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) 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(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(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)=∑(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(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) 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 There may be a combination of input values which will never occur will never occur if they do occur, the output is of no concern. if they do occur, the output is of no concern. The function value for such combinations is called a don't care. 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. 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 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. Treat don't cares as if they are 1s to generate PIs. Delete PI's that cover only don't care minterms. 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). 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. Example Simplify the function f(a,b,c,d) whose K-map is shown at the right. Simplify the function f(a,b,c,d) whose K-map is shown at the right. f = a’c’d+ab’+cd’+a’bc’ f = a’c’d+ab’+cd’+a’bc’or f = a’c’d+ab’+cd’+a’bd’ f = a’c’d+ab’+cd’+a’bd’ The middle two terms are EPIs, while the first and last terms are selected to cover the minterms m 1, m 4, and m 5. The middle two terms are EPIs, while the first and last terms are selected to cover the minterms m 1, m 4, and m 5. dd11 dd00 1011 1010 dd11 dd00 1011 1010 dd11 dd00 1011 1010 ab cd 00 01 11 10 00 01 11 10

6. Another Example Simplify the function g(a,b,c,d) whose K-map is shown at right. Simplify the function g(a,b,c,d) whose K-map is shown at right. g = a’c’+ ab or g = a’c’+ ab or g = a’c’+b’d g = a’c’+b’d 0dd0 1dd1 d0d1 001d 0dd0 1dd1 d0d1 001d 0dd0 1dd1 d0d1 001d ab cd

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. A=w+xz+xy B=x’y+x’z+xyz’ C=y’z’+yz D=z’

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

11. POS Example 0000 1100 0111 1111 ab cd 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 F=AB F=A+B 1

13. Mixed Logic Combinational Circuits

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

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

16. Multiple Output Function E.g. F1=f(a,b,c)=∑(2,6,7) E.g. F1=f(a,b,c)=∑(2,6,7) F2=f(a,b,c)=∑(1,3,7) F2=f(a,b,c)=∑(1,3,7) 111 00011110 0 1 AB C 1 11 00011110 0 1 AB C F2=a’c+bc F1=bc’+ab F2=a’c+abc F1=bc’+abc

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

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

19. E.g. E.g. F 1 (a,b,c,d)=∑m(2,3,5,7,8,9,10,11,13,15) F 1 (a,b,c,d)=∑m(2,3,5,7,8,9,10,11,13,15) F 2 (a,b,c,d)=∑m(2,3,5,6,7,10,14,15) F 2 (a,b,c,d)=∑m(2,3,5,6,7,10,14,15) F 3 (a,b,c,d)=∑m(2,3,5,7,8,9,10,11,13,15) F 3 (a,b,c,d)=∑m(2,3,5,7,8,9,10,11,13,15) Multiple Output Function

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