Speech Acoustics, Formant Structure & Auditory Perception

Speech Acoustics, Formant Structure & Auditory Perception

Speech is the most ecologically significant acoustic signal processed by the human auditory system. Accurate audiological assessment, hearing aid prescription, and auditory rehabilitation require a comprehensive, doctoral-level understanding of speech acoustics—from vocal tract kinematics and formant physics to speech spectrum dynamics and perceptual degradation under sensorineural pathology.

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The Source-Filter Theory of Speech Production

Formulated mathematically by Gunnar Fant (1960), the Source-Filter Theory models speech production as a linear, two-stage acoustic process: a sound source operating independently of an acoustic filter.

1. The Glottal Sound Source ($U[s]$)

• Vocal Fold Vibration: During phonation of voiced sounds, expired pulmonary airflow forces the adducted vocal folds into sustained, self-oscillating mucosal vibration (aerodynamic myoelastic theory / Bernoulli effect).
• Fundamental Frequency ($F_0$): The rate at which the vocal folds open and close per second. $F_0$ determines perceived vocal pitch:
  • Adult Males: $\approx 100 - 130\,\text{Hz}$ (mean $\approx 120\,\text{Hz}$)
  • Adult Females: $\approx 190 - 250\,\text{Hz}$ (mean $\approx 220\,\text{Hz}$)
  • Young Children: $\approx 250 - 350\,\text{Hz}$ (mean $\approx 300\,\text{Hz}$)
• Acoustic Source Spectrum: Glottal pulses generate an acoustic spectrum composed of $F_0$ and an infinite series of integer harmonic multiples ($2F_0, 3F_0, 4F_0, \dots$).
• Glottal Roll-off: The glottal source spectrum naturally loses acoustic energy at a rate of $-12\,\text{dB}$ per octave as frequency increases. High harmonics carry substantially less energy than low harmonics.
• Unvoiced Sound Source: For unvoiced speech sounds (e.g., /s/, /t/, /f/), the glottis remains open, and the acoustic source is not vocal fold vibration, but turbulent, aperiodic noise generated by air forced through supraglottal constrictions.

2. The Supraglottal Vocal Tract Filter ($T[s]$)

• Quarter-Wave Resonator: The vocal tract (pharynx, oral cavity, and nasal cavity) behaves acoustically as a tube open at the lips and closed at the vocal folds. For a uniform tube of length $L$, resonance peaks occur at odd multiples of a quarter-wavelength: $F_n = \frac{(2n - 1)c}{4L}$ where $n$ is the resonance number ($1, 2, 3, \dots$), $c$ is the speed of sound in warm, humid air ($\approx 34,300\,\text{cm/s}$), and $L$ is vocal tract length.
• For an average adult male vocal tract ($L \approx 17.5\,\text{cm}$):
  • $F_1 = \frac{1 \times 34,300}{4 \times 17.5} = \frac{34,300}{70} = 490\,\text{Hz} \approx 500\,\text{Hz}$
  • $F_2 = \frac{3 \times 34,300}{70} = 1470\,\text{Hz} \approx 1500\,\text{Hz}$
  • $F_3 = \frac{5 \times 34,300}{70} = 2450\,\text{Hz} \approx 2500\,\text{Hz}$
• Formants: The natural acoustic resonant frequencies of the vocal tract. The vocal tract does not generate sound; it acts as a frequency-dependent band-pass filter, transmitting frequencies near its formants with minimal impedance while severely attenuating frequencies between formants.

3. Lip Radiation Characteristic ($R[s]$)

• As the sound wave exits the mouth into free space, the opening of the lips acts as an acoustic baffle. This air-to-free-field impedance boundary acts as a high-pass acoustic filter, boosting high frequencies at a rate of $+6\,\text{dB}$ per octave.

4. Net Speech Acoustic Output Spectrum ($P[s]$)

Combining the source, filter, and radiation characteristics yields the net radiated speech output: $P(s) = U(s) \times T(s) \times R(s)$ In terms of overall spectral tilt: $\text{Net Slope} = (-12\,\text{dB/octave glottal source}) + (+6\,\text{dB/octave lip radiation}) = -6\,\text{dB/octave}$ Consequently, the radiated human speech spectrum exhibits an overall downward slope of approximately $-6\,\text{dB}$ per octave above $500\,\text{Hz}$.

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Vowel Formant Acoustics & The Vowel Quadrilateral

Vowels are voiced, continuous, high-energy phonemes produced with an open, unconstricted vocal tract. Distinct vowel identities are created by altering the geometry of the vocal tract via tongue posture, jaw opening, and lip rounding.

