Introduction <ul><li>The Karnaugh map (also known as a Veitch diagram) was invented in 1953 by Maurice Karnaugh, a telecommunications engineer at Bell Labs. It is a graphical way of minimizing Boolean expressions. An expression’s truth table is drawn as a matrix in such a way that each row and each column of the matrix puts minterms that differ in the value of a single variable adjacent to each other. Then, by grouping adjacent cells of the matrix, you can identify product terms that eliminate all complemented literals, resulting in a minimized version of the expression. </li></ul>
Properties <ul><li>Normally, extensive calculations are required to obtain the minimal expression of a Boolean function, but one can use a Karnaugh map instead. </li></ul><ul><li>A Karnaugh map may have any number of variables, but usually works best when there are only a few - between 2 and 6 for example . </li></ul><ul><li>Each square in a Karnaugh map corresponds to a minterm (and maxterm). The picture to the right shows the location of each minterm on the map. </li></ul>
Size of Map <ul><li>In a Karnaugh map with n variables, a Boolean term mentioning k of them will have a corresponding rectangle of area 2 n − k . Common sized maps are of 2 variables which is a 2x2 map; 3 variables which is a 2x4 map; and 4 variables which is a 4x4 map </li></ul><ul><li>For problems involving more than six variables, solving the Boolean expressions is more preferred than the Karnaugh map. </li></ul>
SOP & POS <ul><li>A Karnaugh map can also be drawn for 0’s as well as 1’s </li></ul><ul><li>The SOP (Sum of Product) expression refers to drawing of a Karnaugh map in which high values (1’s) are plotted. The simplified equation comes in SOP form such as (A.B)+(B.C) </li></ul><ul><li>The POS (Product of Sum) expression represents the low (0) values in the Karnaugh Map. The simplified equation comes in POS form like (A+B).(C+D) </li></ul>Pairs, Quads & Octets <ul><li>A group of two adjacent 1’s in the Veitch Diagram is known as Pair. </li></ul><ul><li>The quad is a group of four 1’s that are end to end, or in the form of a square. Two variables and their compliments drop out of the Boolean equation in the quad. </li></ul><ul><li>An octet is a group of eight adjacent 1’s. an octet always eliminates three variables and their complements </li></ul>
Don't cares <ul><li>Karnaugh maps also allow easy minimizations of functions whose truth tables include " don't care " conditions </li></ul><ul><li>They are usually indicated on the map with a hyphen / dash / X in place of the number . </li></ul><ul><li>The value can be a " 0, " " 1, " or the hyphen / dash / X depending on if one can use the " 0 " or " 1 " to simplify the Karnaugh Map more . </li></ul>
Thank You <ul><li>References: </li></ul><ul><li>Digital Fundamentals (Floyd) </li></ul><ul><li>Digital Computer Electronics (Malvino Brown) </li></ul><ul><li>Wikipedia, The free Encyclopedia </li></ul>
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