Sound — sampling and file size
| English | Chinese | Pinyin |
|---|---|---|
| analogue data | 模拟数据 | mó nǐ shù jù |
| digital data | 数字数据 | shù zì shù jù |
| sampling | 采样 | cǎi yàng |
| amplitude | 振幅 | zhèn fú |
| sampling rate | 采样率 | cǎi yàng lǜ |
| sampling resolution | 采样分辨率 | cǎi yàng fēn biàn lǜ |
| quantisation noise | 量化噪声 | liàng huà zào shēng |
Why the phone makes you sound thin
- A telephone call samples your voice 8,000 times a second. A CD samples the same voice 44,100 times a second. Your voice has not changed; the phone simply throws away everything above about 3.4 kHz.
- That is why a phone call is perfectly intelligible and unmistakably thin: speech lives below 3.4 kHz, but the sparkle of a cymbal or a hard
ssound does not. - The choice was not laziness. It was a calculation: sample fast enough to carry speech, and no faster, because every extra sample costs bandwidth on every call in the country.
- This lesson is how a continuous sound becomes numbers, the two settings that control quality and size, and the calculation the exam asks for.
Analogue and digital
- Sound in the real world is a continuous pressure wave: analogue data 模拟数据, which varies smoothly and takes infinitely many values.
- A computer stores digital data 数字数据: discrete numbers. Converting one to the other is sampling 采样.
- Sampling measures the wave's amplitude 振幅, its height, at regular time intervals, and stores each measurement as a binary number.
The smooth wave becomes a row of numbers
What does sampling do?
Sampling turns a continuous analogue wave into discrete digital values. Removing inaudible frequencies is lossy compression, a different step.
The two settings
- Sampling rate 采样率 is the number of samples taken per second, measured in hertz. CD quality is 44.1 kHz; a telephone uses 8 kHz.
- Sampling resolution 采样分辨率 (sample bit depth) is the number of bits used to store each sample's amplitude. CD quality is 16 bits, giving 65,536 possible levels.
- The rate controls which pitches survive; the resolution controls how precisely each measurement is stored.
Sound sampling
y = a sin(bt + c)
Sampling measures a sound wave at regular intervals — a higher rate copies it more truly.
The sampling rate of a digital sound is:
Sampling rate = samples per second (hertz). The bits per sample is the sample resolution.
Put the steps of turning a sound into a digital file in order.
Rate decides how often you measure; resolution decides how finely each measurement is rounded, and the rounding is where quantisation noise comes from.
Sound file size
- Mono is one channel; stereo is two, which doubles the size.
- A 5-second mono clip at 8 kHz and 8 bits: $8000 \times 8 \times 5 \times 1 = 320\,000$ bits, which is 40,000 bytes.
A 5-second mono clip is sampled at $8000\ \text{Hz}$ with 8-bit resolution. What is its size in bits?
size $= 8000 \times 8 \times 5 \times 1 = 320\,000$ bits.
Sound file size in bits = sampling rate x sampling resolution x duration x ____.
Mono is one channel, stereo two. Leaving the channel count out halves every stereo answer.
Worked example: a CD track
- A 10-second stereo clip is recorded at CD quality: 44.1 kHz, 16-bit. Find its size in MiB.
- $44\,100 \times 16 \times 10 \times 2 = 14\,112\,000$ bits.
- Divide by 8 for bytes: $1\,764\,000$. Divide by $1024^2$: 1.68 MiB.
- Every mark is in the working: the four factors, one division by 8, and a stated prefix. Forgetting the two channels halves the answer.
A 10-second stereo clip is sampled at 44100 Hz with 16-bit resolution. How many bits does it need?
44100 x 16 x 10 x 2 = 14,112,000 bits, about 1.68 MiB. Leaving out the two channels halves the answer.
A stereo recording is twice the size of the same mono recording at the same rate and resolution.
Channels multiply the size: two channels, twice the samples. The channel count belongs in the formula.
Changing the settings
- Higher sampling rate: higher pitches are captured, so the sound is more accurate, but the file is larger.
- Higher sampling resolution: each amplitude is stored in finer steps, so there is less quantisation noise 量化噪声, the roughness introduced when a smooth value is rounded to the nearest level; the file is larger.
- Lower of either: a smaller file and an audible loss of quality. As with images, answer in these terms, not "it sounds bad".
Increasing the sampling rate will:
A higher sampling rate captures higher frequencies (better quality) but increases the file size.
A higher sample resolution (more bits per sample) reduces quantisation noise.
More bits give finer amplitude steps, so each sample is closer to the true value — less quantisation noise (but a larger file).
Match each change to its main effect.
Rate decides which pitches survive; resolution decides how precisely each sample is stored.
Worked example: how fast must you sample?
- The sampling rate must be at least twice the highest frequency you want to keep.
- Human hearing reaches about 20 kHz. What is the minimum rate to capture it? $2 \times 20\,000 = 40\,000$ Hz, which is why CD chose 44.1 kHz, a little above the minimum.
- A telephone samples at 8 kHz. What is the highest frequency it can carry? Half of 8 kHz, so 4 kHz, and in practice about 3.4 kHz. Speech fits; music does not.
To capture frequencies up to $20\,000\ \text{Hz}$, what is the minimum sampling rate, in Hz?
The sampling rate must be at least twice the highest frequency: $2 \times 20\,000 = 40\,000\ \text{Hz}$.
A telephone samples at 8000 Hz. What is the highest frequency, in Hz, it can in principle carry?
Half the sampling rate, since the rate must be at least twice the highest frequency kept. In practice a phone carries about 3.4 kHz.
Trading quality against size
- Speech for a voice note: a low rate and resolution are enough, and the file is small.
- Music for distribution: CD settings or better, because listeners hear the difference.
- Live streaming: the rate and resolution are chosen to fit the available bandwidth, then lossy compression removes more.
- A "justify" answer names the setting, the effect on quality and the effect on size, tied to the use.
Marks that slip away
- Rate is samples per second; resolution is bits per sample. The question names one.
- Stereo is two channels. Leaving the channel count out of the formula halves every stereo answer.
- The minimum rate is twice the highest frequency wanted, not the same as it.
- Say quantisation noise, not "it sounds fuzzy". The syllabus term carries the mark.
You've got it
- sampling converts continuous analogue data into digital data by measuring the amplitude at regular intervals
- sampling rate = samples per second (CD 44.1 kHz), sampling resolution = bits per sample (CD 16)
- size in bits = rate $\times$ resolution $\times$ duration $\times$ channels; stereo doubles it
- higher rate or resolution means better accuracy, less quantisation noise and a larger file; the rate must be at least twice the highest frequency kept