William J. Pielemeier
Gregory H. Wakefield
EECS Dept., Univ. of Michigan, 1301 Beal Ave., Ann Arbor, MI 48109
Time resolution, frequency resolution, and superposition requirements present a difficult trade-off in time-frequency representations for musical signals, both for individual instrument and ensemble analysis. Existing methods have significant limitations, particularly at low frequencies. A representation is developed in Cohen's class [L. Cohen, Proc. IEEE 77(7), 941--981 (1989)], which improves on existing methods for a large class of musical signals. The pseudo-Wigner distribution is extended to a proportional-bandwidth form well suited to the frequency resolution required for musical signals and musical frequency analysis, paralleling a constant-Q transform [J. Brown, J. Acoust. Soc. Am 89, 425--434 (1991)]. This has excellent resolution, but fails to provide even limited superposition. A kernel is designed for Cohen's class, based on a musical signal model, providing a limited superposition property while maintaining improved resolution when applied to this distribution. Favorable performance is demonstrated relative to constant-Q methods and spectrograms for both single instrument and ensemble cases, which are beyond the capabilities of phase vocoder and Fourier series-based methods.