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Principles of Combinational Logic Simplification and Minimization Techniques

Learn how to simplify Boolean functions using don't care conditions and minimize functions with examples and practical exercises. Explore Product of Sums simplification and Mixed Logic Combinational Circuits.

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Principles of Combinational Logic Simplification and Minimization Techniques

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  1. CHAPTER 3 Principles of Combinational Logic (Sections 3.4-3.8)

  2. Simplifying Boolean Functions • Exe: F(x,y,z)=∑(0,2,3,4,5,7) F(a,b,c,d)=∑(0,3,4,5,7,11,13,15) F(w,x,y,z)=∑(0,1,4,5,9,11,13,15) F(a,b,c,d)=∑(0,1,2,4,5,6,8,9,12,13,14) F(a,b,c,d)=∑(1,3,4,5,7,8,9,11,15) F(w,x,y,z)=∑(1,5,7,8,9,10,11,13,15)

  3. Don't Care Conditions • There may be a combination of input values which • will never occur • if they do occur, the output is of no concern. • The function value for such combinations is called a don't care. • They are denoted with x or–. Each x may be arbitrarily assigned the value 0 or 1 in an implementation. • Don’t cares can be used to further simplify a function

  4. Minimization using Don’t Cares • Treat don't cares as if they are 1s to generate PIs. • Delete PI's that cover only don't care minterms. • Treat the covering of remaining don't care minterms as optional in the selection process (i.e. they may be, but need not be, covered).

  5. cd Example ab 01 11 00 10 0 1 0 1 00 1 1 0 1 01 • Simplify the function f(a,b,c,d) whose K-map is shown at the right. • f = a’c’d+ab’+cd’+a’bc’ or • f = a’c’d+ab’+cd’+a’bd’ • The middle two terms are EPIs, while the first and last terms are selected tocover the minterms m1, m4, and m5. 0 0 d d 11 10 1 1 d d 0 1 0 1 1 1 0 1 0 0 d d 1 1 d d 0 1 0 1 1 1 0 1 0 0 d d 1 1 d d

  6. Another Example cd ab d 1 0 0 1 d 0 d • Simplify the function g(a,b,c,d) whose K-map is shown at right. • g = a’c’+ abor • g = a’c’+b’d 1 d d 1 0 d d 0 d 1 0 0 1 d 0 d 1 d d 1 0 d d 0 d 1 0 0 1 d 0 d 1 d d 1 0 d d 0

  7. Don't Care Conditions A=f(w,x,y,z)=∑(5,6,7,8,9)+ ∑d(10,11,12,13,14,15) B=f(w,x,y,z)=∑(1.2.3.4.9)+ ∑d(10,11,12,13,14,15) C=f(w,x,y,z)=∑(0,3,4,7,8)+ ∑d(10,11,12,13,14,15) D=f(w,x,y,z)=∑(0,2,4,6,8)+ ∑d(10,11,12,13,14,15)

  8. Don't Care Conditions

  9. Don't Care Conditions A=w+xz+xy B=x’y+x’z+xyz’ C=y’z’+yz D=z’

  10. Product of Sums Simplification • Use sum-of-products simplification on the zeros of the function in the K-map to get F’. • Find the complement of F’, i.e. (F’)’ = F • using DeMorgan’s Theorem.

  11. POS Example cd ab 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 0 • F’(a,b,c,d) = ab’ + ac’ + a’bcd’ • F = (a’+b)(a’+c)(a+b’+c’+d)

  12. Mixed Logic Combinational Circuits Function lable 1 F=A+B F=AB

  13. Mixed Logic Combinational Circuits

  14. Mixed Logic Combinational Circuits • P138: • Ex.3.19 • 3.7.2 Conversion to bubble logic • What is mismatch logic? • Convert to bubble logic. (P140: examples)

  15. Mixed Logic Combinational Circuits • Exe. Write the switching expressions for the following logic circuit. H H=D+C(A’+B’) H

  16. AB AB 00 00 01 01 11 11 10 10 1 1 0 0 1 1 1 1 1 1 Multiple Output Function • E.g. F1=f(a,b,c)=∑(2,6,7) F2=f(a,b,c)=∑(1,3,7) C C F2=a’c+bc F2=a’c+abc F1=bc’+abc F1=bc’+ab

  17. Multiple Output Function F2=a’c+bc F2=a’c+abc F1=bc’+abc F1=bc’+ab

  18. Multiple Output Function Page 146 exp. • F1=f(a,b,c)=∑(2,4,5,6) • F2=f(a,b,c)=∑(2,3,6,7) • F3=f(a,b,c)=∑(2,5,6,7)

  19. Multiple Output Function • E.g. F1(a,b,c,d)=∑m(2,3,5,7,8,9,10,11,13,15) F2(a,b,c,d)=∑m(2,3,5,6,7,10,14,15) F3(a,b,c,d)=∑m(2,3,5,7,8,9,10,11,13,15)

  20. Homework • P152: 10. • P153: 11.a, 11.b • P155: 24.b, 24.d

  21. TO BE CONTINUED

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