Uchechukwu Ofoegbu Temple University

Chapter 2: Combinational SystemsAdapted from Alan Marcovitz’sIntroduction to Logic and Computer Design Uchechukwu Ofoegbu Temple University

Riddle Four switches can be turned on or off. One is the switch for the incandescent overhead light in the next room, which is initially off, but you don't know which. The other three switches do nothing. From the room with the switches in it, you can't see whether the light in the next room is turned on or off. You may flip the switches as often and as many times as you like, but once you enter the next room to check on the light, you must be able to say which switch controls the light without flipping the switches any further. (And you can't open the door without entering, either!) How can you determine which switch controls the light?

Design Process for Combinational Systems • Begins with a verbal description of the intended system, known as the PROBLEM STATEMENT • A block diagram of the system should be developed • The desired objectives and constraints

Illustrations • A system with four inputs, A, B, C, and D, and one output, Z, such that Z = 1 iff three of the inputs are 1. • A single light (that can be on or off) that can be controlled by any one of three switches. One switch is the master on/off switch. If it is off, the lights are off. When the master switch is on, a change in the position of one of the other switches (from up to down or from down to up) will cause the light to change state. • A system to do 1 bit of binary addition. It has three inputs (the 2 bits to be added plus the carry from the next lower order bit) and produces two outputs, a sum bit and a carry to the next higher order position.

Illustrations • A system that has as its input the code for a decimal digit, and produces as its output the signals to drive a seven-segment display, such as those on most digital watches and numeric displays. • A system with nine inputs, representing two 4-bit binary numbers and a carry input, and one 5-bit output, representing the sum. (Each input number can range from 0 to 15; the output can range from 0 to 31.)

Design Steps • Represent each of the inputs and output in binary. • This is sometimes taken care of in the problem statement (ex 1, 3, 5) • Formalize the design specification either in the form of a truth table or of an algebraic expression. • There are 2n input combinations for n inputs in a truth table. • Truth tables are written in binary order to avoid omissions. • If necessary, break the problem into smaller sub-problems before or after creating the truth tables.

Design Steps • Simplify the description. • Most times expressions have to be converted to algebraic forms • Several techniques exist for reducing complexity of algebraic forms • Implement the system with the available components, subject to the design objectives and constraints. • Gates are the most common components • A gate is a network with one output • The less number of gates required, the more desirable the system is, since each signal passing through a gate introduces a delay in the system. • More complex systems can be used in addition to gates (ex. Adders, decoders, e.t.c.).

Don’t Care Conditions • Don’t cares occur when • The output is not specified for all input combinations, so for the remaining input combinations, it doesn’t matter • There are input combinations that don’t occur in the system • Flip-flops – one systems drives another • When we just don’t care • Don’t cares are represented by X in a truth table. • The output of the combination could either be a 1 or a 0.

Developing Truth Tables • A system with four inputs, A, B, C, and D, and one output, Z, such that Z = 1 iff three of the inputs are 1.

A single light (that can be on or off) that can be controlled by any one of three switches. One switch is the master on/off switch. If it is off, the lights are off. When the master switch is on, a change in the position of one of the other switches (from up to down or from down to up) will cause the light to change state.

4. A system that has as its input the code for a decimal digit, and produces as its output the signals to drive a seven-segment display

Developing Truth Tables Example • Truth table for a 1-bit full subtractor with a borrow input, bin, and inputs x, y, that produces an a difference output, d, and a borrow output, bout. • 0 0 • 0 1 • 1 0 • 1 1

Developing Truth Tables Example • Truth table for a 1-bit full subtractor with a borrow input, bin, and inputs x, y, that produces an a difference output, d, and a borrow output, bout.

Switching Algebra • Simplify the description. • Most times expressions have to be converted to algebraic forms • Several techniques exist for reducing complexity of algebraic forms • Implement the system with the available components, subject to the design objectives and constraints. • Gates are the most common components • A gate is a network with one output • The less number of gates required, the more desirable the system is, since each signal passing through a gate introduces a delay in the system. • More complex systems can be used in addition to gates (ex. Adders, decoders, e.t.c.).

Definitions • Literal: • The appearance of a variable or its complement. • Product Term: • one or more literals connected by AND operators. • Standard product term: • Also called minterm • product term that includes each variable of the problem, either uncomplemented or complemented. • Sum of products expression (often abbreviated SOP) • one or more product terms connected by OR operators. • A canonical sum or sum of standard product terms: • a sum of products expression where all of the terms are standard product terms.

