💾Intro to Computer Architecture Unit 2 – Digital Logic and Data Representation

Binary and Number Systems

• Binary number system uses only two digits (0 and 1) to represent numbers and perform arithmetic operations
• Decimal number system (base-10) is the most common number system used in everyday life and consists of digits 0 through 9
• Hexadecimal number system (base-16) uses digits 0 through 9 and letters A through F to represent values
  • Commonly used in computer programming and memory addressing
• Octal number system (base-8) employs digits 0 through 7 and is less frequently used in modern computing
• Binary-coded decimal (BCD) represents each decimal digit with a fixed number of binary bits (usually four)
  • Used in financial calculations to avoid rounding errors
• Converting between number systems is an essential skill in computer architecture
  • Examples include binary to decimal, hexadecimal to binary, and decimal to hexadecimal conversions

Boolean Algebra and Logic Gates

• Boolean algebra is a mathematical system that operates on logical values (true or false) using logical operators (AND, OR, NOT)
• AND gate produces a true output only when all inputs are true
  • Represented by the logical expression: $A \cdot B$ or $A \land B$
• OR gate produces a true output when at least one input is true
  • Represented by the logical expression: $A + B$ or $A \lor B$
• NOT gate (inverter) produces the opposite logical value of its input
  • Represented by the logical expression: $\overline{A}$ or $\lnot A$
• Exclusive OR (XOR) gate produces a true output when exactly one input is true
  • Represented by the logical expression: $A \oplus B$
• NAND and NOR gates are universal gates that can be used to create any other logic gate
• De Morgan's laws describe the relationship between logical operators and their negations
  • $\overline{A \cdot B} = \overline{A} + \overline{B}$ and $\overline{A + B} = \overline{A} \cdot \overline{B}$

Digital Logic Circuits

• Combinational logic circuits produce outputs that depend solely on the current inputs
  • Examples include adders, decoders, and multiplexers
• Sequential logic circuits produce outputs that depend on both the current inputs and the previous state of the circuit
  • Examples include flip-flops, counters, and shift registers
• Half adder is a combinational circuit that adds two single-bit numbers and produces a sum and a carry output
• Full adder is a combinational circuit that adds three single-bit numbers (two operands and a carry-in) and produces a sum and a carry output
• Multiplexer (MUX) is a combinational circuit that selects one of several input signals and forwards it to a single output based on a select signal
• Demultiplexer (DEMUX) is a combinational circuit that takes a single input signal and directs it to one of several outputs based on a select signal
• Flip-flops (e.g., D, JK, and T flip-flops) are basic sequential circuits used for storing state information

Data Representation in Computers

• Unsigned integers are represented using a fixed number of bits, with the leftmost bit representing the most significant bit (MSB) and the rightmost bit representing the least significant bit (LSB)
• Signed integers can be represented using various methods, such as sign-magnitude, one's complement, and two's complement
  • Two's complement is the most common representation for signed integers in modern computers
• Floating-point numbers are represented using a format that includes a sign bit, exponent, and mantissa (significand)
  • IEEE 754 standard defines the most widely used floating-point formats (single-precision and double-precision)
• Characters are typically represented using ASCII (American Standard Code for Information Interchange) or Unicode (e.g., UTF-8) encoding
• Images are represented using formats such as bitmap (BMP), JPEG, and PNG, which store pixel information and metadata
• Audio is represented using formats like WAV and MP3, which store sound wave information and compression data
• Video is represented using formats such as AVI, MP4, and WebM, which store a sequence of images and audio data

Memory and Storage Concepts

• Memory hierarchy organizes storage devices based on their speed, capacity, and cost
  • Registers, cache, main memory (RAM), and secondary storage (e.g., hard drives, SSDs)
• Random Access Memory (RAM) is volatile memory that provides fast read and write access to stored data
  • Static RAM (SRAM) uses flip-flops to store data and is faster but more expensive than Dynamic RAM (DRAM)
  • Dynamic RAM (DRAM) uses capacitors to store data and requires periodic refreshing to maintain data integrity
• Read-Only Memory (ROM) is non-volatile memory that stores permanent data and instructions
  • Examples include PROM, EPROM, and EEPROM
• Cache memory is a high-speed memory located close to the processor that stores frequently accessed data and instructions
  • Levels of cache (L1, L2, L3) with increasing size and latency
• Virtual memory is a technique that allows the operating system to use secondary storage as an extension of main memory
  • Paging and segmentation are common virtual memory management techniques
• Endianness refers to the order in which bytes are arranged in a multi-byte data type
  • Little-endian (least significant byte first) and big-endian (most significant byte first) are the two main endianness formats

Arithmetic and Logic Unit (ALU)

• ALU is a critical component of the processor that performs arithmetic and logical operations on data
• Arithmetic operations include addition, subtraction, multiplication, and division
  • Adders (e.g., ripple carry adder, carry lookahead adder) are used to perform addition and subtraction
  • Multipliers (e.g., array multiplier, Booth's algorithm) are used to perform multiplication
  • Dividers (e.g., restoring divider, non-restoring divider) are used to perform division
• Logical operations include AND, OR, NOT, XOR, and bit shifting
  • Bit shifting operations (e.g., logical shift, arithmetic shift, rotate) are used to move bits left or right within a data word
• ALU also performs comparison operations (e.g., equal to, greater than, less than) for conditional branching and control flow
• Floating-point arithmetic is performed using specialized floating-point units (FPUs) that adhere to the IEEE 754 standard
• SIMD (Single Instruction, Multiple Data) extensions (e.g., SSE, AVX) allow the ALU to perform parallel operations on multiple data elements

Input/Output Systems

• I/O systems enable communication between the computer and external devices
  • Examples include keyboards, mice, displays, printers, and network interfaces
• Memory-mapped I/O treats I/O devices as memory locations, allowing the processor to access them using regular memory read and write instructions
• Port-mapped I/O uses dedicated I/O instructions (e.g., IN, OUT) to access I/O devices through specific ports
• Interrupts are signals generated by I/O devices to notify the processor of events that require attention
  • Interrupt handlers are software routines that process interrupts and perform the necessary actions
• Direct Memory Access (DMA) allows I/O devices to access main memory directly, without the intervention of the processor
  • DMA controllers manage the transfer of data between I/O devices and memory
• Buses (e.g., address bus, data bus, control bus) are communication pathways that connect the processor, memory, and I/O devices
  • Examples include PCI, PCIe, USB, and SATA
• Protocols (e.g., TCP/IP, HTTP, USB) define the rules and formats for data exchange between devices

Practical Applications and Examples

• Digital logic circuits are used in the design of processors, memory systems, and I/O controllers
  • Example: A 4-bit ripple carry adder can be constructed using four full adders connected in series
• Boolean algebra and logic gates are used to optimize and simplify digital circuits
  • Example: A complex logic function can be minimized using Karnaugh maps or Boolean algebra identities
• Data representation is crucial for storing and processing various types of information in computers
  • Example: A 32-bit signed integer using two's complement can represent values from -2^31 to 2^31-1
• Memory hierarchy and storage concepts are essential for designing efficient and cost-effective computer systems
  • Example: A computer with a 64-bit address space, 16 GB of RAM, and a 512 GB SSD can efficiently manage memory and storage resources
• ALU design is critical for the performance and functionality of processors
  • Example: A pipelined ALU can execute multiple instructions simultaneously, improving overall processor performance
• I/O systems enable computers to interact with the external world and perform useful tasks
  • Example: A computer with a USB keyboard, a wireless mouse, and a high-resolution display can provide an effective user interface for various applications