Topic 4_Principle - Standard & Canonical

E LE 3223 D IG ITA L

SY STE M S Standard & Canonical Form 1

Topic 4:

Principle of Combinational Circuit(Standard & Canonical Forms)

ELE 3223 DIGITAL SYSTEMS Standard & Canonical Form 2

Combinational CircuitsDEFINITION:

Logic circuits without feedback from output to input, constructed from a functionality complete gate set are said to be combinational .

Logic circuits that contain no memory (ability to store information) are combinational.

Those that contain memory, including flip-flops are said to be sequential


Mathematical definitionLet X be the set of all input variables and Y set of all output variables.

⇒ The combinational function F operates on input variables set X to produce output variable set Y

Output variables are not fed back to the input.

{ }no xxxx ,..,,, 21

x0

xn

y0

yn

Inputs Outputs..

.

.Combinational

Logic Functions

F

( )XFY =


Analysis of Combinational Logic

Examples of combinational circuits :decoders, encoders, multiplexers, adders, subtractors, multipliers, comparators, etc.

Need to consider the implementation of combinational systems with combinational logic circuits.

Combinational logic deals with the method of “combining” basic gates into circuits that carry out a desired application.


General Logic Design Sequence

Problem StatementTruth-Table ConstructionSwitching Equations WrittenEquations SimplifiedLogic Diagram DrawnDecide on Logic Family for ImplementationLogic Circuit Built


Derivation of Switching EquationLogic can be described in several ways

Truth Table

Logic Diagram

Boolean Equation


Standard Form

Standard Form can be derived from truth table. It can be formed either:

Sum of products (SOP) – based on logic 1or

Products of Sum (POS) – based on logic 0


Combinational LogicEach input variable group that produces a logical 1 in a truth table output column can form a term in an Boolean Expression.

Each term is formed by ANDing input variablesEach AND term is then ORed with other AND terms to complete output Boolean Equation

NOTE : Each AND term (also called a product term) identified one input condition where the output is a logical 1.

abccbabcacbaK +++=


DefinitionsLiteral

A Boolean variable or its complement. e.g. ⇒ and are both literals

Product TermA product term is a literal or the logical product (AND) of multiple literals.e.g. Let be binary variables

⇒ a product term could be

X XX

ZYX ,,

XYZYXX ,,


Sum TermA sum term is a literal or the logical OR of multiple literals. e.g. Let be binary variables ⇒ a sum term couldbe

Sum of ProductsSOP is the logical OR of multiple product terms. Each product term is the AND of binary literals.e.g. is a SOP expression

Products of SumsPOS is the logical AND of multiple OR terms. Each sum term is the OR of binary literals.

e.g. is a POS expression

ZYX ,,ZYXYXX +++ ,,

XYZYZYXX +++

( )( )( )ZYZXYYZYXX ++++


Minterms and MaxtermsMinterm

A minterm is a special case product (AND) term. A minterm is a product term that contains all the input variables (each literal no more than once) that make up a Boolean expression.

MaxtermA maxterm is a special case (OR) term. A maxterm is a sum term that contains all the input variables (each literal no more than once) that make up a Boolean expression.


Minterm ShorthandA minterm has one literal for each input variable, either in its normal or complemented form.

a b c• •a b c• •a b c• •

a b c• •a b c• •

a b c• •

a b c• •a b c• •

= m0

= m1

= m2

= m3

= m4

= m5

= m6

= m7

a b c F0 0 0 00 0 1 10 1 0 10 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0

Note: Binary ordering

A canonical sum-of-products form of an expression consists only of minterms OR’dtogether

F a b c a b c a b c a b c a b cF= • • + • • + • • + • • + • •=∑

( ) ( ) ( ) ( ) ( )

( , , , , ) m + m + m + m + m

F = 1 2 3 5 6

m 1 2 3 5 6


Minterms of Different SizesTwo variables:

a b minterm0 0 a’b’ = m00 1 a’b = m11 0 a b’ = m21 1 a b = m3

Three variables:a b c minterm0 0 0 a’b’c’ = m00 0 1 a’b’c = m10 1 0 a’b c’ = m20 1 1 a’b c = m31 0 0 a b’c’ = m41 0 1 a b’c = m51 1 0 a b c’ = m61 1 1 a b c = m7

Four variables:a b c d minterm0 0 0 0 a’b’c’d’ = m00 0 0 1 a’b’c’d = m10 0 1 0 a’b’c d’ = m20 0 1 1 a’b’c d = m30 1 0 0 a’b c’d’ = m40 1 0 1 a’b c’d = m50 1 1 0 a’b c d’ = m60 1 1 1 a’b c d = m71 0 0 0 a b’c’d’ = m81 0 0 1 a b’c’d = m91 0 1 0 a b’c d’ = m101 0 1 1 a b’c d = m111 1 0 0 a b c’d’ = m121 1 0 1 a b c’d = m131 1 1 0 a b c d’ = m141 1 1 1 a b c d = m15


Canonical Sum of Products

A canonical SOP is a complete set of mintermsthat defines when an output variable is a logical 1.

Each minterm corresponds to the row in the truth table when the output function is 1.


Canonical Product of Sums

A canonical POS is a complete set of maxterms that defines when an output variable is a logical 0.

Each maxterm corresponds to the row in the truth table when the output function is 0.


Canonical FormsCanonical defined as “conforming to a general rule”.

The rule for switching logic in that each term used in a switching equation must contain all of the variables.

Two formats generally exist for expressing switching equations in a canonical form.

Sum of mintermsProduct of maxterms


Canonical FormsCanonical forms are not simplified

Normally the opposite of simplification, containing redundancies.

Use Boolean Theorems to simplify the expressionsto eliminate redundancylower cost of the final logic circuit

Design may require converting to logic realised in one form to another form

TTL NAND gates to ECL NOR gateThus can be better to convert to canonical form before simplification carried out


The Connection: Truth Tables to Functions

a b c F0 0 0 00 0 1 10 1 0 10 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0

a b c• •a b c• •a b c• •

a b c• •a b c• •

Function F is true if any ofthese and-terms are true!

Condition that a is 0, b is 0, c is 1.

OR

Sum-of-Products form

F a b c a b c a b c a b c a b c= • • + • • + • • + • • + • •( ) ( ) ( ) ( ) ( )


Minterm ShorthandA minterm has one literal for each input variable, either in its normal or complemented form.a b c• •

a b c• •a b c• •

a b c• •a b c• •

a b c• •

a b c• •a b c• •

= m0

= m1

= m2

= m3

= m4

= m5

= m6

= m7

a b c F0 0 0 00 0 1 10 1 0 10 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0

Note: Binary ordering

A canonical sum-of-products form of an expression consists only of minterms OR’d together

F a b c a b c a b c a b c a b cF= • • + • • + • • + • • + • •=∑

( ) ( ) ( ) ( ) ( )

( , , , , ) m + m + m + m + m

F = 1 2 3 5 6

m 1 2 3 5 6


Minterms of Different SizesFour variables:

a b c d minterm0 0 0 0 a’b’c’d’ = m00 0 0 1 a’b’c’d = m10 0 1 0 a’b’c d’ = m20 0 1 1 a’b’c d = m30 1 0 0 a’b c’d’ = m40 1 0 1 a’b c’d = m50 1 1 0 a’b c d’ = m60 1 1 1 a’b c d = m71 0 0 0 a b’c’d’ = m81 0 0 1 a b’c’d = m91 0 1 0 a b’c d’ = m101 0 1 1 a b’c d = m111 1 0 0 a b c’d’ = m121 1 0 1 a b c’d = m131 1 1 0 a b c d’ = m141 1 1 1 a b c d = m15

Two variables:a b minterm0 0 a’b’ = m00 1 a’b = m11 0 a b’ = m21 1 a b = m3

Three variables:a b c minterm0 0 0 a’b’c’ = m00 0 1 a’b’c = m10 1 0 a’b c’ = m20 1 1 a’b c = m31 0 0 a b’c’ = m41 0 1 a b’c = m51 1 0 a b c’ = m61 1 1 a b c = m7


Sum-of-Products MinimizationF in canonical sum-of-products form (minterm form):

