Encoder and octal-to-binary (8-to-3) encoder

An encoder is a combinational circuit that converts an active signal on one of its $2^n$ (or fewer) input lines into an n-bit binary code. It does the reverse of a decoder.

8-to-3 line (octal-to-binary) encoder

It has 8 inputs $D_0$–$D_7$ (one per octal digit) and 3 outputs $A_2A_1A_0$. It is assumed that only one input is 1 at any time.

Output expressions

Each output is the OR of the inputs whose code has a 1 in that bit:

$$\begin{aligned} A_2 &= D_4 + D_5 + D_6 + D_7 \\ A_1 &= D_2 + D_3 + D_6 + D_7 \\ A_0 &= D_1 + D_3 + D_5 + D_7 \end{aligned}$$

Circuit diagram

 D0 D1 D2 D3 D4 D5 D6 D7
  │  │  │  │  │  │  │  │
  x  │  │  │  ├──┼──┼──┼──[OR]── A2  (D4,D5,D6,D7)
     │  ├──┼──┼──┼──┼──┼──[OR]── A1  (D2,D3,D6,D7)
     ├──┼──┼──┼──┼──┼──┼──[OR]── A0  (D1,D3,D5,D7)

Three 4-input OR gates are enough; $D_0$ is not connected (no output bit is 1 for it).

Limitations

• If no input is active, the output is 000, the same as for $D_0$; a valid-output line is needed to tell them apart.
• If two inputs are active together, the output is wrong (e.g. $D_3$ and $D_5$ give $011 + 101 = 111$, which reads as 7). A priority encoder solves this by encoding only the highest active input.