In the context of efficient sound synthesis by digital waveguides, we present a novel methodology for joint modeling of string instrument body radiativity and driving-point bridge admittance functions, as obtained from experimental data. From our modeling framework, aimed at simulation of guitar and bowed string sound, here we focus on the body of the instrument and leave aside the strings. First, a modal decomposition of the measured bridge admittance is obtained by means of a novel frequency-domain algorithm for optimization of recursive digital filters in parallel form. Then, from extracted modal parameters, the radiativity, and admittance functions are modeled by projecting measurements over a common modal basis, enforcing passivity of the two-dimensional admittance model by means of semidefinite programming. We propose a formulation that enables the joint realization of bridge reflectance and sound radiativity as a lumped delay line termination in which a single bank of resonant filters is shared among all string reflection and body radiation outputs. Our approach provides efficient means to model two-dimensional (2-D) bridge reflectance, 2-D string-string coupling, sound radiation with an arbitrary number of outputs, and (implicitly) vibrational energy loss from the bridge transmittance to nonradiating modes and dissipation.

We present a technique to iteratively optimize poles of a recursive digital filter in parallel form. Only exposing the poles as the variables to optimize, we employ a linearly constrained gradient descent routine in which the numerical estimation of the error gradient involves first obtaining the zeros by projecting the target response over a basis of responses defined by the pole positions at a given step. Example fits are presented for exponentially decaying white noise and measured violin radiativity filters.

An acoustic reverberator consisting of a network of delay lines connected via scattering junctions is proposed. All parameters of the reverberator are derived from physical properties of the enclosure it simulates. It allows for simulation of unequal and frequency-dependent wall absorption, as well as directional sources and microphones. The reverberator renders the first-order reflections exactly, while making progressively coarser approximations of higher-order reflections. The rate of energy decay is close to that obtained with the image method (IM) and consistent with the predictions of Sabine and Eyring equations. The time evolution of the normalized echo density, which was previously shown to be correlated with the perceived texture of reverberation, is also close to that of the IM. However, its computational complexity is one to two orders of magnitude lower, comparable to the computational complexity of a feedback delay network and its memory requirements are negligible.

We provide a single channel speech enhancement method leveraging the harmonic structure of voiced speech. A sinusoidal model, based on the pitch of the speaker, is used to filter noisy speech and remove any noise components that lie between the harmonics. To remove noise that lie on each harmonic frequency, we use a noise estimation procedure that exploits spectral sparsity of voiced speech. By measuring the power spectrum at frequencies that correspond to the zero crossings of the windowing function, we can estimate the noise levels even in frames that have voiced speech. We also provide a constrained linear least squares formulation to reduce “musical noise” which arises from difficulty in estimating speech and noise power spectral densities. We show that our method yields high perceptual performance over existing methods, and can easily adapt to conditions in which the noise characteristics are constantly changing.

Classical geometric waveforms used in virtual analog synthesis suffer from aliasing distortion when simple sampling is used. An efficient antialiasing technique is based on expressing the waveforms as a filtered sum of time-shifted approximately bandlimited polynomial-spline basis functions. It is shown that by optimizing the coefficients of the basis function so that the aliasing distortion is perceptually minimized, the alias-free bandwidth of classical waveforms can be expanded. With the best of the case examples given here, the generated impulse-train and sawtooth waveform are alias-free up to fundamental frequencies over 10 kHz when the sampling rate is 44.1 kHz.

The first artificial reverberation algorithms were proposed in the early 1960s, and new, improved algorithms are published regularly. These algorithms have been widely used in music production since the 1970s, and now find applications in new fields, such as game audio. This overview article provides a unified review of the various approaches to digital artificial reverberation. The three main categories have been delay networks, convolution-based algorithms, and physical room models. Delay-network and convolution techniques have been competing in popularity in the music technology field, and are often employed to produce a desired perceptual or artistic effect. In applications including virtual reality, predictive acoustic modeling, and computer-aided design of acoustic spaces, accuracy is desired, and physical models have been mainly used, although, due to their computational complexity, they are currently mainly used for simplified geometries or to generate reverberation impulse responses for use with a convolution method. With the increase of computing power, all these approaches will be available in real time. A recent trend in audio technology is the emulation of analog artificial reverberation units, such as spring reverberators, using signal processing algorithms. As a case study we present an improved parametric model for a spring reverberation unit.

An efficient allpass filter design method is introduced to match the dispersion characteristics of vibrating stiff strings. The proposed method designs an allpass filter in cascaded biquad form directly from the target group delay, placing the poles at frequencies at which the group delay area function achieves odd integer multiples of ¿, and fixing the pole radii according to a smoothness parameter. The pole frequencies are seen to be roots of quartic polynomials, and an efficient approximation to the desired roots is provided. Design examples show the method to outperform a previous closed-form design. Furthermore, the proposed method can achieve an arbitrarily wide bandwidth of good approximation by increasing the filter order, as the method is numerically robust and yields stable allpass filters.

The linear coupling of two strings or of a single string vibrating in two orthogonal polarizations leads to two observable phenomena: two-stage decay and beating. In this paper, we present methods for accurately measuring and modeling the lower partials of a recorded guitar tone, where coupling effects are most audible. These estimated parameters are then used for accurate resynthesis in a hybrid modal/waveguide model. We make use of the fact that two-stage decay occurs in analyzed tones to allow direct measurement of sinusoidal decay rates. A traditional iterative optimization algorithm is explored and found to be most effective in the special case when only beating occurs. Sound examples are provided on the Web.

One of the challenges in virtual analog synthesis is avoiding aliasing when generating classic waveforms such as sawtooth and square wave which have theoretically infinite bandwidth in their ideal forms. The human auditory system renders a certain amount of aliasing inaudible, which allows room for finding cost-effective algorithms. This paper suggests efficient algorithms to reduce the aliasing using low-order fractional delay filters in the framework of bandlimited impulse train (BLIT) synthesis. Examining Lagrange, B-spline interpolators and allpass fractional delay filters, optimized methods will be discussed for generating classic waveforms (sawtooth, square, and triangle). Techniques for generating more complicated harmonics such as pulse width modulation, hard-sync, and super-saw are also presented. The perceptual evaluation is performed by comparing the threshold of hearing and masking curve of oscillators with their aliasing levels. The result shows that the BLIT using the computationally efficient third-order B-spline generates waveforms that are perceptually free of aliasing within practically used fundamental frequencies.

The 16 papers in this special issue focus on virtual audio effects and musical instruments.

An efficient approach to the generation of classical synthesizer waveforms with reduced aliasing is proposed. This paper introduces two new classes of polynomial waveforms that can be differentiated one or more times to obtain an improved version of the sampled sawtooth and triangular signals. The differentiated polynomial waveforms (DPW) extend the previous differentiated parabolic wave method to higher polynomial orders, providing improved alias-suppression. Suitable polynomials of order higher than two can be derived either by analytically integrating a previous lower order polynomial or by solving the polynomial coefficients directly from a set of equations based on constraints. We also show how rectangular waveforms can be easily produced by differentiating a triangular signal. Bandlimited impulse trains can be obtained by differentiating the sawtooth or the rectangular signal. An objective evaluation using masking and hearing threshold models shows that a fourth-order DPW method is perceptually alias-free over the whole register of the grand piano. The proposed methods are applicable in digital implementations of subtractive sound synthesis.

This paper presents a procedural approach to derive nonlinear filters from schematics of audio circuits for the purpose of digitally emulating analog musical effects circuits in real time. This work, the first in a two-part series, extends a well-known efficient nonlinear continuous-time state-space formulation for physical modeling of musical acoustics to real-time modeling of nonlinear circuits. Rules for applying the formulation are given, as well as a procedure to derive simulation parameters automatically from circuit netlists. Furthermore, a related nonlinear discrete-time state-space algorithm is proposed to alleviate problems in solving particular circuit configurations. These methods were devised to solve non-convergence problems in the simulation of strongly saturated, nonparametric guitar distortion circuits such as the saturating diode clipper, which is presented as an example derivation. Experimental considerations and sonic performance on various other circuits will be presented in a subsequent paper.

A generative model of a human voice is presented, based on many pseudo-physical considerations. For robustness, observation noise is also included in the model. An EM-algorithm framework for inference and learning is then described. An instance of approximate inference and subsequent learning presented allows an extraction of voice parameter which can be used for structured coding application. This set of parameters allows a great amount of compression as well as the flexibility in making modification to pitch, duration and breathiness, noise-free synthesis compared to other non-parametric approaches.

