Priority encoder and octal priority encoder

A priority encoder is an encoder that, when more than one input is active at the same time, outputs the code of the input with the highest priority and ignores the others. It also has a valid output V that is 1 when at least one input is active, to separate "input $D_0$ active" from "no input active" (both give code 000).

Octal (8-to-3) priority encoder

Inputs $D_0$–$D_7$ (active high), $D_7$ has the highest priority. Outputs $A_2A_1A_0$ (binary code) and V. X = don't care.

Output equations

Each output is 1 for the rows where the highest active input has that bit set. For example, $A_0 = 1$ for highest input 7, 5, 3 or 1. Input 5 counts only if 6 and 7 are 0, and so on. After simplification (a term like $\overline{D_7}D_6$ reduces to $D_6$ because $D_7 + \overline{D_7}D_6 = D_7 + D_6$):

$$\begin{aligned} A_2 &= D_4 + D_5 + D_6 + D_7 \\ A_1 &= D_7 + D_6 + D_3\overline{D_4}\,\overline{D_5} + D_2\overline{D_4}\,\overline{D_5} \\ A_0 &= D_7 + D_5\overline{D_6} + D_3\overline{D_4}\,\overline{D_6} + D_1\overline{D_2}\,\overline{D_4}\,\overline{D_6} \\ V &= D_0 + D_1 + D_2 + D_3 + D_4 + D_5 + D_6 + D_7 \end{aligned}$$

Logic diagram

D4,D5,D6,D7 ─────────────────[OR]── A2

D7 ─────────────────────────┐
D6 ─────────────────────────┤
D3,D4',D5' ──[AND]──────────┼[OR]── A1
D2,D4',D5' ──[AND]──────────┘

D7 ─────────────────────────┐
D5,D6' ──────[AND]──────────┤
D3,D4',D6' ──[AND]──────────┼[OR]── A0
D1,D2',D4',D6' ─[AND]───────┘

D0 ... D7 ───────────────────[OR]── V

Complements $\overline{D_2}, \overline{D_4}, \overline{D_5}, \overline{D_6}$ come from NOT gates.

Example: $D_6 = D_3 = D_1 = 1$, others 0. Then $A_2 = 1$, $A_1 = D_6 = 1$, $A_0 = 0$ (since $\overline{D_6} = 0$ blocks every term except $D_7$). Output = 110 = 6, the highest-priority active input, and V = 1. The IC 74148 is a standard 8-to-3 priority encoder (with active-low inputs and outputs).