ECE 233 Intro to Digital Systems Topics Forming Logic Circuits Sum of Products Product of Sums ECE 233 Intro to Digital Systems 1 Administrative Homework 2 due Monday 12 11 23 Lab 1 checkpoint 2 due tonight by 11 59 PM Try to finish the Tutorial Lab 1 Tomorrow 12 06 23 Lab report due at the beginning of class on Friday 12 08 23 ECE 233 Intro to Digital Systems 2 Review Logic circuits are made up of logic gates Common gates include AND OR and NOT Others include NAND NOR XOR and XNOR A logic circuit or Boolean function is composed of a combination of these gates Truth tables are used to express the function of a logic circuit Show all combination of inputs and the output it produces Example Truth table for an AND gate ECE 233 Intro to Digital Systems 3 Definitions Variable A circuit input Literal A single variable in a term Product term A term where the variables or their complements are AND ed i e ABCD Sum term A term where the variables or their complements are OR ed i e A B C D ECE 233 Intro to Digital Systems 4 Two Level Circuits Any truth table can be expressed as a two level circuit Sum of Products SoP AND OR Circuit A two level function composed of AND gates in the first level and their outputs going into an OR gate Can include inverted inputs such as Example AB AC D Product of Sums PoS OR AND Circuit A two level function composed of OR gates in the first level and their outputs going into an AND gate Example A B A C D ECE 233 Intro to Digital Systems 5 Sum of Products SoP To find a Sum of Products expression 1 Find the rows that produce an output of 1 These are the product terms i e AND these input variables o If a variable is 0 then use the inverted version i e o If a variable is 1 then use the non inverted version i e A 2 OR all of those product terms Example A B 0 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 C 0 1 0 1 0 1 0 1 F 0 1 1 0 0 0 1 1 ECE 233 Intro to Digital Systems 6 In class Activity For the following truth tables find the Sum of Products expression 1 2 BA F 0 0 1 0 1 0 1 0 1 1 1 0 3 BA 0 0 0 1 1 0 1 1 F 0 1 1 0 F XOR A B 0 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 C 0 1 0 1 0 1 0 1 F 1 1 0 0 0 1 0 1 ECE 233 Intro to Digital Systems 7 Product of Sums PoS To find a Product of Sums expression terms i e OR these input variables o If a variable is 0 then use the non inverted version i e A o If a variable is 1 then use the inverted version i e o Note the major difference in finding the sum terms 1 Find the row that produce an output of 0 These are the sum 2 AND all of those sum terms Example BA 0 0 0 1 1 0 1 1 F 1 0 1 0 ECE 233 Intro to Digital Systems 8 In class Activity For the following truth tables find the Product of Sums 1 expression 2 BA F 0 0 1 0 1 1 1 0 0 1 1 0 3 BA 0 0 0 1 1 0 1 1 F 0 1 1 0 F XOR A B 0 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 C 0 1 0 1 0 1 0 1 F 1 1 0 0 0 1 0 1 ECE 233 Intro to Digital Systems 9 Alternative Forms Minterms Minterm A product term AND containing all the variables or their complements each appears once Can be designated with the lower case m i e m0 m1 etc Symbol C 0 1 0 1 0 1 0 1 A B m0 0 0 m1 0 0 m2 1 0 m3 1 0 m4 0 1 m5 0 1 m6 1 1 m7 1 1 ECE 233 Intro to Digital Systems Minterm A B C A B C A BC A BC AB C AB C ABC ABC 10 Alternative Forms Minterms Therefore some Sum of Products can be expressed as a Sum of Minterms The previous SoP examples were made up of minterms Example m0 m2 m 0 2 The complement of a function includes the minterms not included in the original function Example F A B m 0 2 then m 1 3 ECE 233 Intro to Digital Systems 11 Alternative Forms Maxterms Maxterm A sum term OR containing all the variables or their complements each appears once Designated with the upper case M M0 M1 etc Symbol C 0 1 0 1 0 1 0 1 Maxterm A B C A B C A B C A B C A B C A B C A B C A B C A B M0 0 0 M1 0 0 M2 1 0 M3 1 0 M4 0 1 M5 0 1 M6 1 1 M7 1 1 ECE 233 Intro to Digital Systems 12 Alternative Forms Maxterms Therefore some Product of Sums can be expressed as a Product of Maxterms The previous PoS examples were made up of maxterms Example M2 M3 M 2 3 The complement of a function includes the maxterms not included in the original function Example F A B M 0 2 then M 1 3 ECE 233 Intro to Digital Systems 13 Converting Between Forms Can convert one form to the other by using the unused indices Example F X Y Z m 7 6 3 2 M 5 4 1 0 Although a Sum of Minterms is a Sum of Products why aren t all Sum of Products a Sum of Minterms ECE 233 Intro to Digital Systems 14 So why does this matter A circuit or system s behavior can be defined with a truth table With that truth table and the techniques in this lecture a logical expression like a Sum of Products can be produced That expression tells you what combination of logic gates produces the desired behavior i e The actual physical hardware needed to implement the function In short you can now specify design and implement a simple digital system ECE 233 Intro to Digital Systems 15 Summary A truth table can be translated into a two level expression 1 Sum of Products SoP A two level function composed of AND gates in the first level and their outputs going into an OR gate o Product terms can be found by looking at the rows with a 1 2 Product of Sums PoS A two level function composed of OR gates in the first …
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