ECE 598: The Speech Chain Lecture 8: Formant Transitions; Vocal Tract Transfer Function

Today Perturbation Theory: Perturbation Theory: A different way to estimate vocal tract resonant frequencies, useful for consonant transitions A different way to estimate vocal tract resonant frequencies, useful for consonant transitions Syllable-Final Consonants: Formant Transitions Syllable-Final Consonants: Formant Transitions Vocal Tract Transfer Function Vocal Tract Transfer Function Uniform Tube (Quarter-Wave Resonator) Uniform Tube (Quarter-Wave Resonator) During Vowels: All-Pole Spectrum During Vowels: All-Pole Spectrum Q Bandwidth Bandwidth Nasal Vowels: Sum of two transfer functions gives spectral zeros Nasal Vowels: Sum of two transfer functions gives spectral zeros

Topic #1: 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

“ Sensitivity ” Functions

Sensitivity Functions for the Quarter- Wave Resonator (Lips Open)  L /AA//ER/ /IY//W/ Note: low F3 of /er/ is caused in part by a side branch under the tongue – perturbation alone is not enough to explain it.

Sensitivity Functions for the Half- Wave Resonator (Lips Rounded)  L /L,OW//UW/ Note: high F3 of /l/ is caused in part by a side branch above the tongue – perturbation alone is not enough to explain it.

Formant Frequencies of Vowels From Peterson & Barney, 1952

Topic #2: Formant Transitions, Syllable-Final Consonant

Events in the Closure of a Nasal Consonant Vowel Nasalization Formant Transitions Nasal Murmur

Formant Transitions: A Perturbation Theory Model

Formant Transitions: Labial Consonants “the mom” “the bug”

Formant Transitions: Alveolar Consonants “the tug” “the supper”

Formant Transitions: Post-alveolar Consonants “the shoe” “the zsazsa”

Formant Transitions: Velar Consonants “the gut” “sing a song”

Topic #3: Vocal Tract Transfer Functions

Transfer Function “Transfer Function” T(  )=Output(  )/Input(  ) “Transfer Function” T(  )=Output(  )/Input(  ) In speech, it’s convenient to write T(  )=U L (  )/U G (  ) In speech, it’s convenient to write T(  )=U L (  )/U G (  ) U L (  ) = volume velocity at the lips U L (  ) = volume velocity at the lips U G (  ) = volume velocity at the glottis U G (  ) = volume velocity at the glottis T(0) = 1 T(0) = 1 Speech recorded at a microphone = pressure Speech recorded at a microphone = pressure P R (  ) = R(  )T(  )U G (  ) P R (  ) = R(  )T(  )U G (  ) R(  ) = j  f/r = “radiation characteristic” R(  ) = j  f/r = “radiation characteristic”  = density of air  = density of air r = distance to the microphone r = distance to the microphone f = frequency in Hertz f = frequency in Hertz

Transfer Function of an Ideal Uniform Tube Ideal Terminations: Ideal Terminations: Reflection coefficient at glottis: zero velocity,  =1 Reflection coefficient at glottis: zero velocity,  =1 Reflection coefficient at lips: zero pressure,  =  1 Reflection coefficient at lips: zero pressure,  =  1 Obviously, this is an approximation, but it gives… Obviously, this is an approximation, but it gives… T(  ) = 1/cos(  L/c) = …              …  n = n  c/L –  c/2L F n = nc/2L – c/4L 122232…122232…

Transfer Function of an Ideal Uniform Tube Peaks are actually infinite in height (figure is clipped to fit the display)

Transfer Function of a Non-Ideal Uniform Tube Almost ideal terminations: Almost ideal terminations: At glottis: velocity almost zero,  ≈1 At glottis: velocity almost zero,  ≈1 At lips: pressure almost zero,  ≈  1 At lips: pressure almost zero,  ≈  1 T(  ) = 1/(j/Q +cos(  L/c)) … at F n =nc/2L – c/4L,… T(2  F n ) =  jQ 20log 10 |T(2  F n )| = 20log 10 Q

Transfer Function of a Non-Ideal Uniform Tube

Transfer Function of a Vowel: Height of First Peak is Q 1 =F 1 /B 1 T(  ) =  (j  +j2  F n +  B n )(j  j2  F n +  B n ) T(2  F 1 ) ≈ (2  F 1 ) 2 /(j4  F 1  B 1 ) =  jF 1 /B 1 Call Q n = F n /B n T(2  F 1 ) ≈  jQ 1 20log10|T(2  F 1 )| ≈ 20log10Q 1 (2  F n ) 2 +(  B n ) 2 n=1 ∞

