We propose a new analog filter discretization method that is useful for discretizing systems with features near or above the Nyquist limit. A conformal mapping approach is taken, and we introduce the peaking conformal map and shelving conformal map. The proposed method provides a close match to the original analog frequency response below half the sampling rate and is parameterizable, order preserving, and agnostic to the original filter's order or type. The proposed method should have applications to dis-cretizing filters that have time-varying parameters or need to be implemented across many different sampling rates.

High-order dispersive delays are typically implemented as a cascade of second-order allpass sections, making them difficult to modify in real time and prone to numerical issues. Here, such filters are implemented using a modal architecture-a parallel combination of resonant filters tuned to a set of mode frequencies determined by the desired dispersion characteristic. Two configurations are presented, a “dispersive delay filter” in which a single dispersive arrival is produced, and a “dispersive comb filter” in which a series of increasingly dispersive echoes of the input signal are generated. The parallel structure has computational and numerical advantages. Simple, closed-form techniques for designing the needed mode filter frequencies and amplitudes according to the desired dispersion and equalization are presented, and applied to spring reverberator emulation and decorrelation filter design.

This letter describes a numerical algorithm for synthesizing optimal low-sidelobe beampatterns. The pattern synthesis problem is formulated as a constrained optimization that minimizes the spatially weighted energy arriving at the array subject to unit gain in the look direction and a constant sidelobe level. The weighting takes on the form of a sensor correlation matrix, parameterized between uncorrelated sensor noise and correlated signal arrival terms according to a balancing factor. Setting the balancing factor to its extreme values of 0 and 1, produces, respectively, the optimal Riblet-Chebyshev and Delay-and-Sum type beamformers. In between, the method generates a low-sidelobe beamformer with varying beamwidths that provides a trade-off between White Noise Gain and Directivity Index. Simulation examples demonstrate optimal and intermediate designs for uniform and non-uniform sensor spacings.

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.

Hybrid reverberators combine convolutional and feedback delay network (FDN) reverberators to exactly reproduce the psycoacoustically important reverberation impulse response onset while efficiently generating the needed late-field characteristics. A simple method for crossfading between the convolutional and FDN components of a hybrid reverberation is presented. It involves forming the convolutional impulse response as the windowed difference between the desired and FDN impulse responses. In this way, arbitrary windows may be applied to the convolution and FDN components of the hybrid reverberator impulse response in forming the cross-fade. For applications in which the impulse response onset depends on a parameter, a singular value decomposition is used to develop a low-rank approximation to the tabulated reverberation impulse response onsets. The approximation is the combination of a few fixed impulse responses, with parameter-dependent weights. An emulation of the EMT 140 plate reverberator is presented as a sample application.

The short-time Fourier transform (STFT) based spectrogram is commonly used to analyze the time-frequency content of a signal. Depending on window size, the STFT provides a trade-off between time and frequency resolutions. This paper presents a novel method that achieves high resolution simultaneously in both time and frequency. We extend Probabilistic Latent Component Analysis (PLCA) to jointly decompose two spectrograms, one with a high time resolution and one with a high frequency resolution. Using this decomposition, a new spectrogram, maintaining high resolution in both time and frequency, is constructed. Termed the “super-resolution spectrogram”, it can be particularly useful for speech as it can simultaneously resolve both glottal pulses and individual harmonics.

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.

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.

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.

The problem of determining the relative positions of speakers in an array given noisy measurements of inter-speaker ranges is considered. A closed-form position estimator which minimizes a weighted equation error norm is presented. The information inequality is used to bound the position estimate mean square error and to gauge the accuracy of the closed-form estimator, which is shown to be nearly efficient.

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.

A lower bound on mean-square-estimate error is derived as an instance of the covariance inequality by concatenating the generating matrices for the Bhattacharyya and Barankin bounds; it represents a generalization of the Bhattacharyya, Barankin, Cramer-Rao, Hammersley-Chapman-Robbins, Kiefer, and McAulay-Hofstetter bounds in that all of these bounds may be derived as special cases. The bound is applicable to biased estimates' of functions of a multidimensional parameter. Termed the hybrid Bhattacharyya-Barankin bound, it may be written as the sum of the mth-order Bhattacharyya bound and a nonnegative term similar in form to the rth-order Hammersley-Chapman-Robbins bound. It is intended for use when small-error bounds, such as the Cramer-Rao bound, may not be tight; unlike many large-error bounds, it provides a smooth transition between the small-error and large-error regions. As an example application, bounds are placed on the variance of unbiased position estimates derived from passive array measurements. Here, the hybrid Bhattacharyya-Barankin bound enters the large-error region at a larger SNR and, in the large-error region, is noticeably greater than either the Hammersley-Chapman-Robbins or the Bhattacharyya bounds.>

