Stochastic models of glottal pulses from the Rosenberg and Liljencrants-Fant models with unified parameters

Highlights

• Stochastic models proposed to generate jitter.

• Stochastic models based on the Rosenberg model and the LF model unified.

• The variation of the glottal time interval modelled as a stochastic process.

• Generation of intelligible voiced sounds.

• Generation of voice signals with jitter and, consequently, hoarse voices.

Abstract

In voice production the random variation of the glottal cycles in relation to a mean value caused by the (quasi-) periodic movement of the vocal folds generates the jitter phenomenon. Its study is important due to applications as identification of pathologies related to the voice, improvement of the naturalness of synthesized voices and calibration of signal processing algorithms to identify glottal cycles. The objective of this work is to build stochastic models for glottal signals, using the Rosenberg and Liljencrants-Fant models with unified parameters, and considering the glottal time interval as a random variable (total time from the opening up to the closing of the vocal folds in each cycle), simulating the situation where the stiffness of the vocal folds is a stochastic process. Four different models are proposed with generation of jitter phenomenon in glottal signals simulated. Intelligible voiced sounds are obtained, in the case of normal voices, signal with low level of jitter, and also in hoarse voices, when the levels of jitter are higher. All the sounds simulated are available to be listened.

Introduction

Human beings are capable of expressing their feelings, thoughts and wishes through the voice, which is an important means of communication for living in society. The generation of voice, in the case of voiced sounds, starts with the airflow coming from the lungs and forcing the vocal folds to oscillate. After passing through the glottis, where the vocal folds are located, air pulses are generated forming the so-called glottal signal, which is a (quasi-)periodic signal, representing the glottal flow, which will be further filtered and amplified by the vocal tract (portion from the glottis up to the mouth), and finally radiated by the mouth and then generating the sound.

The small random variations in the glottal cycles, due to the asymmetric movement of the vocal folds, is a random phenomenon called jitter. There are different measures of jitter, but one of them, called local jitter, gives the relative percentage of the disturbances in relation to a mean value and its typical value for normal voices is around 1% (Mongia, Sharma, 2014, Wong, Ito, Cox, Titze, 1991). It has been verified empirically that vocal jitter increases for some dysphonic voices (Pinto, Titze, 1990, Schoentgen, De Guchteneere, 1995).

Models of jitter can be used to improve naturalness (even if voices with pathological characteristics are considered) to simulate hoarse voices (Bangayan et al., 1997), to generate voice signals to calibrate signal processing algorithms and to help in detecting glottal cycles (Muta et al., 1988). In addition, models of jitter could suggest or confirm the mathematical form of markers that would characterize perturbed cycle lengths statistically rather than heuristically.

Schoentgen (2001) presented stochastic models of jitter through the simulation of instantaneous frequency causing perturbations of the glottal cycles. Ruinskiy and Lavner (2008) presented an algorithm to transform a normal voice in a hoarse voice. Using spring-mass-damper mechanical models of the vocal folds, Cataldo, Soize, 2016, Cataldo, Soize, 2018 considered a stochastic model for the stiffness of the spring to generate jitter.

In this paper, a voice modification algorithm for transforming a modal voice to a hoarse voice is presented. The algorithm is based on creating jitter, which is known as a random phenomenon. Two mathematical deterministic models of the glottal signal are considered, the Rosenberg model and the Liljencrants-Fant model (LF model) with unified parameters (Henrich et al., 2002) and, taking into account the time variation of glottal intervals as a stochastic process, stochastic models of glottal signals are proposed, with generation of jitter. The second model, the LF model, is more sophisticated and it is a better approach to the real glottal signal. Then, through the Fant source-filter theory, voice signals with jitter are synthesized. Two different power spectral densities (PSDs) are associated to the proposed stochastic models and, consequently, four different models are created for the glottal signals. Simulations are performed and voiced sounds with different levels of jitter obtained. A comparison between the models is performed.

It is important to say that, in general, the quality of the signal generated is related to the mathematical model of the pulse used (Rosenberg or LF), to the bandwidths of the formant frequencies and also to the level of jitter.

In real voices, the jitter is always presented. Then, the naturalness of synthesized voices is better obtained when the phenomenon is included, not only for normal voices but also with hoarse voices, with a greater level of jitter, which can be also associated to pathological cases.

With the model presented, it is possible to generate the random phenomenon that is present in all voice signals. By numerical simulation, a big dataset can be generated for different voice signals, with different levels of jitter, and for different kinds of pathologies. Such a big dataset can be used for training an artificial neural network. The use of artificial neural networks to classify voices with pathological characteristics has been used by some researchers (Mohammed, Adulkareem, Mostafa, Maashi, Zapirain, Oleagordia, Al-Dhief, Alhakami, 2020, Megala, Padmapriya, Jayanthi, Suganya, 2019, Souissi, Cherif, 2016, Lotfi, Mourad, Cherif, 2010).

The proposed stochastic model makes it possible to obtain good speech synthesis with jitter. Although jitter is not a particularly robust clinical measure, it is one of the main parameters used, and this is the main reason for focusing in it. With this stochastic model, it will therefore be possible, in a second phase, to classify pathologies, using also other parameters, as presented in different works in the literature. Some of them are very recent (Bennane, Kacha, 2017, Schoentgen, Aichinger, 2019, Lucero, Pelorson, Hirtum, 2020, Asiaee, Vehdian-azimi, Atashi, Keramatfar, Nourbaksh, 2020, Wei, Du, Wang, 2020, Chiaramonte, Bonfiglio, 2020, Li, Hou, Zhang, Jiang, Gong, 2021).

The proposed model shows a way to create jitter in a voice signal using a consistent mathematical basis, and not simply adding a noise to the signal. Moreover, there is an interesting and robust mathematical formulation to generate jitter. Different power spectral density functions can be tested and the corresponding inverse problem solved to identify parameters related to a specific voice or group of voices, including voices with pathological characteristics.

The ideas discussed here are based on earlier studies about stochastic models of jitter, including not only signal processing but also mechanical models to generate jitter, and these models are based on the fact that the stiffness of the vocal folds is a stochastic process and, as consequence, the jitter is produced. In terms of signal processing, the idea is to reproduce the modelling of the stiffness as a stochastic process but considering the glottal time interval as a stochastic process.

The synthesized sounds are available to the readers (the corresponding link will be made available later in the paper) and it can be verified the intelligibility of the sounds created, corresponding to normal voices and also to hoarse voices, which could characterize any type of pathology.