Vocal tract acoustics using the transmission line matrix (TLM) method

Most traditional theories of speech production are currently based on plane waves and on one-dimensional analysis. It i s however well-known that when the frequency of sound reaches a cut-on frequency, higher acoustical modes start to propagate and can become predominant. It is therefore important to evaluate the effects of these higher modes, especially in order to improve acoustical models of the vocal tract. This paper describes a new numerical method to study the propagation and the radiation of speech sounds, and to compute acoustic characteristics of the vocal tract. This method, named Transmission Line Matrix or Modelling (TLM), has been used for simulating electromagnetic wave propagation and is used here for the first time in acoustics. The TLM method provides time domain solutions in 2D and 3D spaces. The main advantage of this method is the simplicity of formulation and programming for a large range of applications. We first describe the principle on which the TLM method is based. The method as well as the boundary conditions used are validated using classical tests. A systematic study of higher order mode propagation and radiation is then presented. We focus on the influence of some critical parameters such as vocal tract width and location of the sound source. In particular, we show how, using TLM simulation, it is possible to derive modal reflection and transmission characteristics of the vocal tract. A typical example of simulation is presented and discussed. 1. PRINCIPLE OF TLM METHOD Transmission Line Matrix (TLM) or Transmission Line Modelling is a general numerical method suitable for simulating three dimensional electromagnetic fields in complex geometries. A first application to acoustic wave propagation problems has been proposed and validated by El-Masri et al. (1996). The principles of the TLM time domain method have been introduced by Johns & Beurle (1971). Waves are represented by a discrete spatial electrical network model (Transmission line matrix). Voltages and currents in this network are equivalent to electric and magnetic fields in electromagnetic systems. Propagation of electromagnetic fields is simulated by the propagation and scattering of pulses in a network consisting of interconnected ideal transmission lines. At each time step, every node receives incident voltage pulses, and sends scattered pulses, as shown in Figure 1. A scattering matrix determines the relationship between incident and scattered pulses. The scattered pulses at time (t) become incident pulses on adjacent nodes at (t + ∆t). The scattering matrix is computed from transmission lines theory. time (t) time (t + ∆t) Figure 1: Propagation of impulsions in a two dimensional network (after Saguet, 1985). 1.2. Equations of TLM in 3 Dimensional of Space in Acoustics Each node from the network represents a parallel junction of three transmissions lines. The equivalent electric scheme of a scalar basic node is shown in Figure 2.