The Formant Frequency Rules

1. First Formant Frequency ($F_1$):
   • Governed primarily by tongue height (vertical dimension) and pharyngeal cavity volume.
   • Rule: $F_1$ is inversely related to tongue height.
   • High vowels (tongue elevated, large pharyngeal cavity: /i/, /u/) produce low $F_1$ values ($\approx 250 - 350\,\text{Hz}$).
   • Low vowels (tongue depressed, constricted pharynx: /æ/, /ɑ/) produce high $F_1$ values ($\approx 700 - 850\,\text{Hz}$).
2. Second Formant Frequency ($F_2$):
   • Governed primarily by tongue advancement (front/back dimension) and oral cavity length.
   • Rule: $F_2$ is directly proportional to tongue advancement.
   • Front vowels (tongue forward, shortened anterior oral cavity: /i/, /e/) produce high $F_2$ values ($\approx 2000 - 2600\,\text{Hz}$).
   • Back vowels (tongue retracted, elongated oral cavity: /u/, /o/, /ɑ/) produce low $F_2$ values ($\approx 800 - 1100\,\text{Hz}$).
3. Lip Rounding Rule:
   • Protruding and rounding the lips (as in /u/ and /o/) physically elongates the entire vocal tract ($L$).
   • Because $F_n \propto \frac{1}{L}$, lip rounding lowers all formant frequencies ($F_1, F_2, F_3$).

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Consonant Acoustics: Cues for Manner, Place & Voicing

Consonants are characterized by constriction or occlusion of the vocal tract. Although consonants carry only a fraction of the total acoustic energy of speech, they carry over $80 - 90\%$ of speech intelligibility.

1. Stop Consonants (Plosives: /p, b, t, d, k, g/)

• Acoustic Stages: Complete oral closure (silent gap) $\to$ abrupt release of intraoral pressure (transient stop burst, $5 - 20\,\text{ms}$) $\to$ rapid formant transitions into adjacent vowel.
• Voice Onset Time (VOT): The duration between the release of the stop burst and the onset of vocal fold vibration.
  • Voiced stops (/b, d, g/): Short VOT, typically ranging from $-20\,\text{ms}$ (pre-voicing) to $+20\,\text{ms}$.
  • Voiceless stops (/p, t, k/): Long VOT ($+40$ to $+100\,\text{ms}$), characterized by an unvoiced aspiration interval prior to phonation onset.
• Burst Spectra (Place of Articulation):
  • Bilabial (/p, b/): Low-frequency or diffuse-flat spectral energy ($\approx 500 - 1500\,\text{Hz}$).
  • Alveolar (/t, d/): Diffuse high-frequency spectral energy ($\approx 3500 - 5000\,\text{Hz}$).
  • Velar (/k, g/): Compact mid-frequency spectral peak ($\approx 1500 - 2500\,\text{Hz}$).
• Formant Transitions & Locus Theory: The rapid shift in formant frequency from the burst into the vowel reflects vocal tract reshaping. $F_2$ transition locus: bilabial $\approx 700 - 800\,\text{Hz}$; alveolar $\approx 1800\,\text{Hz}$; velar $\approx 3000\,\text{Hz}$.

2. Fricatives (/s, z, ʃ, ʒ, f, v, θ, ð/)

• Produced by continuous turbulent airflow forced through a narrow constriction.
• Sibilants (Strident Fricatives: /s, z, ʃ, ʒ/):
  • Characterized by intense, high-energy acoustic noise.
  • /s, z/: Alveolar constriction creates a tiny anterior cavity, resonating at very high frequencies; primary spectral energy is concentrated above $4000 - 5000\,\text{Hz}$ (often reaching $8000\,\text{Hz}$).
  • /ʃ, ʒ/ ("sh", "measure"): Palato-alveolar constriction with lip rounding creates a larger anterior cavity; spectral energy drops down to $\approx 2500 - 3500\,\text{Hz}$.
• Non-Sibilants (/f, v, θ, ð/):
  • Labiodental and interdental constrictions lack a downstream resonant cavity.
  • Extremely low-energy, diffuse, broadband noise spanning $1000 - 8000\,\text{Hz}$. These phonemes are the most difficult to detect for individuals with hearing loss.

3. Nasals (/m, n, ŋ/)

• Characterized by complete oral closure with an open velopharyngeal port, routing acoustic energy through the tortuous nasal cavity.
• Nasal Murmur: A low-frequency resonance peak occurring at $\approx 250 - 300\,\text{Hz}$, caused by the large, compliant pharyngeal-nasal acoustic volume.
• Antiformants (Antiresonances / Spectral Zeros): Acoustic energy trapped within the occluded oral cavity branch is absorbed rather than radiated, creating sharp troughs (spectral dips) in the output spectrum:
  • Bilabial /m/ (largest cavity): Antiformant at $\approx 750 - 1000\,\text{Hz}$
  • Alveolar /n/: Antiformant at $\approx 1500 - 2000\,\text{Hz}$
  • Velar /ŋ/ (smallest cavity): Antiformant at $> 3000\,\text{Hz}$

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Long-Term Average Speech Spectrum (LTASS) & Speech Dynamic Range

The Long-Term Average Speech Spectrum (LTASS) describes the spectral envelope of conversational speech averaged across a broad sample of continuous talkers and phonemes.

