Encoder

An encoder is the inverse of a Decoder: it takes a one-hot input ($2^n$ lines, exactly one asserted) and produces the $n$-bit binary code identifying which line was asserted.

A 4-to-2 encoder takes 4 input lines $w_0,w_1,w_2,w_3$ and outputs 2-bit code $y_1{y}_0$:

• $w_0=1$ → output $00$
• $w_1=1$ → output $01$
• $w_2=1$ → output $10$
• $w_3=1$ → output $11$

The Boolean equations are simple ORs:

$y_0=w_1+w_3$

$y_1=w_2+w_3$

$y_0$ is high when an odd-indexed input is asserted; $y_1$ is high when an upper-half input is asserted.

In the diagram, $w_0$ isn’t connected to either output gate — when $w_0$ is the only asserted input, both outputs are $0$ ($00$), which is the correct binary code for “input $0$.” This is also the encoder’s fundamental limitation: the output $00$ is produced both when $w_0=1$ and when no input is asserted. The encoder cannot distinguish these cases by itself. That’s why a real encoder almost always carries a separate valid (or “any-input-asserted”) output — without it, you can’t tell “input zero was selected” from “nothing happened.”

Why this is useful

Encoders compress information. A keyboard with 64 keys has 64 individual switches; an encoder converts that to a 6-bit code per keypress, which fits much more comfortably on a bus. Same trick for any peripheral that produces a one-hot signal you need to pack down.

Priority encoder

A plain encoder assumes exactly one input is asserted at a time. If two inputs go high simultaneously, the output is the OR of their codes — usually meaningless.

A priority encoder fixes this. It assigns a priority to each input and outputs the code of the highest-priority input that’s currently asserted. Lower-priority inputs are ignored when a higher-priority one is on.

Common convention: highest input index has highest priority. So if $w_2$ and $w_5$ are both high, the priority encoder outputs the code for $w_5$.

Priority encoders also typically have a “valid” output — high when at least one input is asserted, low when no inputs are asserted (in which case the binary output is meaningless, since “none” doesn’t have a unique code in $n$-bit binary).

Priority encoders show up in interrupt controllers (the highest-priority pending interrupt is the one the CPU should service next), in floating-point normalization (find the highest set bit in the mantissa), and in arbitration circuits.