IC220 Gates Basic building blocks of logic Slide Set #A1: Digital - PowerPoint PPT Presentation

Appendix Goals Establish an understanding of the basics of logic design for future material IC220 • Gates – Basic building blocks of logic Slide Set #A1: Digital Logic • Combinational Logic (Appendix A) – Decoders, Multiplexors, PLAs • Clocks • Memory Elements • Finite State Machines 1 2 Logic Design – Digital Signals Boolean Algebra • One approach to expressing the logic function • Only two valid, stable values • Operators: – False = x  – NOT A – True = Output true if • Vs. voltage levels    – AND: ‘A logical product’ x A B AB – Low voltage “usually” Output true if – High voltage “usually”   x A B – OR : ‘A logical sum’ – But for some technologies may be the reverse Output true if • How can we make a function with these signals?   – XOR x A B 1. Specify equations: Output true if – NAND x  A  B Output true if 2. Implement with – NOR   x A B Output true if 3 4

Gates Example A(1) B(1) G C(0) D(1) Equation: 5 6 EX: A-1 to A-4 Truth Tables Part 1 Truth Tables Part 2 • Alternative way to specify logical functions • Not just for individual gates • List all outputs for all possible inputs • Not just for one output – n inputs, how many entries? A – Inputs usually listed in numerical order F B x  A   x A B G C A x A B x A B C F G 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 0 1 0 1 1 1 0 7 8 1 1 1

Laws of Boolean Algebra Laws of Boolean Algebra  1   0  A A A A • Identity Law      • Associative Law A ( B C ) ( A B ) C      A ( B C ) ( A B ) C  1  • Zero and One Law  0  A 1 A 0 • Distributive Law       A ( B C ) ( A B ) ( A C )       A ( B C ) ( A B ) ( A C )  A   A  A 1 A 0 • Inverse Law • DeMorgan’s Law    A B A B    A B A B    A B B A    A B B A • Commutative Law 9 10 DeMorgan’s Law and Bubble Pushing Bubble Pushing Example       A B A B A B A B 11 12

Representing Combinational Logic 2-Level Logic • Represent ______ logic function(s) – Utilizing just two types of gates Truth Table Boolean Formula (assuming we get NOT for free) – Two forms • Sum of products • Product of sums Circuit – Relationship with truth table • Generate a gate level implementation of any set of For combinational logic, these three: logic functions - are equivalently _____________ • Allows for simple reduction/minimization - straight-forward to ____________ - have no ______________ 13 14 EX: A-11 to A-15 Example Reduction/Minimization • Show the sum of products for the following truth table. • Reduction is important to reduce the size of the circuit that performs • Strategy: _________ all the products where the output is ________ the function. This, in turn, reduces the cost of, and delay through, A B C z the circuit. 0 0 0 0 0 0 1 1 • What? 0 1 0 0 – Less power consumption 0 1 1 0 – Less heat 1 0 0 1 – Less space 1 0 1 1 – Less time to propagate a signal through the circuit 1 1 0 0 – Less points of possible failure 1 1 1 1 • It makes good engineering and economic sense! • z = • Is this optimal? 15 16

Minimization by Hand Karnaugh Maps (k-Maps) A B C z 0 0 0 0 • Sum of Products: Truth Table: 0 0 1 1 • A graphical (pictorial) method used to minimize Boolean 0 1 0 0 0 1 1 0 expressions.             1 0 0 1 z ( A B C ) ( A B C ) ( A B C ) ( A B C ) 1 0 1 1 • Don’t require the use of Boolean algebra theorems and equation 1 1 0 0 manipulations. 1 1 1 1 • A special version of a truth table. • Works with two to four input variables (gets more and more difficult with more variables) • Groupings must be __________________ • Final result is in _____________________ form • Okay to duplicate terms while minimizing 17 18 Karnaugh Maps (k-Maps) Example #1 K-Maps Example #2 • Lets create a k-map table • Suppose we already have this k-Map. Minimize the function. – Borders represent all possible conditions A B C z 0 0 0 0 C D C D CD C D – NOT in counting order 0 0 1 1 0 1 0 0 – Be consistent 0 1 1 0 A B 1 0 0 0 1 0 0 1 • -What are the values for the map? 1 0 1 1 – The values of ___ 1 1 0 0 1 1 1 1 • To reduce, circle our powers of 2! A B 0 0 0 1 AB 0 1 1 0 B C B C BC B C A B 1 1 1 1 A A • Result: • Every “1” must be ____________ by at least one term • Larger blocks in k-Map produce smaller product terms 19 20

EX: A-21 to A-24 Truth Table and Logical Circuit Example Example Circuit z = + + B  A  B A  C C • How does a truth table and subsequent sum of products equation create a logic circuit? A • From the earlier example: z = + + A  B  C B A  C z B • Lets build the logical circuit: – Which gates do we need? – How many inputs do we have? C – How do we connect the circuit? 21 22 Don’t Cares General Skills • Sometimes don’t care about the output. • Make sure you can populate a K-Map from a truth table C D C D CD C D • Make sure you can populate a truth table from a K-Map • Given a circuit, know how to construct a truth table A B X 0 0 0 • Given a truth table, know how to produce a sum-of-products, and how to draw a circuit A B 1 0 0 X • Be able to understand minimization and use it • Know DeMorgan’s Law and other Boolean laws AB 1 1 1 1 A B 1 0 0 0 • Each X can be either a 0 or 1 (helps with minimization) • But in actual circuit, each X will have some specific value 23 24