Chapter 2: Basics of Sound Notes

Basics of Sound

Introduction

• Sound is a wave created by the compression and expansion of air molecules, caused by a physical device like a speaker.
• The speaker vibrates, producing a longitudinal pressure wave, which we perceive as sound.

Sound Wave Characteristics

• Sound waves have properties like reflection, refraction, and diffraction.
  • Reflection: Bouncing of sound waves.
  • Refraction: Change in the speed of sound when entering a medium with a different density.
  • Diffraction: Change in the direction of waves when passing through an opening or barrier.

Characteristics of Sound Waves

• Frequency: Number of cycles a sound wave completes in one second, measured in Hertz (Hz).
  • 1 Hz means 1 vibration per second.
  • Humans can typically hear between 20 Hz and 20 kHz.
  • Frequencies above the audible range are called ultrasound, while those below are called infrasound.
• Amplitude: Maximum displacement of a particle in the wave's path, representing the loudness of the sound, measured in decibels (dB).
  • Larger energy in the sound wave results in a larger amplitude and louder sound.
• Waveform: The shape of the wave, which can be sinusoidal, square, triangular, sawtooth, or irregular.
• Speed of propagation: The speed depends on the medium.
  • In air: $340 \, \text{m/sec}$
  • In water: $1500 \, \text{m/sec}$

How Sound Waves Interact

• A tone is a sound with a single frequency, represented by a sinusoidal wave.
• A note is a composite sound, a mixture of various tones with different amplitudes and frequencies.
• When sound waves meet, they can add to or subtract from each other.

Digital Sound

• Digital sound involves the reproduction and transmission of sound stored in a digital format.
• Conversion Process:
  • Analog-to-digital converter (ADC) converts an analog signal to a digital signal.
  • The digital signal can undergo digital signal processing and be stored or transmitted.
  • Finally, a digital-to-analog converter (DAC) converts the digital signal back to analog.
• For computers to work with audio waves, they must be converted from analog to digital form through sampling and quantization.
  • Sampling: Recording a sample of the audio in digital bits at fractions of a second.
  • Quantization: Converting the sampled amplitude values into discrete levels.

Sample Rate

• Sample rate is the number of times a sample is taken, measured in Hertz (Hz).
  • 1 Hz = 1 sample per second
  • 1 kHz = 1,000 samples per second
  • 1 MHz = 1 million samples per second
• Common sample rates: 11.025 kHz, 22.05 kHz, and 44.1 kHz.
• Higher sample rates result in better quality digitized audio.

Nyquist Sampling Theorem

• The sampling frequency must be at least twice the highest frequency component in the signal.

Sampling Example

• Nyquist rate sampling: $f_s = 2f$
• Oversampling: $f_s = 4f$
• Undersampling: $f_s < 2f$

Example Calculation

• A signal has frequency components from 300 Hz to 1.8 kHz. The minimum sampling rate is calculated as follows:
  • $f_m = \text{Maximum frequency of the signal} = 1.8 \, \text{kHz}$
  • Minimum sampling rate = Nyquist rate = $2 \times f_m = 2 \times 1.8 \, \text{kHz} = 3.6 \, \text{kHz}$

Aliasing

• Aliasing occurs due to insufficient sampling when a signal is digitized.
• If a signal with frequency $f<em>o$ is digitized at a rate of $2f</em>o$, the samples produce a waveform with a frequency of $f_o$.
• If the signal is digitized at a rate of $\frac{4}{3}f_o$, the samples produce a distorted waveform.
• In practice, an ADC's sampling rate should be much higher than twice the maximum signal frequency.

Quantization

• Quantization is the process of rounding off the amplitudes of samples to a number of levels.
• Each sample is matched to the closest level, as each level has a specific value.
• A common approach is rounding the sample to the closest integer.

Quantization Error

• Quantization error leads to information loss, which manifests as noise.

Signal to Noise Ratio (SNR)

• SNR is the ratio of the power of the correct signal to the noise, measuring signal quality.
• Measured in decibels (dB), SNR is defined as:
  $SNR = 20 \log<em>{10} \frac{V</em>{\text{signal}}}{V<em>{\text{noise}}} = 10 \log</em>{10} \frac{V<em>{\text{signal}}^2}{V</em>{\text{noise}}^2}$
  Where $V<em>{\text{signal}}$ and $V</em>{\text{noise}}$ are the amplitudes of the signal and noise, respectively.

Bit Depth

• When a sound wave is sampled, each sample is assigned the amplitude value closest to the original wave’s amplitude.
• Higher bit depth provides more possible amplitude values, producing greater dynamic range and lower noise.
• 8-bit sample allows $2^8 = 256$ values.
• 16-bit sample provides $2^{16} = 65,536$ values.
• The greater the sample size, the better the audio quality.

Bit Depth Examples

• 8-bit depth: 256 levels (Telephony quality)
• 16-bit depth: 65,536 levels (Audio CD quality)
• 24-bit depth: 16,777,216 levels (Audio DVD quality)
• 32-bit depth: 4,294,967,296 levels (Best quality)

File Size

• Audio File Size = bit depth * sample rate * duration (seconds) * number of channels
• Example: 16-bit, 44100 Hz, 10 seconds, 2 channels (stereo)
  $16 \times 44100 \times 10 \times 2 = 14,112,000 \, \text{bits} = 1,764,000 \, \text{bytes} = 1,764 \, \text{kB} = 1.764 \, \text{MB}$