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  1. Fundamentals of Computer SciencePart i2 Lecture 3 Digital Logic

  2. Topics for this lecture • Gates and Boolean logic • AND, OR, NOT, NAND, … • Integrated circuits • SSI,MSI,LSI,VLSI • Memory • Flip-Flop • Arithmetic • Half-adder, Full-adder

  3. Microprocessor core layout AMULET3, being designed at the University of Manchester. Low power, high performance, asynchronous, ARM compatible. Size 3mm x 1mm.

  4. Basics of hardware design • Components in the picture • data path (bottom strip), register bank (centre) • ALU, adder, multiplier, instruction controller, ... • The lowest logical design level • gates built from transistors • circuits built from gates (arithmetic, memory) • Device level, physics... • N/P-type semiconductors (transistors)

  5. Digital circuits • Two values only, 0 and 1 • continuous voltage range (within bounds) • 0 is low signal (voltage range 0 to 1) • 1 is high signal (voltage range 2 to 5) • Gates • electronic devices that compute functions of 0/1 • made from transistors (very fast & small switches)

  6. How a transistor works... • When input voltage below threshold, no current flows through; then output voltage high • When input voltage above threshold, current flows through; then output voltage low Note signal is inverted, hence a NOT gate

  7. The three main gates A, B inputs X output X = f(A,B) (truth table) NOTinvert (negate) single input. AND 1 only if both inputs 1 OR 1 if at least one input 1

  8. Two more gates NAND 0 only if both inputs 1 (inverted AND) NOR 0 if at least one input 1 (inverted OR) Simpler - 2 instead of 3 transistors...

  9. B A A A C Boolean algebra • Can have functions of N variables, eg M = f (A,B,C) • Write AB for A AND B A+B for A OR B for NOT A • Write functions as expressions M = BC + A C + AB + ABC (majority vote) • Laws to identify equivalent functions, eg A + BC = (A+B)(A+C) A = 0

  10. More gates (NAND) Simpler - one type of gate… NAND is complete (any circuit can be created from it) Question: Can you do NOT?

  11. More gates (XOR) XOR is eXclusive OR 1 only if exactlyone input is 1

  12. Integrated circuits (chips) SSI chip 5mm x 5mm Standardised Pins Classification based on number of gates: from Small Scale Integrated (SSI), 1-10 gates, to Very Large Scale Integrated (VLSI), > 100,000 gates

  13. Types of chips • Combinational circuits • Boolean functions, transform inputs to output • Memories • can store bits; contain feedback • Flip-Flop • Arithmetic • Half-adder, Full-adder • Control • data buses, clocks, etc

  14. Clock signals • Clock is a circuit that emits a series of pulses. • Interval between pulses = clock cycle time • Usually detect rising/falling edge

  15. The Flip-Flop • Also called a clocked D latch • Has inputs D and clock, and output Q • When control is on, Q = D • When control is off, output cannot change, & hence D stored

  16. Memories • Registers • N-bit flip-flop gives N-bit register • N bits = one word • Memories • M registers gives M-word memory

  17. Arithmetic • Addition • 1-bit addition yields 1-bit result and 1-bit carry 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 carry 1 • Adders • half-adder = 1-bit adder with 2 bit input • full-adder = 1-bit adder with additional carryinput • N-bit adder requires N full-adders

  18. Half-adder • 2 bits on input • 1 bit sum plus 1 bit carry on output • Cannot handle carry in the middle of the word...

  19. Full-adder Built from twohalf-adders • 2 bits andcarry on input • 1 bit sum plus 1 bit carry on output • Can be strung together into N-bit carry, ripple effect

  20. Summary • Arithmetic Logic Unit • similar to adders • Data bus • transfers data along wires, one wire per bit • using control signals (clocks) • More complex circuits • design in Boolean algebra • build from standard chips or produce VLSI

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