In this paper, a simple family of explicit two-step finite difference methods for solving the classical equation of motion of a thin plate is examined. This family depends on several free parameters, and special attention is paid to the stability properties of these schemes, computational issues, and, in particular, the reduction of directional numerical dispersion. Numerical results, employing frequency warping techniques, are presented

Due to its simplicity and accuracy, quadratic peak interpolation in a zero-padded fast Fourier transform (FFT) has been widely used for sinusoidal parameter estimation in audio applications. We propose, as its natural extension, a method to estimate the first order amplitude and frequency modulation rates of time-varying sinusoidal components, as well as to correct biases in conventional amplitude, frequency and phase estimates. We derive exact formulas for Gaussian windows and obtain approximate formulas for often-used windows by introducing a simple window adaptation scheme. Experimental results show the average estimation biases of the AM and FM rates with a 30 ms Hann window are below 1% for typical AM/FM rates in speech.

The focus of the paper is to hide information in sinusoidal audio representations. Watermarks are embedded by introducing quantization index modulation (QIM) in frequencies of sinusoids. The frequency shifts due to QIM are controlled to be less than one-thirtieth Bark, which requires high accuracy in frequency estimation for watermark decoding. Therefore, the paper presents a hybrid multiple frequency estimator which consists of spectral interpolation followed by iterative square error minimization. The frequency estimator is tested in watermarking both computer generated sounds and actual recordings. The watermarks are intended to survive MP3 compression. Selected experiments are documented and discussed.

Writing on dirty paper, a mathematical model for watermarking, is examined and extended to allow simultaneous watermarking at multiple rates. Based on the extended model, it is argued that the optimal watermarking strategy is to share the power. The argument is further supported by simulations of joint spread spectrum and quantization watermarking.

This paper proposes to watermark parametric representations for synthetic audio. Our watermark system combines quantization index modulation at the encoder and maximum likelihood parameter estimation at the decoder. To guarantee error-free data hiding under expected types of attacks, knowledge of Fisher information and Cramer-Rao bounds is applied to the system design. Experiments show that, merely by quantizing the frequency of sinusoidal tones, one can achieve 50 b/s of data hiding that is robust to perceptually shaped additive attacks such as an MP3 compression.

We present a method to analyze the streaming sound output of a real acoustic violin and infer the control inputs used by the musician. In particular, we extract the following parameters: which note was played, which string it was played on, whether the instrument was bowed or plucked and the location of the bowing or plucking point from a discrete set. The approach we use, pattern classification of the short-time Fourier transform (STFT) frames of the sound signal, is general enough to enable application to almost any musical instrument.

In this paper, some simple families of explicit two-step finite difference methods for solving the wave equation in two and three spatial dimensions are examined. These schemes depend on several free parameters, and can be associated with so-called interpolated digital waveguide meshes. Special attention is paid to the stability properties of these schemes (in particular the bounds on the space-step/time-step ratio) and their relationship with the passivity condition on the related digital waveguide networks. Boundary conditions are also discussed. An analysis of the directional numerical dispersion properties of these schemes is provided, and minimally directionally-dispersive interpolated digital waveguide meshes are constructed.

Digital waveguides are generalized to the multivariable case with the goal of maximizing generality while retaining robust numerical properties and simplicity of realization. Multivariable complex power is defined, and conditions for "medium passivity" are presented. Multivariable complex wave impedances, such as those deriving from multivariable lossy waveguides, are used to construct scattering junctions which yield frequency dependent scattering coefficients which can be implemented in practice using digital filters. The general form for the scattering matrix at a junction of multivariable waveguides is derived. An efficient class of loss-modeling filters is derived, including a rule for checking validity of the small-loss assumption. An example application in musical acoustics is given.

In this paper, we present a system of sines+transients+noise modeling techniques that dynamically switches between parametric representations and transform coding based representations. The sines and noise are represented by parametric models using multiresolution sinusoidal modeling and Bark-band noise modeling, respectively. The transients are modeled by short regions of transform coding. In addition, new methods are presented for selection and quantization of sinusoidal trajectories based on trajectory length and signal-to-masking thresholds. This system is useful for both low bitrate audio coding (20-40 kbps) and compressed-domain processing, such as time-scale modification.

