Karnaugh map kmap can be used to minimize functions of up to 6 variables. Lecture 6 karnaugh map kmap university of washington. Stroud combinational logic minimization 912 2 karnaugh maps kmap alternate forms of 3variable kmaps note endaround adjacency. The top and side for a 6 variable map of the map is numbered in full gray code. Scribd is the worlds largest social reading and publishing site. K maps can be used for expressions with 2, 3, 4, and 5 variables. Karnaugh maps k maps an n variable k map has 2n cells with each cell corresponding to an n variable truth table value. If w0, then 4 variable k map becomes 3 variable k map.
This is not the proper way to group a 5 variables k map. Five variable k map examples electronics and communication. Kmap is directly applied to twolevel networks composed of and and or gates. The labeling of variables on a karnaugh map is arbitrary. Y xyz xyz xyz xyz x xyz xyz xyz xyz z y m 0 m 1 m 3 m 2 xm 4 m 5 m 7 m 6 z yz 00 01 11 10 0 xyz xyz x. The cells are managed in a way so that simplification of a given expression is simply a matter of properly grouping the cells. Now, let us discuss the 5 variable k map in detail. K map is used for minimization or simplification of a boolean expression. Chap 5 ch 1 lecture 5 karnaugh maps algebraic procedures.
Oct 22, 2008 this is a retouched picture, which means that it has been digitally altered from its original version. A karnaugh map kmap is a pictorial method used to minimize boolean expressions without having to use boolean algebra theorems and equation manipulations. If v0, then 5 variable k map becomes 4 variable k map. Construct minterm and maxterm from a compressed kmap. For kvariable maps, this reduction technique can also. Karnaugh map or k map is an alternative way to write truth table and is used for the simplification of boolean expressions. Previous gate questions on kmap, sop and pos expressions 1987 till date 1991.
The karnaugh map boolean algebraic simplification technique. The karnaugh map km or kmap is a method of simplifying boolean algebra expressions. Stroud combinational logic minimization 912 5 k map minimization goals larger groups. K maps are used to simplify boolean functions and expressions without using algebrabased boolean methods. A general representation of a 2 variable k map plot is shown below. It is useful for up to 5 or 6 variables, and is a good tool to help understand the process of logic simplification. When the number of variables increases, the number of the square cells increases. Now fill the cell marked with subscript 0,1,2,4, 5,7,10 and 15 with value 1 as we are dealing with sum of products sop. The fivevariable kmap is in effect two fourvariable kmaps drawn horizontally to form an extension of each other. The number of cells in 5 variable k map is thirtytwo, since the number of variables is 5. Imagine a map like those drawn previously, with an identically sized map stacked directly selection from introduction to digital systems. Karnaugh maps kmaps are a convenient way to simplify boolean expressions. A 1 in the kmap can be used by more than one group. Minimization of boolean functions using karnaugh maps maurice.
The corresponding 5 variable karnaugh map using gray code is shown in fig. Kmaps can be used for expressions with 2, 3, 4, and 5 variables. Five variable karnaugh map problem example duration. When we are simplifying a boolean equation using karnaugh map, we represent the each cell of k map containing the conjunction term with 1. K map cells are arranged such that adjacent cells correspond to truth rows that differ in only one bit position logical adjacency. Product of sums reduction using karnaugh map boolean. Kmaps for sumofproduct design a minimal sumofproduct design can be created as follows. Looping entered variable kmaps follows the same general principles as looping 10 mapsoptimal groupings of 1s and entered variables evs are sought for sop circuits, and optimal groupings of 0s and evs are sought for pos circuits. Define karnaugh and establish the correspondence between karnaugh maps and truth tables and logical expressions. We have mentally stacked the four sub maps to see the group of 4cells corresponding to out cf a magnitude comparator used to illustrate a 6variable kmap compares two binary numbers, indicating if they are equal, greater than, or.
Introduction of kmap karnaugh map in many digital circuits and practical problems we need to find expression with minimum variables. The karnaugh map can also be described as a special arrangement of a truth table. The two halves are distinguished by the fifth variable, which can be equal to 0 or 1. A five variable karnaugh map can be constructed using two four variable karnaugh maps. This very simple kmap demonstrates that an nvariable kmap contains all the combination of the n variables in the k map space. The university of texas at dallas computer science. Expression are most commonly expressed in sum of products form. Since it is difficult to draw a threedimensional object on. Karnaugh maps kmaps an nvariable kmap has 2n cells with each cell corresponding to an nvariable truth table value. Karnaugh map method or kmap method is the pictorial representation of the boolean equations.
This is a retouched picture, which means that it has been digitally altered from its original version. We fill grid of kmap with 0s and 1s then solve it by making groups. The two fourvariable maps are designated as a 0 and a 1, respectively. Karnaugh map k map can be used to minimize functions of up to 6 variables. The cells are arranged in a way so that simplification of a given expression is simply a matter of properly grouping the cells. Kmap cells are labeled with the corresponding truthtable row. Reflection map the 5 variable reflection map that uses gray code refer to section 5. Department of computer science kmaps karnaugh map is an array of cells in which each cell represents a binary value of the input variables. The product term in a canonical sop expression is called a. K map cells are labeled with the corresponding truthtable row. Since function f has 4 variables so we will create a 4 variable k map having 2 4 16 cells. Sum of products reduction using karnaugh map boolean. January 18, 2012 ece 152a digital design principles 4 reading assignment roth cont 5karnaugh maps 5.
