sec5 periodic sounds handout

Periodic and Aperiodic Sounds

1. Periodic Sounds

  • Definitions:

    • Periodic Sounds: Signals that repeat themselves over time.

    • Pure Tones: A specific type of periodic sound characterized by a single frequency. Many periodic sounds contain more complex waveforms and are not pure tones.

  • Reference: Walker et al. (2011). Cortical encoding of pitch: recent results and open questions. Hearing research.

2. Characteristics of Periodic Sounds

2.1 Time Domain

  • Repeating Waveforms: Periodic sounds have waveforms that are consistent over time.

  • Period (T): The time duration for one complete cycle of the waveform.

2.2 Frequency Domain

  • Discrete Spectra: The representation of different frequencies present in the sound.

  • Fundamental Frequency (F0):

    • Defined as the frequency at which the waveform repeats itself.

    • Relation: [ F0 (Hz) = \frac{1}{T (s)} ] (Fourier Transform).

3. Harmonics

  • Components: Each periodic sound consists of multiple frequency components.

  • Harmonic Series:

    • Frequencies are integer multiples of the fundamental frequency (F0).

    • Example:

      • 1st harmonic: F0

      • 2nd harmonic: 2F0

      • n-th harmonic: nF0.

4. Estimating Fundamental Frequency (F0)

  • From the Waveform: F0 can be estimated by determining the period (T).

  • From the Spectrum: F0 can be identified as the Greatest Common Factor (GCF) of all component frequencies present in the sound.

    • Example calculations:

      • 5 components: 300, 400, 500, 600, 700 Hz → F0: 100 Hz.

      • 3 components: 100, 300, 500 Hz → F0: 100 Hz.

5. Levels of Complex Tones

  • Component Level: The sound pressure level of an individual frequency component.

  • Overall Level:

    • For harmonic complexes with equal component levels:[ Overall Level = Component Level + 10 \log_{10}(N) ]Where N = number of components.

    • Example: If component level is 40 dB SPL and N=5,

      • Overall Level = 40 + 10 log10(5) = 47 dB SPL.

6. Aperiodic Sounds

  • Definition: Sounds that do not have a repeating pattern; common in the real world.

    • Types include noises and transients.

7. Gaussian Noise

  • Definition: A type of noise with random amplitude that follows the Gaussian distribution.

  • Characteristics:

    • Long-term average spectrum is flat (all frequencies present).

    • Also referred to as white noise due to coverage of all frequencies.

    • Bandwidth: Covering the entire frequency range, making it a broadband signal.

8. Narrowband Noise

  • Definition: Similar to Gaussian noise but limited to a specific frequency range.

  • Parameters:

    • Bandwidth: Width of the frequency range covered.

    • Spectrum Level: Intensity per 1 Hz range, visualized in spectrum representation.

    • Example passbands:

      • 50 Hz to 350 Hz (300 Hz bandwidth).

      • 100 Hz to 300 Hz (200 Hz bandwidth).

      • 150 Hz to 250 Hz (100 Hz bandwidth).

9. Spectrum Level and Overall Level of Narrowband Noises

  • Overall Level is calculated as: [ Overall Level = Spectrum Level + 10 \log_{10}(Bandwidth) ]

    • Example: If Spectrum Level is 15 dB SPL and bandwidth is 100 Hz,

      • Overall Level = 15 + 10 log10(100) = 35 dB SPL.

10. Center Frequency and Bandwidth

  • Examples:

    • Center frequency = 750 Hz, BW = 300 Hz, Spectrum level = 20 dB SPL, Overall level = 44.8 dB SPL.

    • Center frequency = 350 Hz, BW = 500 Hz, Spectrum level = 30 dB SPL, Overall level = 57 dB SPL.

11. Sound Levels and Calculations

  • Calculating sound levels based on known parameters:

    • For pure tones:

      • Known RMS sound pressure: [ L = 20 \log_{10}(p_{rms}/p_{ref}) ]

    • For discrete spectra:

      • If known peak or pP-P: [ p_{rms} = 0.707 \times p_{peak} ]

    • For narrowband noise with equal-level components: [ L = L_{comp} + 10 \log_{10}(N) ]

12. Summary

  • Periodic Sounds: Defined by repeating waveforms, identified by a fundamental frequency (F0) that relates to the period (T).

  • Levels of periodic sounds are based on component levels and follow logarithmic relationships.

  • Aperiodic Sounds: Contrast to periodic sounds; includes Gaussian noise (white noise) with flat spectra covering all frequencies, and narrowband noise, limited to defined bandwidths.

Periodic and Aperiodic Sounds

1. Periodic Sounds

Definitions:

  • Periodic Sounds: Signals that repeat themselves predictably over time, typically resulting in a consistent auditory experience.

  • Pure Tones: A specialized type of periodic sound characterized by a single, unchanging frequency. Though many periodic sounds can be broken down into pure tones, most possess more complex waveforms that include multiple frequencies.

  • Reference: Walker et al. (2011). Cortical encoding of pitch: recent results and open questions. Hearing research.

