WEEK 2.2 CANONICAL FORMS

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1 WEEK 2.2 CANONICAL FORMS 1

2 Canonical Sum-of-Products (SOP) Given a truth table, we can ALWAYS write a logic expression for the function by taking the OR of the minterms for which the function is a 1. This representation of a function is a sum of minterms and is called a canonical sum-of-products (SOP) representation of the function Examples: Shortcut notation: ECE 124 Digital Circuits and Systems Page 2

3 Sum-of-Products Implementations If implemented with gates, a SOP will always have the following form. A plane of NOT gates (inverters to generate all literals), followed by A plane of AND gates (to implement the minterms), followed by A single OR gate (to take the sum ). ECE 124 Digital Circuits and Systems Page 3

4 Canonical Product-of-Sums (POS) Given a truth table, we can ALWAYS write a logic expression for the function by taking the AND of the maxterms for which the function is a 0. This representation of a function is a product of maxterms and is called a canonical product-of-sums (POS) representation of the function. Examples: Shortcut notation: ECE 124 Digital Circuits and Systems Page 4

5 Product-Of-Sums Gate Implementations If implemented with gates, a POS will always have the form: A plane of NOT gates (inverters to generate all literals), followed by A plane of OR gates (to implement the maxterms), followed by A single AND gate (to take the product ). ECE 124 Digital Circuits and Systems Page 5

6 General Comments There are always two canonical representations for a function, the SOP or the POS. Sometimes, one implementation is simpler than the other implementation (in terms of its cost). SOP and POS implementations are often referred to as 2-level logic implementations. This is because we assume NOT gates at the input are free, so we see that there are two levels of gates (AND-OR for SOP and OR-AND for POS) required to implement the function. ECE 124 Digital Circuits and Systems Page 6

7 Conversion between SOP and POS It is always possible to convert between a POS and SOP representation for a functin. Consider f1= (1,4,7) which can also be expressed as f1 = (0,2,3,5,6). f1 = (1,4,7) = m1+m4+m7 =!(!f1) // double inversion is okay =![ (m0 + m2 + m3 + m5 + m6) ] //!f1 is those minterms not in f1 =![!( (!m0)(!m2)(!m3)(!m5)(!m6) ) ] = (M0)(M2)(M3)(M5)(M6) = (0,2,3,5,6). DeMorgan (!m0)=(m0) DeMorgan Note: Quickly, we can change from minterms (maxterms) to maxterms (minterms) by changing ( ) to ( ) and list those indices of terms missing from the original list. ECE 124 Digital Circuits and Systems Page 7

8 Standard Sum-Of-Products (1) A function described using a canonical SOP (minterms) is by no means minimal. It might require more gates/literals than required. Let us call any AND of literals a product term. We can then express logic functions in Standard Sum-Of-Products form where, instead of minterms, the AND terms are simply product terms. We can start with a canonical SOP and use Boolean algebra to simply the expression into something simpler. ECE 124 Digital Circuits and Systems Page 8

9 Standard Sum-Of-Products (2) Let s consider one of our previous functions in Canonical SOP. If we used Boolean algebra to simplify, we would find that f 2 can also be written as a Standard SOP using a sum of product terms: This is not in minterm form ECE 124 Digital Circuits and Systems Page 9

10 Standard Product-Of-Sums (1) A function described using a canonical POS (maxterms) is by no means minimal. It might require more gates/literals than required. Let us call any OR of literals a sum term. We can then express logic functions in Standard Product-Of-Sums form where, instead of maxterms, the OR terms are simply sum terms. We can start with a canonical POS and use Boolean algebra to simply the expression into something simpler. ECE 124 Digital Circuits and Systems Page 10

11 Example of Standard Product-Of- Sums Forms Let s consider one of our previous functions in Canonical POS. If we used Boolean algebra to simplify, we would find that f 1 can also be written as a Standard POS using a product-of-sum terms: ECE 124 Digital Circuits and Systems Page 11

12 Other Logic Gates Although we can always implement any function we want using AND/OR/NOT, there are other types of logic gates that prove useful. ECE 124 Digital Circuits and Systems Page 12

13 NAND and NOR gates (2-inputs) NAND gate performs a NOT-AND operation. NOR gate performs a NOT-OR operation. NAND/NOR gates can be extended to multiple inputs, but the NAND/NOR gates are not associative (explained later). We should always think of NAND as NOT-AND and NOR as NOT-OR. ECE 124 Digital Circuits and Systems Page 13

14 NAND and NOR gates (n-inputs) Think of multiple input NAND/NOR gates in terms of the operations they perform; i.e., NOT-AND (for a NAND) and NOT-OR (for a NOR). Example: 3-input versions: ECE 124 Digital Circuits and Systems Page 14

15 XOR and NXOR gates (2-inputs) XOR gate (with 2-inputs performs a difference operation ): NXOR gate (with 2-inputs performs a equivalence operation ): XOR/NXOR gates are incredibly useful for arithmetic operations like addition/subtraction/multiplication. These gates can also be extended to multiple inputs, but we need to be clear on their definitions with multiple inputs. ECE 124 Digital Circuits and Systems Page 15

16 XOR gates with multiple inputs. A XOR gate with > 2 inputs performs the odd operation ; the output is a 1 whenever an odd number of inputs are 1. Example: 3-input versions: XOR gates are associative (explained later). ECE 124 Digital Circuits and Systems Page 16

17 NXOR gates with multiple inputs. A NXOR gate with > 2 inputs performs the odd function ; the output is a 1 whenever an even number of inputs are 1. Example: 3-input versions: NXOR gates are non-associative (explained later). ECE 124 Digital Circuits and Systems Page 17

18 Buffer (1-input) Does nothing logically; Used in implementation to boost a signal s strength. ECE 124 Digital Circuits and Systems Page 18

19 Associative and Non-Associative Gates AND/OR gates are associative gates. This means that we can collapse many smaller AND (OR) gates into a single AND (OR) gate with multiple inputs. Example: XOR gates are also associative. Not all types of logic gates are associative. ECE 124 Digital Circuits and Systems Page 19

20 Non-Associative Gates NAND/NOR and NXOR gates are not associative. ECE 124 Digital Circuits and Systems Page 20

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