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Digital Logic CPE231. Digital Fundamentals, Ninth Edition By Floyd Instructor: Eng. Tuqa Manasrah. Ch4: Boolean Algebra and Logic Simplification. Apply the basic laws and rules of Boolean algebra Apply DeMorgan's theorems to Boolean expressions

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digital logic cpe231

Digital Logic CPE231

Digital Fundamentals, Ninth Edition

By Floyd

Instructor: Eng. Tuqa Manasrah

ch4 boolean algebra and logic simplification
Ch4: Boolean Algebra and Logic Simplification
  • Apply the basic laws and rules of Boolean algebra
  • Apply DeMorgan's theorems to Boolean expressions
  • Describe gate networks with Boolean expressions
  • Evaluate Boolean expressions
  • Simplify expressions by using the laws and rules of Boolean algebra
  • Convert any Boolean expression into a sum-of-products (SOP) form
  • Convert any Boolean expression into a product-of-sums (POS) form
  • Use a Karnaugh map to simplify Boolean expressions
  • Use a Karnaugh map to simplify truth table functions
  • Utilize "don't care" conditions to simplify logic functions

Digital Logic - Spring 08

laws and rules of boolean algebra

Term

Boolean function

Laws and rules of Boolean algebra
  • Boolean algebra is the mathematics of digital systems.
  • Boolean function: described by Boolean equation.
  • Boolean equation: express logical relationship between binary variables

Digital Logic - Spring 08

basic laws of boolean algebra
Basic laws of Boolean algebra
  • Commutative law: the order of variables makes no difference
    • X + Y = Y + X
    • XY = YX
  • Associative law: the result is the same regardless of the grouping of the variables.
    • X + (Y + Z) = (X + Y) + Z
    • X(YZ) = (XY)Z
  • Distributive law: express the process of factoring in which a common variable is factored out the product term.
    • XY + XZ = X( Y + Z)
    • X + YZ = (X + Y)(X + Z)

Digital Logic - Spring 08

rules of boolean algebra
Rules of Boolean algebra

Digital Logic - Spring 08

continue
Continue..
  • X + XY = X

= X(1 + Y) …distributive law

= X . 1 … Rule 2

= X … Rule 4

  • X + X’Y = X +Y

= (X + XY) + X’Y … Rule 10

= XX + XY + X’y … Rule 7

= XX + XY + XX’ + X’Y …Rule 8

= (X + X’) (X + Y)…Factoring

= 1 . (X+Y) … Rule 6

= X + Y … Rule 4

  • (X + Y)(X + Z) = X + YZ

Do it …

Digital Logic - Spring 08

demorgan s theorem
Demorgan’s Theorem
  • State the equivalency of NAND & negative-OR:
  • State the equivalency of NOR & negative-AND:
  • Prove using truth table.

Digital Logic - Spring 08

examples
Examples
  • Apply DeMorgans Theorem:
    • (AB’+A’B)’
    • (XYZ)’
    • (X’+Y’+Z’)’
    • ((A+B+C)D)’
    • ((ABC)’+D+E)’
    • (A’B(C+D’)+E)’

Digital Logic - Spring 08

boolean analysis of logic circuits
Boolean Analysis of Logic Circuits
  • Each logic function can be expressed with three methods:
    • Boolean expression
    • Logic circuit
    • Truth table
  • Having a function expressed with one method, you can derive the remaining.

Digital Logic - Spring 08

example
Example
  • The logic circuit shown below implement two logic functions: F1 and F2.
  • Analyze the circuit by deriving the Boolean expressions of F1 and F2.
  • Once the Boolean expression for a given logic circuit has been determined, a truth table that shows the output for all values of the input variables can be developed. Derive the truth table of the logic functions F1, F2.

Digital Logic - Spring 08

simplification using boolean algebra
Simplification using Boolean algebra.
  • The goal is to reduce a given expression to its simplest form, or to a more convenient one for implementation.
  • Use laws, rules and theorems of Boolean algebra to manipulate and simplify an expression

Digital Logic - Spring 08

examples12
Examples
  • A+AB+AB’C = A
  • AB’ + A(B+C)’ + B(B+C)’ = AB’
  • ABC’+A’B’C+A’BC+A’B’C’ = ABC’+A’C+A’B’
  • (AB)’+(AC)’+A’B’C’ = A’ + B’+ C’
  • AB’C(BD+CDE)+AC = A(C’ + B’DE)

Digital Logic - Spring 08

standard forms for boolean algebra
Standard forms for Boolean algebra
  • Standardization makes the evaluation, simplification and implementation of Boolean expressions much more systematic and easier.
  • All Boolean expressions, regardless of their form, can be converted into either of two forms: sum-of-product, product-of-sum.

