Periodicity

Periodicity is the defining characteristic of oscillatory motion—the property that motion repeats itself at regular time intervals. Understanding periodicity is fundamental to analyzing vocal fold vibration, acoustic waveforms, and all oscillatory phenomena in voice science. This section explores the mathematical and graphical representations of periodic behavior, establishing the conceptual foundation for more complex analyses of voice signals.

Definition of Periodicity

A signal or motion is periodic if it repeats itself exactly after a fixed time interval called the period. Mathematically, a function x(t) is periodic if:

x(t + T) = x(t)  for all t

where T is the period—the smallest positive value for which this relationship holds.

Key Properties

Repetition: The same pattern recurs indefinitely. One complete pattern constitutes one cycle.

Regularity: The time interval between repetitions remains constant. Irregular repetition does not constitute true periodicity.

Infinite Extension: Mathematical periodicity extends infinitely in time. Real physical signals have finite duration but may be periodic during their duration.

Uniqueness of Period: While x(t + nT) = x(t) for any integer n, the period T is defined as the smallest time interval satisfying the periodicity condition.

Non-Periodic Signals

Not all signals are periodic. Aperiodic signals include:

Transients (single events, impulses)
Random noise
Speech consonants (typically aperiodic)
Whispered speech
Chaotic signals (quasi-periodic but never exactly repeating)

Normal phonation produces quasi-periodic signals—nearly periodic with small cycle-to-cycle variations (jitter, shimmer).

Period and Frequency

Period and frequency represent inverse mathematical relationships, describing the same physical phenomenon from complementary perspectives.

Period (T)

Definition: Time duration of one complete cycle, measured in seconds (s) or milliseconds (ms).

Physical Meaning: How long it takes for motion to return to the same state with the same velocity.

Measurement: Identify any point in the cycle (e.g., maximum displacement, zero crossing with positive slope) and measure time until the signal returns to that identical state.

Typical Values:

Male speech: T ≈ 10 ms (low pitch)
Female speech: T ≈ 4-5 ms (higher pitch)
Children: T ≈ 3-4 ms (still higher pitch)
Musical notes: T varies from ~2 ms (soprano high C) to ~40 ms (bass low notes)

Frequency (f or F0)

Definition: Number of complete cycles per unit time, measured in Hertz (Hz) or cycles per second.

Relationship to Period:

f = 1/T    or    T = 1/f

Physical Meaning: Rate of repetition. Higher frequency means more rapid oscillation.

Typical Values:

Male speech: f ≈ 100-120 Hz
Female speech: f ≈ 200-220 Hz
Children: f ≈ 250-300 Hz
Musical range: f ≈ 80 Hz (bass) to 1000+ Hz (soprano)

Angular Frequency (ω)

For mathematical analysis, angular frequency is often more convenient:

ω = 2πf = 2π/T

Measured in radians per second (rad/s). This representation simplifies trigonometric expressions in oscillation equations.

Conversions

Common conversions between representations:

Period to Frequency:

If T = 5 ms, then f = 1/0.005 s = 200 Hz

Frequency to Period:

If f = 440 Hz (A4 note), then T = 1/440 s ≈ 2.27 ms

Frequency to Angular Frequency:

If f = 100 Hz, then ω = 2π(100) ≈ 628 rad/s

Graphical Representation of Periodic Signals

Visual representation helps develop intuition about periodic behavior.

Time-Domain Representation

Plotting signal amplitude vs. time reveals periodicity through visual pattern repetition.

Features to Identify:

One complete cycle: From any point to the next occurrence of the identical pattern
Period (T): Horizontal distance spanning one cycle
Amplitude: Vertical extent of oscillation
Waveform shape: Sinusoidal, triangular, square, or complex

Figure 4.4: Examples of periodic waveforms showing (a) simple sinusoidal oscillation and (b) complex periodic waveform with multiple frequency components.

Landmark Selection

To measure period accurately, select consistent landmarks:

Positive-Going Zero Crossing: Point where signal crosses zero while increasing. Often easy to identify precisely.

Maximum (Peak): Highest point in cycle. Clear visually but may have flat top making precise identification difficult.

Minimum (Trough): Lowest point in cycle. Similar considerations as maximum.

Negative-Going Zero Crossing: Signal crosses zero while decreasing.

Consistency matters more than which landmark is chosen—always measure from the same phase of the cycle.

Complex Periodic Waveforms

Real vocal fold vibration produces complex periodic waveforms containing multiple frequency components. Despite complexity, periodicity remains identifiable:

Fundamental period (T0) corresponds to vocal fold vibration rate
Waveform shape may be asymmetric, non-sinusoidal
Pattern repeats every T0 seconds even if shape is complicated
Harmonic components (multiples of fundamental frequency) create complexity while preserving periodicity

Mathematical Description

Various mathematical functions can represent periodic signals.

Sinusoidal Functions

The simplest periodic function:

x(t) = A sin(ωt + φ)

or equivalently:

x(t) = A cos(ωt + φ)

where:

A is amplitude (maximum displacement from equilibrium)
ω is angular frequency (2πf)
φ is phase (determines initial position at t=0)
t is time

Properties:

Period: T = 2π/ω
Frequency: f = ω/(2π)
Pure sinusoids contain single frequency (fundamental only)
Represent simple harmonic motion

Complex Periodic Functions

Fourier’s Theorem states that any periodic function can be expressed as a sum of sinusoids:

x(t) = A₀ + Σ[Aₙ sin(nωt + φₙ)]

where:

A₀ is DC component (average value)
Aₙ are amplitudes of harmonic components
n = 1, 2, 3, … represents harmonic number
ω is angular frequency of fundamental
φₙ are phases of harmonic components

This decomposition underlies Fourier analysis, enabling frequency-domain analysis of complex periodic signals like voice.

Other Periodic Functions

Square Wave: Abrupt transitions between two values. Contains fundamental plus odd harmonics.

Triangle Wave: Linear rise and fall. Contains fundamental plus odd harmonics with amplitudes decreasing as 1/n².

Sawtooth Wave: Linear rise, abrupt drop (or vice versa). Contains all harmonics with amplitudes decreasing as 1/n.

Voice waveforms typically resemble sawtooth or pulse shapes, containing fundamental plus many harmonics.

Periodicity in Vocal Fold Vibration

Vocal fold oscillation during phonation exhibits periodicity with clinically relevant characteristics.

Fundamental Period (T0)

The fundamental period corresponds to one complete cycle of vocal fold vibration:

Opening Phase: Folds move laterally, glottis widens Closing Phase: Folds move medially, glottis narrows Closed Phase: (optional) Folds in contact, glottis closed Return: Folds separate to begin next cycle

T0 typically ranges from 2-15 ms in speech, corresponding to F0 (fundamental frequency) of 65-500 Hz.

Quasi-Periodicity

Real vocal fold vibration is quasi-periodic rather than perfectly periodic:

Jitter: Cycle-to-cycle variation in period. Normal jitter < 1%; pathological conditions show higher jitter.

Shimmer: Cycle-to-cycle variation in amplitude. Normal shimmer < 5%; pathology increases shimmer.

Despite these variations, the signal remains approximately periodic—patterns repeat with small perturbations.

Glottal Flow Waveform

Airflow through glottis exhibits periodic variation:

Peak Flow: Maximum flow occurs during maximal glottal opening Flow Minimum: Minimal or zero flow during glottal closure AC Flow: Oscillatory component with period T0 DC Flow: Average flow rate over many cycles

The shape of the flow waveform affects voice quality and spectral characteristics.

Relationship to Acoustic Signal

The acoustic pressure wave produced by voicing is periodic with same fundamental period as vocal fold vibration:

Fundamental frequency F0 equals vocal fold vibration rate
Harmonics at integer multiples (2F0, 3F0, 4F0, …)
Harmonic amplitudes shaped by vocal tract resonances (formants)
Waveform shape in time domain reflects harmonic content

Measuring Periodicity

Various methods extract period/frequency information from signals.

Manual Measurement

From time-domain waveform:

Identify consistent landmark (e.g., positive-going zero crossing)
Measure time from one occurrence to next
Average over multiple cycles for accuracy

Advantages: Simple, direct, no computation required

Disadvantages: Time-consuming, subject to judgment, limited precision

Autocorrelation

Autocorrelation measures signal similarity to time-shifted version of itself:

R(τ) = ∫ x(t) × x(t+τ) dt

Properties:

Maximum at τ = 0 (signal perfectly correlates with itself)
Second maximum occurs at τ = T (period)
Reveals periodicity even in noisy signals
Works well for quasi-periodic signals

Frequency-Domain Analysis

Fourier Transform converts time-domain signal to frequency domain:

Spectrum: Shows amplitude vs. frequency Fundamental Peak: Appears at F0 Harmonic Peaks: Appear at 2F0, 3F0, etc. Period: T = 1/F0 where F0 is frequency of fundamental peak

Advantages: Reveals all frequency components, separates fundamental from harmonics

Specialized Algorithms

Modern voice analysis uses sophisticated algorithms:

Cepstral Analysis: Identifies periodicity in log spectrum, robust to formant structure

Wavelet Analysis: Time-frequency analysis useful for time-varying pitch

Phase Vocoder: Tracks instantaneous frequency even during pitch changes

These methods handle imperfect periodicity (jitter) and extract F0 contours from speech.

