This article provides insight into the Karnaugh map K-map Boolean algebraic simplification technique via a few examples. It also includes a brief note on the advantages and the disadvantages of K-maps. Digital electronics deals with the discrete-valued digital signals. In general, any electronic system based on the digital logic uses binary notation zeros and ones to represent the states of the variables involved in it.
Thus, Boolean algebraic simplification is an integral part of the design and analysis of a digital electronic system. Although Boolean algebraic laws and DeMorgan's theorems can be used to achieve the objective, the process becomes tedious and error-prone as the number of variables involved increases. This necessitates the use of a suitable, relatively-simple simplification technique like that of Karnaugh map K-mapintroduced by Maurice Karnaugh in The K-map method of solving the logical expressions is referred to as the graphical technique of simplifying Boolean expressions.
K-maps are also referred to as 2D truth tables as each K-map is nothing but a different format of representing the values present in a one-dimensional truth table. Further, each cell within a K-map has a definite place-value which is obtained by using an encoding technique known as Gray code. The specialty of this code is the fact that the adjacent code values differ only by a single bit. This means that each K-map cell can be addressed using a unique Gray Code-Word.
These concepts are further emphasized by a typical celled K-map shown in Figure 1, which can be used to simplify a logical expression comprising of 4-variables A, B, C and D mentioned at its top-left corner.
Here the rows and the columns of the K-map are labeled using 2-bit Gray code, shown in the figure, which assigns a definite address for each of its cells.
We'll start with a given problem statement. Design a digital system whose output is defined as logically low if the 4-bit input binary number is a multiple of 3; otherwise, the output will be logically high.Lista loteria nacional de hoy
The output is defined if and only if the input binary number is greater than 2. The first step in designing any digital system is to have a clear idea of the variables involved in the process, along with their state-values. Further, depending on the problem statement, we have to arrive at the number of output variables and their values for each and every combination of the input literals, which can be conveniently represented in the form of a truth table.
Note that, in addition to the input and output columns, the truth table also has a column which gives the decimal equivalent of the input binary combination, which makes it easy for us to arrive at the minterm or maxterm expansion for the given problem.Esp32 dht11
Thus for the given example. However, sometimes the logical expression which is to be simplified might be directly given in terms of SOP or POS forms. In this case, the requirement for the truth table can be overlooked provided that we express the given expression in its canonical form, from which the corresponding minterms or maxterms can be obtained. From Step 1, we know the number of input variables involved in the logical expression from which size of the K-map required will be decided.
Further, we also know the number of such K-maps required to design the desired system as the number of output variables would also be known definitely. The procedure is to be repeated for every single output variable.
K-map simplification can also be referred to as the "simplification by grouping" technique as it solely relies on the formation of clusters. This can be done by expressing the bits which are common amongst the Gray code-words which represent the cells contained within the considered group.Jacknet rgb sync evga
Another way to describe the process of obtaining the simplified logical expression for a group is to eliminate the variable s for which the corresponding bits appear within the group as both 0 and 1. Finally, all these group-wise logical expressions need to be combined appropriately to form the simplified Boolean equation for the output variable.
The same procedure must be repeated for every output variable of the given problem. Having obtained the simplified logical expression, we can decide on the type and the number of gates required to realize the expected logic for every output bit, which further results in the complete design of the desired system. The digital system designed to realize the full adder in terms of sum and carry outputs in POS form is shown by Figure Karnaugh map method or K-map method is the pictorial representation of the Boolean equations and Boolean manipulations are used to reduce the complexity in solving them.
As it is evaluated from the truth table method, each cell in the K-map will represent a single row of the truth table and a cell is represented by a square. The cells in the k-map are arranged in such a way that there are conjunctions, which differs in a single variable, are assigned in adjacent rows. The K-map method supports the elimination of potential race conditions and permits the rapid identification.
By using Karnaugh map technique, we can reduce the Boolean expression containing any number of variables, such as 2-variable Boolean expression, 3-variable Boolean expression, 4-variable Boolean expression and even 7-variable Boolean expressions, which are complex to solve by using regular Boolean theorems and laws.
It will look like see below image. The possible min terms with 2 variables A and B are A. The following table shows the positions of all the possible outputs of 2-variable Boolean function on a K-map. When we are simplifying a Boolean equation using Karnaugh map, we represent the each cell of K-map containing the conjunction term with 1.
After that, we group the adjacent cells with possible sizes as 2 or 4. In case of larger k-maps, we can group the variables in larger sizes like 8 or The groups of variables should be in rectangular shape, that means the groups must be formed by combining adjacent cells either vertically or horizontally.
Diagonal shaped or L-shaped groups are not allowed. The following example demonstrates a K-map simplification of a 2-variable Boolean equation. Here the lower right cell is used in both groups. After grouping the variables, the next step is determining the minimized expression. By reducing each group, we obtain a conjunction of the minimized expression such as by taking out the common terms from two groups, i.
For a 3-variable Boolean function, there is a possibility of 8 output min terms. The general representation of all the min terms using 3-variables is shown below. A typical plot of a 3-variable K-map is shown below. It can be observed that the positions of columns 10 and 11 are interchanged so that there is only change in one variable across adjacent cells. This modification will allow in minimizing the logic.
Up to 8 cells can be grouped in case of a 3-variable K-map with other possibilities being 1,2 and 4. The largest group size will be 8 but we can also form the groups of size 4 and size 2, by possibility.
In the 3 variable Karnaugh map, we consider the left most column of the k-map as the adjacent column of rightmost column. So the size 4 group is formed as shown below. So the group of size 4 is reduced as the conjunction Y. To consume every cell which has 1 in it, we group the rest of cells to form size 2 group, as shown below. The 2 size group has no common variables, so they are written with their variables and its conjugates. In this equation, no further minimization is possible.
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Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. The K map does not necessarily fail for higher dimensions. The problem is that it is so difficult to visualize for more than five variables. A 4 variable K-map is 2 dimensional and easy to visualize.
A 5 variable is three dimensional, but is still manageable from a visualization standpoint, because the 2 states of the 5th variable only require visually moving from one plane to the next, without moving in the x or y directions of either plane. Just getting equations correct with more than 5 variables is difficult enough using the K map, much less considering an optimum set of terms "core" prime implicants and "choice" prime implicants.
A truth table is a list of every possible output for every possible input. As you can imagine, these can get quite large. From the truth table, you can directly get a boolean expression via sum of products, but this is also a rather large expression.
Karnaugh maps take truth tables and provide a visual way to produce a much simpler formula for expressing the same logic. That said, any time you produce a truth table, you can and probably should produce a K-Map to simplify the logic. A k map use reduces the number of logic gates to be used to solve an logic expression and hence reduces the error.
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Or are there special occasion where I should use one over the other? HostileFork says dont trust SE 30k 7 7 gold badges 85 85 silver badges bronze badges.Vxrail g series
It's been years but I'll guess : I think a Karnaugh map helps lead to minimal required logic for the problem. It's a special case of a truth table. Active Oldest Votes. Chris Chris 3 3 silver badges 8 8 bronze badges. The disadvantage of k map : It is not suitable for computer reduction. It is not suitable when the number of variables involved exceed four.
Care must be taken to field in every cell with the relevant entry, such as a 0, 1 or don't care terms. Iwo Kucharski 3, 2 2 gold badges 42 42 silver badges 60 60 bronze badges.
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The Karnaugh Map Boolean Algebraic Simplification Technique
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K-Map (Karnaugh Map)
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