## Combinational Logic Functions

Read this chapter, which describes the design of several components using logic gates, including adders, encoders and decoders, multiplexers, and demultiplexers. This chapter also mentions ladder logic. If you are not familiar with ladder logic, you may also optionally read Chapter 6.1 and 6.2 as a reference. Note that ladder logic is not generally used in computer design and may be omitted. For the one input decoder, note that for a 0 input, the D0 output is a 1 and when the input is a 1, the D1 output is a 1. All the other decoders work the same way. One output line is a 1and the rest are 0's, indicating which binary number has been placed on the input lines. So an input in binary of the number 6 would cause D6 to be a 1.

### Encoder

#### What is an Encoder?

An encoder is a circuit that changes a set of signals into a code. Let’s begin making a 2-to-1 line encoder truth table by reversing the 1-to-2 decoder truth table.

This truth table is a little short. A complete truth table would be

One question we need to answer is what to do with those other inputs? Do we ignore them? Do we have them generate an additional error output? In many circuits, this problem is solved by adding sequential logic in order to know not just what input is active but also which order the inputs became active.

#### Encoder Design Applications

A more useful application of combinational encoder design is a binary to 7-segment encoder. The seven segments are given according to:

Our truth table is:

Deciding what to do with the remaining six entries of the truth table is easier with this circuit. This circuit should not be expected to encode an undefined combination of inputs, so we can leave them as “don’t care” when we design the circuit. The equations were simplified with Karnaugh maps.

#### Equation Collection Summary

The collection of equations is summarized here:

#### The Resulting Circuit Diagram

The circuit is:

#### The Resulting Ladder Diagram

And the corresponding ladder diagram: