CHAPTER2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, KMap and QuineMcCluskey


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1 CHAPTER2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, KMap and QuineMcCluskey 2. Introduction Logic gates are connected together to produce a specified output for certain specified combinations of input variables, with no storage involved, the resulting circuits is called combinational logic. In combinational logic the output variable are the all times dependent on the combination of input variables. A combinational circuit consists of input variables, logic gates and output variables the logic gates accept signals generate output signals. This process transforms binary information from the given input data to the required output [8]. The combinational circuits accept in input binary and generate output variables depending on the logical combination. Logic minimization uses a variety of techniques to obtain the simplest gatelevel implementation of a logic function. The heart of digital logic design is the Boolean algebra (Boole, 954). A few decades later C.E.Shannon showed how the Boolean algebra can be used in the design of digital circuits (Shannon, 938). Using Boolean laws it is possible to minimize digital logic circuits (Huntington, 94). Since minimization with the use of Boolean laws is neither systematic nor suitable for computer implementation, a number of algorithms were proposed in order to overcome the implementation issue [47][48]. Karnaugh proposed a technique for simplifying Boolean expressions using an elegant visual technique, which is actually a modified truth table intended to allow minimal sumof products (SOP) and 6
2 productofsums (POS) expressions to be obtained (Karnaugh, 953). The Karnaugh Map (KMap) based technique breaks down beyond six variables. The Quine McCluskey (Q M) method is a computerbased technique for simplification and the modified QuineMcCluskey (M QM) method is a very simple and systematic technique for minimizing Boolean functions. To minimize a Boolean expression by simplifying the logic function, we can reduce the original number of digital components (gates) required to implement digital circuits. Therefore by reducing the number of gates, the chip size and the cost will be reduced, and the speed will be increased. Boolean algebra is constructed by connecting the Boolean constants and variables with the Boolean operations. These Boolean expressions are also known as Boolean formulas and we use Boolean expression to describe Boolean function. If the Boolean expression (A + B') C is used to describe the function f, then it is written as f (A, B, C) = (A + B') C or f = (A + B') C Based on the structure of Boolean expression it can be categorized in different formulas. One such categorization is the normal formulas. Let us consider the four variable Boolean functions. f (A, B, C, D) = A + B'C + ACD' In this Boolean function the variables are appeared either in a complemented or an uncomplemented form. Each occurrence of variables in either a complement or an uncomplimentary form is called a literal. Thus the above function consists of six literals they appear in the product of the terms. A product term is defined as either a literal or a product of literals three product terns namely AB'C and ACD' function. 7
3 f (A, B, C, D) = (B +D')(A + B' + C)(A'+ C) The above Boolean function consists of seven literals here they appear in the sum terms. A sum term is defined as either a literal a sum of literals. Three sum terms namely (B + D'), (A+ B'+C) and (A'+ C) the literals can be arranged in two ways Sum of product form (SOP) and Product of sum form (POS) 2.2 Sum of product forms The word sum and product are derived from the symbolic representation of the OR and AND function by + and. But we realize that these are not arithmetic operation in the usual sense. A product term is any group of literals that are AND together for example A, XY and so on. A sum term is any group of literals that are OR together. Such as A+B+C, X+Y and so on.. f (A, B, C) = A + A B'C' 2. f (P, Q, R, S) = P'Q + Q R +R S Each of these sums of products expressions consists of two or more product terms (AND) that are OR together each product tern consists of one or more literals appearing in either complemented or uncomplemented form. For example in the sum of products expression A + A B'C' the first product term contains literals A B and C in their uncomplemented form. The second product term contains literals B and C in their complemented form. The sum of product form is also known as disjunctive normal form or disjunctive normal formula. 8