2. Characteristics of Periodic Sounds

2.1 Time Domain

  • Repeating Waveforms: The waveforms associated with periodic sounds repeat at regular intervals, establishing a sense of rhythm.

  • Period (T): The time duration required for one complete cycle of the waveform; this can influence how the sound is perceived, with shorter periods resulting in higher frequencies.

2.2 Frequency Domain

  • Discrete Spectra: The visual representation of the different frequencies that make up the sound, allowing analysis of complex sounds.

  • Fundamental Frequency (F0): The lowest frequency of a periodic sound. It is defined as the frequency at which the waveform repeats itself; this is pivotal in music and acoustics since it determines the perceived pitch of the sound.

  • Relation: Fundamental Frequency can be mathematically expressed as:

    $F0 (Hz) = \frac{1}{T (s)}$(This relationship is often explored in the context of a Fourier Transform analysis.)

3. Harmonics

  • Components: Each periodic sound consists of multiple frequency components, contributing to its timbre or color.

  • Harmonic Series: Frequencies are integer multiples of the fundamental frequency (F0).

    • Examples:

      • 1st harmonic: F0

      • 2nd harmonic: 2F0

      • n-th harmonic: nF0

4. Estimating Fundamental Frequency (F0)

  • From the Waveform: The fundamental frequency (F0) can be estimated by analyzing the period (T) of the waveform through tools such as oscilloscopes.

  • From the Spectrum: The fundamental frequency can be identified as the Greatest Common Factor (GCF) of all the component frequencies present in the sound.

  • Example Calculations:

    • For 5 components: 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz → F0: 100 Hz.

    • For 3 components: 100 Hz, 300 Hz, 500 Hz → F0: 100 Hz.

5. Levels of Complex Tones

  • Component Level: This refers to the sound pressure level of each individual frequency component within a complex tone — it’s foundational for understanding sound intensity.

  • Overall Level:For harmonic complexes with equal component levels:

  • Overall Level can be calculated using the formula:

    Overall Level = Component Level + 10 log10(N)Where N = number of components.

  • Example: If the component level is 40 dB SPL and N = 5, thenOverall Level = 40 + 10 log10(5) = 47 dB SPL.

6. Aperiodic Sounds

  • Definition: Sounds that do not exhibit a repeating pattern. These sounds are common in the real world and can be perceived as noise.

  • Types: Includes various forms such as random noise, speech transients, and irregular environmental sounds.

7. Gaussian Noise

  • Definition: A specific type of noise characterized by having random amplitudes that follow the Gaussian distribution, resulting in a smooth envelope in the time domain.

  • Characteristics:

    • The long-term average spectrum is flat, meaning all frequencies are present, contributing to the textural fullness of the sound.

    • Often referred to as white noise due to its coverage of the full audible frequency range.

  • Bandwidth: Thus defined as making it a broadband signal, encompassing all frequencies.

8. Narrowband Noise

  • Definition: Similar to Gaussian noise but is confined to a specific frequency range, which may lead to different perceptual effects when experienced.

  • Parameters:

    • Bandwidth: Indicates the width of the frequency range that the noise covers.

    • Spectrum Level: Reflects the intensity per 1 Hz across its frequency range, visualized in spectrum representations.

  • Example passbands:

  • 50 Hz to 350 Hz (300 Hz bandwidth).

  • 100 Hz to 300 Hz (200 Hz bandwidth).

  • 150 Hz to 250 Hz (100 Hz bandwidth).

9. Spectrum Level and Overall Level of Narrowband Noises

  • The overall level of narrowband noise can be calculated using the formula:

Overall Level = Spectrum Level + 10 log10(Bandwidth)

  • Example: If the Spectrum Level is 15 dB SPL with a bandwidth of 100 Hz, thenOverall Level = 15 + 10 log10(100) = 35 dB SPL.

10. Center Frequency and Bandwidth

  • Examples:

  • Center frequency = 750 Hz, Bandwidth (BW) = 300 Hz, Spectrum Level = 20 dB SPL, Overall Level = 44.8 dB SPL.

  • Center frequency = 350 Hz, BW = 500 Hz, Spectrum Level = 30 dB SPL, Overall Level = 57 dB SPL.

11. Sound Levels and Calculations

  • Calculating sound levels: Requires consideration of known parameters regarding the sound in question.

    • For pure tones: The formula for the known RMS sound pressure level is expressed as:$L = 20 log_{10}(\frac{p_{rms}}{p_{ref}})$

    • For discrete spectra: When dealing with known peak or peak-to-peak values:$p_{rms} = 0.707 \times p_{peak}$.

    • For narrowband noise with equal-level components: The total sound level is calculated as:$L = L_{comp} + 10 log_{10}(N)$

12. Summary

  • Periodic Sounds: Characterized by repeating waveforms and identified by a fundamental frequency (F0) related to the period (T).

  • Levels of periodic sounds are determined by component levels and adhere to logarithmic relationships, essential for understanding sound intensity.

  • Aperiodic Sounds: Contrast periodic sounds with their lack of rhythmic patterns; include Gaussian noise (white noise) with flat spectra covering all audible frequencies and narrowband noise that is restricted to defined bandwidths, yielding distinct auditory characteristics.