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Computing Modified Bessel functions with large Modulation Index for Sound Synthesis Applications
Abstract
Ordinary Bessel functions are a common function used when examining the spectral properties of frequency modulated signals, particularly in sound synthesis applications. Recently, it was shown that modified Bessel functions can also be used for sound synthesis. However, to limit the impact of aliasing distortion when using these functions, it is essential to set an upper limit on the frequency-dependent modulation index used when computing these functions. However, it can be impossible to do this beyond a certain threshold when using standard mathematical software tools such as Matlab, or the scientific toolbox of the Python language, because of numerical overflow issues. This short paper presents an approach to overcome this limitation using the MaxStar algorithm. Results are also presented to demonstrate the usefulness of this solution.
... In this paper we consider a fractional variant of the classical modified Bessel equation of order ν. As is well-known, the modified Bessel functions have many important applications for example in probability, physics and biology [24,27,30]. We investigate a modified Bessel equation in which the integer-order derivatives are replaced by Riemann-Liouville fractional derivatives. ...
In this note we propose a fractional generalization of the classical modified Bessel equation. Instead of the integer-order derivatives we use the Riemann-Liouville version. Next, we solve the fractional modified Bessel equation in terms of the power series and provide an asymptotic analysis of its solution for large arguments. We find a leading-order term of the asymptotic formula for the solution to the considered equation. This behavior is verified numerically and shows high accuracy and fast convergence. Our results reduce to the classical formulas when the order of the fractional derivative goes to integer values.
Techniques for the generation of bandlimited signals for applica-tion to digital implementations of subtractive synthesis have been researched by a number of authors. This paper hopes to contrib-ute to the variety of approaches by proposing a technique based on Frequency Modulation (FM) synthesis. This paper presents and explains the equations required for bandlimited pulse genera-tion using modified FM synthesis. It then investigates the rela-tionships between the modulation index and the quality of the reproduction in terms of authenticity and aliasing for a sawtooth wave. To determine the performance of this technique in com-parison to others two sets of simulation results are offered: the first computes the relative power of the non-harmonic compo-nents, and the second uses the Perceptual Evaluation of Audio Quality (PEAQ) algorithm. It is shown that this technique com-pares well with the alternatives. The paper concludes with sug-gestions for the direction of future improvements to the method.
The problem of practical realization of the optimal fixed-delay symbol-by-symbol detection algorithm is investigated. A new structure is developed which is fully parallel with a speedup factor of 2/sup D+1/ (compared to serial implementation), where D is the delay constraint. Through systematic reformulations of the algorithm, a number of simplifications are performed that avoid the complex and slow computations of exponentials and a large number of multiplications, all at the expense of a small lookup table and a number of simple operations involving additions, comparisons, and table lookups. A number of approximations are applied to this simplified parallel symbol (SPS) detector which lead to the derivation of suboptimal detectors. An interesting suboptimal detector so derived is identical in computations to the Viterbi detector. A comparison of the SPS detector and Viterbi detector shows that the former has a slightly better performance at low values of signal-to-noise ratio (SNR) and the latter has a lower complexity at higher values of SNR; otherwise, the two detectors are comparable in performance and complexity.< >
One common requirement in signal processing systems is the removal of a dc bias from a signal and a good dc blocking algorithm is the desirable tool for signal processing. There were two blockers introduced, namely, nonlinear phase fixed-point dc blocker and a general linear-phase dc blocker network, which are prove to be useful in various DSP applications. The nonlinear-phase although computational efficient, dc blocking filter achieves ideal operation when output data quantization is used. It also described that an alternate dc blocking filter exhibits a linear-phase frequency response at the expense of larger data memory.
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