4 2.3 Product of Sum forms A product of sums is any groups of sum terms AND together some examples of this form are:. f (A, B, C) = (A+ B) (B'+ C) 2. f (P, Q, R, S) = (P + Q). (R + S'). (P+ S) Each of the product sums expression consists of two or more sum terms (OR) that are ANDed together each sum term consists of one or more literals appearing in either complemented or uncomplemented form. The product of sum form is also known as conjunctive normal form or conjunctive normal formula Standard sop form or minterm canonical forms We can realize that in the sop from all the individual terms do not involve al literals. For example in expression AB +A' the first product terms do not contain literals C. If each term in SOP form contains all the literals then the sop form known as standard or canonical sop form. In the expression A' + A+ A' + AB'C, all the literals are present in each product terms in other words we can say that a sum of products is a standard sum of products if every product term involves every literals or its complement. One standard sum of products expression is: f (A, B, C) = AB'C+ A + A'' 2.4. Standard pos form or maxterm canonical forms If each term in pos form contains all the literals then the pos form is known as standard or canonical pos form. Each individual term in the standard pos is called maxterm. Therefore canonical pos form is also known as maxterm canonical form. In 9
5 other words we can say that a product of sums is a standard or canonical product of sums if every sum term involves every literals or its complement. One standard product of sums expression is f (A, B, C) = (A+B+C) + (A+B'+C) 2.5 Converting expressions in to standard sop or pos forms Sum of products form can be converted to standard sum of products by ANDing the terms in the expression with terms formed by ORing the literal and its complement which are not present in that term. For example for a three literal expressions with literals A, B and C, if there is a term AB, and C is missing, then we form term (C+C) and AND it with AB therefore we get AB (C+C') = A + A' Steps to convert sop to standard SOP form: Step : finding the missing literal in each product term. Step 2: AND each product term having missing literals with term form by ORing the literals and its complement. Step 3: Expanding the term by applying distributive law and reorder the literals the literals in the product terms. Step 4: Reduce the expression by omitting repeated product terms if any because A + A = A Illustration: Converting the given expression in standard sop form: 2
6 f (A, B, C) = AC + AB + Literals A is missing Literals C is missing Literals B is missing AND product term with missing literals + its complement f (A, B, C) = AC(B+B') + AB(C+ C') +(A+ A') Expand: f (A, B, C) =ACB + ACB' + A + A'+ A + A' RECORDER: f (A, B, C) = A + AB'C + A+ A'+A + A' 2.6 M Notations (Minterms and Maxterms) Each individual term in standard SOP form is called minterm and each individual term in standard POS form is called maxterm the concept of minterms and maxterms allows us to introduce a very convenient shorthand notation to express logical function it gives the minterm and maxterm for a three literals variable logical function. Where the number of minterm as well as maxterms is 8. In general, for an n  variables logical function there are n minterm and an equal number of maxterm. As shown in Table 2. each minterm is represented by m i and each maxterm is represented as M i, where the subscript i is the decimal number equivalent of the natural binary number. With these shorthand notations logical function can be represented as follows: 2
7 . f (A, B, C) = A'B'C'+A'B'C+A'+A' = m o + m + m 3 + m 6 = m (,, 3, 6) Variables A B C Minterm(mi) A' B' C' = m o A' B' C = m A' B C' = m 2 A' B C = m 3 A B' C' = m 4 A B' C = m 5 A B C' = m 6 A B C = m 7 Maxterm(Mi) A + B + C = M A + B + C' = M A + B' + C = M 2 A + B' + C' = M 3 A' + B + C = M 4 A' + B + C' = M 5 A' + B' + C = M 6 A' + B' + C' = M 7 Table 2. Minterms and Maxterms for three variables 2. f (A, B, C) = (A + B + C') (A + B'+ C') (A' + B' + C) = M + M 3 + M 6 = M (, 3, 6,) The logical expression can be represented in the truth table form. To write logic expression in standard SOP or POS form corresponding to a given truth table. The logic expression corresponding to a given truth table can be written in a standard sum of products form by writing one product term for each input combination that produces an output of. These product terms are together to create the standard sum of products [23][24]. The product terms are expressed by writing complement of a variable when it appears as an input, and the variable itself appears as an input. Considering, the following truth table 2.2. The product corresponding to input 22
