Representing sound digitally

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課程筆記

Sound as a Wave

Sound is a wave of pressure variations that travels through a medium (like air).
A microphone (a transducer) converts sound waves into an analog electrical signal that mirrors the wave's shape.
Analog audio retains its wave-like characteristics throughout storage and playback, but it is susceptible to noise and distortion.
In contrast, digital audio represents sound as numerical samples, which can be stored and processed without degradation.

An analogue signal varies continuously; a digital one stores a finite set of discrete values.

Digital Representation of Sound

Digital audio encodes the sound wave as a continuous sequence of numerical samples.
Each sample measures the amplitude (loudness) of the wave at a specific moment in time.
The process of converting an analog signal to digital is done by an analog-to-digital converter (ADC).
The most common method is pulse-code modulation (PCM), which records the amplitude as a binary number.

Sampling measures the amplitude of a sound wave at regular moments to produce discrete sample points.

Sampling Rate

The sampling rate is the number of samples taken per second, measured in hertz (Hz).
For example, CD audio uses a sampling rate of 44.1 kHz, meaning 44,100 samples per second.
A higher sampling rate captures more detail of the wave, especially high frequencies, but results in larger file sizes.
According to the Nyquist–Shannon sampling theorem, the sampling rate must be at least twice the highest frequency to accurately reconstruct the sound.

Sampling more often reconstructs a changing wave more faithfully; sparse sampling can mislead.

Bit Depth

Bit depth (or resolution) is the number of bits used to represent each sample's amplitude.
CD audio uses 16-bit resolution, allowing 65,536 possible amplitude levels.
A higher bit depth gives a more accurate representation of amplitude, reducing quantization noise.
Higher bit depth also increases file size because each sample stores more data.

Quantisation rounds each sample to the nearest available level; more levels keep more detail.

Trade-offs: Quality vs. File Size

Increasing sampling rate and bit depth improves sound quality but increases file size.
Lower sampling rates and bit depths reduce file size but can lose detail and introduce distortion.
For example, a low sampling rate may miss high-frequency sounds, and a low bit depth can make quiet sounds less distinct.
Choosing settings depends on the purpose: high quality for music production, lower for voice recordings or streaming.

Playback and Reconstruction

To play back digital audio, a digital-to-analog converter (DAC) converts the digital samples back into an analog signal.
The DAC reconstructs a continuous wave by 'joining the dots' between samples, following the Nyquist–Shannon theorem.
The analog signal is then amplified and sent to a loudspeaker, which converts it back into sound waves.
Digital audio allows infinite copies without quality loss, unlike analog copies which degrade.

Sound is captured, converted to digital, stored, converted back, and played.

Advantages of Digital Audio

Digital audio can be stored, edited, and transmitted easily using computers and digital devices.
It is immune to noise and distortion during storage and transmission, as long as errors are corrected.
Error correction (channel coding) ensures bit accuracy, e.g., Eight-to-fourteen modulation on CDs.
Digital formats enable compression (like MP3) to reduce file sizes for streaming and storage.

Real-World Applications

Modern music distribution relies on digital audio, with services like iTunes selling digital files.
Streaming services (Apple Music, Spotify, YouTube) provide temporary access to digital audio over the internet.
Digital audio is used in podcasts, sound effects, and all forms of recorded music.
Before digital, music was sold as physical records and tapes, which degraded with copying.

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練習題

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1.Digital audio represents sound as a continuous analog waveform.
Easy
True or false?
2.What does an analog-to-digital converter (ADC) do?
Easy
- AConverts a digital signal to an analog signal
- BConverts an analog signal to a digital signal
- CAmplifies an analog signal
- DCompresses a digital audio file
3.Which of the following are used in digital audio systems to reduce file size? (Select all that apply)
Medium
- AMP3
- BAAC
- CFLAC
- DPCM
- EDAC
4.Analog audio signals are immune to noise and distortion.
Easy
True or false?
5.What is the Nyquist frequency for a system with a sampling rate of 40 kHz?
Medium
- A10 kHz
- B20 kHz
- C40 kHz
- D80 kHz
6.Match each term with its correct description.
Medium
- Sample rate
- Bit depth
- ADC
- DAC
- Number of samples taken per second
- Number of bits used per sample
- Converts analog signal to digital
- Converts digital signal to analog
7.Put the steps in the digital audio playback process in the correct order.
Medium
- Digital signal is stored or transmitted
- DAC converts digital to analog
- Analog signal is amplified
- Loudspeaker produces sound
8.What is the sampling rate in samples per second?
Medium
- A44,100
- B441
- C4,410
- D441,000

