Digital Electronics: Number Systems and Logic Gates

• Digital electronics deals with signals that are discrete levels
• Microprocessors are digital integrated circuits that perform computing functions

Number Systems

• A number system is represents numbers
• Radix, or base, indicates unique digits in a number system
• Common number systems:
  • Decimal (base 10)
  • Binary (base 2)
  • Octal (base 8)
  • Hexadecimal (base 16)
• Decimal uses digits 0-9
• Binary uses digits 0-1
• Octal uses digits 0-7
• Hexadecimal uses digits 0-9 and A-F

Binary Arithmetic

• 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 rules:
  • 0 * 0 = 0
  • 0 * 1 = 0
  • 1 * 0 = 0
  • 1 * 1 = 1
• Binary division follows the same principles as decimal division using subtraction and comparison

Logic Gates

• Logic gates form the foundation of digital circuits
• They conduct logical operations on binary inputs, producing a binary output
• Common logic gates:
  • AND: Output is 1 only if all inputs are 1
  • OR: Output is 1 if at least one input is 1
  • NOT: Output is the inverse of the input (1 becomes 0, 0 becomes 1)
  • NAND: Output is the inverse of AND (NOT AND)
  • NOR: Output is the inverse of OR (NOT OR)
  • XOR: Output is 1 if inputs are different
  • XNOR: Output is 1 if inputs are the same

Boolean Algebra

• Boolean algebra is a system used to simplify digital circuits.
• It uses variables (A, B, C, etc.) represent logic signals
• Operations:
  • AND: Represented by a dot (.) or implied multiplication (A.B or AB)
  • OR: Represented by a plus (+) sign (A + B)
  • NOT: Represented by an overbar (A')
• Laws/Theorems:
  • Commutative: A + B = B + A, A.B = B.A
  • Associative: (A + B) + C = A + (B + C), (A.B).C = A.(B.C)
  • Distributive: A.(B + C) = A.B + A.C, A + (B.C) = (A + B).(A + C)
  • Identity: A + 0 = A, A.1 = A
  • Inverse: A + A' = 1, A.A' = 0
  • DeMorgan's Theorems: (A + B)' = A'.B', (A.B)' = A' + B'
• Boolean algebra simplifies logic expressions, reduces the number of gates required, and also optimizes circuit design

Combinational Circuits

• Output in combinational circuits depends only on current input values
• They do not have memory or feedback
• Examples:
  • Adders: Perform binary addition (half adder, full adder)
  • Subtractors: Perform binary subtraction (half subtractor, full subtractor)
  • Multiplexers (MUX): Select one of several inputs and forward it to a single output
  • Demultiplexers (DEMUX): Direct a single input to one of several outputs
  • Encoders: Convert a decimal value to a binary code
  • Decoders: Convert a binary code to a decimal value, activating a specific output line

Sequential Circuits

• Sequential circuits' output depends on current and past inputs (stored state)
• They use memory elements like flip-flops to store information
• Output relies on the input and the current state
• The next state relies on the input and the current state

Flip-Flops

• Flip-flops store one bit of information, and are memory elements in sequential circuits
• Types:
  • SR Flip-Flop: Set-Reset flip-flop
  • JK Flip-Flop: Versatile flip-flop
  • D Flip-Flop: Data flip-flop
  • T Flip-Flop: Toggle flip-flop
• Flip-flops are triggered by a clock signal to synchronize state changes

Registers

• Registers store bits of data using groups of flip-flops
• They operate as temporary storage for data within a digital system
• Types:
  • Serial In Serial Out (SISO)
  • Serial In Parallel Out (SIPO)
  • Parallel In Serial Out (PISO)
  • Parallel In Parallel Out (PIPO)

Counters

• Counters count pulses/clock signals using sequential circuits
• Types:
  • Asynchronous (Ripple) Counters: Output of one flip-flop triggers the next
  • Synchronous Counters: All flip-flops triggered simultaneously by the clock
  • Up/Down Counters: Count in ascending or descending order
  • Mod-N Counters: Count up to a specific number (N) and then reset

Microprocessors

• Microprocessors have a central processing unit (CPU) on a single integrated circuit
• They fetch instructions from memory, decode them, and execute them
• Components:
  • Arithmetic Logic Unit (ALU): Performs arithmetic and logical operations
  • Control Unit: Fetches, decodes, and executes instructions and controls data flow
  • Registers: Store data and addresses
  • Cache Memory: small, fast memory for frequently accessed data
• Key characteristics:
  • Clock Speed: Determines the rate at which instructions are executed
  • Word Size: Number of bits the processor can handle at once (e.g., 8-bit, 16-bit, 32-bit, 64-bit)
  • Instruction Set Architecture (ISA): Defines the instructions the processor can execute
  • Addressing Modes: How the processor accesses memory locations

Memory

• Used to store instructions and data that the microprocessor uses
• Types of Memory:
  • RAM (Random Access Memory): Volatile memory (data is lost when power is off). Includes SRAM and DRAM
  • ROM (Read Only Memory): Non-volatile memory (data is retained when power is off). Includes PROM, EPROM, and EEPROM
  • Cache Memory: Small, fast memory that stores frequently accessed data

Input/Output (I/O)

• Microprocessors interact with the external world through I/O devices
• Keyboards, displays, sensors, and actuators are examples of I/O devices
• I/O interfaces handle communication between the microprocessor and I/O devices.
• Methods for I/O data transfer:
  • Programmed I/O: The CPU directly controls the I/O operations
  • Interrupt-Driven I/O: The I/O device signals the CPU when it needs attention
  • Direct Memory Access (DMA): Allows I/O devices to transfer data directly to/from memory without CPU intervention