8 23 combination is A'', the product corresponding to input combination is A', and product corresponding to input combination is A. Thus the standard sum of product form is F (A, B, C) = A'' + A'+A' = m 2 +m 3 +m 6 A B C Y Table No.2.2. Minterms and Maxterms for three variables (m 2,m 3 and m 6 ) A B C Y Table No.2.3 Minterms and Maxterms for three variables (M 2 and M 5 )
9 The logic expression corresponding to a truth table can be also written in a standard product of sums form by writing one sum term output is. The sum terms are expressed by writing complement of a variable when it appears as an input and the variables itself when it appears as an input is as shown in table 2.3. The sum corresponding to input combination is A+ B'+C thus the standard product of sums corresponding to input is A'+B+C'. Thus the standard product of sums form is as follows, f (A, B, C) = (A+B'+C) (A'+B+C') = M 2 + M Karnaugh Map minimization The simplification or minimization of any digital circuit is an important activity in digital circuit design. To simplify the circuit, the designer tries to find another circuit that produces the same output as the original but with less number of gates. The main objective of this process is to keep the number of digital gates as minimum as possible and thus to get minimal cost solution. There are various methods of simplification such as Boolean algebra, Karnaugh maps, Tabulation method, Computer Aided Design etc. All these methods use the simplification of Boolean function that represents the digital logic. Another important point is that the KMap simplification is manual technique and simplification process is heavily depends on the human abilities. To meet this requirement, W. V. Quine and E.J. McCluskey developed an exact tabular method to simplify the Boolean expressions. This method is called the QuineMcCluskey or tabular method. 24
10 Maurice Karnaugh developed the Karnaugh map in 953. This technique is quite easy and fast in comparison with Boolean algebra. Karnaugh maps work well for up to six input variables. A Karnaugh map consists of many arrays of rectangles or boxes arranged in rows and columns. The size of the Karnaugh map with n Boolean variables is equal to 2 n. The size for maps of 2 variables is a 2x2 map (four boxes), for 3 variables it is a 2x4 map, and for 4 variables it is a 4x4 map and so on. The Boolean variables are arranged in an order according to the principles of gray code where only one variable changes in adjacent squares. Each square represents a minterm (sometimes a maxterm) corresponding to the truth table. A minterm is a Boolean expression consisting of a product term of variables (or their complimented form). The minterms are identified by associating numbers to them like m, m,.m n. For simplifying an input expression, the adjacent minterms are identified and a group of 2 (Pair), 4 (Quad) or 8 (Octet) adjacent minterms are formed. The minterms can only form a group if they are adjacent horizontally and vertically and not diagonally. The groups should be as large as possible and overlapping of any minterm on two or more groups is allowed. Similarly the wrap around minterms is also allowed for forming a group. If a term and its compliment both appear in a group, delete both from the resultant product term. Finally the Boolean expression of the remaining terms can be obtained. For simplification of Boolean expression by Boolean algebra we need better understanding of Boolean laws, rules and theorems and during the process of simplification we have to predict each successive step. Boolean algebra alone is the simplest possible expression on the other hand the map method gives us a systematic 25
11 approach for simplifying a Boolean expression and the map method first proposed by veitch and modified by karnaugh hence it is known as the veitch diagram or the karnaugh map[48]. Product terms are assigned to the cells of a karnaugh map by labeling each row and each column of the map with a variable and its complement or with a combination of variables and complement. The product term corresponding to a given cell is then the product of all variables in the row and column, where the cells is located to label the rows and columns of a, 2, 3 and 4 variable map and the product terms corresponding to each cell. B' B B'C' B'C ' A' A' A' A'B' A'B A' A'B'C' A'B'C A' A'' A A A AB' AB A AB'C' AB'C A A'  Variable map (2 Cells) 2 Variable map (4 Cells) C'D' C'D CD CD' 3 Variable map (8 Cells) A'B' A'B'C'D' A'B'C'D A'B'CD A'B'CD' A'B A''D' A''D A'D A'D' AB A'D' A'D AD AD' AB' AB'C'D' AB'C'D AB'CD AB'CD' 4 Variable map Fig. 2. Outlines of, 2, 3 and 4 variable maps with product terms 26
