1. ECE 331 – Digital System Design
Standard Forms for
Boolean Expressions
(Lecture #4)

2. Standard Forms for Boolean Expressions
Sum-of-Products (SOP)
Derived from the Truth table for a function by considering those rows for which F = 1.
The logical sum (OR) of product (AND) terms.
Realized using an AND-OR circuit.
Product-of-Sums (POS)
Derived from the Truth table for a function by considering those rows for which F = 0.
The logical product (AND) of sum (OR) terms.
Realized using an OR-AND circuit.
ECE Digital System Design

3. Sum-of-Products (SOP)
ECE Digital System Design

4. ECE 331 - Digital System Design
Minterms
A minterm, for a function of n variables, is a product term in which each of the n variables appears once.
Each variable may appear in its complemented or uncomplemented form.
For a given row in the Truth table, the corresponding minterm is formed by
Including variable xi, if xi = 1
Including the complement of xi, if xi = 0
ECE Digital System Design

5. ECE 331 - Digital System Design
Minterms
ECE Digital System Design

6. ECE 331 - Digital System Design
Sum-of-Products
Any function F can be represented by a sum of minterms, where each minterm is ANDed with the corresponding value of the output for F.
F = S (mi . fi)
where mi is a minterm
and fi is the corresponding functional output
Only the minterms for which fi = 1 appear in the expression for function F.
F = S (mi) = S m(i)
Denotes the logical
sum operation
shorthand notation
ECE Digital System Design

7. ECE 331 - Digital System Design
Sum-of-Products
The Canonical Sum-of-Products for function F is the Sum-of-Products expression in which each product term is a minterm.
The expression is unique
However, it is not necessarily the lowest-cost
Synthesis process
Determine the Canonical Sum-of-Products
Use Boolean Algebra (and K-maps) to find an optimal, functionally equivalent, expression.
ECE Digital System Design

8. ECE 331 - Digital System Design
Sum-of-Products
AND
sum term f x 1 2 3
X3.X2'
X3.X2' + X1.X3'
X1.X3' OR
AND
product term
Product Term = Logical ANDing of literals
Sum Term = Logical ORing of product terms
ECE Digital System Design

9. ECE 331 - Digital System Design
Sum-of-Products
Use the Distributive Laws to multiply out a Boolean expression.
Results in the Sum-of-Products (SOP) form.
F = (A + B).(C + D).(E)
F = (A.C + A.D + B.C + B.D).(E)
F = A.C.E + A.D.E + B.C.E + B.D.E
Product terms are
of single variables
not in SOP form
H = A.B.(C + D) + ABE
ECE Digital System Design

10. Product-of-Sums (POS)
ECE Digital System Design

11. ECE 331 - Digital System Design
Maxterms
A Maxterm, for a function of n variables, is a sum term in which each of the n variables appears once.
Each variable may appear in its complemented or uncomplemented form.
For a given row in the Truth table, the corresponding Maxterm is formed by
Including the variable xi, if xi = 0
Including the complement of xi, if xi = 1
ECE Digital System Design

12. ECE 331 - Digital System Design
Maxterms
ECE Digital System Design

13. ECE 331 - Digital System Design
Product-of-Sums
Any function F can be represented by a product of Maxterms, where each Maxterm is ANDed with the complement of the corresponding value of the output for F.
F = P (Mi . f 'i)
where Mi is a Maxterm
and f 'i is the complement of the corresponding functional output
Only the Maxterms for which fi = 0 appear in the expression for function F.
F = P (Mi) = P M(i)
Denotes the logical
product operation
shorthand notation
ECE Digital System Design

14. ECE 331 - Digital System Design
Product-of-Sums
The Canonical Product-of-Sums for function F is the Product-of-Sums expression in which each sum term is a Maxterm.
The expression is unique
However, it is not necessarily the lowest-cost
Synthesis process
Determine the Canonical Product-of-Sums
Use Boolean Algebra (and K-maps) to find an optimal, functionally equivalent, expression.
ECE Digital System Design

15. ECE 331 - Digital System Design
Product-of-Sums OR
product term f x 2 1 3
X1 + X3
(X1+X3) . (X2'+X3')
X2' + X3'
AND OR
sum term
Sum Term = Logical ORing of literals
Product Term = Logical ANDing of sum terms
ECE Digital System Design

16. ECE 331 - Digital System Design
Product-of-Sums
Use the Distributive Laws to factor a Boolean expression.
Results in the Product-of-Sums (POS) form.
F = V.W.Y + V.W.Z + V.X.Y + V.X.Z
F = (V).(W.Y + W.Z + X.Y + X.Z)
F = (V).(W + X).(Y + Z)
Sum terms are
of single variables
not in POS form
H = (A+B).(C+D+E) + CE
ECE Digital System Design

