Assessment of Tracks of Resonance Frequencies of the Vocal Tract

Abstract

A new method for estimating formant frequency tracks of the vocal tract for arbitrary speech segments is proposed. The method uses the ratio of two Fourier transforms of a speech signal with special exponential-type windows depending on some parameter. This ratio is used for specific points in time and is considered as a function of frequency and parameter. By analyzing, for several parameter values, the distribution of minimum points (in terms of frequency) for the phase of this ratio and/or a similar distribution of extreme points for its amplitude, it is possible to estimate formant frequencies from the peaks of these distributions. A mathematical study is presented that substantiates this approach. A series of numerical experiments were carried out on the processing of synthetic and real speech signals, which confirmed the performance capabilities of the proposed formant evaluation method. In particular, in experiments with synthesized vowels, it was found that the error in estimating their resonance frequencies is small and stable with respect to additive noise up to a signal-to-noise ratio of 5 dB. For real speech, the method makes it possible to calculate the formant frequency tracks for both sounds with vocal excitation and for voiceless fricatives, aspirated plosives, and whispered speech.

APPENDIX

Here we present the derivation of the formulas for the functions $A(\omega ,r),\,\,\Phi (\omega ,r)$, as well as $A(\omega )$ and $\Phi (\omega )$ used in the proposed method of formant analysis.

We denote designate $y(r,\omega )(t) = {{S}_{r}}(\omega ,t)$, where the function ${{S}_{r}}(\omega ,t)$ is determined by formula (1). This notation emphasizes that this value depends on time, and on quantities $r,\omega$, it also depends on the parameters. In addition to frequency ω, we use “close” frequencies $\omega + \delta \omega$ ($\left| {\delta \omega } \right| \ll \omega$). We determine the relation

$$\begin{gathered} \alpha (r,\omega )(t) = \frac{{y(r,\omega + \delta \omega )(t)}}{{y(r,\omega )(t)}} = 1 + \frac{{\Delta y}}{y}, \\ \Delta y = y(r,\omega + \delta \omega ) - y(r,\omega ). \\ \end{gathered} $$

Then, taking into account the “smallness” of quantity $\delta \omega$ and formula (1), we obtain

$$\begin{gathered} \Delta y = \,y(r,\omega + \delta \omega ) - y(r,\omega ) \approx \frac{{\partial y}}{{\partial \omega }}(r,\omega )\delta \omega \\ = \delta \omega \int\limits_0^\infty {{{W}_{r}}(t - \tau )x(\tau )( - j\tau ){{e}^{{ - j\omega \tau }}}d\tau } \\ = j\delta \omega \int\limits_0^\infty {{{W}_{r}}(t - \tau )x(\tau )(t - \tau ){{e}^{{ - j\omega \tau }}}d\tau } \\ - \,\,j(\delta \omega )t\int\limits_0^\infty {{{W}_{r}}(t - \tau )x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } \\ = j\delta \omega \left( {\int\limits_0^\infty {{{W}_{{1r}}}(t - \tau )x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } } \right. \\ \left. { - \,\,t\int\limits_0^\infty {{{W}_{r}}(t - \tau )x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } } \right), \\ \end{gathered} $$

where ${{W}_{{1r}}}(z) = z{{W}_{r}}(z)$ Hence

$$\begin{gathered} \alpha (r,\omega )(t) = 1 + \frac{{\Delta y}}{y} \approx 1 - j\delta \omega t \\ + \,\,j\delta \omega \frac{{\int\limits_0^\infty {{{W}_{{1r}}}(t - \tau )x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}{{\int\limits_0^\infty {{{W}_{r}}(t - \tau )x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }} \\ = 1 - j\delta \omega t + j\delta \omega \frac{{\int\limits_0^t {(t - \tau ){{e}^{{ - (1 + r)(t - \tau )}}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}{{\int\limits_0^t {{{e}^{{ - (1 + r)(t - \tau )}}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }} \\ = 1 - j\delta \omega \frac{{\int\limits_0^t {\tau {{e}^{{ - (1 + r)(t - \tau )}}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}{{\int\limits_0^t {{{e}^{{ - (1 + r)(t - \tau )}}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}. \\ \end{gathered} $$

In what follows we will be interested in the quantity ${{A}_{\alpha }}(r,\omega ,t) = ln\left| {\alpha (r,\omega )(t)} \right|$ and phase ${{\Phi }_{\alpha }}(r,\omega ,t)$ relationship $\alpha (r,\omega )(t)$. For brevity, we call these quantities the amplitude and phase functions associated with the relation $\alpha (r,\omega )(t)$. To find them, we use the formula

$$\begin{gathered} \ln \alpha (r,\omega )(t) = ln\left| {\alpha (r,\omega )(t)} \right| + j{{\Phi }_{\alpha }}(r,\omega ,t) \\ = {{A}_{\alpha }}(r,\omega ,t) + j{{\Phi }_{\alpha }}(r,\omega ,t). \\ \end{gathered} $$