Spectral Distribution & The "Speech Banana"

• Energy Peak: Maximum acoustic energy in the LTASS occurs between $400\,\text{Hz}$ and $500\,\text{Hz}$, driven by the high fundamental frequencies and high-intensity $F_1$ energy of vowels.
• High-Frequency Roll-off: Above $500\,\text{Hz}$, the overall spectral envelope rolls off at approximately $-9\,\text{dB}$ per octave. By $4000\,\text{Hz}$, speech energy is approximately $20 - 25\,\text{dB}$ lower than at $500\,\text{Hz}$.
• The Speech Dynamic Range: At any individual 1/3-octave frequency band, the acoustic level of conversational speech fluctuates across a total dynamic range of $30\,\text{dB}$:
  • Peak conversational energy reaches $+12\,\text{dB}$ above the root-mean-square (RMS) level.
  • Trough (minimum) consonant energy extends to $-18\,\text{dB}$ below the RMS level.
  • Audiometric translation: If average conversational speech is presented at $65\,\text{dB SPL}$, the instantaneous acoustic components fluctuate between $47\,\text{dB SPL}$ and $77\,\text{dB SPL}$. In hearing aid fittings, the entire $30\,\text{dB}$ dynamic range must be rendered audible and comfortable. (The +12/−18 dB split is the classic Articulation Index convention; the ANSI S3.5-1997 SII models the same $30\,\text{dB}$ range as $\pm 15\,\text{dB}$ around the band speech level.)

The Speech Intelligibility Index (SII / AI)

The Speech Intelligibility Index (ANSI S3.5) is a computational metric (ranging from 0.0 to 1.0) that correlates directly with speech intelligibility by calculating the proportion of the $30\,\text{dB}$ speech dynamic range that is audible across discrete frequency bands, multiplied by a frequency importance function:

• The Power vs. Intelligibility Paradox:
  • Frequencies below $500\,\text{Hz}$ contribute approximately $60 - 70\%$ of the acoustic power of speech, but provide only $5 - 10\%$ of intelligibility.
  • Bands at and above $1000\,\text{Hz}$ carry only a small share of the acoustic power, yet hold about $70\%$ of the SII importance weighting (roughly $60\%$ in the bands from 1000 to 4000 Hz).
  • The single most critical octave band for speech recognition is centered at $2000\,\text{Hz}$, contributing roughly $26\%$ of the total SII weighting (octave-band importance of about $0.26$ in ANSI S3.5).

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Sensorineural Hearing Loss & Upward Spread of Masking

The physical acoustic properties of speech explain the classic symptomatology observed in patients with sensorineural hearing loss (SNHL).

High-Frequency SNHL & Phonemic Confusion

In typical age-related (presbycusis) or noise-induced hearing loss, cochlear hair cell damage is concentrated in the high-frequency basal turn.

• Patients retain normal low-frequency hearing ($< 500\,\text{Hz}$), fully preserving vowel detection and perceived loudness.
• Loss of audibility above $1500\,\text{Hz}$ eliminates the weak high-frequency cues required to identify voiceless fricatives (/s/, /z/, /f/, /θ/) and stop consonant burst spectra (/t/, /k/).
• This creates the quintessential clinical complaint: "I can hear people talking just fine, but I can't understand what they are saying."

Upward Spread of Masking

In normal ears, cochlear non-linearity and sharp tuning prevent low-frequency sounds from masking higher frequencies. In sensorineural hearing loss, two pathological mechanisms trigger severe upward spread of masking:

1. Loss of Cochlear Tuning: Outer hair cell loss broadens auditory filter bandwidths (ERBs), allowing low-frequency basilar membrane vibrations to bleed into higher frequency channels.
2. Basilar Membrane Wave Mechanics: Intense low-frequency vowel energy (e.g., /ɑ/ or /o/ at $75\,\text{dB SPL}$) sends a high-amplitude traveling wave sweeping past the basal cochlea. In the damaged cochlea, this intense basal stimulation completely masks the fragile, low-intensity consonant transitions (e.g., /s/ at $45\,\text{dB SPL}$).

Clinical Countermeasures: To limit upward spread of masking, audiologists avoid excessive low-frequency gain, use multichannel compression and noise reduction so that loud low-frequency noise does not raise gain across the whole spectrum, and fit directional or remote microphones to improve the signal-to-noise ratio.