Prof. YingLi Tian Sept. 10, 2012 Department of Electrical Engineering The City College of New York The City University of New York (CUNY) Lecture 4: Implementation AND, OR, NOT Gates and Compliment 1 EE210: Switching Systems

TA’s Students who didn’t receive TA’s , please send an to Mr. Zhang, by putting subject: “EE210 ” Mr. Chenyang Zhang Course website: ee.ccny.cuny.edu/www/web/yltian/EE2100.html 2

Outlines Quick Review of the Last Lecture AND, OR, NOT Gates Switching Algebra Properties of Switching Algebra Definitions of Algebraic Functions Implementation AND, OR, NOT Gates Complement (NOT) Truth table to algebraic expressions 3

Definition of Switching Algebra OR -- a + b (read a OR b) AND -- a · b = ab (read a AND b) NOT -- a´ (read NOT a) 4

SOP and POS A sum of products expression (often abbreviated SOP) is one or more product terms connected by OR operators. ab´ + bc´d + a´d + e´ ---- ?? terms, ?? literals A product of sums expression (POS) is one or more sum terms connected by AND operators. SOP: x´y + xy´ + xyz POS: (x + y´)(x´ + y)(x´ + z´) A literal is the appearance of a variable or its complement. A term is one or more literals connected by AND, OR, operators.

Gate Implementation 6 P2b: a(bc) = (ab) c These three implementations are equal.

Implementation of functions with AND, OR, NOT Gates -- 1 Given function: f= x´yz´ + x´yz + xy´z´ + xy´z + xyz Two-level circuit (maximum number of gates which a signal must pass from the input to the output) 7

Implementation of functions with AND, OR, NOT Gates -- 2 (1)x´yz´ + x´yz + xy´z´ + xy´z + xyz (2) x´y + xy´ + xyz (3) x´y + xy´ + xz (4) x´y + xy´ + yz

Function: x´y + xy´ + xz, when only use uncomplemented inputs: Implementation of functions with AND, OR, NOT Gates -- 3

Multi-level circuit 10 Function? (see Page50)

Commonly used terms DIPs – dual in-line pin packages (chips) ICs – integrated circuits SSI – small-scale integration (a few gates) MSI – medium-scale integration (~ 100 gates) LSI -- large-scale integration VLSI – very large-scale integration GSI – giga-scale integration 11

Examples Need a 3-input OR (or AND), and only 2- input gates are available Need a 2-input OR (or AND), and only 3- input gates are available 12

Positive and Negative Logic Use 2 voltages to represent logic 0 and 1 For example: Low: Volt; High: >2.1Volt; Transition state: Volt Positive logic: High voltage  1, Low voltage  0 Negative logic: Low voltage  1, High voltage  0

The Complement (NOT) DeMorgan: P11a: (a + b)´ = a´ b´ P11b: (ab)´ = a´ + b´ P11aa: (a + b + c …)´ = a´ b´ c´ … P11bb: (abc…)´ = a´ + b´ + c´ + … Note: (ab)´ ≠ a´ b´ (a + b)´ ≠ a´ + b´ ab + a´ b´ ≠ 1 14

Find the complement of a given function Repeatedly apply DeMorgan’s theorem 1. Complement each variable (a to a´ or a´ to a) 2. Replace 0 by 1 and 1 by 0 3. Replace AND by OR, OR by AND, being sure to preserve the order of operations See Example 2.5 (Page53) and Example 2.6 (page 54). 15

Example of Complement 16 f = wx´y + xy´ + wxz f ´ = (wx´y + xy´ + wxz)´ = (wx´y)´(xy´)´(wxz)´ = (w´+x+y´)(x´+y)(w´+x´+z´)

f is 1if a = 0 AND b = 1 OR if a = 1 AND b = 0 OR if a = 1 AND b = 1 f is 1 if a´ = 1 AND b = 1 OR if a = 1 AND b´ = 1 OR if a = 1 AND b = 1 f is 1if a´b = 1 OR if ab´ = 1 OR if ab = 1 f = a´b + ab´ + ab = a + b (OR) Truth Table to Algebraic Expressions

f (A, B, C) = ∑m(1, 2, 3, 4,5) = A´B´C + A´BC´ + A´BC + AB´C´+ AB´C ff´f´ To obtain f (A, B, C), add all minterms with output = 1 (SOP): f ´(A, B, C) = ∑m(0, 6, 7) = A´B´C´ + ABC´ + ABC A standard product term, also minterm is a product term that includes each variable of the problem, either uncomplemented or complemented.

f = (f ´ )´= (A + B + C)(A´+B´+C)(A´+B´+C´) ff´f´ A standard sum term, also called a maxterm, is a sum term that includes each variable of the problem, either uncomplemented or complemented. POS:

20 f (A, B, C) = A´B´C + A´BC´ + A´BC + AB´C´+ AB´C = A´B´C + A´B + AB´ = A´(B´C + B) + AB´ = A´C + A´B + AB´ = B´C + A´B + AB´ To simplify: f ´(A, B, C) = A´B´C´ + ABC´ + ABC = A´B´C´ + AB See page56 for details. P9a: ab + ab´ = a P10a: a + a´ b = a + b P8a: a (b + c) = ab + ac P10a: B + C

Truth Table with don’t care Include them as a separate sum. 21 f (a, b, c) = ∑m(1, 2, 5) + ∑d(0, 3) abcf 000X X

Number of different functions of n variables

Announcement: Review Chapter HW2 is out today, due on 9/12. Next class (Chapter ): NAND, NOR, Exclusive-OR (EOR) Gates Simplification of Algebraic Expressions 23