Nonlinear source-filter coupling in phonation: theory

A theory of interaction between the source of sound in phonation and the vocal tract filter is developed. The degree of interaction is controlled by the cross-sectional area of the laryngeal vestibule (epilarynx tube), which raises the inertive reactance of the supraglottal vocal tract. Both subglottal and supraglottal reactances can enhance the driving pressures of the vocal folds and the glottal flow, thereby increasing the energy level at the source. The theory predicts that instabilities in vibration modes may occur when harmonics pass through formants during pitch or vowel changes. Unlike in most musical instruments (e.g., woodwinds and brasses), a stable harmonic source spectrum is not obtained by tuning harmonics to vocal tract resonances, but rather by placing harmonics into favorable reactance regions. This allows for positive reinforcement of the harmonics by supraglottal inertive reactance (and to a lesser degree by subglottal compliant reactance) without the risk of instability. The traditional linear source-filter theory is encumbered with possible inconsistencies in the glottal flow spectrum, which is shown to be influenced by interaction. In addition, the linear theory does not predict bifurcations in the dynamical behavior of vocal fold vibration due to acoustic loading by the vocal tract.     

Observation and analysis of in vivo vocal fold tissue instabilities produced by nonlinear source-filter coupling: a case study

Different source-related factors can lead to vocal fold instabilities and bifurcations referred to as voice breaks. Nonlinear coupling in phonation suggests that changes in acoustic loading can also be responsible for this unstable behavior. However, no in vivo visualization of tissue motion during these acoustically induced instabilities has been reported. Simultaneous recordings of laryngeal high-speed videoendoscopy, acoustics, aerodynamics, electroglottography, and neck skin acceleration are obtained from a participant consistently exhibiting voice breaks during pitch glide maneuvers. Results suggest that acoustically induced and source-induced instabilities can be distinguished at the tissue level. Differences in vibratory patterns are described through kymography and phonovibrography; measures of glottal area, open/speed quotient, and amplitude/phase asymmetry; and empirical orthogonal function decomposition. Acoustically induced tissue instabilities appear abruptly and exhibit irregular vocal fold motion after the bifurcation point, whereas source-induced ones show a smoother transition. These observations are also reflected in the acoustic and acceleration signals. Added aperiodicity is observed after the acoustically induced break, and harmonic changes appear prior to the bifurcation for the source-induced break. Both types of breaks appear to be subcritical bifurcations due to the presence of hysteresis and amplitude changes after the frequency jumps. These results are consistent with previous studies and the nonlinear source-filter coupling theory.     

Nonlinear behavior of vocal fold vibration: The role of coupling between the vocal folds

Coupling between the vocal folds is one of the nonlinear mechanisms allowing regulation and synchronization of mucosal vibration. The purpose of this study was to establish that modulations such as diplophonia and abnormalities observed in vocal signals that may be observed in some cases of laryngeal pathology can be considered as nonlinear behavior due to the persistence of some physical interaction (coupling). An experimental model using excised porcine larynx was designed to create tension asymmetry between the vocal folds and to obtain vocal signals with modulations. Signals were analyzed by spectral analysis and the phase portrait method. Results were compared with computer-generated synthetic signals corresponding to nonlinear combinations of sinusoid signals. Under these conditions, evidence of nonlinear behavior was detected in 85% of experimental signals. These findings were interpreted as a demonstration of vocal fold interaction. Based on these findings, the authors conclude that (1) coupling must be taken into account in physical models of laryngeal physiology, and that (2) methods of nonlinear dynamics may be used for objective voice analysis.     

Vestibular vocal fold behavior during phonation in unilateral vocal fold paralysis

Voice quality in patients with vocal fold paralysis can be affected by several factors, such as the position of the paralyzed vocal fold, its degree of atrophy, the configuration of its free edge, and the level differences between both vocal folds. Depending on the related vocal deficiency the patient will attempt to compensate using different maneuvers, such as increment of vocal tract and neck muscle contraction to improve glottal closure. This is probably one of the reasons why ventricular folds are frequently requested. The objective of this study is to analyze the behavior of the homolateral and contralateral vestibular folds to delineate patterns of vestibular motion during sustained phonation, in cases of unilateral vocal fold paralysis.     