F a b c a b c a b c a b c a b c= • • + • • + • • + • • + • •( ) ( ) ( ) ( ) ( )

Use algebraic manipulation to make a simpler sum-of-products form

Use commutativity to reorder to group similar terms

)()()()()()( cbacbacbacbacbacbaF ••+••+••+••+••+••=

))(())(())(( cbaabacccbaaF •++•++•+=Use distributivityto factor out common terms

Duplicate term - OK

Use x’+x = 1 identity

)()()( cbbacbF •+•+•=

We will find a better method (K-maps) later…


Example of Canonical Conversion SOP

Identify the missing variables in each AND termfor ⇒

for ⇒

for ⇒

( ) bccabacbafY ++== ,,

ba c ( ) cbacbaccbaba +=+→

ca b ( ) cbacabbbcaca +=+→

bc a ( ) bcaabcaabcbc +=+→


Example of Canonical Conversion ⇒ Canonical SOP form:

Two terms the same ⇒ , thus final expression is

bcaabccbacabcbacbaY +++++=

AAA =+

bcaabccbacabcbaY ++++=


Example of Canonical Conversion SOP

Identify the missing variables in each termfor ⇒

for ⇒

( ) zyxwzyxwfG +== ,,,

xw zy &( )( )

zyxwzxywzyxwxyzwzzyyxwxw

+++++→

zy wx &

( )( )wxzywxzywxzyxwzy

wwxxzyzy+++

++→ rr

wxzywxzywxzyxwzyzyxwzxywzyxwxyzwG +++++++=


Example of Canonical Conversion POS

Identify the missing variables in each OR termfor ⇒

for ⇒

( ) ))((,, cbbacbafT ++==

)( ba + c( ) ))(( cbacbaccbaba ++++=++→+

))(()( cbacbaaacb ++++→++

a)( cb +

))()()(( cbacbacbacbaT ++++++++=


Example of Canonical Conversion ⇒ Canonical POS form:

Two terms the same ⇒ , thus final expression is

AAA =.

))()()(( cbacbacbacbaT ++++++++=

))()(( cbacbacbaT ++++++=


Generation of Switching Equations from Truth-Table

What happens when we have a large number of minterms or maxterms ?

Switching equations can be written more conveniently by using minterm or maxterm numerical designation.

where decimal equivalent value for the term can be written directly.

bcaabccabcbacbaP ++++=


Generation of Switching Equations

If decoded each of the minterms based on binary weighting of each variable and produce a list of decimal minterms, the result would be

( )∑= 3,7,6,4,5P

bcaabccabcbacbaP ++++=


Product-of-Sums from a Truth TableA 0 0 0 0 1 1 1 1

B 0 0 1 1 0 0 1 1

C 0 1 0 1 0 1 0 1

F 0 0 0 1 1 1 1 1

F 1 1 1 0 0 0 0 0

Find an expressionfor F’ (the complement)

CBACBACBAF ++=

Complement both sides…CBACBACBAF ++=

)()()( CBACBACBAFCBACBACBAF

++•++•++=

••= Use DeMorgan’s Law to re-express as product-of sums


MaxtermsA 0 0 0 0 1 1 1 1

B 0 0 1 1 0 0 1 1

C 0 1 0 1 0 1 0 1

F 0 0 0 1 1 1 1 1

F 1 1 1 0 0 0 0 0

F A B C A B C A B C= + + • + + • + +( ) ( ) ( )

Maxterms

To find a Product-of-Sums form for a truth tableMake one maxterm for each row in which the function is zeroFor each maxterm, each variable appears once

In its complemented form if it is one in the rowIn its regular form if it is zero in the row


Maxterm ShorthandProduct of Sums

A B C Maxterms

A + B + C = M7

A + B + C = M6

A + B + C = M5

A + B + C = M4

A + B + C = M3

A + B + C = M2

A + B + C = M1

A + B + C = M00 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

F in canonical maxterm form:

F A B C A B C A B CF M M MF

= + + • + + • + += • •= ∏

( ) ( ) ( )0 1 2

M(0, 1, 2)


Generation of Switching Equations

A canonical POS is representation by

( )3,7,6,4,5π=P

Documents

Topic 4_Principle - Standard & Canonical