Acoustic Tube Modeling (I) 虞台文. Content Introduction Wave Equations for Lossless Tube Uniform Lossless Tube Lips-Radiation Model Glottis Model One-Tube.

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Acoustic Tube Modeling (I) 虞台文

Content Introduction Wave Equations for Lossless Tube Uniform Lossless Tube Lips-Radiation Model Glottis Model One-Tube Vocal Tract Model Exercises

Acoustic Tube Modeling (I) Introduction

Vocal Tract

Acoustic Tube Derivation Lips Glottis

Assumptions Lips Glottis Consists of M interconnected sections of equal length, and each section is of uniform area. The traverse dimension of each section is small enough compared with a wave length so that the sound propagation though an individual section can be treated as a plane wave. Sections are rigid so that internal losses due to wall vibration, viscosity, and heat conduction are negligible. The model is linear and uncoupled from glottis. The effects of the nasal tract can be ignored.

Discrete Area Functions Lips Glottis Area Lips Glottis

Acoustic Tube Modeling (I) Wave Equations for Lossless Tube

dx  ( 密度 ) V ( 體積 ) A ( 面積 ) A+dA 壓力 (p) 壓力 (p+dp) m=  V ( 質量 ) F v ( 速度 ) u ( 容積速度 ) Eliminate higher order terms

Wave Equations for Lossless Tube dx  ( 密度 ) V ( 體積 ) A ( 面積 ) A+dA v ( 速度 ) u ( 容積速度 ) p ( 壓力 ) v+dv u+du Mass Continuity Condition

Wave Equations for Lossless Tube dx  ( 密度 ) V ( 體積 ) A ( 面積 ) v ( 速度 ) u ( 容積速度 ) p ( 壓力 ) v+dv u+du Mass Continuity Condition

Vocal Tract A(x, t) x=0x=lx=l GlottisLips u(x,t)u(x,t) p(x,t)p(x,t)

Acoustic Tube Modeling (I) Uniform Lossless Tube

Uniformly Lossless Tube x=0x=lx=l

Uniformly Lossless Tube x=0x=lx=l

Uniformly Lossless Tube x=0x=lx=l

Pressure vs. Volume Flow x=0x=lx=l u(x,t)u(x,t) u+(tx/c)u+(tx/c) u (t+x/c)u (t+x/c)

Pressure vs. Volume Flow

0 Characteristic Impedance of the tube.

x=0x=lx=l u+(t)u+(t) u+(tl/c)u+(tl/c) u(t)u(t) u(t+l/c)u(t+l/c) u+(tx/c)u+(tx/c) u(t+x/c)u(t+x/c)  + u(x,t)u(x,t) + + Z p(x,t)p(x,t) Pressure vs. Volume Flow

x=0x=lx=l u+(t)u+(t) u+(tl/c)u+(tl/c) u(t)u(t) u(t+l/c)u(t+l/c) u+(tx/c)u+(tx/c) u(t+x/c)u(t+x/c)  + u(x,t)u(x,t) + + Z p(x,t)p(x,t) Pressure vs. Volume Flow 壓力受順流與逆流強度和而改變

Acoustic Tube Modeling (I) Lips-Radiation Model

Boundary Condition (Lips) Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips ZLZL ZLZL p L (t)=p T (l, t) u L (t)=u T (l, t) Radiation Impedance Assumed Z L (j  ) is real

Boundary Condition (Lips) Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips ZLZL ZLZL p L (t)=p T (l, t) u L (t)=u T (l, t) Assumed Z L (j  ) is real

Boundary Condition (Lips) Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips ZLZL ZLZL p L (t)=p T (l, t) u L (t)=u T (l, t)

Boundary Condition (Lips) Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips ZLZL ZLZL p L (t)=p T (l, t) u L (t)=u T (l, t)

Boundary Condition (Lips) Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips ZLZL ZLZL p L (t)=p T (l, t) u L (t)=u T (l, t) 1+  L Delay LL In case Z L  0,  L = 1 In case Z L  0,  L = 1

Acoustic Tube Modeling (I) Glottis Model

Boundary Condition (Glottis) uG(t)uG(t) ZGZG ZGZG Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips

Boundary Condition (Glottis) uG(t)uG(t) ZGZG ZGZG Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips p G (t)=p T (0, t) Assumed Z G (j  ) is real

Boundary Condition (Glottis) uG(t)uG(t) ZGZG ZGZG Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips p G (t)=p T (0, t) Assumed Z G (j  ) is real

Boundary Condition (Glottis) uG(t)uG(t) ZGZG ZGZG Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips p G (t)=p T (0, t) Assumed Z G (j  ) is real

Boundary Condition (Glottis) uG(t)uG(t) ZGZG ZGZG Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips In case Z G >> 0,  G =1 In case Z G >> 0,  G =1 Delay

Acoustic Tube Modeling (I) One-Tube Vocal Tract Model

One-Tube Model uG(t)uG(t) ZGZG ZGZG Tube (Vocal Tract) Tube (Vocal Tract) GlottisLips ZLZL ZLZL uL(t)uL(t) 1+  L Delay(  ) LL 1+  G GG

Impulse Response 1+  L Delay(  ) LL 1+  G GG (t)(t) va(t)va(t) Soonest Response By Reflection & Propagation

Impulse Response 1+  L Delay(  ) LL 1+  G GG (t)(t) va(t)va(t)

Impulse Response 1+  L Delay(  ) LL 1+  G GG (t)(t) va(t)va(t) 1+  L Delay(  ) Delay(2  ) LL 1+  G GG

Impulse Response l=17.5 cm c=350 m/sec =500 Hz  = l/c = 0.5 msec 

Impulse Response l=17.5 cm c=350 m/sec =500 Hz  = l/c = 0.5 msec  For nature vowel, resonance frequencies (formants) were approximately 500, 1500, 2500, 3500 Hz.

Digital Simulation for One-Tube Model z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 LL GG 1+  L (1+  G )/2 u G [n]= u G (nT) u L [n]= u L (nT) 1+  L Delay(  ) LL 1+  G GG How many sections are required?

Digital Simulation for One-Tube Model Assume  L =1,  G =1. zMzM zMzM 1 uG[n]uG[n] uL[n]uL[n] z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 LL GG 1+  L (1+  G )/2 u G [n]= u G (nT) u L [n]= u L (nT)

Digital Simulation for One-Tube Model zMzM zMzM 1 uG[n]uG[n] uL[n]uL[n] z-plane

Digital Simulation for One-Tube Model z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 LL GG 1+  L (1+  G )/2 u G [n]= u G (nT) u L [n]= u L (nT) How many sections are required? Voice Band 20~3400 Hz Sampling rate 8000 Hz T = 1/8000 = msec 0.5 msec Glottis Lips 4

Acoustic Tube Modeling (I) Exercises

Exercise z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 LL GG 1+  L (1+  G )/2 u G [n]= u G (nT) u L [n]= u L (nT) M sections Find the transfer function of the above system.

Computer Simulation z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 z1z1 LL GG 1+  L (1+  G )/2 u G [n]= u G (nT) u L [n]= u L (nT) 4 sections Using different  G and  L and feeding periodic impulse trains with different periods to the system to generate sounds. Plot the generated waveforms.