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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.

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Presentation on theme: "Acoustic Tube Modeling (I) 虞台文. Content Introduction Wave Equations for Lossless Tube Uniform Lossless Tube Lips-Radiation Model Glottis Model One-Tube."— Presentation transcript:

1 Acoustic Tube Modeling (I) 虞台文

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

3 Acoustic Tube Modeling (I) Introduction

4 Vocal Tract

5

6 Acoustic Tube Derivation Lips Glottis

7 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.

8 Discrete Area Functions Lips Glottis Area Lips Glottis

9 Acoustic Tube Modeling (I) Wave Equations for Lossless Tube

10 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

11 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

12 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

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

14 Acoustic Tube Modeling (I) Uniform Lossless Tube

15 Uniformly Lossless Tube x=0x=lx=l

16 Uniformly Lossless Tube x=0x=lx=l

17 Uniformly Lossless Tube x=0x=lx=l

18 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)

19 Pressure vs. Volume Flow

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21

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23

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25 0 Characteristic Impedance of the tube.

26 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

27 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

28 Acoustic Tube Modeling (I) Lips-Radiation Model

29 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

30 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

31 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)

32 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)

33 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

34 Acoustic Tube Modeling (I) Glottis Model

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

36 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

37 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

38 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

39 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

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

41 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

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

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

44 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

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

46 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.

47 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?

48 Digital Simulation for One-Tube Model Assume  L =1,  G =1. zMzM zMzM 11 1 2 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)

49 Digital Simulation for One-Tube Model zMzM zMzM 11 1 2 1 uG[n]uG[n] uL[n]uL[n] z-plane

50 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 = 0.125 msec 0.5 msec Glottis Lips 4

51 Acoustic Tube Modeling (I) Exercises

52 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.

53 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.

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