Inverse filter analysis of common harmonic structure on Specmurt using Riemann’s ζ function

Abstract

In this paper, based on the interesting relationship between the log‐spaced δ function sequence and Riemann’s ζ function, the analytical properties of the inverse filter of the common harmonic structure on specmurt analysis are discussed. Specmurt [S. Sagayama et. al., in Proc. SAPA (2004)] is a simple and efficient technique for the multi‐pitch analysis of polyphonic music signals. If all tones have the same harmonic pattern, the power spectrum on the log‐scaled frequency can be regarded as the convolution of the common harmonic structure and the distribution of fundamental frequencies. Based on the model, overtones are effectively suppressed by the inverse filtering of the common harmonic structure in specmurt. Thus, for the stable processing, analytic properties of the inverse filter are significant. Our new finding is that when the common harmonic structure is expressed as a log‐spaced δ function sequence with a particular kind of decay, the Fourier transform is exactly equal to Riemann’s ζ function. Interpreting several properties of Riemann’s ζ function in a signal processing context gives us the new perspectives like an explicit representation or cascaded decomposition of the inverse filter.