When a POS expression is mapped, it can easily be converted to the equivalent SOP form directly from the Karnaugh map. Also, given a mapped SOP expression, an equivalent POS expression can be derived directly from the map. This provides a good way to compare both minimum forms of an expression to determine if one of them can be implemented with fewer gates than the other.
Converting From SOP to POS Form Using K-map
The procedure is as follows:
M E T H O D O L O G Y
1. Map the SOP expression
2. For an SOP expression, all the cells that do not contain 1s contains 0s. Group all the cells containing 0.
3. Write sum term for each of the group. This will provide the minimized POS expression.
Converting From POS to SOP form using K-map
It is also possible to employ a K-map to convert an equation from it’s POS form to its SOP form. The procedure is as follows:
M E T H O D O L O G Y
1. Map the POS expression
2. For a POS expression, all the cells that do not contain 0s contains 1s. Group all the cells containing 1.
3. Write product term for each of the group. This will provide the minimized SOP expression.
EXAMPLE 4.27
Write down the simplified Boolean expression in (a) sum of products form and (b) products of sums form for:
(i) Y A B C D_ , , , i=Sm 1 4 6 9 10 11 14 15_ , , , , , , , i
Chapter 4 Minimization Techniques Page 231
SOLUTION :
(i) Given Boolean expression is
Y =Sm 1 4 6 9 10 11 14 15_ , , , , , , , i
Given function is in standard SOP form. To represent it on K-map, we place 1s in the cells corresponding to minterms present in the function. Remaining cells are filled with 0s as shown.
(a) To write the simplified expression in sum of products form, we have to form groups of adjacent 1’s as shown in the K-map. Note that there are 2 pairs and 1 quad of adjacent 1’s in the given K-map.
(b) To write the simplified expression in product of sums form, we have to form groups of 0’s as shown. There are 4 possible pairs of adjacent 0’s. The simplified Boolean expression is
, , ,
Y A B C D_ i=_A+ +B C Bi_ + +C D Ai_ + +B C Ai_ + +B Di
(ii) Given Boolean expression is
, , ,
Y A B C D_ i =PM 0 1 3 5 6 7 9 10 11 12 13 15_ , , , , , , , , , , , i
Given function is in standar POS form. To represent this on K-map, place 0’s in the cells corresponding to maxterms present in the function. Remaining cells are filled with 1s as shown in side column. (a) To write the simplified expression in sum of products form, we have to form groups of adjacent 1’s as shown in Figure below.
The simplified Boolean expression:
, , ,
Chapter 4 Minimization Techniques Page 233
EXAMPLE 4.28
Minimize the following function using K-map (a) f A B C D_ , , , i=Sm_0 1 2 5 8 15, , , , , i+d_6 7 10, , i
(b) f A B C D_ , , , i=Sm_2 8 9 10 12 13, , , , , i+d_7 11, i
(c) f A B C D_ , , , i=Sm_7 9 11 12 13 14, , , , , i+d_3 5 6 15, , , i
SOLUTION :
(a) f A B C D_ , , , i =Sm_0 1 2 5 8 15, , , , , i+d_6 7 10, , i
The function is given in terms of minterms and don’t care conditions. K-map representation of the given function is shown. To obtain a minimize SOP expression, we form groups of adjacent 1’s and also include don’t cares if they can be used in grouping. There are 2 pairs and 1 quad of adjacent 1’s and don’t cares as shown.
Minimized POS expression
, , ,
f A B C D_ i =B D+A C D+BCD
(b) f A B C D_ , , , i =Sm_2 8 9 10 12 13, , , , , i+d_7 11, i
The K-map for the given function is shown below. To obtain a minimize SOP expression, we form groups of adjacent 1’s and also include don’t cares if they can be used in grouping. There is 1 pair and 1 quad of adjacent 1’s and don’t cares as shown.
Minimized SOP expression.
, , ,
f A B C D_ i =AC+BCD
Alternate Grouping:
If we group adjacent 1’s don’t care in another way as shown below, we would not get the minimized function.
, , ,
f A B C D_ i =AC+BCD+AB
(c)f A B C D_ , , , i =Sm_7 9 11 12 13 14, , , , , i+d_3 5 6 15, , , i
K-map for the given function is as shown. To obtain a minimize SOP expression, we form groups of adjacent 1’s and also include don’t
Page 234 Minimization Techniques Chapter 4
Digital Electronics by Ashish Murolia and RK Kanodia For More Details visit www.nodia.co.in cares if they can be used in grouping. There are 3 quads of adjacent
1’s and don’t cares as shown. Minimized SOP expression
, , ,
f A B C D_ i =AB+AD+CD
There may be another way of grouping, out no. of terms will remain same in the minimized expressions.