Digital Logic - Spring 08

sum of products sop form
Sum-of-Products (SOP) form
  • A product term consists of the product of literals (variables or their complements).
  • When two or more products are summed the results is an SOP.
  • E.g. AB + ABC , A’B + A’BC’+ AC
  • SOP expression can contain a single variable term.
  • In SOP a single overbar cannot extend over more than one variable ( )

Digital Logic - Spring 08

continue15
Continue…
  • Implementation:
    • AND/OR
    • NAND/NAND
  • Conversion from general expressions:
    • Apply Boolean algebra techniques.
    • E.g. AB + B(CD+EF), ((A+B)’+C)’
  • The domain of a Boolean expression is the set of variables contained in the expression either complemented or not.
  • SOP may not include the domain of variable in all products. (A’BC + ABD’ + AB’CD)
  • A standard SOP expression is a one in which all the variables in the domain appear in each product term. (A’BC + AB’C + ABC’ + A’B’C’)
  • A non standard SOP can be converted to the standard one by using the Boolean algebra rule (x + x’ =1).
  • Convert wx’y + x’yz’ + wxy’ to the standard sop form.

Digital Logic - Spring 08

the product of sum pos from
The Product-of-Sum (POS) from
  • A sum term consists of the sum of literals.
  • When two or more sum terms are multiplied the results is an POS.
  • E.g. (A+B)(A+B+C), (A’+B)(A’+B+C’)(A+C)
  • POS expression can contain a single variable term.
  • In POS a single overbar cannot extend over more than one variable ( )

Digital Logic - Spring 08

continue17
Continue…
  • Implementation:
    • OR/AND
    • NOR/NOR
  • POS may not include the domain of variable in all terms. (A’+B+C)(A+B+D’)(B’+C+D)
  • A standard SOP expression is a one in which all the variables in the domain appear in each product term. (A’+B+C)(A+B+C’)(A+B+C’)(A’+B’+C’)
  • A non standard POS expression can be converted to the standard form by using the Boolean algebra rule 8 (xx’=0) and rule12((X+Y)(X+Z)= X+YZ).
  • Convert (a+b’)(b+c) to the standard POS form.

Digital Logic - Spring 08

boolean expressions and truth table
Boolean expressions and Truth Table
  • A SOP expression is equal to 1 if at least one of the product terms is equal to 1.
  • A POS expression is equal to 0 if at least one of the sum terms is equal to 0.
  • E.g. determine the truth table for the expressions:
    • F1= A’BC’ + AB’C
    • F2= (A+B’+C)(A+B+C’)(A’+B’+C’)
  • Determine the standard expression from the truth table:
    • SOP
      • List the binary values of the inputs for which the output is 1.
      • Convert the binary values to the corresponding product term.(1  corresponding variable, 0corresponding variable complement)
    • POS
      • List the binary values of the inputs for which the output is 0.
      • Convert the binary values to the corresponding sum term.(0  corresponding variable, 1corresponding variable complement)

Digital Logic - Spring 08

definition minterm
Definition: Minterm
  • Product term in which all variables appear once (complemented or not).
  • For the variables X, Y and Z example minterms: X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ
  • Each minterm represents exactly one combination of the binary variables in a truth table.
  • For n variables, there will be 2n minterms.
  • Minterms are labeled from minterm 0, to minterm 2n-1

m0 , m1 , m2 , … , m2n-2 , m2n-1

  • For n = 3, we have

m0 , m1 , m2 , m3 , m4 , m5 , m6 , m7

Digital Logic - Spring 08

definition maxterm
Definition: Maxterm
  • Sum term in which all variables appear once (complemented or not)
  • For the variables X, Y and Z the maxterms are: X+Y+Z , X+Y+Z’ …. , X’+Y’+Z’
  • Minterms and maxterms with the same subscripts are complements:
  • Example:

Digital Logic - Spring 08

standard form of f sum of minterms
Standard Form of F:Sum of Minterms
  • The same as standard SOP
  • OR all of the minterms of truth table for which the function value is 1
  • F = m0 + m2 + m5 + m7

Digital Logic - Spring 08

complement of f
Complement of F
  • Not surprisingly, just sum of the other minterms
  • In this case

F’ = m1 + m3 + m4 + m6

Digital Logic - Spring 08

product of maxterms
Product of Maxterms
  • The same as standard POS
  • F = m0 + m2 + m5 + m7
  • Can express F as AND of all Maxterms of rows that should evaluate to 0

or

Digital Logic - Spring 08

the karnaugh map
The Karnaugh Map
  • A systematic method for simplifying Boolean expressions.
  • If properly used, will produce the simplest SOP or POS expression.
  • Graphical representation of truth table.
  • A box for each minterm
    • So 2 variables, 4 boxes
    • 3 variable, 8 boxes
    • And so on

Digital Logic - Spring 08

k map from truth table examples
K-Map from Truth TableExamples
  • There are implied 0s in empty boxes

Digital Logic - Spring 08

function from k map
Function from K-Map
  • Can generate function from K-map

Simplifies to X + Y (in a moment)

Digital Logic - Spring 08

a 3 variable k map
A 3-variable K-Map
  • Eight minterms
  • Look at encoding of columns and rows

Digital Logic - Spring 08

a 4 variabel k map
A 4-variabel K-Map
  • At limit of K-map

Digital Logic - Spring 08

cell adjacency
Cell Adjacency
  • Cells are arranged so that there is only a single variable change between adjacent cells.
  • Each cell is adjacent to the cells that are immediately next to it on any of its four sides.
  • A cell is not adjacent to the cells that are diagonally touch any of its corners.
  • Adjacency is cylindrical.
  • Note that Z’ wraps from left edge to right edge.

