an alternative to the conventional representation by ordinary differential equations and are suitable for computer implementation. This article extends that concept to multidimensional (MD) systems, normally described by partial differential equations (PDEs). Transfer function modeling is presented for scalar and for vector PDEs. Vector PDEs contain multiple dependent output variables, e.g., a potential and a flux quantity. This facilitates the direct formulation of boundary and interface conditions in their physical context. It is shown how carefully constructed transformations for the space variable lead to transfer function models for scalar and vector PDEs. They are the starting point for the derivation of discrete models by standard methods for one-dimensional systems. The presented functional transformation approach is suitable for a number of technical applications, like electromagnetics, optics, acoustics and heat and mass transfer.

Physical modeling of drums by transfer function methods

Multidimensional (MD) physical systems are usually given in terms of partial differential equations (PDEs). Similar to one-dimensional systems, they can also be described by transfer function models (TFMs). In addition to including initial and boundary conditions as well as excitation functions exactly, the TFM can also be discretized in a simple way. This leads to suitable implementations for digital signal processors. Therefore it is possible to implement physics based digital sound synthesis algorithms derived from TFMs in real-time. This paper extends the previously presented solution for vibrating strings with one spatial dimension to two-dimensional drum models.

Multirate simulations of string vibrations including nonlinear fret-string interactions using the functional transformation method

The functional transformation method (FTM) is a well-established mathematical method for accurate simulations of multidimensional physical systems from various fields of science, including optics, heat and mass transfer, electrical engineering, and acoustics. This paper applies the FTM to real-time simulations of transversal vibrating strings. First, a physical model of a transversal vibrating lossy and dispersive string is derived. Afterwards, this model is solved with the FTM for two cases: the ideally linearly vibrating string and the string interacting nonlinearly with the frets. It is shown that accurate and stable simulations can be achieved with the discretization of the continuous solution at audio rate. Both simulations can also be performed with a multirate approach with only minor degradations of the simulation accuracy but with preservation of stability. This saves almost 80% of the computational cost for the simulation of a six-string guitar and therefore it is in the range of the computational cost for digital waveguide simulations.

Solution of vector partial differential equations by transfer function models

Transfer function models for the description of physical systems have recently been introduced to the field of multidimensional digital signal processing. They provide an alternative to the conventional representation by partial differential equations (PDE) and are suitable for computer implementation. This paper extends the transfer function approach to vector PDEs. They arise from the physical analysis of multidimensional systems in terms of potential and flux quantities. Expressing the resulting coupled PDEs in vector form facilitates the direct formulation of boundary and interface conditions in their physical context. It is shown how a carefully constructed transformation for the space variable leads to transfer function models for vector PDEs. They are the starting point for the derivation of discrete models by standard methods for one-dimensional systems. The presented functional transformation approach is suitable for a number of technical applications, like electromagnetics, optics, acoustics and heat and mass transfer.

Partial differential equation models for continuous multidimensional systems

The description of the continuous and discrete multidimensional (MD) models in current use has not yet reached the same state of maturity as for one-dimensional systems. To proceed in that direction, we investigate the connections between certain discrete partial differential equation (PDE) models (finite-difference models, transfer function models, MD wave digital filters). The starting points are potential-flux models that are the standard form of physics-based continuous MD systems. It is shown how certain matrix operations lead to various popular PDE models. The properties of the associated matrices provide the link to the discrete PDE models mentioned above. These investigations are presented in general form along with examples for an electrical transmission line and for acoustic wave propagation.

Digital sound synthesis by physical modelling

After recent advances in coding of natural speech and audio signals, the synthetic creation of musical sounds is also gaining importance. Various methods for waveform synthesis are currently used in digital instruments and software synthesizers. A family of new synthesis methods is based on physical models of vibrating structures (string, drum etc.) rather than on descriptions of the resulting waveforms. This article describes various approaches to digital sound synthesis in general and discusses physical modelling methods in particular. Physical models in the form of partial differential equations are presented. Then it is shown, how to derive discrete-time models which are suitable for real-time DSP implementation. Applications to computer music are given as examples.

Stable systems for nonlinear discrete sound synthesis with the functional transformation method

Discrete systems for digital sound synthesis derived with the functional transformation method (FTM) from physical models have recently been presented. For linear systems, the FTM solves the partial differential equation (PDE) describing the vibrating structure analytically. The algorithms obtained after discretization of the analytical solution preserve the inherent physical stability of the continuous system. For nonlinear physical models, as they occur in real musical instruments, the direct application of the FTM leads to an implicit continuous equation. This paper shows that discretization results in an explicit solution. Furthermore, the stability problems occurring after discretization of the nonlinear system are fixed by instantaneous energy considerations. For the example of a slapped transversal oscillating tightened string with frequency dependent loss terms an efficient algorithm is derived. It can also be applied to other types of interaction between a resonating body and its environment.