week 2.2 canonical forms - university of waterloobasir/ece124/week2-2.pdf · 2012-01-15 ·...

WEEK 2.2 CANONICAL FORMS

1

Canonical Sum-of-Products (SOP)

ECE 124 Digital Circuits and Systems

• 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:

001 100 111

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”).


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:

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”).


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.


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).

• 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.


DeMorgan

(!m0)=(M0) DeMorgan

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.


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 f2 can also

be written as a Standard SOP using a sum of product terms:

This is not in minterm form

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.


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 f1 can also

be written as a Standard POS using a product-of-sum terms:

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.


NAND and NOR gates (2-inputs)

NAND gate performs a “NOT-AND” 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”.

• NOR gate performs a “NOT-OR” operation.

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:


XOR and NXOR gates (2-inputs)

• XOR gate (with 2-inputs performs a “difference 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.

• NXOR gate (with 2-inputs performs a “equivalence operation”):

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.



• XOR gates are associative (explained later).

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.



• NXOR gates are non-associative (explained later).

Buffer (1-input)

• Does nothing logically; Used in implementation to “boost” a signal’s strength.


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.

Non-Associative Gates

NAND/NOR and NXOR gates are not

associative.


week 2.2 canonical forms - university of waterloobasir/ece124/week2-2.pdf · 2012-01-15 ·...

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