12 From the above Fig 2., when we move from one cell to next along any row or from one cell to the next along any column only one variable in the product term changes. The only change that occurs in moving along the bottom row from AB to AB' is the change from B to B'. Similarly the only change that occurs in moving down the right columns along each row and column to the single change rule. The gray code has same properties; hence gray code is used to label the rows and columns of KMap. 2.8 Grouping cells for simplification In the last section we have discussed representation of Boolean function on the karnaugh map. We have also seen that minterms are marked by s and maxterms are marked by s. Once the Boolean function is plot on the karnaugh map we have to use grouping technique to simplify the Boolean function. The grouping is nothing but combining terms in adjacent cells. Two cells are said to be adjacent if they conform the single change rule. The simplification is achieved by grouping adjacent s or s in grouping of 2I where I =,2.n and n is the number of variables. 2.9 Grouping two adjacent ones (pair) The karnaugh map contains a pair of s that are horizontal adjacent to each other. The equation shown below, the first term represents A' B' C and the second term represents A' B C. In these two terms only the B variable appears in both normal and complemented from. Thus these two terms can be combined to give a result that eliminates the B variable since it appears in both uncomplemented and complemented form. This can be proved as follows. Y = A'B'C + A'B C = A' C (B' + B) = A' C 27
13 This same principle holds true for any pair of vertically or horizontal adjacent s shows an example of two vertically adjacent s. These two can be combined to eliminate a variable since it appears in both its uncomplemented and complemented forms. This gives the result, Y= A'B C + A B C = B C A B'C' B'C ' A' A'C A Fig. 2.2 (a) Horizontal Adjacent 's A'C A B'C' B'C ' A' A Fig. 2.2 (b) Vertically Adjacent A A' B'C' B'C ' AC' A Fig. 2.2 (c) Adjacent Corners AC' 28
14 In a Karnaugh map the leftmost column and rightmost column are considered to be adjacent. Thus, the two s in these columns with a common row can be combined to eliminate one variable. This is illustrated in Fig 2.2. Here variable B has appeared in both its complemented and uncomplemented forms and hence eliminated as follows. The another illustration here two s from top row and bottom row of some column are combined to eliminate variables A, since in a KMap the top row and bottom row are considered to be adjacent. Y = A B' C' + A B C' = A C'(B' + B) = A C' CD C'D' C'D CD CD' AB A'B' A'B AB AB' B'C'D Fig. 2.3 s Group B'C'D 29
15 Y = A' B' C' D + A B' C'D = B' C' D (A' + A) = B' C' D Group A' C A B'C' B'C A' ' A Group 2 Fig. 2.4 s Group of A'C& Y = A' B' C + A'B C + A B C = A' B' C + A' B C + A' B C + A B C = A' C (B' + B) + B C (A' + A) = A' C + B C Karnaugh map has two over lapping pairs of s in the map. Fig 2.4 shows that we can share one term between two pairs. Where three groups of pairs can be formed but only two pairs are enough to include all s present the KMap in such case third pair is not required. Grouping four adjacent ones (Q quad) 3
16 In a KMap we can group four adjacent s, the resultant group is called quad. Shows the four s are horizontal adjacent vertically adjacent. A KMap contains four s in a square and they are considered adjacent to each other, also adjacent because as mentioned earlier the top and bottom rows are considered to be adjacent to each other and the leftmost and rightmost columns are also adjacent to each other. From the above KMap we can notice that when a quad is combined two variables are eliminated thereby we have following terms with 4 variables. Thus overlapping groups as mentioned earlier one term can be shared between two or more groups. 2. Simplification of sum of product expression (minimal sums) We have discussed how combination of pair s quads and octets on a KMap can be used to obtain a simplified expression. A pair of s eliminate one variable a quad of s eliminate two variables and an octets of s eliminates two variables three variables in general. When a variable appears in both complemented and uncomplemented from within a group that variables is eliminated from the resultant expression variables that are same in all with the group must appear in both final expression each group gives us a product term and summation of all product term implies the function and hence is an implicate of the function all the implicates of a function determined using a KMap are the prime implicants. From the above discussion we can outline generalized procedure to simplify Boolean expression as follows.. Plot the KMap and place s adjacent in those cells corresponding to the s in the truth table or sum of product expression place s in other cells. 3
17 2. Check the KMap for adjacent s and encircles those s which are not adjacent to any other s these are called isolated s. 3. Check for those s which are adjacent to only and which are encircle such pairs. 4. Check for quads and octets of adjacent s even if it contain some s that have already been encircled while doing this make sure that there are minimum number of groups. 5. Combine any pairs necessary to include any s that have not yet been grouped. 6. Form the simplified expression by summation product terms of all groups. To verify the procedure, we consider the following expression. Minimize the expression Y = AB'C + A'B'C + A' + AB'C' + A'B'C' The following is the method: Step : KMap to three variables and it is plotted according to the given expression. Step 2: There are no isolated s A A' B'C' B'C ' A Fig. 2.5 KMap for three variables 32
18 Step3: in the cell 3 is adjacent to in the cell. Fig 2.6 pair is combined and referred to as group. Step 4: There are no octets but there is a quad cell,, 4 and 5 from a quad this quad is combined and referred to as group 2. A B'C' B'C ' A' A A' C Fig. 2.6 Group 2 Step 5: All s have already been grouped. Step 6: Each group generates a term in the expression for Y in group B variable is eliminated A and C are eliminated and we get, A'C A B'C' B'C ' A' B' A Fig. 2.7 Group Formation Y=A' C+B' 33
19 2. Essential prime implicants After grouping the cell the sum terms which appear in the k amp are called prime implicate group it is observe that some cells may appear in only one prime implicate group while other cells may appear in more than one prime implicant group these group cells, 4, 9, appear in only one prime implant group these cells are grouped these calls are called essential cells and corresponding prime implicate are called essential prime implicant. 2.2 Simplification of product of sums expression In the above discussion we have considered the Boolean expression in sum of products from and grouped 2, 4, and 8 adjacent ones to get the simplified Boolean expression in the in the same form in practice the designer should examine both the sum of products and product of sums reduction to assertion which is more simplified we have already seen the representation of products of sums on the k amp once the expression is plotted on KMap once the expression is plotted on the KMap of zero each of zero result a sum term and it is nothing but the prime implicate. The technique for using maps for POS reduction is simple step by step process.. Plot the ka map and place s in those cells corresponding to the s in the truth table or maxterm in the products of sum expression. 2. Checking the 3. KMap for adjacent s and encircle s which are not adjacent to any other s these are called isolated s. 4. Check for those s which are adjacent to s to only other and encircle such pairs. Checking for quad and octets of adjacent s even if it contains some s that have already been encircled while doing this make sure there are minimum number of groups. 34
20 5. Combine any pairs necessary to include any s that have not yet been grouped 6. Form the simplified SOP expression for F by summing product terms of all the groups. (NOTE: The simplified expression is in the complemented form because we have grouped s to simplify the expression) 7. Use De Morgan s theorem on F to produce the simplified expression in POS form. example. To get familiar with these steps we will illustrate through the following Minimize the expression Y = (A + B + C') (A + B' + C') ( A'+ B'+ C') (A' + B + C) (A+B+C) Solution: (A + B + C') = M, ( A + B' + C') = M 3 ( A' + B' + C') =M 7 (A' + B + C) =M 4, (A + B + C) =M Step : Fig. 2.8 (a) shows the KMap for three variables and it is plotted according to given maxterms. A' B'C' B'C ' A' A Fig. 2.8 (a) KMap for three variables 35
21 Step 2: There are no isolated s. Step 3: in the cell 4 is adjacent only to and in the cell 7 is adjacent only to in the cell 3. These two pairs are combined and referred to as group and group 2 respectively shown in fig. 2.8(b). Step 4: There are no quads and octets. A A' B'C' B'C ' A Fig. 2.8 (b) Group and Group 2 Step 5: The in the cell can be combined with in the cell 3 to Form a pair. This pair is referred to as group 3. Step 6: In group and in group 2, A is eliminated, where as in Group 3 variable B is eliminated shown in fig. 2.8(c) and we get Y = + + AC A B'C' B'C A' ' A'C A B'C' Fig. 2.8 (c) Group 3 variable 36
