Images, Sound and Data Representation

Every type of data stored on a computer is ultimately represented as a pattern of binary bits. The same bit pattern can mean different things depending on context:

• 0100 0001 as ASCII = 'A' (the letter A)
• 0100 0001 as unsigned binary = $65_{10}$ (the number 65)
• 0100 0001 as part of an image = a dark-grey pixel value

Analogue vs digital:

	Analogue	Digital
Values	Continuous range (infinite precision)	Discrete set of values
Signal	Smooth waveform	Steps between defined levels
Examples	Vinyl records, old CRT signals, thermometers	MP3 files, PNG images, digital thermometers

ADC/DAC: an analogue-to-digital converter (ADC) samples an analogue signal at regular intervals and converts each sample to a digital value. A digital-to-analogue converter (DAC) converts stored digital values back to an analogue signal (e.g. for audio output through speakers).

A bitmap (raster image) stores an image as a 2D grid of pixels (picture elements). Each pixel stores a colour value.

Key properties:

Property	Definition	Example
Resolution	Number of pixels (width × height)	1920 × 1080 = 2 073 600 pixels
Colour depth	Bits per pixel	24-bit → $2^{24}$ ≈ 16.7 million colours
DPI	Dots per inch (print resolution)	300 dpi for print; 72–96 dpi for screen

File size formula (uncompressed, ignoring metadata):

$\text{Size (bits)}=\text{width}\times \text{height}\times \text{colour depth}$

Worked example — a 1920 × 1080 image with 24-bit colour:

$1920\times 1080\times 24=49766400\text{ bits}=6220800\text{ bytes}\approx 5.9\text{ MiB}$

Metadata stored alongside the image: width, height, colour depth, file format, compression type, and potentially GPS coordinates, camera settings, etc.

A vector image is stored as a list of geometric objects, each described by its properties rather than individual pixels.

Object types (primitives): point, line, arc, circle, rectangle, polygon, bezier curve

Each object has properties:

• Position ($x$, $y$), dimensions (width, height)
• Fill colour, stroke colour, stroke width
• Transparency, rotation, layering order

Example — a simple vector graphic:

Circle: centre=(100, 100), radius=50, fill=#FF0000, stroke=#000000, width=2
Rectangle: x=200, y=50, w=120, h=80, fill=#0000FF
Line: from=(10, 10), to=(300, 200), stroke=#008000, width=3

Vector vs bitmap comparison:

	Vector	Bitmap
Scaling	Infinite — no quality loss	Quality degrades at high zoom (pixelation)
File size	Small for simple shapes	Large for detailed images
Best for	Logos, diagrams, fonts, maps	Photographs, complex images with many colours
Editing	Easy to move/resize objects	Difficult — editing pixels individually

Sound is a pressure wave. To store it digitally:

1. Sampling: an ADC measures the amplitude of the sound wave at regular time intervals
2. Quantisation: each sample is rounded to the nearest representable value (determined by bit depth)

Key parameters:

Parameter	Description	Higher value →
Sampling rate	Samples per second (Hz)	Better frequency reproduction; larger file
Bit depth (sample resolution)	Bits per sample	More amplitude levels; better dynamic range; larger file
Channels	Mono (1) or stereo (2)	Stereo doubles file size

Nyquist theorem: the sampling rate must be at least twice the highest frequency to be captured. CD audio uses 44 100 Hz (samples/sec) to capture frequencies up to 22 050 Hz (the upper limit of human hearing ≈ 20 000 Hz).

File size formula:

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

Worked example — CD-quality audio (44 100 Hz, 16-bit, stereo), 3 minutes:

$44100\times 16\times 180\times 2=254016000\text{ bits}\approx 30\text{ MiB}$

MIDI (Musical Instrument Digital Interface) stores instructions for making music, not recorded audio.

A MIDI file contains event messages such as:

NOTE_ON  channel=1  note=60 (middle C)  velocity=80
NOTE_OFF channel=1  note=60  time=500ms
PROGRAM_CHANGE channel=1  instrument=0 (Grand Piano)

MIDI vs audio files:

	MIDI	Audio (WAV/MP3)
What is stored	Instructions (note, pitch, duration, velocity)	Sampled audio waveform
File size	Very small (kilobytes)	Large (megabytes)
Change tempo	Yes — just change timing	Requires re-encoding
Change pitch	Yes — transpose notes	Requires re-encoding
Change instrument	Yes — change program number	Not possible
Sounds like original performance	No — depends on playback hardware	Yes
Requires instrument sounds	Yes (synthesiser or soundfont)	No

MIDI is used in music production, electronic instruments, and sequencers. It is not suitable for recording spoken audio or live acoustic performances.

1. Using the wrong file size formula

Bitmap file size: width × height × colour depth (in bits). Sound file size: sampling rate × bit depth × duration × channels. Confusing the two or using decimal units (KB = 1000 bytes) when the question expects binary units (KiB = 1024 bytes) loses marks.

2. Claiming vector images are always smaller

Vectors are smaller than bitmaps for simple geometric shapes, but a photograph in vector format (if it were possible) would require millions of tiny shapes — larger than a bitmap. The comparison depends on image complexity.

3. Confusing sampling rate and bit depth

Sampling rate affects how accurately frequencies are captured (how often you measure). Bit depth affects how many amplitude levels can be stored (how precisely you measure each sample). They are independent parameters.

4. Stating MIDI records sound

MIDI does not record audio. It records instructions. The actual sound is generated at playback time by a synthesiser. Two playback devices may produce very different sounds from the same MIDI file.