What is Sum Of Product (SOP) Form?

In the sum of the product form of representation, The product num is logical and operation of the different input variables where the variables could be in the true form or in the complemented form.

The Sum of Product (SOP) form refers to a Boolean expression where several product terms (involving AND operations) are added up. This is perhaps the most popular form used in the design and optimization of digital systems. In SOP format, each of those elements is a combination (ANDing) of some variables (or their negations); all of them are combined by means of disjunction (ORing). This is very helpful in, for instance, the creation of logic circuits from truth tables or achieving algorithmic simplifications of logic functions.

Example:

A.B, A.B̅.C (example of product term)

In SOP sum refers to logical OR Operation. Therefore, in this sum of product form of expression, we perform logical or operations on different product terms. Therefore it is known as the Sum of Product form.

Example:

A+ B̅.C + A.C

as we can see in above example the product terms(A , B̅.C , A.C) which are created by multiplying input variables are are summed with each other .

The SOP form can be in either canonical form or non-canonical form.

1. Non-Canonical SOP Form

In this form each product term between may or may not contain all the variables of the function.

Example:

F(A,B,C) = A+ B̅.C + A.C

as we can see in above example the function have variables A, B, C but we are not including each variable in each product term. in first product term (A) we have not included B & C. In second product term we (B̅.C) have not included A. While in last product term we have not included B.

2. Canonical SOP Form

In canonical SOP form each product term contains all the variables of the function, where variables in each product term can be in true form or complemented form.

Example :

F(A,B) = Ā.B + A.B̅

As we can see in above example each product term contain all the variables which are present in function. In first one(Ā.B) A is present in complementary form while B is in true form. In second one A is present in true form while B is in complementary form.

Writing Boolean Expression for Given Truth Table in SOP Form

Suppose following is the truth table for an electronic circuit:

The SOP form is written when the function if high. i.e we will take conditions only in which the output is one. In given truth table the columns with F = 1 are the rows with high output. Now we will write function in the form of given variables which are A, B and C. If we have 1 in the cell of given variable we will write it as it is , and if we have 0 in the given cell then we will write it's complement.

F = Ā.B.C̅ + A.B̅.C̅ + A.B̅.C + A.B.C

This is the POS in canonical form for the given truth table. Each expressions which lies in between the + are known as Minterms.

Example :

A.B.C, A.B̅.C

Therefore we can also write it as

F(A, B, C) = M2 + M4 + M5 + M6 + M7
F(A,B,C) = ∑m(2, 4, 5, 6, 7)

we can further reduce it to non-canonical SOP form by using boolean algebra.

F = Ā.B.C̅ + A.B̅.C̅ + A.B̅.C + A.B.C
F = Ā.B.C̅ + A.B̅.[C̅ + C] + A.B.[C̅ + C]
F = Ā.B.C̅ + A.B̅ + A.B
F = Ā.B.C̅ + A.[B̅ + B]
F = A + B.C̅

This is how we can get non-canonical SOP form of Boolean expression.

Creating a Truth Table with SOP Boolean Expressions

Suppose following is the SOP expression for which we will make truth table now

F(A, B, C) = A'B' + AB + AC

1. As we have three input values (A, B, C) we will create 2^3 = 8 rows in truth table to include all input combinations.

2. for each input combination we will put values of input in expression and calculate the output:

F(A, B, C) = A'B' + AB + AC
F(0, 0, 0) = 0'0' + 00 + 00 = 0 + 0 + 0 = 0
F(0, 0, 1) = 0'0' + 00 + 01 = 0 + 0 + 0 = 0
F(0, 1, 0) = 0'1' + 01 + 00 = 0 + 0 + 0 = 0
F(0, 1, 1) = 0'1' + 01 + 01 = 0 + 0 + 0 = 0
F(1, 0, 0) = 1'0' + 10 + 10 = 0 + 0 + 0 = 0
F(1, 0, 1) = 1'0' + 10 + 11 = 0 + 0 + 1 = 1
F(1, 1, 0) = 1'1' + 11 + 10 = 0 + 1 + 0 = 1
F(1, 1, 1) = 1'1' + 11 + 11 = 0 + 1 + 1 = 1

3. Create truth table form given input combinations and calculated output.

Advantages of SOP

• Simple to derive: Since truth tables are often used for generating Sankey diagrams, the implementation is straightforward using basic logic gates including AND and OR.
• In case of Complexity: There may be cases where the simpler formulation of a certain equation in a digital system design is not possible since a simpler version is required but a Boolean equivalent transformation does exist.
• Uniformity in Presentation: SOP offers a way to encode Boolean functions in a consistent manner. This is helpful when creating devices that utilize combinational logic.
• Easy Implementation: The design of the SOP can be easily constructed in a digital circuit by the use of AND-OR gates, which is advantageous in the design of hardware.

Disadvantages of SOP

• Can lead to complex circuits: In scenarios where minimization is not employed, the SOP representation can lead to expansive and complex circuits, leading to increased hardware expenditure as well as power usage.
• Not Suitable for Large Functions: As the number of variables in the function increases, the number of product terms also increases; therefore the expression as well as the corresponding circuit becomes bigger.
• Not Always the Simplest Version Possible: Of some functions, for instance, SOP optimized to the minimizing level is usually an ineffective option in representing a particular function than other options like the POS.

Applications of SOP

• Digital Circuit Design: The SOP form is utilized very often in the implementation of combinational circuits like multiplexers, encoders and decoders among others.
• Logic Minimization: Most of the logic minimization approaches ranging from Karnaugh mapping to Quine-McCluskey uses the standard sum of product for better construction of digital circuits.
• Programmable Logic Arrays (PLAs): In PLAs, which is a configurable device, the logic is defined with the help of SOP expressions.
• Design Finite State Machine: When designing finite state machine, the transition logic between the different states is expressed in the form of SOP.