Spectral Line Broadening with Transform Domain Additive Synthesis Adrian Freed adrian@cnmat.berkeley.edu CNMAT, UC Berkeley, 1750 Arch Street, Berkeley, CA 94709, (510) 643 9990 x 308 Abstract After a survey of inverse transform methods for the efficient synthesis of narrow-band and broad-band signals, a novel spectral line broadening technique is introduced for synthesis of pitch modulated noise signals. This new transform-domain approach is compared to the time-domain oscillator method with respect to their relative efficiency on modern processors Introduction: Noise in Musical Instrument Sounds The term “noise” is used to describe the perception of a multitude of features of sounds from musical instruments, for example: • Dense modes, e.g., cymbals • Additive “noise” from turbulence in blown instruments such as the flute or consonants in the voice. • Impulses from short-term interactions such as hammer strikes, string plucks, key and tone hole closure and openings. • Bandwidth broadening from non-linear mechanisms such as piano dampers, harpsichord quills, tampoura and the sarod jawari bridge. • Correlated or convolutional noise in blown instruments where a reed (or vocal fold) gates or modulates a turbulent noise source. This is also observed in bowed instruments and flue pipes. • Impulse bursts as found in maracas, cabasa, and washboard. • Non-linear oscillator noise generated within the oscillator itself (chaos). The Sum of Sinusoid+Residual models of McAulay/Quatieri, Serra/Smith, Depalle/Rodet, et al., have proved useful for modeling and coding short musical tones. The assumption of these models is that the residual is colored independently of sinusoidal parameter estimates. This assumption is invalid for most musical instruments so inadequate fusion of re-synthesized noise and sinusoidal components is often observed. This is especially troublesome when transformations are applied such as time scaling and pitch shifting (Laroche, 1993, Laroche and Dolson, 1997, Laroche, et al., 1993). The problem is that all forced oscillators (bowed strings, voice, reeds, trumpets, flue pipes, etc.) generate nearly-periodically modulated noise, not additive noise. A combination of a better understanding of the physics of these oscillatory mechanisms (Rodet, 1993, Rodet, 1995) and new methods in higher order statistics (Brillinger and Irizarry, 1998, Dubnov and Rodet, 1997), wavelets (Goodwin and Vetterli, 1996) and time series (Irizarry, 1998) are leading to better tools for multi- level decomposition of sounds into transient events, pitched and unpitched oscillations, convolutional noise and colored noise. These new models require efficient, real-time noise synthesis algorithms. This paper contributes an efficient implementation of one such algorithm for noise synthesis: spectral line broadening. Line Broadening Modulating the phase of a sinusoidal carrier with a random signal results in a narrow band noise source. This spectral broadening process has been used for decades in spread-spectrum radio frequency (RF) communications systems where it is usually implemented directly in the time domain. Musical applications of line broadening were explored by Risset and Wessel in the 1970’s (Risset and Wessel, 1982). With appropriate parameters for the noise amplitudes, sounds synthesized using spectral line broadening processes are perceived as similar to the noise found in voice and musical instruments such as flutes and flue pipes. Since the two noise generating mechanisms are quite different, it is interesting to consider what features the mechanisms have in common that may explain a similar percept. In the voice and aformentioned wind instruments, the noise process is the result of turbulence, the amplitude of which is dependent on air velocity, which is modulated by the nearly periodic primary oscillator. The fundamental frequency and partial amplitudes are not greatly influenced by the turbulence. This independence is a feature of the spectral line broadening process because of the use of a zero mean random phase modulation. In physical systems the amplitude of the primary oscillator and turbulent noise are both proportional to driving energy. The amplitude parameter of the line broadening spectral synthesis process conveniently adjusts the amplitude of both elements. This parameterization is a convenient starting point for more sophisticated musical instrument models that dose noise and partial energy according to frequency and driving force.