Transfer Function of a Vowel: Bandwidth of a Peak is B n T(  ) =  (j  +j2  F n +  B n )(j  j2  F n +  B n ) T(2  F 1 +  B 1 ) ≈ (2  F 1 ) 2 /((j4  F 1 )(  B 1 +  B 1 )) =  jQ 1 /2 At f=F B n, |T(  )|=0.5Q n 20log 10 |T(  )| = 20log 10 Q 1 – 3dB (2  F n ) 2 +(  B n ) 2 n=1 ∞

Amplitudes of Higher Formants: Include the Rolloff T(  ) =  (j  +j2  F n +  B n )(j  j2  F n +  B n ) At f above F 1 T(2  f) ≈ (F 1 /f) T(2  F 2 ) ≈ (  jF 2 /B 2 )(F 1 /F 2 ) 20log10|T(2  F 2 )| ≈ 20log 10 Q 2 – 20log 10 (F 2 /F 1 ) 1/f Rolloff: 6 dB per octave (per doubling of frequency) (2  F n ) 2 +(  B n ) 2 n=1 ∞

Vowel Transfer Function: Synthetic Example L 1 = 20log 10 (500/80)=16dB L 2 = 20log 10 (1500/240) – 20log 10 (F 2 /F 1 ) = 16dB – 9.5dB L 3 = 20log 10 (2500/600) – 20log 10 (F 3 /F 1 ) – 20log 10 (F 3 /F 2 ) B 2 = 240Hz B 1 = 80Hz B 3 = 600Hz? (hard to measure because rolloff from F 1, F 2 turns the F 3 peak into a plateau) F 4 peak completely swamped by rolloff from lower formants

Shorthand Notation for the Spectrum of a Vowel T(s) =  (s  s n )(s  s n *) s = j  s n =  B n +j2  F n s n * =  B n  j2  F n s n s n * = |s n | 2 = (2  F n ) 2 +(  B n ) 2 T(0) = 1 20log 10 |T(0)| = 0dB snsn*snsn* n=1 ∞

Another Shorthand Notation for the Spectrum of a Vowel T(s) =  (1-s/s n )(1-s/s n *) 1 n=1 ∞

Topic #4: Nasalized Vowels

Vowel Nasalization Nasalized Vowel Nasal Consonant

Nasalized Vowel P R (  ) = R(  )(U L (  )+U N (  )) U N (  ) = Volume Velocity from Nostrils P R (  ) = R(  )(T L (  )+T N (  ))U G (  ) = R(  )T(  )U G (  ) T(  ) = T L (  ) + T N (  )

Nasalized Vowel T(s) = T L (s)+T N (s) = (1-s/s Ln )(1-s/s Ln *) + (1-s/s Nn )(1-s/s Nn *) = (1-s/s Ln )(1-s/s Ln *)(1-s/s Nn )(1-s/s Nn *) 1/s Zn = ½(1/s Ln +1/s Nn ) s Zn = n th spectral zero T(s) = 0 if s=s Zn 11 2(1-s/s Zn )(1-s/s Zn *)

The “Pole-Zero Pair” 20log 10 T(  ) = 20log 10 (1/(1-s/s Ln )(1-s/s Ln *)) + 20log 10 ((1-s/s Zn )(1-s/s Zn *)/(1-s/s Nn )(1-s/s Nn *)) = original vowel log spectrum + log spectrum of a pole-zero pair

Additive Terms in the Log Spectrum

Transfer Function of a Nasalized Vowel

Pole-Zero Pairs in the Spectrogram Nasal Pole Zero Oral Pole

Summary Perturbation Theory: Perturbation Theory: Squeeze near a velocity peak: formant goes down Squeeze near a velocity peak: formant goes down Squeeze near a pressure peak: formant goes up Squeeze near a pressure peak: formant goes up Formant Transitions Formant Transitions Labial closure: loci near 250, 1000, 2000 Hz Labial closure: loci near 250, 1000, 2000 Hz Alveolar closure: loci near 250, 1700, 3000 Hz Alveolar closure: loci near 250, 1700, 3000 Hz Velar closure: F2 and F3 come together (“velar pinch”) Velar closure: F2 and F3 come together (“velar pinch”) Vocal Tract Transfer Function Vocal Tract Transfer Function T(s) =  s n s n */(s-s n )(s-s n *) T(s) =  s n s n */(s-s n )(s-s n *) T(  =2  Fn) = Q n = F n /B n T(  =2  Fn) = Q n = F n /B n 3dB bandwidth = B n Hertz 3dB bandwidth = B n Hertz T(0) = 1 T(0) = 1 Nasal Vowels: Nasal Vowels: Sum of two transfer functions gives a spectral zero between the oral and nasal poles Sum of two transfer functions gives a spectral zero between the oral and nasal poles Pole-zero pair is a local perturbation of the spectrum Pole-zero pair is a local perturbation of the spectrum