Data Representation

How data is represented in digital systems.

In digital systems, data is represented using binary format, which consists of only two states: 0 and 1. These binary digits, or bits, form the foundation of all digital data representation. 

1. Binary Representation (Bits)

• Bits (Binary Digits): A bit is the smallest unit of data in digital systems and can have a value of either 0 or 1. Computers use bits to represent everything they process.
• Byte: A group of 8 bits is called a byte. It is the basic unit of information in many systems, often used to represent a character (e.g., ASCII character encoding).
• Larger Units: Beyond bits and bytes, larger units include kilobytes (KB), megabytes (MB), gigabytes (GB), and terabytes (TB), which measure data storage and transmission.

2. Number Representation

• Unsigned Binary Numbers: These represent positive whole numbers in binary form.

  • Example: $1011_2=11\text{ in decimal .}$

• Signed Binary Numbers: These represent both positive and negative numbers using systems like two's complement.

  • Example: In an 8-bit system, $10000001_2$ represents -127 in two's complement.

• Fixed-Point Representation: Used to represent numbers with fractions (e.g., 12.75) in binary. The fractional part is represented by assigning bits after the decimal point in a fixed position.

• Floating-Point Representation: Used for very large or very small numbers, typically following the IEEE 754 standard. Numbers are represented in scientific notation (sign, exponent, mantissa).

  • Example: $-1.101 \times 2^5$represents a number in floating-point format.

3. Character Representation

• ASCII (American Standard Code for Information Interchange): This encoding scheme assigns a unique binary number to each character (letters, numbers, symbols). Each character is represented using 7 or 8 bits.

  • Example: The letter 'A' is represented by 01000001 (65 in decimal).

• Unicode: A more comprehensive character set that can represent characters from many languages and symbols, extending beyond the 8-bit ASCII limit.

  • Example: The character 'A' in Unicode is $U+0041$, represented in binary as (01000001)2​.

4. Image Representation

• Bitmaps (Raster Graphics): Images are represented as grids of pixels, where each pixel is assigned a binary value corresponding to its color. In black-and-white images, 1 bit per pixel is used, whereas grayscale or color images use more bits (e.g., 8, 16, 24 bits per pixel).

  • Example: A pixel in a 24-bit RGB image might be represented as three 8-bit values for red, green, and blue (e.g., 11111111 00000000 00000000 for pure red).

• Vector Graphics: Instead of pixels, vector graphics use mathematical equations to represent lines, curves, and shapes, allowing for scaling without loss of quality. These graphics are often used in SVG (Scalable Vector Graphics) files.

5. Audio Representation

• Pulse Code Modulation (PCM): Digital audio is represented by sampling an analog audio signal at regular intervals (the sampling rate) and converting these samples into binary numbers. The higher the bit depth, the more accurately the audio can be represented.

  • Example: A 16-bit PCM audio file samples the amplitude of the audio wave 44,100 times per second (44.1 kHz).

• Compressed Formats: Audio can also be compressed into formats like MP3 or AAC, which use algorithms to reduce file size while maintaining sound quality by discarding less audible parts of the sound.

6. Video Representation

• Frames and Pixels: Digital video is essentially a series of images (frames) displayed rapidly in sequence. Each frame is represented as a bitmap of pixels.
  • Example: A video at 1080p resolution contains 1920x1080 pixels per frame, with each pixel represented by a binary value for color.
• Compression: Video files are typically large, so compression techniques like MPEG, H.264, or H.265 are used to reduce file sizes. Compression algorithms balance quality and size by reducing redundant information between frames.

7. Boolean Logic Representation

• Boolean values (True/False or 1/0) are fundamental in representing logical conditions in digital systems. They are used in logic gates, which perform operations like AND, OR, and NOT to process data.
  • Example: The result of a Boolean expression like (A AND B) might be stored as 1 or 0 depending on the input values.

8. Instructions and Programs

• Machine Code: At the lowest level, instructions for the CPU are represented in binary, known as machine code. Each instruction, such as adding two numbers or moving data between memory locations, is encoded in binary.
  • Example: The machine code instruction to add two numbers might be represented as 11000011.
• Assembly Language: This is a more human-readable form of machine code but still closely related to the binary representation of instructions.

9. Data Compression

• Lossless Compression: Techniques like Huffman Coding and Run-Length Encoding compress data by finding patterns and replacing repeating data with shorter representations.

  • Example: In Huffman Coding, frequently used characters are assigned shorter binary codes.

• Lossy Compression: Used in media formats (like JPEG for images, MP3 for audio), lossy compression reduces file size by discarding less critical data.

  • Example: JPEG reduces image quality by approximating the color values of pixels, resulting in smaller file sizes.

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Summary Table: Data Representation in Digital Systems

Type of Data	Representation Method
Numbers	Binary, Two’s complement for signed numbers, Fixed-point, Floating-point
Characters	ASCII (7-bit or 8-bit), Unicode (variable length, up to 32 bits)
Images	Bitmaps (raster graphics), Vector graphics (equations), color depth (e.g., 24-bit RGB for color)
Audio	PCM (Pulse Code Modulation), compressed formats (MP3, AAC)
Video	Frames as bitmaps, compressed formats (MPEG, H.264)
Logical Values	Boolean logic (True/False or 1/0), used in logic gates and CPU instructions
Instructions	Machine code (binary instructions for the CPU), Assembly language (human-readable low-level instructions)
Compressed Data	Lossless (Huffman Coding), Lossy (JPEG for images, MP3 for audio)

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Conclusion

In digital systems, data is represented in binary form, where sequences of 0s and 1s encode everything from numbers and characters to audio, video, and instructions. Understanding how data is represented enables the development and optimization of digital systems, as well as the ability to efficiently store, process, and transmit data. The binary system’s versatility and efficiency make it the foundation of modern computing.