Comparison of vocal tract formants in singing andnonperiodic phonation

The skilled use of nonperiodic phonation techniques in combination with spectrum analysis has been proposed here as a practical method for locating formant frequencies in the singing voice. The study addresses the question of the degree of similarity between sung phonations and their nonperiodic imitations, with respect to both frequency of the first two formants as well as posture of the vocal tract. Using magnetic resonance imaging (MRI), linear predictive coding (LPC), and spectrum analysis, two types of nonperiodic phonation (ingressive and vocal fry) are compared with singing phonations to determine the degree of similarity/difference in acoustic and spatial dimensions of the vocal tract when these phonation types are used to approximate the postures of singing. In comparing phonation types, the close similarity in acoustic data in combination with the relative dissimilarity in spatial data indicates that the accurate imitations are not primarily the result of imitating the singing postures, but have instead an aural basis.     

Modeling the role of nonhuman vocal membranes in phonation

Although the mammalian larynx exhibits little structural variation compared to sound-producing organs in other taxa (birds or insects), there are some morphological features which could lead to significant differences in acoustic functioning, such as air sacs and vocal membranes. The vocal membrane (or "vocal lip") is a thin upward extension of the vocal fold that is present in many bat and primate species. The vocal membrane was modeled as an additional geometrical element in a two-mass model of the larynx. It was found that vocal membranes of an optimal angle and length can substantially lower the subglottal pressure at which phonation is supported, thus increasing vocal efficiency, and that this effect is most pronounced at high frequencies. The implications of this finding are discussed for animals such as bats and primates which are able to produce loud, high-pitched calls. Modeling efforts such as this provide guidance for future empirical investigations of vocal membrane structure and function, can provide insight into the mechanisms of animal communication, and could potentially lead to better understanding of human clinical disorders such as sulcus vocalis.     

Dependence of phonation threshold pressure on vocal tract acoustics and vocal fold tissue mechanics

Analytical and computer simulation studies have shown that the acoustic impedance of the vocal tract as well as the viscoelastic properties of vocal fold tissues are critical for determining the dynamics and the energy transfer mechanism of vocal fold oscillation. In the present study, a linear, small-amplitude oscillation theory was revised by taking into account the propagation of a mucosal wave and the inertive reactance (inertance) of the supraglottal vocal tract as the major energy transfer mechanisms for flow-induced self-oscillation of the vocal fold. Specifically, analytical results predicted that phonation threshold pressure (Pth) increases with the viscous shear properties of the vocal fold, but decreases with vocal tract inertance. This theory was empirically tested using a physical model of the larynx, where biological materials (fat, hyaluronic acid, and fibronectin) were implanted into the vocal fold cover to investigate the effect of vocal fold tissue viscoelasticity on Pth. A uniform-tube supraglottal vocal tract was also introduced to examine the effect of vocal tract inertance on Pth. Results showed that Pth decreased with the inertive impedance of the vocal tract and increased with the viscous shear modulus (G") or dynamic viscosity (eta') of the vocal fold cover, consistent with theoretical predictions. These findings supported the potential biomechanical benefits of hyaluronic acid as a surgical bioimplant for repairing voice disorders involving the superficial layer of the lamina propria, such as scarring, sulcus vocalis, atrophy, and Reinke's edema.     

Voice production model integrating boundary-layer analysis of glottal flow and source-filter coupling

A voice production model is created in this work by considering essential aerodynamic and acoustic phenomena in human voice production. A precise flow analysis is performed based on a boundary-layer approximation and the viscous-inviscid interaction between the boundary layer and the core flow. This flow analysis can supply information on the separation point of the glottal flow and the thickness of the boundary layer, both of which strongly depend on the glottal configuration and yield an effective prediction of the flow behavior. When the flow analysis is combined with the modified two-mass model of the vocal fold [Pelorson et al. (1994). J. Acoust. Soc. Am. 96, 3416-3431], the resulting acoustic wave travels through the vocal tract and a pressure change develops in the vicinity of the glottis. This change can affect the glottal flow and the motion of the vocal folds, causing source-filter coupling. The property of the acoustic feedback is explicitly expressed in the frequency domain by using an acoustic tube model, allowing a clear interpretation of the coupling. Numerical experiments show that the vocal-tract input impedance and frequency responses representing the source-filter coupling have dominant peaks corresponding to the fourth and fifth formants. Results of time-domain simulations also suggest the importance of these high-frequency peaks in voice production.     

