Deconvolution of Log-Frequency Spectrum

Regarding $v(x)$ as the observed power spectral density function of a multipitch signal, the fundamental frequency distribution $u(x)$ can be restored via the deconvolution of the observed spectrum $v(x)$ with the common harmonic structure pattern $h(x)$ (in other words, inverse filtering $v(x)$ in respect to $h(x)$):

$$u(x)=h^{-1}(x)*v(x).$$ (2)

In the (inverse) Fourier domain, this relation is written as a division:

$$U(y)=\displaystyle\frac{V(y)}{H(y)},$$ (3)

where $U(y)$, $H(y)$ and $V(y)$ are the (inverse) Fourier transform of $u(x)$, $h(x)$ and $v(x)$, respectively. The fundamental frequency pattern $u(x)$ is then restored by

$$u(x)={\mathcal F}[\ U(y)\ ].$$ (4)

This process is briefly illustrated in Fig 3. The process is done over every short-time analysis frame and thus we finally obtain a piano-roll-like visual representation.

図 3: The overview of specmurt method.