Waveforms

Periodic and stochastic audio signals

Lorenz Schwarz
Karlsruhe University of Arts and Design (HfG)

Winter Semester 2024/25
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Complex vibratory systems

While a spring–mass system produces a single sinusoidal vibration,
real vibrating systems (strings, air columns, membranes, plates)
produce many simultaneous vibration modes.

Modes and spectra in real sounds

Pure sine waves are rare in the real world:

• Real sound sources excite many modes, producing complex spectra.
• Each normal mode is a sinusoid.
• Superposition of modes gives the harmonic series and timbre.

Standing waves on a vibrating string

On a vibrating string, waves travel both ways, interfere, and form standing waves with nodes (no motion) and antinodes (max motion):

Normal modes on a string

Normal modes are standing-wave patterns that fit the string's boundary conditions.

→ Their frequencies are integer multiples of the fundamental.

Mode frequencies

Allowed frequencies are integer multiples of the fundamental :

• = mode (harmonic) number
• = string length
• = wavelength of the -th mode
• = frequency of the -th harmonic
• = wave speed on the string

Inharmonic partials in real sounds

Plates, gongs, bells, drumhead membranes and other real world sound sources have inharmonic partials.

→ Inharmonic Partials that are not integer multiples of the fundamental frequency.

Periodic functions

A periodic function in audio describes a waveform that repeats its shape at regular time intervals. Understanding these fundamental shapes and their spectral properties is essential for sound synthesis.

1. Sine wave
2. Sawtooth wave
3. Triangle wave
4. Square wave
   • Pulse wave

Sine wave

Symmetrical and curved rise and fall with no abrupt changes:

• has no overtones (single frequency, fundamental only)
• serves as a building block of periodic signals (additive synthesis)
• rarely exists alone in nature
• resembles a pipe sound, like a flute or an organ
• used often as a test tone to assess signal integrity

▶ Sine 400 Hz

Sawtooth wave

A sawtooth is characterized by a linear rise followed by an abrupt drop:

• also called ramp ('up' or 'down')
• ramp down: shifting the phase of the even harmonics by 180°
• rich and full, great for powerful synth bass and lead sounds

▶ Sawtooth 400 Hz

                                     ramp up                   ramp down

Harmonic spectrum of a sawtooth

Triangle wave

Continuous, linear rise and fall between its maximum and minimum values, forming a symmetric triangle:

• closer to a sine wave

▶ Triangle 400 Hz

Harmonic spectrum of a triangle wave

Square wave

The signal spends equal time at the maximum (high) and minimum (low) levels, making it a symmetrical waveform with a 50% duty cycle (T_ON = T_OFF):

• Square waves are often described as sounding "hollow" or "nasal". This means that they are good for creating wind instruments, like a clarinet.

▶ Square wave 400 Hz

Harmonic spectrum of a square wave

Pulse wave

A pulse wave is a non-sinusoidal periodic signal characterized by abrupt alternation between two amplitude levels: a maximum (T_ON) and a minimum (T_OFF):

• Durations of the high and low states differ.

→ Asymmetrical form of a square wave.

Pulse width and duty cycle D

The duty cycle (D) is the percentage of a waveform's period (T_ON) during which the signal is in the "high" or "on" state (value of 1 for a square wave), calculated as the ratio of the on time to the total period (T_ON + T_OFF).

• Ratio of the pulse width to the total period

Duty cycle

→ The duty cycle determines the harmonic spectrum of the pulse wave.

▶ Play examples

Pulse width and timbre

Changing the duty cycle alters the harmonic structure and perceived timbre of a pulse wave.

• Narrowing the duty cycle from 50 % produces a thinner, more nasal sound
• Very narrow pulses create a characteristic reed-like quality
• Commonly used for string- and brass-like synth sounds

→ Modulating the duty cycle over time (PWM) creates dynamic, evolving timbres

Pulse Width Modulation (PWM)

PWM (Pulse Width Modulation) is a type of signal modulation that converts an analog signal into a binary-coded signal by varying the duty cycle of a pulse wave in direct proportion to the amplitude of the analog signal.

Applications of PWM:

• Class-D amplifiers, light dimmers/LED brightness control
• Variable-speed control for computer fans, and servo motor
• Sound synthesis

Gibbs phenomenon

Digital waveforms

In digital systems, frequency components above the Nyquist frequency (half the sampling rate) are mirrored back into the audible range, creating new, non-harmonic frequencies.

This effect is called aliasing.

→ Steep low-pass filtering before sampling minimizes aliasing (band-limited synthesis)

▶ Square wave sweep without band-limiting

▶ Square wave sweep with band-limiting

Stochastic signals

In contrast to periodic signals, stochastic signals (noise) are random and non-repeating, and are described primarily by their spectral distribution rather than their waveform shape.

1. White noise
2. Pink noise
3. Brownian/Red noise
4. Blue/Azure noise
5. Violet/Purple noise

Types of power-law colored noise

The term colored noise refers to signals whose power distribution across frequencies is roughly similar to the corresponding spectra of visible light.

→ An imprecise analogy inspired by filtering white light.

White noise

White noise, analogous to white light which contains all spectral components, has equal energy distributed across all frequencies with a constant power spectral density.

Applications:

• Audio synthesis
• Testing and calibrating audio equipment

Grey noise

While white noise has physically equal energy at all frequencies, grey noise has perceptually equal loudness.

→ Inverse of the equal-loudness curve (A-weighting) compensates for the human ear’s varying sensitivity across the frequency spectrum.

Pink noise

Pink noise has equal power per octave, meaning its power decreases as frequency increases.

• Power spectrum is inversely proportional to frequency ()
• Found in natural and biological systems
• Sounds like a steady waterfall or rainfall
• Also called flicker noise in electronics

Application:

• Commonly used for acoustic measurements and sound system calibration.

Brownian noise ( noise)

Brownian noise, also known as Brown noise or Red noise, has a power density that decreases by 6 dB per octave (or 20 dB per decade), emphasizing lower frequencies.

• It approximates the random patterns of Brownian motion.
• Named after Robert Brown, who discovered Brownian motion in 1827.

Blue/Azure noise

Blue noise's power density increases by 3 dB per octave as the frequency increases.

• Inverse of pink noise.
• Proportional to frequency.

Application:

• Dithering

Violet/Purple noise

Power density increases by 6 dB/octave with frequency.

• Differentiated white noise
• Opposite of brown noise

Additive synthesis

Build complex sounds by adding sine waves together

• Requires many oscillators (one per partial)
• Controlling timbre changes over time is complex
• Computationally expensive

→ Additive synthesis: conceptually simple, practically expensive

Subtractive sound synthesis

Subtractive synthesis starts with rich, periodic waveforms (like sawtooth or square) and removes frequencies using filters.

Signals for measurement and analysis

Beyond sound synthesis, certain signals are designed specifically for measuring and analyzing acoustic systems, such as room reverberation and loudspeaker response.

Dirac delta function ( distribution)

A mathematical function with infinite amplitude at a single point and infinitely small duration.

Application:

• Impulse response measurement

Impulse and unit sample

Sine sweep (chirp)

A sine sweep uses a sinusoid with an increasing frequency to excite an acoustic system, enabling the calculation of its impulse response.

Application:

• Impulse response measurement

Copyright and Licensing

Original content: © 2025 Lorenz Schwarz
Licensed under CC BY 4.0. Attribution required for all reuse.

Includes: text, diagrams, illustrations, photos, videos, and audio.

Third-party materials: Copyright respective owners, educational use.

Contact: lschwarz@hfg-karlsruhe.de

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