Use of a bilinear conformal map to achieve a frequency warping nearly identical to that of the Bark frequency scale is described. Because the map takes the unit circle to itself, its form is that of the transfer function of a first-order allpass filter. Since it is a first-order map, it preserves the model order of rational systems, making it a valuable frequency warping technique for use in audio filter design. A closed-form weighted-equation-error method is derived that computes the optimal mapping coefficient as a function of sampling rate, and the solution is shown to be generally indistinguishable from the optimal least-squares solution. The optimal Chebyshev mapping is also found to be essentially identical to the optimal least-squares solution. The expression 0.8517[arctan(0.06583fs)]/sup 1/2/-0.916 is shown to accurately approximate the optimal allpass coefficient as a function of sampling rate f/sub s/ in kHz for sampling rates greater than 1 kHz. A filter design example is included that illustrates improvements due to carrying out the design over a Bark scale. Corresponding results are also given and compared for approximating the related "equivalent rectangular bandwidth (ERB) scale" of Moore and Glasberg (ACTA Acustica, vo.82, p.335-45, 1996) using a first-order allpass transformation. Due to the higher frequency resolution called for by the ERB scale, particularly at low frequencies, the first-order conformal map is less able to follow the desired mapping, and the error is two to three times greater than the Bark-scale case, depending on the sampling rate.

We describe a computationally efficient method of generating more accurate sinusoidal parameters {amplitude, frequency, phase} from a wideband polyphonic audio source in a multiresolution, non-aliased fashion. This significantly improves upon previous work of sinusoidal modeling that assumes a single-pitched monophonic source, such as speech or an individual musical instrument, while using approximately the same number of sinusoids. In addition to a more general analysis, we can now perform high-quality modifications such as time-stretching and pitch-shifting on polyphonic audio with ease.

The feedback delay network (FDN) has been proposed for digital reverberation, The digital waveguide network (DWN) is also proposed with similar advantages. This paper notes that the commonly used FDN with an N/spl times/N orthogonal feedback matrix is isomorphic to a normalized digital waveguide network consisting of one scattering junction joining N reflectively terminated branches. Generalizations of FDNs and DWNs are discussed. The general case of a lossless FDN feedback matrix is shown to be any matrix having unit-modulus eigenvalues and linearly independent eigenvectors. A special class of FDNs using circulant matrices is proposed. These structures can be efficiently implemented and allow control of the time and frequency behavior. Applications of circulant feedback delay networks in audio signal processing are discussed.

The restoration of a band-limited signal which has undergone amplitude clipping is studied. This problem can be viewed as that of recovery from signal drop-outs (missing samples over an interval of time), with the extrapolated signal constrained to lie outside the clipping interval during the drop out. If the signal is oversampled, and the clipping threshold moderate, a unique reconstruction often results from application of signal matching and bandwidth constraints. More generally, however, candidate reconstructions are seen to lie on or inside a polyhedron in the space of sampled signals. In contrast to the case of extrapolation through missing samples, upper and lower limits typically can be placed on the reconstructed signal at every sample point. In light of this finding, methods for choosing a unique reconstruction are discussed. Finally, the case of noise is considered.>

The problem of estimating the location of a radiating source from range difference measurements taken from a passive, stationary array is discussed. A new closed-form source location estimator, termed the Spherical Interpolation Estimator, is presented and analyzed. The location estimates produced by the Spherical Interpolation Estimator are approximate minimizers of a weighted equation error norm, and are shown to approach the maximum likelihood source location estimator.

This paper proves global convergence of the Constant-Modulus Algorithm (CMA) for the case of a real channel when the model order is equal to or greater than that of the channel (the so-called "model-complete" case). The analysis is based on an exact fourth-order Taylor series representation for the cost function minimized asymptotically by the CMA.

Two algorithms are proposed for the continuous tracking of multipath delay. The methods use an adaptive delay line interpolated by a first-order filter. The interpolation coefficient is explicitly estimated using the recursive Gauss-Newton algorithm.

A digital resampling method is proposed which allows non-uniform and time-varying resampling. The method is based on interpolated look-up in a large table of filter coefficients. One filter table handles all conversion factors. Formulas are given for determining the size of look-up table needed for a given precision requirement.

An adaptive notch filter is derived by using a general prediction error framework. The proposed infinite impulse response filler has a special structure that guarantees the desired transfer characteristics. The filter coefficients are updated by a version of the recursive maximum likelihood algorithm. The convergence properties of the algorithm and its asymptotic behavior are discussed, and its performance is evaluated by simulation results.