Landmark-Based Speech Recognition:
Spectrogram Reading,Support Vector Machines,
Dynamic Bayesian Networks,and Phonology
Mark [email protected]
University of Illinois at Urbana-Champaign, USA
Lecture 2: Acoustics of Vowel and Glide Production
• One-Dimensional Linear Acoustics– The Acoustic Wave Equation– Transmission Lines– Standing Wave Patterns
• One-Tube Models– Schwa– Front cavity resonance of fricatives
• Two-Tube Models– The vowel /a/– Helmholtz Resonator– The vowels /u,i,e/
• Perturbation Theory– The vowels /u/, /o/ revisited– Glides
1. One-Dimensional Acoustic Wave Equation
and Solutions
Acoustics: Constitutive Equations
Acoustic Plane Waves: Time Domain
Acoustic Plane Waves: Frequency Domain
Tex
Solution for a Tube with Constant Area and Hard Walls
2. One-Tube Models
Boundary Conditions
L0
Resonant Frequencies
Standing Wave Patterns
Standing Wave Patterns: Quarter-Wave Resonators
Tube Closed at the Left End, Open at the Right End
Standing Wave Patterns: Half-Wave Resonators
Tube Closed at Both Ends Tube Open at Both Ends
Schwa and Invv (the vowels in “a tug”)
F1=500Hz=c/4L
F2=1500Hz=3c/4L
F3=2500Hz=5c/4L
Front Cavity Resonances of a Fricative
/s/: Front Cavity Resonance = 4500Hz 4500Hz = c/4L if Front Cavity Length is L=1.9cm
/sh/: Front Cavity Resonance = 2200Hz 2200Hz = c/4L if Front Cavity Length is L=4.0cm
3. Two-Tube Models
Conservation of Mass at the Juncture of Two Tubes
A1
A2 = A1/2
U1(x,t) U2(x,t)= 2U1(x,t)
Total liters/second transmitted = (velocity) X (tube area)
Two-Tube Model: Two Different Sets of Waves
Incident Wave P1+
Reflected Wave P1- Incident Wave P2-
Reflected Wave P2+
Two-Tube Model: Solution in the Time Domain
Two-Tube Model in the Frequency Domain
Approximate Solution of the Two-Tube Model, A1>>A2
Approximate solution: Assume that the two tubes are completely decoupled, so that the formants include - F(BACK CAVITY) = c/4 LBACK
- F(FRONT CAVITY) = c/4LFRONT
LBACK
LFRONT
The Vowels /AA/, /AH/
LBACK
LFRONT
LBACK=8.8cm F2= c/4LBACK = 1000Hz
LFRONT=12.6cm F1= c/4LFRONT = 700Hz
Acoustic Impedance
Z(,j)
0
Z(,j)
0
Low-Frequency Approximations of Acoustic Impedance
Helmholtz Resonator
-Z1(,j) =
0
Z2(,j)
0
The Vowel /i/
Back Cavity = Pharynx Resonances: 0Hz, 2000Hz, 4000HzFront Cavity = Palatal Constriction Resonances: 0Hz, 2500Hz, 5000Hz
Back Cavity Volume = 70cm3
Front Cavity Length/Area = 7cm-1
1/2√MC = 250Hz
Helmholtz Resonance replaces all 0Hz partial-tube resonances.
250Hz
2000Hz2500Hz
The Vowel /u/: A Two-Tube Model
Back Cavity = Mouth + Pharynx Resonances: 0Hz, 1000Hz, 2000HzFront Cavity = Lips Resonances: 0Hz, 18000Hz, …
Back Cavity Volume = 200cm3
Front Cavity Length/Area = 2cm-1
1/2√MC = 250Hz
Helmholtz Resonance replaces all 0Hz partial-tube resonances.
250Hz
1000Hz
2000Hz
The Vowel /u/: A Four-Tube Model
Two Helmholtz Resonators = Two Low-Frequency Formants! F1 = 250Hz F2 = 500Hz
F3 = Pharynx resonance, c/2L = 2000Hz
250Hz 500Hz
2000Hz
Pharynx
VelarTongueBodyConstriction Mouth
Lips
4. Perturbation Theory
Perturbation Theory(Chiba and Kajiyama, The Vowel, 1940)
A(x) is constant everywhere, except for one small perturbation.
Method: 1. Compute formants of the “unperturbed” vocal tract. 2. Perturb the formant frequencies to match the area perturbation.
Conservation of Energy Under Perturbation
Conservation of Energy Under Perturbation
“Sensitivity” Functions
Sensitivity Functions for the Quarter-Wave Resonator (Lips Open)
L
/AA/ /ER/ /IY/ /W/
Sensitivity Functions for the Half-Wave Resonator (Lips Rounded)
L
/L,OW/ /UW/
Formant Frequencies of Vowels
From Peterson & Barney, 1952
Summary• Acoustic wave equation easiest to solve in frequency domain, for
example:– Solve two boundary condition equations for P+ and P-, or– Solve the two-tube model (four equations in four unknowns)
• Quarter-Wave Resonator: Open at one end, Closed at the other– Schwa or Invv (“a tug”)– Front cavity resonance of a fricative or stop
• Half-Wave Resonator: Closed at the glottis, Nearly closed at the lips– /uw/
• Two-Tube Models– Exact solution: use reflection coefficient– Approximate solution: decouple the tubes, solve separately
• Helmholtz Resonator– When the two-tube model seems to have resonances at 0Hz, use, instead,
the Helmholtz Resonance frequency, computed with low-frequency approximations of acoustic impedance
– /iy/: F1 is a Helmholtz Resonance– /uw/ and /ow/: Both F1 and F2 are Helmholtz Resonances
• Perturbation Theory– Perturbed area Perturbed formants– Sensitivity function explains most vowels and glides in one simple chart