17. ECE 331 - Digital System Design
SOP and POS
Any function F may be implemented as either a Sum- of-Products (SOP) expression or a Product-of-Sums (POS) expression.
Both forms of the function F can be realized using logic gates that implement the basic logic operations.
However, the two logic circuits realized for the function F do not necessarily have the same cost.
Objective: minimize the cost of the designed circuit
Compare the cost of the SOP realization with that of the POS realization
ECE Digital System Design

18. Converting between SOP and POS
The sum-of-products (SOP) form of a Boolean expression can be converted to its corresponding product-of-sums (POS) form by factoring the Boolean expression.
The product-of-sums (POS) form of a Boolean expression can be converted to its corresponding sum-of-products (SOP) form by multiplying out the Boolean expression.
ECE Digital System Design

19. ECE 331 - Digital System Design
Dual
The dual of a Boolean expression is formed by changing AND to OR, OR to AND, 0 to 1, and 1 to 0.
Alternately, it can be determined by complementing the entire Boolean expression, and then complementing each of the literals.
The SOP and POS are duals of one another.
ECE Digital System Design

20. Logic Circuit Implementations
ECE Digital System Design

21. ECE 331 - Digital System Design
Logic Gates
AND and OR Gates
2-input gates realized with 6 CMOS transistors
3-input gates realized with 8 CMOS transistors
NAND and NOR Gates
2-input gates realized with 4 CMOS transistors
3-input gates realized with 6 CMOS transistors
Therefore, it is more cost efficient to design logic circuits using NAND and NOR gates.
ECE Digital System Design

22. “Redrawing” the NAND Gate
bubble denotes inversion
Inverter (NOT gate) x 1 x x 1 1 x x 2 2 x 2
(a) x x = x + x 1 2 1 2
Remember, this is an application of DeMorgan's Theorem
ECE Digital System Design

23. “Redrawing” the NOR Gate
bubble denotes inversion
Inverter (NOT gate) x 1 x x 1 1 x x 2 x 2 2
(b) x + x = x x 1 2 1 2
Remember, this is an application of DeMorgan's Theorem
ECE Digital System Design

24. ECE 331 - Digital System Design
SOP using NAND Gates
Converting from AND-OR to NAND-NAND
Draw the AND-OR logic circuit for the SOP expression.
Add bubbles (inversion)
At the output of each AND gate
At the corresponding inputs of the OR gate
Two bubbles on the same signal cancel (A'' = A)
All gates in the logic circuit are NAND gates
Two different representations for the NAND gate
74xx08 Quad 2-input NAND Gate
ECE Digital System Design

25. NAND Gate Realization of SOPx x 1 1 x x 2 2 x x 3 3 x x 4 4 x x 5 5
NAND gate
F = X1.X2 + X3.X4.X5 x 1 x 2
SOP Expression x 3 x 4 x 5
ECE Digital System Design

26. ECE 331 - Digital System Design
POS using NOR Gates
Converting from OR-AND to NOR-NOR
Draw the OR-AND logic circuit for the POS expression.
Add bubbles (inversion)
At the output of each OR gate
At the corresponding inputs of the AND gate
Two bubbles on the same signal cancel (A'' = A)
All gates in the logic circuit are NOR gates
Two different representations for the NOR gate
74xx02 Quad 2-input NOR Gate
ECE Digital System Design

27. NOR Gate Realization of POSx x 1 1 x x 2 2 x x 3 3 x x 4 4 x x 5 5
NOR gate
F = (X1+X2).(X3+X4+X5) x 1 x 2
POS Expression x 3 x 4 x 5
ECE Digital System Design

28. Implement the function f(x1, x2, x3) = S m(2,3,4,6,7)
Example:
Implement the function
f(x1, x2, x3) = S m(2,3,4,6,7)
using only NAND gates.
ECE Digital System Design

29. ECE 331 - Digital System Designf
(a) SOP implementation
(b) NAND implementation x2 x3 x1
F = X2 + X1.X3'
ECE Digital System Design

30. Implement the function f(x1, x2, x3) = S m(2,3,4,6,7)
Example:
Implement the function
f(x1, x2, x3) = S m(2,3,4,6,7)
using only NOR gates.
ECE Digital System Design

31. ECE 331 - Digital System Designx1 f
(a) POS implementation
(b) NOR implementation x3 x2
F = (X1+X2).(X2+X3')
ECE Digital System Design