Untangling Phase and Time in Monophonic Sounds
- 1
Abstract
We are looking for a mathematical model of monophonic sounds with independent time and phase dimensions. With such a model we can resynthesise a sound with arbitrarily modulated frequency and progress of the timbre. We propose such a model and show that it exactly fulfils some natural properties, like a kind of timeinvariance, robustness against non-harmonic frequencies, envelope preservation, and inclusion of plain resampling as a special case. The resulting algorithm is efficient and allows to process data in a streaming manner with phase and shape modulation at sample rate, what we demonstrate with an implementation in the functional language Haskell. It allows a wide range of applications, namely pitch shifting and time scaling, creative FM synthesis effects, compression of monophonic sounds, generating loops for sampled sounds, synthesise sounds similar to wavetable synthesis, or making ultrasound audible.
- S. Owre, N. Shankar, J. M. Rushby and D. W. J. Stringer- Calvert, “The Prototype Verification System,” PVS System Guide, 2001.
- H. Thielemann, “Optimally Matched Wavelets,” PhD. Thesis, Universit?t Bremen, March 2006.
- G. Strang, “Eigenvalues of and Convergence of the Cascade Algorithm,” IEEE Transactions on Signal Processing, Vol. 44, 1996, pp. 233-238.
- I. Daubechies and W. Sweldens, “Factoring Wavelet Transforms into Lifting Steps,” Journal of Fourier Analysis and Applications, Vol. 4, No. 3, 1998, pp. 245-267.
- R. W. Hamming, “Digital Filters,” Signal Processing Series, Prentice Hall, Upper Saddle River, January 1989.
- S. P. Jones, “Haskell 98 Language and Libraries, the Revised Report,” 1998. http://www.haskell.org/definition/
- H. Thielemann, “Audio Processing Using Haskell,” DAFx: Conference on Digital Audio Effects, G. Evangelista and I. Testa, Eds., Federico II University of Naples, Italy, October 2004, pp. 201-206.
- S. P. Jones, R. Leshchinskiy, G. Keller and Manuel M. T. Chakravarty, “Harnessing the Multicores: Nested Data Parallelism in Haskell,” IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’08), 2008.
- D. C. Massie, “Wavetable Sampling Synthesis,” in Applications of Digital Signal Processing to Audio and Acoustics, Mark Kahrs and Karlheinz Brandenburg, Eds., pp. 311–341. Kluwer Academic Press, 1998.
- J. M. Clowning, “The Synthesis of Complex Audio Spectra by Means of Frequency Modulation,” Journal of the Audio Engineering Society, Vol. 21, No. 7, 1973, pp. 526-534.
- B. Lang, “Einbettungsverfahren für Netzwerkgleichungen,” Ph.D. Thesis, Universit?t Bremen, Germany, November 2002.
- U. Z?lzer, Ed., “DAFx: Digital Audio Effects,” John Wiley and Sons Ltd., Hoboken, February 2002.
- S. Roucos and A. M. Wilgus, “High Quality Timescale Modification for Speech,” Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 1985, pp. 493-496.
- J. Makhoul and A. El-Jaroudi, “Time-Scale Modification In Medium To Low Rate Speech Coding,” Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 1986, pp. 1705-1708.
- S. Disch and U. Z?lzer, “Modulation and Delay Line Based Digital Audio Effects,” Proceedings DAFx-99: Workshop on Digital Audio Effects, Trondheim, December 1999, pp. 5-8.