Then, under the conditions $\left| {\delta \omega } \right|t \ll 1,\,$ $\left| {\delta \omega } \right| \ll \omega$, i.e., for $\left| {\frac{{\Delta y}}{y}} \right| \ll 1$, we have

$$\begin{gathered} \ln \alpha (r,\omega )(t) = ln\left( {1 + \frac{{\Delta y}}{y}} \right) \approx \frac{{\Delta y}}{y} \Rightarrow \\ {{A}_{\alpha }}(r,\omega ,t) \approx \operatorname{Re} \left\{ {\frac{{\Delta y}}{y}} \right\},\,\,\,{{\Phi }_{\alpha }}(r,\omega ,t) \approx \operatorname{Im} \left\{ {\frac{{\Delta y}}{y}} \right\}. \\ \end{gathered} $$

Hence,

$$\begin{gathered} {{A}_{\alpha }}(r,\omega ,t) \approx - \delta \omega \operatorname{Re} \left\{ {j\frac{{\int\limits_0^t {\tau {{e}^{{ - (1 + r)(t - \tau )}}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}{{\int\limits_0^t {{{e}^{{ - (1 + r)(t - \tau )}}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}} \right\}, \\ {{\Phi }_{\alpha }}(r,\omega ,t) \approx - \delta \omega \operatorname{Im} \left\{ {j\frac{{\int\limits_0^t {\tau {{e}^{{(1 + r)\tau }}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}{{\int\limits_0^t {{{e}^{{(1 + r)\tau }}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}} \right\}. \\ \end{gathered} $$

The amplitude and phase functions associated with $\alpha (r,\omega )(t)$, for an amplitude-limited signal $x(t)$ stabilize at $t \to \infty$; i.e., as we say below, at the end of the speech segment:

$$\begin{gathered} A(\omega ,r) = ln\left| {\alpha (r,\omega )(\infty )} \right| = - \delta \omega \operatorname{Re} \left\{ {U(\omega ,r)} \right\}, \\ \Phi (\omega ,r) = {{\Phi }_{\alpha }}(r,\omega ,\infty ) = - \delta \omega \operatorname{Im} \left\{ {U(\omega ,r)} \right\}, \\ \end{gathered} $$

(A1)

where

$$\begin{gathered} U(\omega ,r) \\ = j\left\{ {\int\limits_0^\infty {\tau {{e}^{{(1 + r)\tau }}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } } \right\}{{\left\{ {\int\limits_0^\infty {{{e}^{{(1 + r)\tau }}}x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } } \right\}}^{{ - 1}}}. \\ \end{gathered} $$

(A2)

Expressions (A1) and (A2) will be used for formant analysis of general speech signals $x(t)$. A prerequisite for this is a numerical study of their behavior for different $r \leqslant - 1$ for simple signals (see Section 3), as well as an analytical study of their limiting values $A(\omega )$ and $\Phi (\omega )$ for $r \to - 1$; i.e., quantities

$$\begin{gathered} A(\omega ) = {{A}_{\alpha }}( - 1,\omega ,\infty ) \approx \delta \omega \operatorname{Re} \left\{ {U(\omega , - 1)} \right\} \\ = \delta \omega \operatorname{Re} \left\{ { - j\frac{{\int\limits_0^\infty {\tau x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}{{\int\limits_0^\infty {x(\tau ){{e}^{{ - j\omega \tau }}}d\tau } }}} \right\} = \delta \omega \operatorname{Re} \left\{ {\frac{{F'[x](\omega )}}{{F[x](\omega )}}} \right\}, \\ \end{gathered} $$

$$\begin{gathered} \Phi (\omega ) = {{\Phi }_{\alpha }}( - 1,\omega ,\infty ) \approx \delta \omega \operatorname{Im} \left\{ {U(\omega , - 1)} \right\} \\ = \delta \omega \operatorname{Im} \left\{ {\frac{{F{\kern 1pt} '[x](\omega )}}{{F[x](\omega )}}} \right\} = \delta \omega \operatorname{Im} \left\{ {\left[ {\ln F[x](\omega )} \right]{\kern 1pt} '} \right\}. \\ \end{gathered} $$

Here, $F[x](\omega ) = \int_0^\infty {x(\tau ){{e}^{{ - j\omega \tau }}}d\tau }$ is the Fourier transform of the signal. This study is presented in Section 3.