Medial surface dynamics of an in vivo canine vocal fold during phonation

Quantitative measurement of the medial surface dynamics of the vocal folds is important for understanding how sound is generated within the larynx. Building upon previous excised hemilarynx studies, the present study extended the hemilarynx methodology to the in vivo canine larynx. Through use of an in vivo model, the medial surface dynamics of the vocal fold were examined as a function of active thyroarytenoid muscle contraction. Data were collected using high-speed digital imaging at a sampling frequency of 2000 Hz, and a spatial resolution of 1024 x 1024 pixels. Chest-like and fry-like vibrations were observed, but could not be distinguished based on the input stimulation current to the recurrent laryngeal nerve. The subglottal pressure did distinguish the registers, as did an estimate of the thyroarytenoid muscle activity. Upon quantification of the three-dimensional motion, the method of Empirical Eigenfunctions was used to extract the underlying modes of vibration, and to investigate mechanisms of sustained oscillation. Results were compared with previous findings from excised larynx experiments and theoretical models.     

Nonlinear phenomena in contemporary vocal music.

Complex and multiphonic voice signals of vocal improvisors are analyzed within the framework of nonlinear dynamics. Evidence is given that nonlinear phenomena are extensively used by performers associated with contemporary music. Narrow-band spectrograms of complex vocalizations are used to visualize the appearance of nonlinear phenomena (spectral bifurcation diagrams). Possible production mechanisms are discussed in connection with previous research, personal performance and pedagogical experience. Examples for period doubling, biphonation and irregular aperiodic phonation in vocal sonorities of contemporary vocal improvisors are given, and glottal whistle production encompassed with biphonation and triphonation is shown. Furthermore, coincidences of harmonics-formant matching associated with abrupt transitions to subharmonics and biphonation in the vocal output are provided. This also shows the recurrent use of nonlinear phenomena by performers. It is argued that mechanisms such as source-tract coupling or vocal fold desynchronization due to asymmetry are used in a reproducible way for musical tasks.     

Nonlinear statistical coupling

By considering a nonlinear combination of the probabilities of a system, a physical interpretation of Tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed. The escort probability is interpreted as the coupled probability, with Q=1−q defined as the degree of nonlinear coupling between the statistical states. Positive values of Q have coupled statistical states, a larger entropy metric, and a maximum coupled-entropy distribution of compact-support coupled-Gaussians. Negative values of Q have decoupled statistical states and for −2<Q<0 a maximum coupled-entropy distribution of heavy-tail coupled-Gaussians. The conjugate transformation between the heavy-tail and compact-support domains is shown to be for coupled-Gaussian distributions. This conjugate relationship has been used to extend the generalized Fourier transform to the compact-support domain and to define a scale-invariant correlation structure with heavy-tail limit distribution. In the present paper, we show that the conjugate is a mapping between the source of nonlinearity in non-stationary stochastic processes and the nonlinear coupling which defines the coupled-Gaussian limit distribution. The effects of additive and multiplicative noise are shown to be separable into the coupled-variance and the coupling parameter Q, providing further evidence of the importance of the generalized moments.     

Sensitivity of vocal fold vibratory modes to their three-layer structure: implications for computational modeling of phonation

The sensitivity of the eigenmodes and eigenfrequencies of the human vocal fold to its three-layer structure is studied using finite-element modeling. The study covers a variety of three-dimensional vocal fold models ranging from an idealized, longitudinally uniform structure to a physiologically more realistic, longitudinally varying structure. Geometric parameters including the thickness of the ligament and cover layers as well as the ligament length are varied systematically. The results indicate that vocal fold vibratory modes are quite insensitive to the longitudinal variation in the thickness of the three layers as well as the variation in ligament length. However, significant overall changes in thickness of each layer can produce noticeable changes in these modes. The implications of these findings on computational modeling of phonation are discussed.     

