Digital System Overview and Number Systems

Introduction to Digital Systems

  • Instructor: Muhammad Alfian Amrizal, DIKE UGM
  • Topics covered: Introduction to digital system, Information Representation, Number Systems, Arithmetic Operations, Binary Coding

Digital System Overview

  • Definition: A digital system processes discrete inputs to produce discrete outputs.
    • Discrete Inputs: Digital signals that the system accepts.
    • System State: Internal state of the system.
    • Discrete Outputs: The results produced by the system.

Types of Digital Systems

  • Combinational Logic System: No state present.
  • Sequential Systems: Have states that are updated at discrete intervals.
    • Synchronous Sequential System: Updates states at fixed time intervals.
    • Asynchronous Sequential System: Updates states at any time.

Example of Digital Systems

  • Digital Counter: Accepts Count Up and Reset inputs.
    • Outputs a visual display reflecting the count.
  • Digital Computer: Components include memory, control unit, and input/output peripherals.
    • Inputs from devices like keyboard, mouse and outputs to displays or speakers.

Embedded Systems

  • Definition: Computers integrated into other devices.
  • Examples: Microcontrollers, Digital Signal Processors.

Information Representation in Digital Systems

  • Variables in digital systems are represented using discrete (binary) values.
    • Common Representation: Physical quantities such as voltage or magnetic field.

Number Systems

  • General information: Base 10 (Decimal) used commonly versus others like binary.
  • Positional vs Non-Positional Number Systems:
    • Non-Positional: Value determined by symbols (e.g., Roman numerals).
    • Positional: Value determined by digit position and base (e.g., Decimal, Binary).
    • E.g., a number in base rr: ((A{n-1} A{n-2} … A1 A0.A{-1} A{-2} … A{-m})r$ leads to calculations involving powers of $r$.
Binary Numbers
  • Base 2 system: only uses digits 0 and 1, called bits.
  • Significance:
    • LSB = Least Significant Bit (rightmost)
    • MSB = Most Significant Bit (leftmost)
  • Example: Binary 11010 = 1*2^4 + 1*2^3 + 0*2^2 + 1*2^1 + 0*2^0 = 26

Powers of Two

  • Base 2 examples and significant numbers:
    • 210=10242^{10} = 1024 (Kilo), 220=1,048,5762^{20} = 1,048,576 (Mega), 230=1,073,741,8242^{30} = 1,073,741,824 (Giga)

Conversions

  • Decimal to Binary: Repeatedly divide by 2 and note the remainders in reverse order.
  • Fraction to Binary: Multiply by 2 and record integer parts sequentially until reaching 0 (e.g., $0.6875
    ightarrow 0.1011$ in binary).
  • Base Conversions: Generally convert to decimal first, then to the desired base.

Common Bases Overview

  • Binary: 0, 1
  • Octal (Base 8): 0-7
  • Hexadecimal (Base 16): 0-9 and A-F for 10-15.
  • Conversion methods involve changing representations while maintaining value integrity.

Binary Codes

  • Binary Coded Decimal (BCD) refers to encoding the decimal digits separately in binary.
  • Gray Code: A binary code system where two successive values differ by one bit. Useful for applications such as optical sensors.

Conclusion and References

  • Main reference for the course: Manno, M.M., Kime, C.R. & Martin, T., “Logic and Computer Design Fundamentals”, 5th Edition.
  • Additional resources or real-world applications could include videos demonstrating specific digital systems in action.