It was assumed that the pressureat the lips is zero and the volume velocity source is ideal  no energy loss at the input and output. For radiation impedance: Morse and Ingard (1986), Beranek (1954), Flanagan (1972). Lip radiation is modelled by a piston in an infinite wall. The acoustic impedance is a resistance R r and an inductor L r in parallel. For the infinite wall R r = 1289 / 9  2 and L r =8a / 3  c Boundary Effects

The impedance equation can be converted to a differential equation via Laplace transform. Portnoff (1973) numerically simulated the above equation coupled to the wave equation.  Broader bandwidths and lowering of the resonances are observed. Higher frequencies are affected most. (This can be seen by considering Z r  0 when   0 (small)  p(l,t) = 0 but for large ,  L r >>R r  Z r = R r ) Boundary Effects

Glottal Source and Impedance Flanagan-Isızaka (1978) proposed a linear model for the impedance. This is a differential equation in the time-domain. The solution of this equation together with the wave equation yields broadening of the bandwidths at low frequencies. This can be seen from Z g = R g + j  L g becomes Z g  R g for small  ; (purely resistive). Overall Frequency Response introduces a highpass filter effect

Summarizing the results 1)Resonances are due to vocal tract. Resonant frequenciesshift as a result various losses. 2)Bandwidths of lower resonances are controlledby wall vibration and glottal impedance loss. 3)Bandwidths of higher resonances are controlled by radiation, viscous and thermal losses.

(Energy loss is assumed to be only at the lips.) For the k th tube (*) Boundary conditions are Model of Concatenated Tubes

The boundary conditions and (*) yield : Let  k = l k /c Using in (2) Model of Concatenated Tubes

(4) – (3)  Let  Both equations contain a component due to reflection and one component due to transmission. Model of Concatenated Tubes

Lips: Suppose radiation impedance is real. At the N th tube Since There is no backward going wave from free space. Boundary Relations

Therefore Outgoing wave from the lips: Let the outer space be represented by an infinite length tube of cross section A N+1, and in particular, if (Also, if Z r (  ) = 0  r L =1, A N+1    no radiation from the lips.

Glottis: Suppose the glottal impedance is purely real, Z g =R g In particular, if glottal impedance is modeled by a tube of cross section A 0 and A 0 is chosen such that By chosig A k s properly it is possible to approximate formant bandwidths. Boundary Relations

Consider a vocal-tract model consisting of N lossless concatenated tubes with total length l. length of a tube =  x = l/N propagation time in a tube =  =  x/c Let v a (t) be the impulse samples of the impulse response with 2  intervals. At the end of the tube v a (t) can be written as: Since the samples up to time N  will be zero. The Laplace transform and the frequency response are The frequency response is periodic with 2  /2  ; V a (  + 2  /2  ) = V a (  ) Discrete Time Realization

If the impulse response of the system is not band limited then aliasing will occur. The discrete-time frequency response can be obtained by  =  /2  Discrete Time Realization

The effect of aliasing can be reduced by decreasing the individual tube lengths and hence the sampling period. It can be shown that the transfer function of the discrete-time model has the form Discrete Time Realization When A k > 0 all poles are inside the unit circle.

Ex: Let l=17.5 cm, c=350 m/sec. Find N to cover a bandwidth of 5000 Hz. (Assume that vocal tract impulse response and excitation are bandlimited to 5000 Hz.) Soln:  /2  is the cutoff bandwidth Hz   rad/sec  /2  =    = 1/20000 N = l/c  = 10 Since N is the order of the system, there are 5 complex conjugate poles  There can be 5 resonances in the given bandwidth. Discrete Time Realization

Comparison of numerical simulation of differential equations and concatenated tube model (N=10) Discrete Time Realization Reflection coefficientsTube cross sections

Simulation Concatenated tubes