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Number Systems and Logic Functions in Digital Design

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  1. Overview • Last lecture: • Abstraction and hierarchy in digital design • Combinational vs. sequential systems • High-level examples of combinational and sequential designs • The whole course in 2 examples! • In section • Number systems • Binary; 2’s complement • Today • Number systems (very quick review) • Combinational logic • Logic functions and truth tables CSE 370 – Winter 2002 - Number syst.; Logic functions- 1

  2. The basics binary numbers • Base conversion (binary, octal, decimal, hexadecimal) • Positional number system • 1012=510 (1 * 22 + 0 * 21+ 1 * 20) • 1478=10310 (1 * 82 + 4 * 81+ 7 * 80) Octal (base 8) • 2C116=70510 (2 * 162 + 12 * 161+ 1 * 160) hexadecimal (base 16) • Do it also for fractions • Conversion between binary/octal/hex • Binary: 10011110001 • Octal: 10 | 011 | 110 | 001=23618 • Hex: 100 | 1111 | 0001=4F116 • Addition and subtraction are trivial, but worth practicing • See Katz, appendix A CSE 370 – Winter 2002 - Number syst.; Logic functions- 2

  3. The basics base conversion • Conversion from decimal to binary/octal/hex: divide by the base • The digits are generated from least significant (rightmost) to most significant (leftmost) • Why does it work (recall positional system) CSE 370 – Winter 2002 - Number syst.; Logic functions- 3

  4. Number systems • How do we write negative binary numbers? • We want same number of positive and negative representations (or as close to same as possible) • How do we represent zero? • How do we recognize easily if a number is positive or negative? • Historically: Three approaches • Sign and magnitude • Ones complement • Twos complement • Twos complement makes addition and subtraction easy • Used almost universally in present-day systems CSE 370 – Winter 2002 - Number syst.; Logic functions- 4

  5. Approach 1: Sign and magnitude • The most-significant bit (msb) is the sign digit • 0  positive • 1 negative • The remaining bits are the number’s magnitude • Problem 1: Two representations for zero • 0 = 0000 and also –0 = 1000 • Problem 2: Arithmetic is cumbersome CSE 370 – Winter 2002 - Number syst.; Logic functions- 5

  6. Approach 2: Ones complement • Negative number: Bitwise complement of positive number • 0011  310 • 1100 –310 • Solves the arithmetic problem • Remaining problem: Two representations for zero • 0 = 0000 and also –0 = 1111 CSE 370 – Winter 2002 - Number syst.; Logic functions- 6

  7. Only one zero! msb is the sign digit 0  positive 1 negative Negative number: Bitwise complement plus one 0011  310 1101 –310 Number wheel Note: one more negative number – 1 0 – 2 + 1 1111 0000 1110 0001 – 3 + 2 1101 0010 – 4 + 3 1100 0011 1011 0100 – 5 + 4 1010 0101 – 6 + 5 1001 0110 1000 0111 – 7 + 6 – 8 + 7 Approach 3: Twos complement CSE 370 – Winter 2002 - Number syst.; Logic functions- 7

  8. Twos complement (con’t) • Complementing a complement restores the original number • Arithmetic is easy • We ignore the carry • Same as a full rotation around the wheel • Subtraction = negation and addition • Easy to implement in hardware CSE 370 – Winter 2002 - Number syst.; Logic functions- 8

  9. – 1 – 1 0 0 – 2 – 2 + 1 + 1 1111 1111 0000 0000 1110 1110 0001 0001 – 3 – 3 + 2 + 2 1101 1101 0010 0010 – 4 – 4 + 3 + 3 1100 1100 0011 0011 1011 1011 0100 0100 – 5 – 5 + 4 + 4 1010 1010 0101 0101 – 6 – 6 + 5 + 5 1001 1001 0110 0110 1000 1000 0111 0111 + 6 + 6 – 7 – 7 – 8 – 8 + 7 + 7 Overflow • Conditions: Sign bit changes • Summing two positive numbers gives a negative result • Summing two negative numbers gives a positive result 6 + 4  –6 –7 – 3  +6 CSE 370 – Winter 2002 - Number syst.; Logic functions- 9

  10. Combinational logic topics • Logic functions, truth tables, and switches • NOT, AND, OR, NAND, NOR, XOR, . . . • minimal set • Axioms and theorems of Boolean algebra • proofs by re-writing • proofs by perfect induction • Gate logic • networks of Boolean functions • time behavior • Canonical forms • two-level • incompletely specified functions • Simplification • Boolean cubes and Karnaugh maps • two-level simplification CSE 370 – Winter 2002 - Number syst.; Logic functions- 10

