EXERCISES
4.3 Uniform Tube Model
4.3.3 Boundary Effects
2N−2 equations in 2N unknown pressure and velocity variables. The two additional equa- tions necessary for a unique solution are supplied by the boundary conditions at the lips and glottis.
The resulting frequency responseVa()= U (l,)Ug() of the numerical simulation is shown in Figure 4.10a (pressure and velocity computed at 96 samples in space along thexvariable) for a uniform tube with yielding walls and no other losses, terminated in a zero pressure boundary condition [26]. The tube is 17.5 cm in length and 5 cm2 in cross section. The yielding wall parameters are given by Flanagan [8] to bemw = 0.4 gm/cm2,bw = 6500 dyne-sec/cm3, and kw = 0. The result is markedly different from the lossless case. Because of energy loss due to the wall vibration, the poles of the transfer function are no longer on thej axis and so the bandwidth is nonzero. In addition, the resonant frequencies have slightly increased. Finally, these effects are most pronounced at low frequencies because the inertial mass of the wall results in less wall motion at high frequencies.
Viscosity and Thermal Loss —The effects of both viscous and thermal loss can be represented by modification of Equations (4.20) and (4.21) with the introduction of a resistive term, rep- resenting the energy loss due to friction of air particles along the wall, and a conductive term, representing heat loss through the vibrating walls, respectively [8],[26]. The resulting coupled equations are solved numerically for the steady-state condition, again using a central difference approximation to the spatial partial derivative [26]. With viscous and thermal loss only, effects are less noticeable than with wall vibration loss, being more pronounced at high frequencies where more friction and heat are generated. The addition of viscosity and thermal conduction to the presence of vibrating walls yields slight decreases in resonant frequencies and some broad- ening of the bandwidths. Figure 4.10b gives the result of Portnoff’s numerical simulation with all three losses for the uniform tube with zero pressure termination [26].
1st 2nd 3rd 4th 5th
504.6 1512.3 2515.7 3518.8 4524.0
53.3 40.8 28.0 19.0 13.3 Frequency Bandwidth Formant
Uniform Cross Section:
Length = 17.5 cm Area = 5.0 cm2
56 52 48 44 40 36 32 28 24 20 16 12 8 4 – 4
0 3000
Frequency (Hz) (a)
4000 5000 0
1st 2nd 3rd 4th 5th
502.5 1508.9 2511.2 3513.5 4518.0
59.3 51.1 41.1 34.5 30.8 Frequency Bandwidth Formant
0 1000
1000 2000 2000 3000
Frequency (Hz) (b)
4000 5000 0
4 8 12 16 20 24 28 32 36 40
1st 2nd 3rd 4th 5th
473.5 1423.6 2372.3 3322.1 4274.5
62.3 80.5 114.5 158.7 201.7 Frequency Bandwidth Formant
0 1000 2000 3000 Frequency (Hz)
(c)
4000 5000 0
4 8 12 16 20 24 28 32
– 4
20 logVa(Ω) 20 logVa(Ω) 20 logVa(Ω)
Figure 4.10 Frequency response of uniform tube with (a) vibrating walls with p(l,0) = 0; (b) vibrating walls, and viscous and thermal loss withp(l,0) = 0; (c) vibrating walls, viscous and thermal loss, and radiation loss [26],[28]. 3 dB bandwidths are given.
SOURCE: M.R. Portnoff,A Quasi-One-Dimensional Digital Simulation for the Time-Varying Vocal Tract [26]. ©1973, M.R. Portnoff and the Massachusetts Institute of Technology. Used by permission.
131
Vocal Tract
Glottis Lips
Piston in Infinite Wall ug(t)
u(0, t) u(l, t)
Rg Lg Zg
p(0, t) p(l, t) Rr Lr
Zr
+ +
– –
Figure 4.11 Glottal and lip boundary conditions as impedance loads, given by linearized serial and parallel electric circuit models, respectively. Piston in infinite wall model for radiation from lips is illustrated.
Rr (energy loss via sound propagation from the lips) in parallel with a radiation inductanceLr (inertial air mass pushed out at lips):
Zr() = P (l, ) U (l, )
= 1 1
Rr + j L1 r
= j LrRr
Rr +j Lr. (4.24)
For the infinite baffle, Flanagan has given values ofRr =128/9π2andLr =8a/3π cwhere ais the radius of the opening andcthe speed of sound.
Equation (4.24) can be represented in the time domain by a differential equation that is coupled to the wave equation solution of sound propagation within the vocal tract. As with other energy losses, Portnoff [26] has numerically simulated the effect of this coupling for the steady-state vocal tract condition. The frequency responseVa()= U (l,)Ug() with loss from the radiation load, as well as from vibrating walls, viscosity, and thermal conduction, is shown in Figure 4.10c. Consequences of the radiation load are broader bandwidths and a lowering of the resonances, the major effect being on the higher frequencies because radiation loss is greatest in this region. We can see this effect through Equation (4.24). For very small≈0,
Zr ≈ 0
so that the radiation load acts as a short circuit with p(l, t ) ≈ 0. For very large with Lr Rr
Zr ≈ j LrRr j Lr = Rr
4.3 Uniform Tube Model 133
and so the radiation load takes on the resistive component at high frequencies. Because the energy dissipated in radiation is proportional to the real part of the complex impedance Zr [2],[28] and because the real part ofZr monotonically increases with the resistance Rr (the reader should confirm this monotonicity), we deduce that the greatest energy loss, and thus the greatest formant bandwidth increase, from radiation is incurred at high frequencies.
Our final boundary condition is the glottal source and impedance. This is the most difficult addendum to our simple uniform tube because the glottal volume velocity has been shown to be nonlinearly related to the pressure variations in the vocal tract. We return to the true complexity of the glottal impedance later in this chapter; for now it suffices to use a simplification to obtain a flavor for the glottal impedance effect. By simplifying a nonlinear, time-varying two-mass vocal fold model for predicting glottal airflow (Figure 3.5 in Chapter 3), Flanagan and Ishizaka [9] proposed a linearized time-invariant glottal impedance as a resistanceRgin series with an inductanceLg:
Zg() = Rg +j Lg. (4.25)
As illustrated in Figure 4.11, this impedance is placed in parallel with an ideal volume ve- locity source ug(t ) that, for the case of voiced speech, is a typical glottal airflow velocity function as given in Chapter 3. Applying Kirchoff’s current law for electric circuits to the glottal source and impedance in Figure 4.11, we find the modified boundary condition is given by
U (0, ) = Ug()− P (0, ) Zg()
which in the time domain is a differential equation coupled to the partial differential wave equa- tion and other differential loss equations described above. Portnoff [26] performed a numerical simulation of this more realistic glottal boundary condition and found for steady-state a broad- ening of bandwidths at low resonances; this is consistent with Equation (4.25) becauseZg() approaches an open circuit for high frequencies (Zg() ≈ j Lg), and approaches a pure resistance for low frequencies (Zg()≈Rg).
We have up to now found the frequency response relating volume velocity at the lips, U (l, ), to input volume velocity at the glottis,Ug(). In practice, we measure thepressure at the lips with a pressure-sensitive microphone transducer. The pressure-to-volume velocity frequency response can be found as [28]
H () = P (l, ) Ug()
= P (l, ) U (l, )
U (l, ) Ug()
= Zr()Va() (4.26)
where the radiation impedanceZr()= P (l,)U (l,), from Figure 4.11 and Equation (4.24), intro- duces a highpass filter effect onto the frequency responseVa().