EXAMPLE 4.29
Minimize the following function using K-map. (a) f A B C D_ , , , i=PM_0 8 10 11 14, , , , ,i+d_6i
(b) f A B C D_ , , , i=PM_2 8 11 15, , , i+d_3 12 14, , i
(c) f A B C D_ , , , i=PM_0 2 6 11 13 15, , , , , i+d_1 9 10 14, , , i
SOLUTION :
(a) f A B C D_ , , , i =PM_0 8 10 11 14, , , , i+d_6i
The function is given in terms of maxterms and don’t care condition. The binary values of maxterms and don’t care terms appearing in the function are as below.
M0=A+ + +B C D 0000_ i M8=A+ + +B C D 1000_ i M10=A+ + +B C D 1010_ i M11=A+ + +B C D 1011_ i M14=A+ + +B C D 1110_ i M6=A+ + +B C D 0110_ i Given function is in standard POS form with don’t cares. To represent this on K-map, we place 0s in the cells corresponding to above maxterms and X in the cell corresponding to don’t cares as shown. To obtain a minimized POS expression, we form groups of adjacent 0’s and also consider don’t cares if they are useful in grouping. There are 3 possible pairs of adjacent 0’s including some don’t cares, as shown in the K-map.
Minimized POS expression.
, , ,
f A B C D_ i =_B+ +C D Ai_ + +B C Ai_ + +C Di
Alternate Grouping:
There could be another way of grouping adjacent 0’s and don’t care, but the no. of terms will remain same in the minimized POS expressions.
Minimized POS expression
, , ,
f A B C D_ i =_B+ +C D Bi_ + +C D Ai_ + +B Ci
(b) f A B C D_ , , , i =PM_2 8 11 15, , , i+d_3 12 14, , i
First we represent the given POS function on K-map. The binary values of maxterms and don’t care terms appearing in the function
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EXAMPLE 4.30
Write down the simplified Boolean expression in (a) Sum-of-Products form and (b) Product-of-sum for the following functions.
(i) f A B C D_ , , , i=Sm_6 7 9 10 13, , , , i+d_1 4 5 11 15, , , , i
(ii) f A B C D_ , , , i=PM_0 3 4 11 13, , , , i:d_2 6 8 9 10, , , , i
SOLUTION :
(i) Given Boolean function is
, , ,
f A B C D_ i =Sm_6 7 9 10 13, , , , i+d_1 4 5 11 15, , , , i
The given function is expressed in terms of minterms and don’t care condition. We represent the given function on K-map by filling cells corresponding to minterms present in the function with 1. Cells corresponding to don’t care term are filled with X and remaining cells contain 0.
(a) To write the simplified expression in sum-of-Product form, we have to form groups of adjacent 1’s considering don’t cares also which are useful in grouping, as shown in K-map. There are 2 quads and 1 pair of adjacent 1’s including some don’t cares as shown.
Minimized SOP expression
, , ,
f A B C D_ i =AB+C D+ABC
(b) To obtain minimized expression in POS form, we form groups of adjacent 0’s along with some don’t cares which are useful in grouping. There are 2 quads and 1 pair of adjacent 0’s in the given K-map as shown.
Minimized POS expression
, , ,
f A B C D_ i =_A+B Ci_ +D Ai_ + +B Ci
(ii) f A B C D_ , , , i =PM_0 3 4 11 13, , , , i:d_2 6 8 9 10, , , , i
The given function is expression in terms of maxterms and don’t care conditions. We represent the given function on K-map by filling cells corresponding to maxterms present in the function by 0. Cells corresponding to don’t care term are filled with X and remaining cells are filled with 1s.
Chapter 4 Minimization Techniques Page 237
(a) To write the simplified expression in sum-of-product form, we have to make groups of adjacent 1’s considering don’t cares also which are useful in grouping, as shown in K-map. There are 2 quads and 1 pair of adjacent 1’s including some don’t cares as shown.
Minimized SOP expression
, , ,
f A B C D_ i =AD+BC+A C D
(b) To obtain minimized expression in POS form, we form groups of adjacent 0’s along with some don’t cares which are useful in grouping. There are 2 quads and 1 pair of adjacent 0’s in the given K-map as shown.
Minimized POS expression.
, , ,
f A B C D_ i =_A+D Bi_ +C Ai_ + +C Di