A method for characterizing a signal's narrowband content is presented. The scenario of interest is one in which the narrowband components are nonstationary, but persist for long periods of time, such as is found in many passive sonar and vibration analysis applications. This so-called frequency tracking problem-generally treated as a one-dimensional problem-is seen to reduce to that of extracting line structures from gray-scale images called spectrograms. Accordingly, the frequency tracking method presented is simply a line extraction technique based on standard image processing and machine vision methods and tuned to the type of line features and noise background present in spectrograms. Initial experiments using real sonar data indicate that the method outperforms currently used methods and has comparable computational cost.>

The simultaneous estimation of time-of-arrival, damping, frequency, and amplitude parameters of sums of exponentials in white Gaussian noise is cast in terms of a constrained optimization problem. The constraints derive from smoothness conditions at the signal onset (i.e. the signal starts at zero and one of its derivatives may also be zero at the signal onset). Without such constraints, time-of-arrival estimates are shown to be, at best, limited to sampling times, and in the case of the data window beginning after the signal onset, all time-of-arrival estimates are equally valid. Applying the initial condition constraints, an error norm is derived which varies continuously with the time-of-arrival. For this constrained error norm, maximum likelihood estimator performance is demonstrated through simulation of the response of two-pole highpass and bandpass systems to a stepped sinusoid.>

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.>

A lower bound on mean square estimate error is derived as an instance of the covariance inequality by concatenating the generating matrices for the Bhattacharyya and Barankin bounds; it represents a generalization of the Bhattacharyya (1946), Barankin (1949), Cramer-Rao (1945), Hammersley-Chapman-Robbins (1950, 1951), Kiefer (1952), and McAulay-Hofstetter (1971) bounds in that all of these bounds can be derived as special cases. The bound is applicable to biased estimates of vector-valued functions of a vector-valued parameter. Termed the hybrid Bhattacharrya-Barankin bound, it can be written as the sum of the mth-order Bhattacharrya bound and a nonnegative correction term dependent on a set of r so-called test points. It is intended for use when small-error bounds, such as the Cramer-Rao bound, may not be tight; unlike many large-error bounds, it is relatively easy to compute while providing a smooth transition between the small- and large-error regions.>

Estimating the position of a source given the signals received at a sensor array is a basic problem in the fields of acoustics and geophysics. Of interest is how to place the sensors so that the source can be most accurately localized. The Cramer-Rao bound is used as a gauge of the accuracy of source position estimators. A simple geometric interpretation of the Cramer-Rao bound is developed, and subject to geometric constraints, sensor arrangements minimizing bound variance are found. Carter's optimal arrays yielding minimum range, bearing, and position bound variance subject to the constraint that the sensors lie along a line segment are reproduced without tedious algebraic manipulations or computer-aided maximization. The bound variance of the optimal arrays is seen to be roughly a factor of three smaller than that of commonly used uniformly spaced arrays.>

Information inequality is used to place lower bounds on the variance of unbiased spectrum-based range estimates. Bounds are also presented for the case of imprecisely known source spectrum shape and signal propagation model. A closed-form range estimator is developed for the case of deep ocean acoustic waves, and is shown to approach the information inequality for signals with large in-band signal-to-noise ratio. It is shown that spectrum-based range estimates are more accurate than those based on signal arrival times for sources far (at least a few array lengths) from a sensor array.>

The problem of estimating parameters theta that determine the mean mu ( theta ) of a Gaussian-distributed observation X is considered. It is noted that the maximum-likelihood (ML) estimate-in this case, the least-squares estimate-has desirable statistical properties but can be difficult to compute when mu ( theta ) is a nonlinear function of theta . An estimate formed by combining ML estimates based on subsections of the data vector X is proposed as a computationally inexpensive alternative. It is shown that this alternative estimate, termed the divide-and-conquer estimate, has ML performance in the small-error region when the data vector X is appropriately subdivided. As an example application, an inexpensive range-difference-based position estimator is derived and shown by Monte-Carlo simulation to have small-error-region mean-square error equal to the Cramer-Rao bound.>

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.