1. Multiple Output SOP Minimization

2. Multiple-Output Minimization Frequently, practical logic design problems require minimization of multiple-output functions all of which are functions of the same input variables. This is such a tedious task that we relegate it to a computer program, eg, Espresso in the SIS package we see later in the course. Here, we will show what needs to be considered in multiple- output minimization, but advise that all such work be performed with the aid of a computer, ie, use a CAD tool.

3. Example of Multiple-output Minimization To illustrate multiple-output minimization, consider the following three output expressions, each of three variables:

4. Minimizing f 1 f 1 = B’C’ + AB’ + AC’ + A’BC

5. Minimizing f 2 f 2 = A’B’C + BC’ + AB + AC’

6. Minimizing f 3 f 3 = A’C + AB’ + B’C + AC’

7. Shared Prime Implicants

8. Using Shared PIs The object is to minimize each of the three functions in such a way as to retain as many shared terms between them as possible, thus optimizing the combinational logic of this system. Hence, we now need to look at the shared terms.

9. AND-ed functions: f 1.f 2 f 1. f 2 = AC’

10. AND-ed functions: f 2.f 3 f 2. f 3 = AC’ + A’B’C

11. AND-ed functions: f 3.f 1 f 3. f 1 = AC’ + AB’ + A’BC

12. AND-ed functions: f 1.f 2.f 3 f 1. f 2. f 3 = AC’

13. Summarizing Product Terms u The original functions are: u f 1 = B’C’ + AB’ + AC’ + A’BC u f 2 = A’B’C + BC’ + AB + AC’ u f 3 = A’C + AB’ + B’C + AC’ u The product terms, which must be included in the optimized expressions, are: u f 1. f 2. f 3 = AC’ - common to all three. u f 1. f 2 = AC’ u f 2. f 3 = AC’ + A’B’C u f 3. f 1 = AC’ + AB’ + A’BC

14. Including Shared PI: AC’ f 1 = AC’ f 2 = AC’ f 3 = AC’

15. Including Shared PI: A’B’C f 1 = AC’ f 2 = AC’ + A’B’C f 3 = AC’ + A’B’C

16. Including Shared PI: AB’ f 1 = AC’ + AB’ f 2 = AC’ + A’B’C f 3 = AC’ + A’B’C + AB’

17. Including Shared PI: A’BC f 1 = AC’ + AB’ + A’BC f 2 = AC’ + A’B’C f 3 = AC’ + A’B’C + AB’ + A’BC

18. Including Remaining PIs f 1 = AC’ + AB’ + A’BC + B’C’ f 2 = AC’ + A’B’C + AB + BC’ f 3 = AC’ + A’B’C + AB’ + A’BC

19. What have we learnt? Multiple-output minimization is not for the faint hearted. You should be able to find reasonably good solutions from 5-variable Kmaps. Good understanding of these principles will help you to understand how software for SOP minimization works, coming very soon For any practical problem, use a suitable CAD package. The principles illustrated above are used to create efficient programs for multiple-output minimization.