Landmark-Based Speech Recognition:

Spectrogram Reading,Support Vector Machines,

Dynamic Bayesian Networks,and Phonology

Mark Hasegawa-Johnsonjhasegaw@uiuc.edu

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