Data Representation Summary A1 chap 3

Number System and Representation

  • Digital devices use a binary system, while people usually use a decimal system.

Units of Data

  • Bit: The most basic unit of data (0 or 1).
  • Byte: 1 B = 8 bits
  • Kilobyte: 1 KB = 2102^{10} B = 1024 B
  • Megabyte: 1 MB = 2102^{10} KB = 1024 KB
  • Gigabyte: 1 GB = 2102^{10} MB = 1024 MB
  • Bitrate/Data transfer rate units:
    • bps: One bit per second.
    • kbps: 1 kbps = 10310^3 bps = 1000 bps
    • Mbps: 1 Mbps = 10310^3 kbps = 1000 kbps
    • Gbps: 1 Gbps = 10310^3 Mbps = 1000 Mbps
    • Tbps: 1 Tbps = 10310^3 Gbps = 1000 Gbps

Binary Numbers

  • Used in data representation with digits "0" and "1".
  • Conversion to Decimal: Expand the binary number by place values of powers of 2.
  • Conversion from Decimal: Repeatedly divide by 2, arrange remainders from last to first, followed by the final quotient.

Calculation of Binary Numbers

  • Arithmetic operations are similar to decimal numbers.

Hexadecimal Numbers

  • Base-16 number system; each digit represents 4 bits (24=162^4 = 16).
  • Digits: 0-9 and A-F (A=10, B=11, C=12, D=13, E=14, F=15).
  • Conversion to Decimal: Expand the hexadecimal number by place values of powers of 16.
  • Conversion from Decimal: Repeatedly divide by 16, arrange remainders (in hexadecimal) from last to first, followed by the final quotient.

Combination Lock Example

  • If a combination lock has 3 digits, and each digit can be 1-9 or A-F, then total combinations = 15 × 15 × 15 = 3375.

Unsigned and Signed Integers

  • Unsigned Integers: Non-negative integers (positive and zero).
  • Signed Integers: Integers that can be positive, negative, or zero.

Two’s Complement Representation

  • Method for representing signed integers.
  • The two’s complement of X represents -X.
  • Conversion to Decimal: Expand the representation using negative powers of 2 for the leftmost bit and positive powers for the rest.
  • Finding Two's Complement:
    1. Invert all bits (one’s complement).
    2. Add 1 to the one’s complement.

Overflow Error

  • Occurs when the result of a calculation is out of the range representable by the given number of bits.
  • Unsigned Integers: Occurs during addition and subtraction when the result exceeds the maximum representable value.
  • Signed Integers: Can also occur during addition and subtraction.

Character Encoding

  • Systems for representing characters in digital form.
  • ASCII: For English, usually uses 7 bits to represent a character.
  • Chinese Encodings: Big-5 (Traditional Chinese), Guobiao (GB) code (Simplified Chinese).
  • Unicode: Represents multiple languages, uses variable-width encoding (1 to 4 bytes).

Barcode

  • Consists of bars and spaces of different widths representing a text string, often a product ID.

Quick Response (QR) Code

  • Two-dimensional barcode made up of black dots on a white background, representing a URL, phone number, or email.
  • Advantages over Barcodes: Scannable from any direction, higher error correction capability, larger character limit.

Analog vs Digital Data

  • Analog Data: Continuous values (e.g., sound waves).
  • Digital Data: Discrete symbols (0 and 1; e.g. text documents).
  • Digitalization: Conversion of analog data to digital data through Discretisation and Quantisation.

Multimedia

  • Files are often compressed to facilitate transmission and save storage space.
  • Lossless Compression: No data lost, original file retrievable.
  • Lossy Compression: Some data lost, saves more storage space.
  • Compression Ratio = (Original File Size) / (Compressed File Size)

Text

  • Plain Text: Stores characters only, smaller file size.
  • Formatted Text: Stores characters and formatting data, larger file size.
  • File Formats:
    • TXT: Plain text.
    • DOC/DOCX: Formatted text (Microsoft Word).
    • PDF: Formatted text (cross-platform).

Image

  • Bitmap Images: Stored as pixels, editable, file size = colour depth × resolution.
  • Vector Images: Stored as mathematical formulae, smaller file size, do not lose quality when scaled.
  • Colour depth is the number of bits used to represent the color of a pixel.
  • Resolution is the total number of pixels in an image.

Audio

  • Attributes: Bit depth, sampling rate, number of channels.
  • File size of an uncompressed audio file: Bit depth × number of channels × sampling rate × duration.

Video

  • Sequences of images (frames) shown within a duration; may contain audio.
  • Frame rate: Number of frames shown per second (fps).
  • File size of an uncompressed video file: color depth × resolution × frame rate × duration + size of the audio track