  11. Logic functions and truth tables X Y 0 0 1 1 • Buffer (identity) X • NOT X X’ • AND X.Y XY • OR X+Y Y X X Y 0 1 1 0 X Y X X Y Z 0 0 0 0 1 0 1 0 0 1 1 1 Z Y X Y Z 0 0 0 0 1 1 1 0 1 1 1 1 X Z Y CSE 370 – Winter 2002 - Number syst.; Logic functions- 11

  12. Logic functions and truth tables (ct’d) X Y Z 0 0 1 0 1 1 1 0 1 1 1 0 X • NAND X.Y XY • NOR X+Y • XOR X  Y Z Y X Y Z 0 0 1 0 1 0 1 0 0 1 1 0 X Z Y X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 X Z Y CSE 370 – Winter 2002 - Number syst.; Logic functions- 12

  13. Possible logic functions of two variables • There are 16 possible functions of 2 input variables: • in general, there are 2**(2**n) functions of n inputs X F Y • X Y 16 possible functions (F0–F15)0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 10 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 • 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 • 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 X not X Y not Y X xor Y X = Y X and Y X nand Y X or Y X nor Y CSE 370 – Winter 2002 - Number syst.; Logic functions- 13

  14. Cost of different logic functions • Different functions are easier or harder to implement • each has a cost associated with the number of switches (transistors) needed • 0 (F0) and 1 (F15): require 0 switches, directly connect output to low/high • X (F3) and Y (F5): require 0 switches, output is one of inputs • X' (F12) and Y' (F10): require 2 switches for "inverter" or not-gate • X nor Y (F4) and X nand Y (F14): require 4 switches (cf. Katz page 677) • X or Y (F7) and X and Y (F1): require 6 switches • X = Y (F9) and X  Y (F6): require 16 switches • thus, because NOT, NOR, and NAND are the cheapest they are the functions we implement the most in practice CSE 370 – Winter 2002 - Number syst.; Logic functions- 14

  15. Minimal set of functions • All logic functions can be implemented from NOT, AND, and OR • Can we do it with only 2 of the 3? • Can we implement all logic functions from NOT, NOR, and NAND? • For example, implementing X and Yis the same as implementing not (X nand Y) • In fact, we can do it with only NOR or only NAND • NOT is just a NAND or a NOR with both inputs tied together X’ = X nand X X’ = X nor X • NAND and NOR are "duals",that is, it’s easy to implement one using the other X nand Y not ( (not X) nor (not Y) ) X nor Y not ( (not X) nand (not Y) ) CSE 370 – Winter 2002 - Number syst.; Logic functions- 15

  16. From (logic) expressions to gates • More than one way to map expressions to gates • e.g., Z = A' • B' • (C + D) = (A' • (B' • (C + D))) T2 T1 use of 3-input gate A Z A B T1 B Z C C T2 D D CSE 370 – Winter 2002 - Number syst.; Logic functions- 16

  17. What is the optimal gate realization? • We can use the theorems and axioms of Boolean algebra to optimize our designs • Design goals vary depending on the technology. For example: • Reduce the number of inputs? (variables, e.g., A, X and their complements, e.g., A are also called literals) • Reduce number of gates? • Reduce number of levels? • More options (e.g., limited fan-in and fan-out) • How do we explore the trade-offs? • CAD tools • Logic minimization: Reduce number of gates and complexity • Logic optimization: Minimization vs. speed and delay CSE 370 – Winter 2002 - Number syst.; Logic functions- 17

  18. The Mathematics: Boolean algebra • A Boolean algebra is an algebraic structure that consists of • a set of elements B; B = {0, 1} • binary operations { + , • }; + is logical OR, • is logical AND • and a unary operation { ' }; ' is logical NOT • such that the following axioms (laws) hold: 1. the set B contains at least two elements, a, b, such that a  b2. closure: a + b is in B a • b is in B3. commutativity: a + b = b + a a • b = b • a4. associativity: a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c5. identity: a + 0 = a a • 1 = a6. distributivity: a + (b • c) = (a + b) • (a + c) a • (b + c) = (a • b) + (a • c)7. complementarity: a + a' = 1 a • a' = 0 CSE 370 – Winter 2002 - Number syst.; Logic functions- 18

  19. Logic functions and Boolean algebra • Any logic function that can be expressed as a truth table can be written as an expression in Boolean algebra using the operators: ', +, and • X Y X • Y0 0 00 1 01 0 0 1 1 1 X Y X' X' • Y0 0 1 00 1 1 11 0 0 0 1 1 0 0 X Y X' Y' X • Y X' • Y' ( X • Y ) + ( X' • Y' )0 0 1 1 0 1 10 1 1 0 0 0 01 0 0 1 0 0 0 1 1 0 0 1 0 1 ( X • Y ) + ( X' • Y' )  X = Y Boolean expression that is true when the variables X and Y have the same value and false, otherwise X, Y are Boolean algebra variables CSE 370 – Winter 2002 - Number syst.; Logic functions- 19

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