WEEK 2.2 CANONICAL FORMS


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1 WEEK 2.2 CANONICAL FORMS 1
2 Canonical SumofProducts (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 sumofproducts (SOP) representation of the function Examples: Shortcut notation: ECE 124 Digital Circuits and Systems Page 2
3 SumofProducts 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 ProductofSums (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 productofsums (POS) representation of the function. Examples: Shortcut notation: ECE 124 Digital Circuits and Systems Page 4
5 ProductOfSums 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 2level logic implementations. This is because we assume NOT gates at the input are free, so we see that there are two levels of gates (ANDOR for SOP and ORAND 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 SumOfProducts (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 SumOfProducts 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 SumOfProducts (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 ProductOfSums (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 ProductOfSums 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 ProductOf 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 productofsum 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 (2inputs) NAND gate performs a NOTAND operation. NOR gate performs a NOTOR 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 NOTAND and NOR as NOTOR. ECE 124 Digital Circuits and Systems Page 13
14 NAND and NOR gates (ninputs) Think of multiple input NAND/NOR gates in terms of the operations they perform; i.e., NOTAND (for a NAND) and NOTOR (for a NOR). Example: 3input versions: ECE 124 Digital Circuits and Systems Page 14
15 XOR and NXOR gates (2inputs) XOR gate (with 2inputs performs a difference operation ): NXOR gate (with 2inputs 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: 3input 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: 3input versions: NXOR gates are nonassociative (explained later). ECE 124 Digital Circuits and Systems Page 17
18 Buffer (1input) Does nothing logically; Used in implementation to boost a signal s strength. ECE 124 Digital Circuits and Systems Page 18
19 Associative and NonAssociative 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 NonAssociative Gates NAND/NOR and NXOR gates are not associative. ECE 124 Digital Circuits and Systems Page 20
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