Digital Electronics: Number Systems and Conversions

• Digital electronics involves digital signals.
• Digital circuits use discrete, discontinuous values for electrical signals.
• Digital techniques represent and process data in computers, automated systems, and other digital devices.
• Course code BCS 114 relates to digital electronics in computer science.

Number Systems

• A number system defines how numbers are represented.
• Common number systems in digital electronics:
  • Decimal (base-10)
  • Binary (base-2)
  • Octal (base-8)
  • Hexadecimal (base-16)
• The decimal system uses ten digits (0-9).
• The binary system uses two digits (0 and 1).
• The octal system uses eight digits (0-7).
• The hexadecimal system uses sixteen digits (0-9 and A-F).

Conversions

• Decimal to Binary conversion: repeatedly divide the decimal number by 2 and record the remainders.
• Binary to Decimal conversion: sum the products of each bit and its corresponding power of 2.
• Binary to Octal conversion: group the binary digits into sets of three, starting from the right.
• Octal to Binary conversion: convert each octal digit to its 3-bit binary equivalent.
• Binary to Hexadecimal conversion: group the binary digits into sets of four, starting from the right.
• Hexadecimal to Binary conversion: convert each hexadecimal digit to its 4-bit binary equivalent.

Binary Arithmetic

• Binary arithmetic involves performing arithmetic operations using binary numbers.
• Binary addition rules:
  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 0 = 1
  • 1 + 1 = 10 (0 with a carry of 1)
• Binary subtraction rules:
  • 0 - 0 = 0
  • 1 - 0 = 1
  • 1 - 1 = 0
  • 0 - 1 = 1 (with a borrow of 1)
• Binary multiplication is similar to decimal multiplication.
• Binary division is similar to decimal division.

Logic Gates

• Logic gates are basic building blocks of digital circuits that perform logical operations on one or more inputs to produce a single output.

• Common logic gates:

  • AND gate
  • OR gate
  • NOT gate
  • NAND gate
  • NOR gate
  • XOR gate
  • XNOR gate

• AND gate outputs 1 only if all inputs are 1.

• OR gate outputs 1 if at least one input is 1.

• NOT gate inverts the input signal.

• NAND gate outputs 0 only if all inputs are 1 (inverse of AND).

• NOR gate outputs 1 only if all inputs are 0 (inverse of OR).

• XOR gate outputs 1 if the inputs are different.

• XNOR gate outputs 1 if the inputs are the same.

Boolean Algebra

• Boolean Algebra deals with Boolean values (0 and 1) and logical operations.
• Boolean Algebra is used to analyze and simplify digital circuits.
• Basic Boolean operations:
  • AND (represented by .)
  • OR (represented by +)
  • NOT (represented by ')
• Key Boolean algebra laws:
  • Commutative Law: A + B = B + A, A . B = B . A
  • Associative Law: (A + B) + C = A + (B + C), (A . B) . C = A . (B . C)
  • Distributive Law: A . (B + C) = A . B + A . C, A + (B . C) = (A + B) . (A + C)
  • Identity Law: A + 0 = A, A . 1 = A
  • Inverse Law: A + A' = 1, A . A' = 0
  • Idempotent Law: A + A = A, A . A = A
  • Absorption Law: A + (A . B) = A, A . (A + B) = A
  • DeMorgan's Theorem: (A + B)' = A' . B', (A . B)' = A' + B'

Simplification Techniques

• Boolean expressions are simplified using Boolean algebra laws.
• Karnaugh Maps (K-Maps) are a graphical method for simplifying Boolean expressions.
• K-Maps are useful for expressions with up to four variables.
• K-Map simplification involves grouping adjacent cells containing 1s to form larger groups.
• The larger the group, the simpler the resulting expression.

Combinational Circuits

• Combinational circuits' output depends only on the current input values.
• Examples of combinational circuits:
  • Adders
  • Subtractors
  • Multiplexers (MUX)
  • Demultiplexers (DEMUX)
  • Encoders
  • Decoders
• Adders perform binary addition.
• Subtractors perform binary subtraction.
• Multiplexers select one of several input signals and pass it to the output.
• Demultiplexers route a single input signal to one of several outputs.
• Encoders convert a decimal value to a binary code.
• Decoders convert a binary code to a decimal value.

Sequential Circuits

• Sequential circuits' output depends on both current input values and the past history of inputs.
• Sequential circuits have memory elements that store past information.
• Key types of sequential circuits:
  • Flip-flops
  • Registers
  • Counters
• Flip-flops are basic memory elements that store one bit of information.
• Registers are groups of flip-flops used to store multiple bits.
• Counters are sequential circuits that count the number of clock pulses.

Flip-Flops

• Flip-flops are fundamental building blocks of sequential circuits.
• Common types of flip-flops include:
  • SR flip-flop
  • JK flip-flop
  • D flip-flop
  • T flip-flop
• SR flip-flop has Set and Reset inputs.
• JK flip-flop is a versatile flip-flop that avoids the invalid state of the SR flip-flop.
• D flip-flop transfers the input D to the output Q on the clock edge.
• T flip-flop toggles its output on each clock pulse.

Registers

• Registers are used to store binary information.
• Registers consist of a group of flip-flops.
• Registers are used for data storage, shift registers, and parallel data transfer.

Counters

• Counters are sequential circuits that count the number of clock pulses.
• Types of counters:
  • Asynchronous (ripple) counters
  • Synchronous counters
  • Up counters
  • Down counters
• Asynchronous counters have flip-flops that are not clocked simultaneously.
• Synchronous counters have flip-flops that are clocked simultaneously.
• Up counters increment the count value.
• Down counters decrement the count value.

Memory Devices

• Memory devices are used to store digital information.
• Types of memory:
  • RAM (Random Access Memory)
  • ROM (Read Only Memory)
• RAM is volatile memory that can be read from and written to.
• ROM is non-volatile memory that is typically read-only.
• RAM is used for temporary data storage.
• ROM is used for permanent data storage, like firmware.

Explore digital electronics, focusing on number systems like decimal, binary, octal, and hexadecimal. Learn the basics of number representation and conversions between different number systems used in digital circuits and computer science (BCS 114).