22 Step 7: Y' = (B'C' + + A'C)' = (B'C')' ()' (A'C)' = (B' + C'') (B' + C') (A''+ C') = (B + C) (B' + C') (A + C') To directly write the expression for Y by using De Morgan s theorem for each minterm as follows: Y' = B'C' + +A'C Y= (B + C) (B' + C') (A + C') 2.3 Don t Care Conditions In some logic circuits, certain input conditions never occur, therefore the corresponding outputs never appears. In such cases the output level is not defined, it can be either HIGH or LOW. These output levels are indicated by X or d in the truth table 2.4 and are called don t care outputs or don t care conditions or incompletely specified functions. Let us see the output levels in the truth table are defined for input conditions from to. For remaining two conditions of input, output is not defined, hence these are called don t care conditions shown in this truth table 2.4. A circuit designer is free to make the output for any don t care condition either a or a in order to produce the simplest output expression. 37
23 A B C Y X X Table 2.4 Don t Care Conditions 38
24 2.3. Describing Incomplete Boolean Function We describe the Boolean function using either a minterm canonical formula or a maxterm canonical formula. In order to obtain similar type expressions for incomplete Boolean functions we use additional term to specify don t care conditions in the original expression. This is illustrated in the following examples. In expression, f (A, B, C) = M (, 2, 4) + d (, 5) minterms are, 2and 4. The additional term d (, 5) is introduced to specify the don t care conditions. These terms specifies that outputs for minterms and 5 are not specified and hence these are don t care conditions. Letter d is used to indicate don t care conditions in the expression. The above expression indicates how to represent don t care conditions in the minterm canonical formula. For example, f (A, B, C) = M (2, 5, 7) + d (, 3) Don t Care Conditions in Logic Design In this section, we see the incompletely specified Boolean function. Let us see the logic circuit for an even parity generator for 4bit D number. The Table 2.5 shows the truth table for evenparity generator. The truth table shows that the output for last six input conditions cannot be specified, because such input conditions do not occur when input is in the D form. 39
25 A B C D P Table 2.5 Don t Care Conditions in Logic Design 4
26 2.4 Limitations of Karnaugh Map The map method of simplification is convenient as long as the number of variables does not exceed five to six. As the number of variables it is difficult to make judgments about which combinations form the minimum expression. In case of complex problem with 7, 8, of even variables it is almost an impossible task to simplify expression by mapping method. 2.5 Five Variable KMap A 5Variable KMap requires 2 5 = 32 Cells, but adjacent cells are difficult to identify on a single 32cell map. Therefore, two 6cell K maps are generally used. If the variables are A, B, C, D and E two identical 6cell maps containing B, C, D and E can be constructed. One map is then used for A and the other for B as shown in Fig. 2.9(a) and Fig. 2.9(b) respectively. It is important to note that in order to identify the adjacent grouping in the five variable map, we must imagine the two maps superimposed on one another; not hinged or mirror imaged. Every cell in one map is adjacent to the corresponding cell in the other map, because only one variable changes between such corresponding cells. Thus, every row on one map is adjacent to the corresponding columns. Also, the rightmost and leftmost columns within each 6 cell map are adjacent, just as they are in any 6 cell map, as are the top and bottom rows. 4
27 A'B' DE C'D' C'D CD CD' 3 2 A'B AB AB' Fig. 2.9 (a) 6Cells Map DE C'D' C'D CD CD' A'B' A'B AB AB' Fig. 2.9 (b) 32 Cells Map 42
28 2.6 Conclusion The digital circuits can be represented and analyzed using the Boolean functions. KMap in fact a visual diagram of representing all possible ways a Boolean function may be expressed. Logic minimization uses a variety of techniques to obtain the simplest gatelevel implementation of a logic function. The heart of digital logic design is the Boolean algebra and how the Boolean algebra can be used in the design of digital circuits. 43
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