Perceived phonatory effort and phonation threshold pressure across a prolonged voice loading task: a study of vocal fatigue

Although the problem of vocal fatigue is not uncommon in people with voice disorders, research on objective quantifiable indicators of vocal fatigue is limited. It has been suggested that a speaker's perception of increased phonatory effort associated with periods of prolonged voice use is related to increased lung pressure required to initiate and sustain phonation. The purpose of this study was to examine the relationship among perceived phonatory effort (PPE), which was used as a subjective index of vocal fatigue, and phonation threshold pressure (PTP), a quantifiable measure defined as the minimal lung pressure required to initiate and sustain vocal fold oscillation. PTP and PPE were recorded before, during, and after five adult male and five adult female speakers engaged in a prolonged oral reading task designed to induce vocal fatigue. The results supported a direct, moderately strong relationship between PTP and PPE, particularly when PTP was measured during speech produced at comfortable and low-speaking pitch levels. No gender effects were found. PTP returned to baseline levels within 1 hour after the fatiguing task. PPE returned to baseline within 1 day. The data support the use of PTP as an objective index of vocal fatigue.     

Nonlinear oscillation of pathological vocal folds during vocalization

The extended two-mass model is adopted to analyze the nonlinear oscillation of pathological vocal folds during vocalization. Redundant tissue or area in laryngeal patients is modeled as a massless rigid connected to the upper mass of the vocal folds, and a parameter Q is introduced to represent the change of glottal configurations and tension imbalance between the left and right sides of vocal folds. Numerical simulations demonstrate that the pathological vocal-fold decreases the threshold of Q to generate nonlinear vocal oscillation, indicating the improvement of the sensitivity of vocal folds to asymmetries and enhancing the coupling between two sides. Furthermore, the pathological vocal-fold can lower the fundamental frequency and eliminate high-order harmonics, For example, the fundamental frequency decreases from 119.94 Hz to 84.95 Hz when Q=0.58 and the sub-glottal pressure 1450 Pa. However, there are no prominent effects on the amplitudes of sub-harmonic and low-order harmonics.     

On the relation between the phonation threshold lung pressure and the oscillation frequency of the vocal folds

This Letter presents an extension of a previous equation for the phonation threshold pressure by Titze [I. R. Titze, J. Acoust. Soc. Am. 83, 1536-1552 (1988)]. The extended equation contains the vocal-fold oscillation frequency as an explicit factor. It is derived from the mucosal wave model of the vocal folds by considering the general case of an arbitrary time delay for the mucosal wave to travel the glottal height. The results are illustrated with a numerical example, which shows good qualitative agreement with experimental measures.     

Verification of two minimally invasive methods for the estimation of the contact pressure in human vocal folds during phonation

The contact pressure on the vocal fold surface during high pitch or amplitude voice production is believed to be one major source of phonotrauma. Models for the quantitative estimate of the contact pressure may be valuable for prevention and treatment. Various indirect and minimally invasive approaches have been purported to estimate contact pressure. But the accuracy of these methods has not yet been objectively verified in controlled laboratory settings. In the present study, two indirect approaches for the estimation of the contact pressure were investigated. One is based on a Hertzian impact model, and the other on a finite element model. A probe microphone was used for direct measurements of the contact pressure and verifications of the indirect approaches. A silicone replica of human vocal folds was used as a test bed. Consistent contact pressure estimations were obtained using all three methods. The advantages and disadvantages of each approach for eventual clinical applications are described.     

Nonlinear mode coupling in doubly resonant frequency doublers

The concept of a doubly resonant frequency doubler can be used for a variety of experiments concerning both classical phenomena like efficient frequency doubling at low power levels and quantum effects like squeezed states of light or Quantum Non Demolition (QND) measurements. In many of these experiments the strength of the nonlinear coupling of fundamental and second-harmonic modes is of crucial importance. First we treat the general theory for the calculation of the coupling parameter, which depends not only on properties of the nonlinear material but also on resonator geometry and some optical phases. On this basis we discuss in detail the situation for two different monolithic resonator geometries, namely a linear (standing-wave) and a ring (travelling-wave) cavity. Finally we compare theoretical predictions for these resonators to the experimentally achieved results.     

Nonlinear directional coupling between plasmonic slot waveguides

A numerical investigation of nonlinear switching in plasmonic directional couplers made of two dielectric slab waveguides with metallic claddings is presented. We assume Kerr-nonlinear dielectric and study the influence of geometrical parameters, metallic losses, and metal nonlinearities on coupler characteristics. We observe a general trade-off between losses and nonlinearity levels required for the switching operation. Underlying physical mechanisms that affect the coupler performance are discussed. The obtained results can be useful in design and optimization of the nonlinear plasmonic couplers.