Glottal Source Estimation using an Automatic Chirp Decomposition

The deconvolution of speech into its vocal tract and glottis contributions is an important topic in speech processing. Explicitly isolating both components allows to model them independently. While techniques for modeling the vocal tract are rather well-established, it is not the case for the glottal source representation. However the characterization of this latter has been shown to be advantageous in speaker recognition [1], speech disorder analysis [2], speech recognition [3] or speech synthesis [4]. These reasons justify the need of developing algorithms able to robustly and reliably estimate and parametrize the glottal signal.

Some works addressed the estimation of the glottal contribution directly from speech waveforms. Most approaches rely on a first parametric modeling of the vocal tract and then remove it by inverse filtering so as to obtain the glottal signal estimation. In [5], the use of the Discrete All-Pole (DAP) model is proposed. The Iterative Adaptive Inverse Filtering technique (IAIF) described in [6] isolates the source signal by iteratively estimating both vocal tract and source parts. In [7], the vocal tract is estimated by Linear Prediction (LP) analysis on the closed phase. As an extension, the Multicycle closed-phase LPC (MCLPC) method [8] refines its estimation on several larynx cycles. In a fundamentally different point of view, we proposed in [9] a non-parametric technique based on the Zeros of the Z-Transform (ZZT). ZZT basis relies on the observation that speech is a mixed-phase signal [10] where the anticausal component corresponds to the vocal folds open phase, and where the causal component comprises both the glottis closure and the vocal tract contributions. Basically ZZT isolates the glottal open phase contribution from the speech signal, by separating its causal and anticausal components. In [11], a comparative evaluation between LPC and ZZT-based decompositions is led, giving a significant advantage for the second technique.

This paper proposes an extension to the traditional ZZT-based decomposition technique. The new method aims at separating both causal and anticausal contributions by computing the Zeros of a Chirp Z-Transform (ZCZT). More precisely, the Z-transform is here evaluated on a contour possibly different from the unit circle. As a result, we will see that the estimation is much less sensitive to the Glottal Closure Instant (GCI) detection errors. In addition, a way to automatically determine an optimal contour is also proposed.

The paper is structured as follows. Section 2 reminds the principle of the ZZT-based decomposition of speech. Its extension making use of a chirp analysis is proposed and discussed in Section 3. In Section 4, a comparative evaluation of both approaches is led on both synthetic and real speech signals. Finally we conclude in Section 5.

For a series of $N$ samples $(x(0),x(1),...x(N-1))$ taken from a discrete signal $x(n)$, the $ZZT$ representation is defined as the set of roots (zeros) $(Z_{1},Z_{2},...Z_{N-1})$ of the corresponding Z-Transform $X(z)$:

The spectrum of the glottal source open phase is then computed from zeros outside the unit circle (anticausal component) while zeros inside it give the vocal tract transmittance modulated by the source return phase spectrum (causal component). To obtain such a separation, the effects of the windowing are known to play a crucial role [12]. In particular, we have shown that a Blackman window centered on the Glottal Closure Instant (GCI) and whose length is twice the pitch period is appropriate in order to achieve a good decomposition.

The Chirp Z-Transform (CZT), as introduced by Rabiner et al [13] in 1969, allows the evaluation of the Z-transform on a spiral contour in the Z-plane. Its first application aimed at separating too close formants by reducing their bandwidth. Nowadays CZT reaches several fields of Signal Processing such as time interpolation, homomorphic filtering, pole enhancement, narrow-band analysis,…

As previously mentioned, the ZZT-based decomposition is strongly dependent on the applied windowing. This sensitivity may be explained by the fact that ZZT implicitly conveys phase information, for which time alignment is known to be crucial [14]. In that article, it is observed that the window shape and onset may lead to zeros whose topology can be detrimental for accurate pulse estimation. The subject of this work is precisely to handle with these zeros close to the unit circle, such that the ZZT-based technique correctly separates the causal (i.e minimum-phase) and anticausal (i.e maximum-phase) components.

For this, we evaluate the CZT on a circle whose radius $R$ is chosen so as to split the root distribution into two well-separated groups. More precisely, it is observed that the significant impulse present in the excitation at the GCI results in a gap in the root distribution. When analysis is exactly GCI-synchronous, the unit circle perfectly separates causal and anticausal roots. On the opposite, when the window moves off from the GCI, the root distribution is transformed. Such a decomposition is then not guaranteed for the unit circle and another boundary is generally required. Figure 1 gives an example of root distribution for a natural voiced speech frame for which an error of 0.6 ms is made on the real GCI position. It is clearly seen that using the traditional ZZT-based decomposition ($R=1$) for this frame will lead to erroneous results. In contrast, it is possible to find an optimal radius leading to a correct separation.

In order to automatically determine such a radius, let us have the following thought process. We know that ideally the analysis should be GCI-synchronous. When this is not the case, the chirp analysis tends to modify the window such that its center coincides with the nearest GCI (to ensure a reliable phase information). Indeed, evaluating the chirp Z-transform of a signal $x(t)$ on a circle of radius $R$ is equivalent to evaluating the Z-transform of $x(t)\cdot{}exp(log(1/R)\cdot{}t)$ on the unit circle. It can be demonstrated that for a Blackman window $w(t)$ of length $L$ starting in $t=0$:

the radius $R$ necessary to modify its shape so that its new maximum lies in position $t^{*}$ ($<L$) is expressed as:

In particular, we verify that $R=1$ is optimal when the window is GCI-centered ($t^{*}=\frac{L}{2}$) and, since we are working with two-period long windows, the optimal radius does not exceed $exp(\pm\frac{50\pi}{17L})$ in the worst cases (the nearest GCI is then positioned in $t^{*}=\frac{L}{4}$ or $t^{*}=\frac{3L}{4}$). As a means for automatically determining the radius allowing an efficient separation, the sorted root moduli are inspected and the greatest discontinuity in the interval $[exp(-\frac{50\pi}{17L}),exp(\frac{50\pi}{17L})]$ is detected. Radius $R$ is then chosen as the middle of this discontinuity, and is assumed to optimally split the roots into minimum and maximum-phase contributions.

This Section gives a comparative evaluation of the following methods:

• •

  the traditional ZZT-based technique: $R=1$,

• •

  the proposed ZCZT-based technique: $R$ is computed as explained at the end of Section 3 (see Fig. 2),

• •

  the ideal ZCZT-based technique: $R$ is computed from Equation 3 where the real GCI location $t^{*}$ is known. This can be seen as the ultimate performance one can expect from the ZCZT-based technique.

Among others, it is emphasized how the proposed technique is advantageous in case of GCI location errors.

This paper proposed an extension of the ZZT-based technique we proposed in [9]. The enhancement consists in evaluating the Z-transform on a contour possibly different from the unit circle. For this we considered, in the Z-plane, circles whose radius is automatically determined by detecting a discontinuity in the root distribution. It is expected that such circles lead to a better separation of both causal and anticausal contributions. Results obtained on synthetic and real speech signals report an advantage for the proposed ZCZT-based technique, mainly when GCIs are not accurately localized. As future work, we plan to characterize the glottal source based on the proposed framework.

Thomas Drugman is supported by the “Fonds National de la Recherche Scientifique” (FNRS).