Definitions • A Minimum Sum of Products expression: • one of those SOP expressions for a function that has the fewest number of product terms. • If there is more than one expression with the fewest number of terms, then minimum is defined as one or more of those expressions with the fewest number of literals. • (1) xyz + xyz + xyz + xyz + xyz 5 terms, 15 literals • (2) xy + xy + xyz 3 terms, 7 literals • (3) xy + xy + xz 3 terms, 6 literals • (4) xy + xy + yz 3 terms, 6 literals

Simplification xyz + xyz + xyz + xyz + xyz 5 terms, 15 literals = (x’yz’+x’yz)+(xyz + xyz) + xyz associative p2 = x’y(z+z’) + xy’(z+z’) + xyz distributive p8 = x’y.1 +xy’.1 +xyz complement p5 = x’y + xy’ + xyz identity p3 Down to three terms and seven literals

Simplification Reduce the number of literals by adding a second copy of xy’z (or x’yz), based on P6a – indempotency. xyz + xyz + xyz + xyz + xyz + xy’z5 terms, 15 literals = (x’yz’+x’yz)+(xyz + xyz) + (xyz + xy’z) associative p2 = x’y + xy + xz adjacency p9 Down to three terms and six literals

Simplification

Definitions • Sum Term:one or more literals connected by OR operators. • Standard sum term: • also called a maxterm • a sum term that includes each variable of the problem, either uncomplemented or complemented. • Product of sums expression (POS): • one or more sum terms connected by AND operators. • Canonical product or product of standard sum terms: • a product of sums expression where all of the terms are standard sum terms. • SOP: xy + xy + xyz • POS: (x + y)(x + y)(x + z) • both: x + y + z or xyz • neither: x(w + yz) or z + wxy + v(xz + w)

Simplification of Functions in Maxterm Form g = (w’ + x’ + y) (w + x’+ y + z’) = x’ + y + w’(w + z’) distributive (P8b) = x’ + y + w’z’ simplification (p10b) = (x’ + y + w’) (x’ + y + z’) distributive (P8b)

Implementation of Logic Gates f = x’y + xy’ + xz Minimum sum of product implementation of f. Circuit with only uncomplemented inputs.

Implementation of Logic Gates

Implementation of Logic Gates • d = n’+c(h+s) (n+c’(h+s)) • d = (n’c) + (hs) + n(hs) + c • d = n’c(h+s) + n(c’+(hs)) • D = n’c(ns) + n(hs)c =

De Morgan’s Theorem P11a. (a+b)’ = a’b’ P11b. (ab)’ = a’+b’ Please note: (a’+b’) != a’+b’ (ab)’ != a’b’ Proof of DeMorgan’s theorem. De Morgan’s Theorem is generally used to find the complement of an expression

De Morgan’s Theorem - Example Find the complement of f = wx’y +xy’ +wxz Note that f is in SOP so f’ will be in POS • We could apply DeMorgan’s Theorem repeatdely • f’ = (wx’y +xy’ +wxz)’ • f’ = (wx’y)’(xy’)’(wxz)’ • f’ = (w’x+y’)(x’+y)(w’+x’+z’) • Or we could follow these set of rules • Complement each variable • Replace every AND by Or and every OR by AND – but make sure you keep the order of operation

De Morgan’s Theorem - Example Find the complement of f = ab +b’c + c’d • f’ = (a’ + b’+ b + c’ + c + d’) • f’ = (a’ + b’+ b + c’ + c + d’)’ • f’ = a’b’+bc’+cd’ • f’ = (a’ + b’)( b + c’) (c + d’) • f’ = (a’ + b’)’( b + c’)’(c + d’)’

In groups • 1 • 2e,f • 3f • 5 • 8 • 12a • 13c • 14 • 18

From the truth table to Algebraic Expressions f = ab + ab + ab

From the truth table to Algebraic Expressions

Example f(A,B,C) = Σm(1,2,3,6,7) = ? • A’B’C’ + AB’C’ + AB’C • B + C + D + G + H • A’B’C + A’BC’ + A’BC + ABC’+ABC • ABC’ + AB’C + AB’C’+ A’B’C + A’b • ABC + A’BC+A’BC’ f’(A,B,C) = ? • Σm(1,2,3,6,7)’ • Σm(0,4,5) • Σm(A’B’C’ + AB’C’ + AB’C)

Don’t Cares f(A,B,C) = Σm(2,3,6,7)+ Σd(1,5)

In groups • 20 • 22 • 23 • 27

Uchechukwu Ofoegbu Temple University