The Karnaugh Map Boolean Algebraic Simplification Technique

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The specialty of this code is the fact that the adjacent code values differ only by a single bit. The output is defined if and only if the input binary number is greater than 2.

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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. We'll start with a given problem statement.

Gray Coding Further, each cell within a K-map has a definite place-value which is obtained by using an encoding technique known as Gray code.

This means that for the K-map shown in Step 4, the overall simplified output expression is A few more examples elaborating K-map simplification process are shown below.

This article provides insight into the Karnaugh map K-map Boolean algebraic simplification technique via a few examples.

The Karnaugh Map Boolean Algebraic Simplification Technique

A typical but empty Karnaugh map with 16 cells 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. Select and Populate K-Map 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.

This necessitates the use of a suitable, relatively-simple simplification technique like that of Karnaugh map K-mapintroduced by Maurice Karnaugh in 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.

Boolean expressions for each group are to be expressed as sum-terms and not as product-terms. Obtain Boolean Expression for the Output The product-terms obtained for individual groups are to be combined to form sum-of-product SOP form which yields the overall simplified Boolean expression.

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For example in the figure shown below there are two groups with two and one number of 'ones' in them Group 1 and Group 2, respectively. Hence the next term is B. Sum-terms of all individual groups are to be combined to obtain the overall simplified Boolean expression in product-of-sums POS form.

The elements around the edges of the table are considered to be adjacent and can be grouped together. Note that the number of cells in the group must be equal to an integer power to 2, i.

K-Map Simplification - Digital Circuits Questions and Answers - Sanfoundry

Maxterm Solution of K Map The method to be followed in order to obtain simplified maxterm solution using K-map is similar to that for minterm solution except minor changes listed below. K-map cells are to be populated by 'zeros' for each sum-term of the expression instead of 'ones'.

Form the Groups K-map simplification can also be referred to as the "simplification by grouping" technique as it solely relies on the formation of clusters. In the given example: 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.

Diagonal grouping of 'ones' is not permitted. Introduction Digital electronics deals with the discrete-valued digital signals. This means that each K-map cell can be addressed using a unique Gray Code-Word.

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.

Obtain Boolean Expression for Each Group Express each group interms of input variables by looking at the common variables seen in cell-labelling. 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.

It also includes a brief note on the advantages and the disadvantages of K-maps. Grouping is to be carried-on for 'zeros' and not for 'ones'. A Typical K-Map The K-map method of solving the logical expressions is referred to as the graphical technique of simplifying Boolean expressions.

The procedure is to be repeated for every single output variable.

Grouping is to done either horizontally or vertically or interms of squares or rectangles. The number of 'ones' in a group must be a power of 2 i. 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.