Digital Logic - Spring 08

also wraps toroidal topology
Also Wraps (toroidal topology)

Digital Logic - Spring 08

k map sop minimization
K-Map SOP minimization
  • A minimized SOP expression contains the fewest possible terms with the fewest possible variables per term. And can be implemented with fewer logic gates than a standard expression.
  • Mapping
    • From truth table
    • From standard SOP (Sum of Minterms)
    • From a nonstandard SOP
  • simplification

Digital Logic - Spring 08

mapping examples
Mapping Examples
  • Map the following standard SOP expressions:
    • A’B’C + A’BC’ + ABC’ + ABC
    • A’BC + AB’C +AB’C’
    • A’B’CD + A’BC’D’+ ABC’D + A’B’C’D + AB’CD’
    • A’BCD’ + ABCD’ + ABC’D’ + ABCD
  • Map the following nonstandard SOP expressions:
    • X’ + XY’ + XYZ’
    • YZ + X’Z’
    • X’Y’ + WX’ + WXY’ + WX’YZ’ + W’X’Y’Z + WX’YZ
    • W + Y’Z + WYZ’ + W’XYZ’

Digital Logic - Spring 08

simplification of sop expressions
Simplification of SOP expressions
  • The minimum SOP expression is obtained by grouping the 1’s and determining the minimum expression from the map.
  • Rules of grouping:
    • A group must contain a power of 2 cells, i.e. either 1,2,4,8, or 16.
    • Each cell must be adjacent to one or more cells in the group, but all cells not need to be adjacent.
    • Include the largest possible number of 1’s in a group.
    • Each 1 on the map must be included in at least one group.

Digital Logic - Spring 08

continue34
Continue..
  • Determining the minimum product term of each group:
    • For a 3-variable map
      • One cell -> 3 literals
      • Rectangle of 2 cell s -> 2 literals
      • Rectangle of 4 cell s -> 1 literal
      • Rectangle of 8 cell s -> Logic 1
    • For a 4-variable map
      • One cell -> 4 literals
      • Rectangle of 2 cell s -> 3 literals
      • Rectangle of 4 cell s -> 2 literal
      • Rectangle of 8 cell s -> 1 literal
      • Rectangle of 16 cell s -> Logic 1

Digital Logic - Spring 08

examples35
Examples

instead of

Solve this one

Digital Logic - Spring 08

slide36

Overlap is OK.

  • No need to use full m5-- waste of input

Digital Logic - Spring 08

k map pos minimization
K-map POS minimization
  • A 0 is placed in k-map for each sum term.
  • E.g. for A+ B’ + C, a 0 is placed in cell 010.
  • Cell with no zeros are for which the expression is 1.
  • Mapping procedure:
    • Determine the binary value for each sum term in standard SOP.
    • In the k-map place zeros in the corresponding cells.

Digital Logic - Spring 08

simplification of pos expressions
Simplification of POS expressions
  • Group 0s instead of 1s.
  • Follow the same rules as grouping 1s.
  • Conversion between POS and SOP for the sake of implementation with fewer gates.
    • Use the k-map to optimize a SOP expression: (W+X’+Y+Z’)(W’+X+Y’+Z’)(W’+X’+Y’+Z)(W’+X’+Z’)

Digital Logic - Spring 08

examples39
Examples:
  • Map the following expressions on a k-map:
    • A’BC + AB’C + AB’C’
    • A’BCD’+ABCD’+ABC’D’+ABCD
    • BC+A’C’
    • A+ C’D+ACD’+A’BCD’
    • (A+B’+C’+D)(A+B+C+D’)(A+B+C+D)(A’+B+C’+D)
  • Use the k-map to minimize the following expressions:
    • XY’Z+XYZ’+X’YZ+X’YZ’+XY’Z’+XYZ
    • W’X’Y’Z’+WX’YZ+WX’Y’Z+W’YZ+WX’Y’Z’
    • (X+Y’+Z)(X+Y’+Z’)(X’+Y’+Z)(X’+Y+Z)

Digital Logic - Spring 08

don t care conditions
Don’t care conditions
  • Some input combinations are not allowed never occurred treated as don’t care.
  • For each don’t care term, X is placed in the cell.
  • The Xs can be treated as 1s in case of grouping.

Digital Logic - Spring 08

examples41
Examples
  • Optimize the expression with don’t care conditions:
    • F(a,b,c)=∑m(1,2,4), d(a,b,c)= ∑m(0,3,6,7)
    • F(a,b,c,d)=∑m(0,1,7,13,15), d(a,b,c,d)= ∑m(2,6,8,9,10)
    • F(a,b,c,d)=∑m(2,4,9,12,15), d(a,b,c,d)= ∑m(3,5,6,13)
    • F(w,x,y,z)=∏M(3,11,13,15), d(w,x,y,z)= ∑m(0,2,5,8,10,14)

Digital Logic - Spring 08

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