Clinical Significance

Periodicity characteristics provide diagnostic information about voice function.

Normal Periodicity

Healthy phonation shows:

Consistent period across cycles (low jitter)
Stable fundamental frequency
Smooth F0 contours during speech
Harmonics visible in spectrum

Pathological Alterations

Voice disorders may produce:

Increased Jitter: Period irregularity from:

Asymmetric vocal fold vibration
Neurological instability
Tissue asymmetry or lesions

Subharmonics: Spectral peaks at fractional multiples of F0:

Indicates period doubling (bifurcation)
Seen in some pathological voices
Creates rough or harsh quality

Biphonation (Diplophonia): Two simultaneous periods:

Left and right folds vibrating at different rates
Creates two F0 components in spectrum
Results from significant left-right asymmetry

Aperiodicity: Loss of periodicity:

Severe pathology preventing regular oscillation
Spectrum shows noise instead of harmonic peaks
Voice sounds breathy, rough, or aphonic

Period and Pitch Perception

While physically distinct, period/frequency strongly correlate with pitch perception.

Frequency-Pitch Relationship

Low Frequencies: Long periods (10-15 ms) perceived as low pitch (male voices, bass instruments)

High Frequencies: Short periods (2-4 ms) perceived as high pitch (female/child voices, soprano, flute)

Doubling Frequency: Corresponds to one octave increase in pitch (musical interval of 12 semitones)

Complications

Pitch perception is not purely frequency-based:

Missing Fundamental: Removing F0 does not eliminate pitch perception; brain infers fundamental from harmonics

Timbre Effects: Spectral envelope (formants) affects perceived pitch slightly

Context Effects: Surrounding pitches influence perception of a given tone

Nonetheless, F0 (or equivalently, T0) provides the primary cue for pitch perception.

Summary

Periodicity describes the regular repetition of oscillatory motion at fixed time intervals. The period (T) defines the time for one complete cycle, while frequency (f = 1/T) specifies the repetition rate in cycles per second. Periodic signals can be represented graphically as repeating waveforms in the time domain or as discrete spectral peaks in the frequency domain.

Vocal fold vibration during phonation produces quasi-periodic oscillation with fundamental period T0 (corresponding to fundamental frequency F0) that determines perceived pitch. Real voice signals show small cycle-to-cycle variations (jitter and shimmer) rather than perfect periodicity. Clinical analysis of periodicity characteristics—including jitter, spectral harmonics, and fundamental frequency stability—provides valuable diagnostic information about vocal fold function and voice disorders.

Key Takeaways

✅ Periodicity means a signal repeats exactly after a fixed time interval called the period (T)
✅ Frequency (f) and period (T) have an inverse relationship: f = 1/T
✅ Angular frequency ω = 2πf provides convenient representation for mathematical analysis
✅ Fourier’s theorem allows any periodic function to be expressed as a sum of sinusoidal components
✅ Vocal fold vibration produces quasi-periodic oscillation with fundamental period T0 corresponding to F0
✅ Normal voice shows low jitter (period variation) and shimmer (amplitude variation)
✅ Pathological voices may exhibit increased jitter, subharmonics, biphonation, or aperiodicity
✅ Fundamental frequency F0 (= 1/T0) provides the primary cue for pitch perception

Periodicity

Periodicity

Definition of Periodicity

Key Properties

Non-Periodic Signals

Period and Frequency

Period (T)

Frequency (f or F0)

Angular Frequency (ω)

Conversions

Graphical Representation of Periodic Signals

Time-Domain Representation

Landmark Selection

Complex Periodic Waveforms

Mathematical Description

Sinusoidal Functions

Complex Periodic Functions

Other Periodic Functions

Periodicity in Vocal Fold Vibration

Fundamental Period (T0)

Quasi-Periodicity

Glottal Flow Waveform

Relationship to Acoustic Signal

Measuring Periodicity

Manual Measurement

Autocorrelation

Frequency-Domain Analysis

Specialized Algorithms

Clinical Significance

Normal Periodicity

Pathological Alterations

Period and Pitch Perception

Frequency-Pitch Relationship

Complications

Summary

Key Takeaways

Related Topics

Further Reading