Frequency-Zooming ARMA Modeling for Analysis of Noisy String Instrument Tones
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Frequency-Zooming ARMA Modeling for Analysis of Noisy String Instrument Tones Paulo A. A. Esquef Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology, P.O. Box 3000, FIN-02015 HUT, Espoo, Finland Email: [email protected]
Matti Karjalainen Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology, P.O. Box 3000, FIN-02015 HUT, Espoo, Finland Email: [email protected]
¨ ¨ Vesa Valim aki Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology, P.O. Box 3000, FIN-02015 HUT, Espoo, Finland Pori School of Technology and Economics, Tampere University of Technology, P.O. Box 300, FIN-28101 Pori, Finland Email: [email protected] Received 31 May 2002 and in revised form 5 March 2003 This paper addresses model-based analysis of string instrument sounds. In particular, it reviews the application of autoregressive (AR) modeling to sound analysis/synthesis purposes. Moreover, a frequency-zooming autoregressive moving average (FZ-ARMA) modeling scheme is described. The performance of the FZ-ARMA method on modeling the modal behavior of isolated groups of resonance frequencies is evaluated for both synthetic and real string instrument tones immersed in background noise. We demonstrate that the FZ-ARMA modeling is a robust tool to estimate the decay time and frequency of partials of noisy tones. Finally, we discuss the use of the method in synthesis of string instrument sounds. Keywords and phrases: acoustic signal processing, spectral analysis, computer music, sound synthesis, digital waveguide.
1. INTRODUCTION It has been known for quite a long time that a free vibrating body may generate a sound that is composed of damped sinusoids, assuming valid the hypothesis of small perturbations and linear elasticity [1]. This behavior has motivated the use of a set of controllable sinusoidal oscillators to artificially emulate the sound of musical instruments [2, 3, 4]. As for analysis purposes, tools like the short-time Fourier transform (STFT) [5] and discrete cosine transform (DCT) [6] have been widely employed since these transformations are based on projecting the input signal onto an orthogonal basis consisting of sine or cosine functions. An appealing idea, which is also based on resonant behavior of vibrating structures, consists in letting the resonant behavior be parametrically modeled by means of resonant filters (all-pole or pole-zero) excited by a source signal. For short duration excitation signals and filters parameterized by a few coefficients, such a source-filter model implies a compact representation for sound sources. Furthermore, para-
metric modeling of linear and time-invariant systems finds applications in several areas of engineering and digital signal processing, such as system identification [7], equalization [8], and spectrum estimation [9]. The moving-average (MA), the autoregressive (AR), and autoregressive movingaverage (ARMA) models are among the most widely used ones. Indeed, there exists an extensive l
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