## 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.

### Decoder

A decoder is a circuit that changes a code into a set of signals. It is called a decoder because it does the reverse of encoding, but we will begin our study of encoders and decoders with decoders because they are simpler to design.

#### Types of Decoders

##### Line Decoder

A common type of decoder is the line decoder which takes an n-digit binary number and decodes it into 2^{n}data lines. The simplest is the 1-to-2 line decoder. The truth table is

A is the address and D is the dataline. D_{0} is NOT A and D_{1} is A. The circuit looks like

##### 2-to-4 Line Coder

Only slightly more complex is the 2-to-4 line decoder. The truth table is

Developed into a circuit it looks like

##### Larger Line Decoders

Larger line decoders can be designed in a similar fashion, but just like with the binary adder there is a way to make larger decoders by combining smaller decoders. An alternate circuit for the 2-to-4 line decoder is

Replacing the 1-to-2 Decoders with their circuits will show that both circuits are equivalent. In a similar fashion a 3-to-8 line decoder can be made from a 1-to-2 line decoder and a 2-to-4 line decoder, and a 4-to-16 line decoder can be made from two 2-to-4 line decoders.

You might also consider making a 2-to-4 decoder ladder from 1-to-2 decoder ladders. If you do it might look something like this:

For some logic it may be required to build up logic like this. For an eight-bit adder we only know how to sum eight bits by summing one bit at a time. Usually it is easier to design ladder logic from boolean equations or truth tables rather than design logic gates and then “translate” that into ladder logic.

A typical application of a line decoder circuit is to select among multiple devices. A circuit needing to select among sixteen devices could have sixteen control lines to select which device should “listen”. With a decoder only four control lines are needed.