# Binary representation of sound

> Computer Science · CIE A-Level 9618
> Source: https://www.owlsprep.com/study/cie-9618-u1-binary-representation-of-sound/

This module explains how continuous analogue sound waves are converted into binary for digital storage. You will learn to calculate sound file size and understand how sampling parameters affect quality and storage requirements, a core Paper 1 topic for CIE 9618.

**Prerequisites:** [Binary number systems](https://www.owlsprep.com/study/cie-9618-u1-binary-number-systems/); [Units of digital information](https://www.owlsprep.com/study/cie-9618-u1-units-of-information/)

## Learning objectives

- Explain the process of converting analogue sound to digital binary
- Define sampling rate and bit depth and their impact on quality and file size
- Calculate the file size of uncompressed digital sound
- Compare mono and stereo sound representations

## Analogue-to-Digital Conversion Process

**Analogue-to-Digital Conversion (ADC)** — The process of converting a continuous analogue sound wave into discrete binary values for digital storage

*Example:* ADC is used when recording voice on a smartphone

Sound originates as a continuous pressure wave that becomes a continuous electrical signal when captured by a microphone. To store this in a computer, it must be converted to discrete binary values through three core stages:

1. Sampling: Measure the wave amplitude at regular time intervals
2. Quantization: Round each measured amplitude to the nearest allowed discrete level
3. Encoding: Convert each quantized level to a fixed-length binary value

**Worked example:** Outline the steps ADC uses to convert a 1-second sound clip to binary

1. 1. Split the 1-second continuous wave into regular time intervals based on the chosen sampling rate, e.g. 10 samples per second = 10 intervals.
2. 2. Measure the amplitude of the wave at each interval.
3. 3. Assign each measured amplitude to the closest available quantized level from a fixed set.
4. 4. Encode each quantized level as a binary number, then store all binary values in memory.

> **Exam tip:** If asked to list ADC stages, remember S-Q-E: Sampling, Quantization, Encoding.

## Key Parameters: Sampling Rate & Bit Depth

**Sampling Rate** — The number of samples of amplitude taken per second, measured in Hz or kHz

*Notation:* $f_s$

*Example:* 44.1 kHz = 44100 samples per second

**Bit Depth** — The number of bits used to store the amplitude value of one individual sample

*Example:* 16-bit depth gives $2^{16} = 65536$ possible amplitude levels

Both parameters directly impact sound quality and file size. A higher sampling rate captures more high-frequency detail and reduces aliasing distortion. A higher bit depth captures more subtle volume variations and reduces quantization noise.

> **info**
>
> Nyquist's theorem states that the sampling rate must be at least twice the highest frequency in the original sound to accurately reconstruct it. Human hearing tops out at ~20 kHz, so 44.1 kHz is the standard for audio CDs.

**Worked example:** How does increasing sampling rate from 22 kHz to 44 kHz affect sound quality and file size, all other parameters equal?

1. 1. Doubling the sampling rate means twice as many samples are taken per second.
2. Effect on quality: More samples capture more detail of the original continuous wave, especially high frequencies, leading to better sound quality with less distortion.
3. Effect on file size: Twice as many samples means total file size will double.

## Calculating Uncompressed Sound File Size

To calculate the total size of an uncompressed sound file, multiply all four core parameters together:

$$\text{Total size (bits)} = \text{channels} \times \text{duration (seconds)} \times \text{sampling rate (Hz)} \times \text{bit depth}$$

After calculating total size in bits, convert to the required unit. CIE 9618 uses the SI decimal standard (1 kB = 1000 bytes, 1 MB = 1000 kB) unless stated otherwise.

**Worked example:** Calculate the size of a 2-minute mono sound file with 32 kHz sampling rate and 16-bit depth, give your answer in MB.

1. 1. Convert duration to seconds: $2 \times 60 = 120$ seconds
2. 2. Mono = 1 channel, substitute into formula:
3. $$\text{Total bits} = 1 \times 120 \times 32000 \times 16 = 61440000 \text{ bits}$$
4. 3. Convert bits to bytes: divide by 8: $61440000 / 8 = 7680000$ bytes
5. 4. Convert to MB: divide by $10^6$: $7680000 / 1000000 = 7.68$ MB

> **Exam tip:** Always check the required unit for the final answer. A common mistake is leaving the answer in bits when the question asks for MB.

## Mono vs Stereo Sound

Mono sound uses one single audio channel, recorded and played back through all speakers. Stereo sound uses two separate channels, one for left and one for right speaker, to create a spatial 3D effect. All other parameters equal, stereo sound is exactly twice the size of mono sound.

**Worked example:** A stereo sound file is 12 MB. What is the size of the same recording saved as mono, all other parameters unchanged?

1. 1. Stereo = 2 channels, mono = 1 channel. File size is proportional to number of channels.
2. 2. Mono size = stereo size / 2 = 12 / 2 = 6 MB

## Common pitfalls

- **Wrong:** Forgetting to convert duration from minutes/hours to seconds
  - Why it fails: Sampling rate is measured in samples per second, so using minutes directly gives a result 60x too small
  - Correct: Always convert duration to seconds first: multiply minutes by 60, hours by 3600
- **Wrong:** Confusing the effects of sampling rate and bit depth on quality
  - Why it fails: Students often mix up which parameter affects frequency vs volume detail
  - Correct: Sampling rate = how often you measure (affects high frequency detail), bit depth = how accurate each measurement is (affects volume dynamic range)
- **Wrong:** Using 1024 instead of 1000 for unit conversion in CIE exams
  - Why it fails: CIE 9618 follows the SI decimal standard for file size prefixes unless explicitly stated otherwise
  - Correct: Use 1 kB = 1000 bytes, 1 MB = 1000000 bytes for all standard questions
- **Wrong:** Forgetting to multiply by number of channels for stereo sound
  - Why it fails: Many students only calculate for 1 channel even when the question specifies stereo
  - Correct: Always check the number of channels first, multiply by 2 for stereo, 6 for 5.1 surround, etc.

## Cheatsheet

| Parameter | Definition | Effect on File Size | Effect on Quality |
| --- | --- | --- | --- |
| Sampling Rate | Samples per second (Hz) | Proportional: higher = larger | Higher = more high frequency detail |
| Bit Depth | Bits per sample | Proportional: higher = larger | Higher = more volume levels, less noise |
| Number of Channels | Separate audio tracks | Proportional: more = larger | Stereo adds spatial effect vs mono |
| File Size Formula | $C \times D \times f_s \times b$ bits | C = channels, D = duration (s) | $f_s$ = sampling rate, b = bit depth |

## What's next

Understanding binary representation of sound gives you a foundation for working with all types of digital multimedia and data compression. The trade-off between file size and sound quality you learned here is critical for real-world applications like streaming, music production, and mobile audio. All compressed audio formats, such as MP3, build on the fundamental encoding concepts covered in this topic, so mastering these basics will help you understand compression techniques next.

- [Binary Representation of Images](https://www.owlsprep.com/study/cie-9618-u1-binary-representation-of-images/)
- [Lossy and Lossless Data Compression](https://www.owlsprep.com/study/cie-9618-u1-data-compression/)
- [Communication](https://www.owlsprep.com/study/cie-9618-u2-overview/)

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