Page 47 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes
As an example, let us take a look at the exclusive-OR (XOR) gate. (Figure 3.4 for convenience).
Figure 3.4: XOR Gate Schematic Symbol and Truth Table Revisited

Exclusive-OR gate
A
Output
B

A B Output
0 0 0
0 1 1
1 0 1
1 1 0
It is not necessarily obvious what kind of logic circuit would satisfy the truth table. However,
a simple method for designing such a circuit is found in a standard form of Boolean expression
called the Sum-Of-Products, or SOP, form. As you might suspect, a Sum-Of-Products Boolean
expression is literally a set of Boolean terms added (summed) together, each term being a
multiplicative (product) combination of Boolean variables. An example of an SOP expression
would be something like this: ABC + BC + DF, the sum of products “ABC”, “BC”, and “DF”.
Sum-Of-Products expressions are easy to generate from truth tables. All we have to do is examine
the truth table for any rows where the output is “high” (1), and write a Boolean product term that
would equal a value of 1 given those input conditions. In Figure 3.4, rows 2 and 3 have output
high. The product term corresponding to row 2 would be A’B since the term would have a value
of 1 if and only if A = 0 and B = 1. Similarly, the product term corresponding to row 3 would be
AB’. Now, we join our Boolean products together by addition to create a single Boolean expression
for the truth table as a whole. The ∆ is the symbol for XOR.
A ⊕ B = A′B + AB′

Now, we can easily translate the right-hand side of the equation above into a circuit, (refer to Figure 3.5).
Figure 3.5: Simplified Gate-Level Schematic of XOR

An alternative to generating a Sum-Of-Products expression to account for all the “high” (1) output
conditions in the truth table is to generate a Product-Of-Sums, or POS, expression, to account
for all the “low” (0) output conditions instead. For the XOR gate above, a POS expression is
(A’ + B’). (A + B).

Both the Sum-Of-Products and Products-Of-Sums standard Boolean forms are powerful tools when
applied to truth tables. They allow us to derive a Boolean expression and, ultimately, an actual
logic circuit from nothing but a truth table, which is a written specification for what we want a
logic circuit to do. To be able to go from a written specification to an actual circuit using simple,
deterministic procedures means that it is possible to automate the design process for a digital
circuit. In other words, a computer could be programmed to design a custom logic circuit from a

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