1. CEC 220 Digital Circuit Design SOP and POS forms Friday, January 23 CEC 220 Digital Circuit Design Slide 1 of 17

2. Lecture Outline Friday, January 23 CEC 220 Digital Circuit Design Sum of Products / Product of Sums Multiplying out and Factoring Exclusive OR operation Simplification Proving equations (Validity) Slide 2 of 17

3. SOP and POS forms Sum of Products (SOP) Friday, January 23 CEC 220 Digital Circuit Design SUMs PRODUCTs A sum of only products An AND – OR circuit!!! Slide 3 of 17

4. SOP and POS forms Product of Sums (POS) Friday, January 23 CEC 220 Digital Circuit Design PRODUCTs SUMs A product of only sums An OR – AND circuit!!! Slide 4 of 17

5. SOP and POS forms SOP and POS Friday, January 23 CEC 220 Digital Circuit Design Slide 5 of 17

6. SOP and POS forms Converting between SOP and POS Friday, January 23 CEC 220 Digital Circuit Design POS SOP x  (y + z) = x  y + x  z x + y  z = ( x + y)  (x + z) Distributive Law Distributive Law (Dual) SOP POS Slide 6 of 17

7. NAND and NOR Gates NAND Gate Friday, January 23 CEC 220 Digital Circuit Design ABC 001 011 101 110 What happens if we “push” the bubble (from output to input)? Slide 7 of 17

8. NAND and NOR Gates NOR Gate Friday, January 23 CEC 220 Digital Circuit Design ABC 001 010 100 110 What happens if we “push” the bubble (from output to input)? Slide 8 of 17

9. NAND and NOR Gates Converting SOP to NAND – NAND Circuits Friday, January 23 CEC 220 Digital Circuit Design Implementing SOP expressions using NAND gates: An AND – OR circuit!!! A NAND – NAND circuit!!!  Slide 9 of 17

10. SOP and POS forms Converting POS to NOR – NOR Circuits Friday, January 23 CEC 220 Digital Circuit Design Implementing POS expressions using NOR gates: An OR – AND circuit!!! An NOR – NOR circuit!!!  Slide 10 of 17

11. Exclusive OR operation Exclusive-OR Gate Friday, January 23 CEC 220 Digital Circuit Design ABC 000 011 101 110 A B C Slide 11 of 17

12. Exclusive OR operation Exclusive-OR Gate Friday, January 23 CEC 220 Digital Circuit Design ABC 001 010 100 111 A B C Sometimes referred to as the “equivalence” gate Slide 12 of 17

13. Exclusive OR operation Some Exclusive-OR Operations Friday, January 23 CEC 220 Digital Circuit Design Slide 13 of 17

14. Review of Simplification Theorems Friday, January 23 CEC 220 Digital Circuit Design Slide 14 of 17

15. Proving Equations (Validity) Friday, January 23 CEC 220 Digital Circuit Design To prove the validity of an equation we can:  Construct a truth table for each side of the eqn o Need to observe all possible inputs and associated outputs  Apply theorems to either or both sides o Can apply duality to both the LHS and the RHS o Can apply DeMorgan’s theorem (i.e. push bubbles)  Could show that the circuit implementation of each side yields the same output o For possible ALL inputs Slide 15 of 17

16. Examples Friday, January 23 CEC 220 Digital Circuit Design Slide 16 of 17

17. Next Lecture Friday, January 23 CEC 220 Digital Circuit Design Combining Numbers with Logic Creating Boolean Equations Combinational Logic & Truth Tables Minterm and Maxterm Expansions Slide 17 of 17