Note that the position for a cell may change if the labeling is different. Imagine a map like those drawn previously, with an identically sized map stacked directly on top of the other. Modeling, synthesis, and simulation using vhdl book. However, the real challenge is 5 and 6 variable kmaps. There are two possibilities of grouping 16 adjacent min terms. K map is directly applied to twolevel networks composed of and and or gates. The diagram below illustrates the correspondence between the karnaugh map and the truth table for the general case of a two variable problem. Karnaugh map or kmap is an alternative way to write truth table and is used for the simplification of boolean expressions. For sop put 1s in blocks of kmap respective to the minterms 0s elsewhere. Create the kmap, entering a 1 or 0 in each square according to the desired logic function. Gate input cost karnaugh maps 4variable examples kmap with dont care kmap pos forms 5variable examples kmap. We have mentally stacked the four sub maps to see the group of 4cells corresponding to out cf a magnitude comparator used to illustrate a 6variable kmap compares two binary numbers, indicating if they are equal, greater than, or less than each other on three respective outputs.
Each cell of an nvariable kmap has n logically adjacent cells. Karnaugh map or kmap is introduced by a telecom engineer, maurice karnaugh at bell labs in 1953, as a refined technique of edward veitchs veitch diagram and it is a method to simplify or reduce the complexities of a boolean expression. Kmap is used for minimization or simplification of a boolean expression. A function f which has maximum decimal value of 15, can be defined and simplified by a 4 variable karnaugh map. In the above all k maps, we used exclusively the min. If variable x has value 0 for some squares in the group and value 1 for the others, then that variable does not appear in the sum term. Solving 5 variables karnaugh map grouping electrical. If variable x has value 0 for all squares in the group, then the literal x is in the sum term. Aug 08, 2015 the following table shows the positions of all the possible outputs of 2 variable boolean function on a k map. Any boolean expression or function comprising of 5 variables can be solved using the 5 variable kmap. A kmap is a truth table graph, which aids in visually simplifying logic. The way the kmap works is by grouping the numbers that their binary representation has a hamming distance 1 only 1 bit difference in the image you posted. Five variable karnaugh map there are several different formats of a 5 variable k map of which we discuss two most popular ones in this book.
Drawing kmaps terms which differ in only one variable are placed adjacent to each other. For any entered variable kmap, thinking of or actually sketching the submaps can help identify the correct encoding for the entered variables. The older version of the five variable k map, a gray code map or reflection map, is shown above. The karnaugh map km or k map is a method of simplifying boolean algebra expressions. Thus, the map a 0 represents cells designated from 00000. The answer is no more than six inputs for most all designs, and five inputs for the average logic design. If variable x has value 1 for all squares in the group, then the literal x is in the sum term. A graphical technique for simplifying an expression.
Lecture 6 karnaugh maps kmaps kmaps with dont cares 2 karnaugh map k map flat representation of boolean cubes easy to use for 2 4 dimensions harder for 5 6 dimensions virtually impossible for 6 dimensions use cad tools help visualize adjacencies onset elements that have one variable changing are adjacent 3 karnaugh map. Lets first look why a kmap is not practical for functions with more than 4 variables. Maurice karnaugh introduced it in 1953 as a refinement of edward veitchs 1952 veitch chart, which actually was a rediscovery of allan marquands 1881 logical diagram aka marquand diagram but with a focus now set on its utility for switching circuits. Minimization of boolean functions using karnaugh maps. We can minimize boolean expressions of 3, 4 variables very easily using kmap without using. A threevariable karnaugh map for a threevariable expression with inputs x, y, z, the arrangement of minterms is more tricky. In above boolean table, from 0 to 15, a is 0 and from 16 to 31, a is 1. However, the real challenge is 5 and 6 variable k maps.
In the above all kmaps, we used exclusively the min. A kmap can be thought of as a special version of a truth table. Now fill the cell marked with subscript 0,1,2,4,5,7,10 and 15 with value 1 as we are dealing with sum of products sop. Note that truth table row numbers can be mapped to cells in the submaps by reading the kmap index codes, starting with the supermap code, and appending the submap code. Steps to solve expression using kmapselect kmap according to the number of variables.
Again, as we did with 3 variable k map, carefully note the numbering of each cell. Another way to label the kmap use whichever you like. A karnaugh map or k map is a graphic means of depicting boolean functions. The algebraic approach we have used previously is also used to analyze complex. Lets first look why a k map is not practical for functions with more than 4 variables. If v0, then 5 variable kmap becomes 4 variable kmap. I have an 8 variable input, and i remember that i should try and make the selections a big. The way the k map works is by grouping the numbers that their binary representation has a hamming distance 1 only 1 bit difference in the image you posted. Now fill the cell marked with subscript 0,1,2,4,5,7,10 and 15 with value 0 as we are dealing with product of sums pos. Now fill the cell marked with subscript 0,1,2,4, 5,7,10 and 15 with value 0 as we are dealing with product of sums pos.
The four variable function f is given in terms of minterms as f. Karnaugh map location of minterms in kmaps simplification using. There is only one possibility of grouping 32 adjacent min terms. A function f which has maximum decimal value of 31, can be defined and simplified by a 5 variable karnaugh map.
Difficult to tell when you have arrived at a minimum solution. Ececoe 02 2 karnaugh maps karnaugh map definitions a karnaugh map is a twodimensional truthtable. Previous gate questions on kmap, sop and pos expressions. Since function f has 4 variables so we will create a 4 variable kmap having 2 4 16 cells. Kmap cells are arranged such that adjacent cells correspond to truth rows that differ in only one bit position logical adjacency.
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