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UNIT II Gate Level Minimization

UNIT II Gate Level Minimization The map method , four variable & Five variable K-map, POS & SOP Simplification, Don’t care conditions, NAND & NOR Implementation, Other two level Implementation, Ex-or Function

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UNIT II Gate Level Minimization

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  1. UNIT II Gate Level Minimization The map method, four variable & Five variable K-map, POS & SOP Simplification, Don’t care conditions, NAND & NOR Implementation, Other two level Implementation, Ex-or Function Tabular Method- Simplification of Boolean function using tabulation Method.

  2. Gate Level Minimization Introduction • The simplification of Boolean function is very important as it saves the hardware required and hence the cost for design of specific Boolean function. • Previously we have seen the solving of Boolean expression using Boolean laws , rules and theorems which is some what difficult.

  3. In order to overcome this we go for Map Method • The map method gives a systematic approach for simplifying a boolean expression. • The map method was first proposed by Veitch and modified by Karnaugh hence it is known as Veitch Diagram or Karnaugh map. And simply it is called as K-map.

  4. K-map: • The k-map is a chart or a graph , composed of an arrangement of adjacent cells, each representing a particular combination of variables in sum or product forms. • K-map consists of boxes called cells . Each cell represents one of the 2n possible products that can be formed from n variables. • Thus a 2-variable map consists of 22=4 cells, • 3-variable map consists of 23=8 cells, • 4-variable map consists of 24=16 cells.

  5. Example 1. 2-variable map consists of 22=4 cells 2-variable Karnaugh maps are trivial but can be used to introduce the methods you need to learn. The map for a 2-input OR gate looks like this: A 0 1 B 1 0 A 1 1 1 B A+B

  6. Example: Y=A’B’C+A’BC =A’C 2. 3-variable map consists of 23=8 cells, BC 01 A 00 11 10 Grouping: 1.Pair(2) 2.Quad(4) 3.Octet(8) 0 1

  7. 3. 4-variable map consists of 24=16 cells

  8. 4. 5-variable map consists of 25=32cells A=0 A=1 01 DE DE 00 11 10 00 01 11 10 BC 00 BC 00 01 01 11 11 10 10 52

  9. 4. 5-variable map consists of 25=32cells

  10. 5VariableK-map CDE AB 000001011010110 111 101100 00 01 11 10 0 7 5 2 6 4 1 3 12 8 15 11 9 13 10 14 28 29 24 25 31 27 26 30 20 16 17 23 21 19 18 22 51

  11. Note: In the k-map the rows and columns are represented in the gray code for short hand notation instead of variables.

  12. Don’t care conditions • In some logic circuits certain input conditions never occur , there fore the corresponding output never appears. • In such a cases output is not defined . It can be either High or Low . • These output levels are indicated by X or d in the truth tables and are called don’t care outputs or don’t care conditions or incompletely specified functions.

  13. Describing of Incomplete specified Boolean Functions • For incomplete Boolean functions we use additional term to specify don’t care conditions in the original expression. • Ex: f(A,B,C)=Σm(0,2,4)+d(1,5) • f(A,B,C)=ΠM(2,5,7)+d(1,3)

  14. Minimization of Incomplete specified Boolean Functions • Note: • (i) A circuit designer is free to make the output for any don’t care condition either a “0” or “1” in order to produce the simplest output expression. • (ii) We can’t group only don’t care variables.

  15. Limitations of k-map • The k-map simplification is convenient as long as the number of variables does not exceed 5 or 6. • As the no . of variables increases it is difficult to obtain minimal expression. • It is almost impossible task for 7,8 ..... Variables. • (ii) k-map simplification is manual technique and simplification process heavily depends on the human abilities.

  16. Quine-Mc cluskeyor Tabular method: • To meet the disadvantages of k-map simplification Quine-Mc cluskey gives a new method of simplification of large boolean functions. It is known as Tabular method or Quine-Mc clskey method. • Note: The basic principal of this method is observing one bit differ positions in the binary formation of given minterms.

  17. Algorithem to generate prime implicants: • List all the minterms in binary. • Arrange the minterms according to no . of ones. • Compare each binary no with every term, they differ only one position . Put a check mark( ) and copy the term in the next Colum with “-” in the position that they differ. • Apply the same process for the resultant Colum and continue the cycles until no further elimination of literals.

  18. Prime implicant chart: • To obtain minimum no of prime implicants (Essential prime implicants) we are using the prime implicant chart. • Search for single dot column and those prime implicants by putting cheek mark. • Search for multi dot columns one by one if the corresponding minterm is already taken in the final expression ignore the minterm and go to next multi dot column . List all the minterms in binary.

  19. List all the prime implicants. • Select the minimum no of prime implicants which cover all the minterms.

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