複習筆記

## Representing Sound Digitally Sound is a wave. To store it on a computer, we convert it into numbers using **sampling**. ### How Sampling Works - An **analog-to-digital converter (ADC)** measures the **amplitude** (loudness) of the sound wave at regular intervals. - Each measurement is a **sample**. The samples are stored as numbers. - For playback, a **digital-to-analog converter (DAC)** turns the numbers back into an analog signal that can be amplified and sent to a loudspeaker. ### Key Settings: Sample Rate and Bit Depth - **Sample rate**: how many samples are taken per second. Measured in hertz (Hz). - CD audio uses **44,100 samples per second (44.1 kHz)**. - **Bit depth**: how many bits are used for each sample. This determines the precision of each amplitude measurement. - CD audio uses **16-bit resolution** per channel. ### Quality vs. File Size - **Higher sample rate** → more samples → better captures high frequencies → larger file. - **Higher bit depth** → more possible amplitude values → better dynamic range and less noise → larger file. - **Lower settings** → smaller file but lower quality. ### Nyquist Theorem - To accurately record a frequency, the sample rate must be at least **twice** that frequency (the **Nyquist frequency**). - Frequencies above half the sample rate must be filtered out before conversion to avoid **aliasing** distortion. ### Advantages of Digital Audio - **No generation loss**: copying digital audio does not degrade quality, unlike analog copies. - **Error-free**: digital signals can be protected from noise and distortion; disturbances only cause errors if they are large enough to misinterpret a symbol. - **Easy manipulation**: digital audio can be edited, compressed, and transmitted easily. ### Compression - File sizes can be reduced using **audio data compression** (e.g., MP3, AAC, FLAC). - This makes streaming and storage more efficient.

Sampling: Measuring the amplitude of a sound wave at regular intervals to convert it into digital form.
Sample rate: The number of samples taken per second, measured in hertz (Hz). CD audio uses 44,100 samples per second (44.1 kHz).
Bit depth: The number of bits used to represent each sample, determining the precision of amplitude measurement. CD audio uses 16-bit resolution.
Analog-to-digital converter (ADC): A device that converts an analog electrical signal into a digital signal by sampling.
Digital-to-analog converter (DAC): A device that converts a digital signal back into an analog signal for playback.
Nyquist frequency: Half the sampling rate; the maximum frequency that can be accurately represented without aliasing.
Aliasing: Distortion caused by frequencies higher than the Nyquist frequency being incorrectly represented during sampling.
Pulse-code modulation (PCM): A method used to digitally represent sampled analog signals, commonly used in digital audio.
Audio data compression: Techniques (e.g., MP3, AAC, FLAC) used to reduce the file size of digital audio.
Generation loss: The degradation of signal quality that occurs when making copies of analog audio; digital audio does not suffer from this.

字卡

What is digital audio?

Digital audio is a representation of sound recorded in, or converted into, digital form, where the sound wave is encoded as numerical samples in a continuous sequence.

In CD audio, how often are samples taken?

Samples are taken 44,100 times per second (44.1 kHz).

What is the bit resolution of CD audio?

CD audio has 16-bit resolution for each stereo channel.

What does ADC stand for and what does it do?

ADC stands for analog-to-digital converter. It converts an analog electrical signal into a digital signal.

What does DAC stand for and what does it do?

DAC stands for digital-to-analog converter. It converts a digital signal back into an analog signal for playback.

What is pulse-code modulation (PCM)?

PCM is a method used to convert an analog signal to a digital signal by sampling the amplitude at regular intervals.

What is the sampling rate?

The sampling rate is the number of samples taken per second, measured in hertz (Hz).

What is bit depth (resolution)?

Bit depth is the number of bits used to represent each sample, determining the precision of the amplitude measurement.

How does increasing the sample rate affect quality and file size?

Increasing the sample rate improves the accuracy of the recorded frequencies (higher frequencies can be captured) but also increases file size.

How does increasing the bit depth affect quality and file size?

Increasing bit depth improves the dynamic range and reduces quantization error, but also increases file size.

What is the Nyquist frequency?

The Nyquist frequency is half the sampling rate. Frequencies above this cannot be accurately represented and can cause aliasing.

What is an anti-aliasing filter?

An anti-aliasing filter is used before conversion to remove frequencies higher than the Nyquist frequency to prevent aliasing distortion.

What is aliasing?

Aliasing is distortion caused when frequencies above the Nyquist frequency are incorrectly represented as lower frequencies during sampling.

How does playback reconstruct the original sound wave?

A DAC converts the digital samples back into an analog signal, and according to the Nyquist–Shannon sampling theorem, the original band-limited signal can be accurately reconstructed.

What is the advantage of digital audio over analog audio in terms of copying?

Digital audio allows an infinite number of copies to be made without any degradation of signal quality, unlike analog which suffers generation loss.

What is channel coding?

Channel coding is a technique used to encode a digital signal for error correction during storage or transmission, maintaining bit accuracy.

What is eight-to-fourteen modulation?

It is the channel code used for the audio compact disc (CD).

Name two common audio compression formats.

MP3 and AAC are common compressed formats; FLAC is a lossless format.

What is sample-rate conversion?

It is the process of changing the sampling rate of a digital signal, including upsampling and downsampling, to match a common rate.

How does digital audio reduce distribution costs?

Digital audio can be distributed as data files over the internet, reducing the need for physical copies like records and tapes.

What is the role of a transducer like a microphone in analog audio?

A microphone converts physical sound waves into electrical representations of those waveforms.

What is the main difference between analog and digital audio signals?

Analog signals are continuous waveforms susceptible to noise and distortion, while digital signals are discrete numerical samples that can be error-free.

What happens if a disturbance occurs in a digital system?

Disturbances do not result in error unless they are large enough to misinterpret a symbol or disturb the sequence of symbols.

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