Encoders and Priority Encoders

Start here A decoder turns a code into “one line on”. An encoder does the opposite: it sees which line is on and gives back its number in binary. Think of a quiz-show buzzer panel: the player whose buzzer is pressed gets their seat number shown on the screen.

• 2ⁿ inputs → n outputs.
• It assumes exactly one input is 1 at a time, and outputs that input’s number in binary.

Equations: $Y_{1} = I_{2} + I_{3}$ and $Y_{0} = I_{1} + I_{3}$.

Why? $Y_{1}$ is the “2s” bit. It is 1 for inputs 2 and 3. $Y_{0}$ is the “1s” bit. It is 1 for inputs 1 and 3.

The problem: what if two inputs are 1 together, like $I_{1}$ and $I_{2}$? Then $Y_{1} = 1$ and $Y_{0} = 1 \to$ output 11 = 3, which is wrong! Neither 1 nor 2 was chosen.

A priority encoder fixes this: if many inputs are 1, the highest-numbered input wins. The lower ones are ignored.

$X =$ “don’t care”: it doesn’t matter whether that input is 0 or 1, because a higher input has already won.

Equations:

• $Y_{1} = I_{3} + I_{2}$ (the winner is 2 or 3)
• $Y_{0} = I_{3} + \overline{I_{2}} \cdot I_{1}$ (the winner is 3, or it’s 1 and input 2 is not on)

Why the $\overline{I_{2}}$ in $Y_{0}$? $I f I_{2}$ is on, input 2 wins, and 2 = 10 has $Y_{0} =$ 0. So $I_{1}$ can only set $Y_{0}$ when $I_{2}$ is off.

Worked example (2026 T2 end-term, answer 10) Inputs $\left(L_{3} , L_{2} , L_{1} , L_{0}\right) = \left(0 , 1 , 1 , 1\right)$, with $Y_{1} = L_{3} + L_{2}$ and $Y_{0} = L_{3} + \overline{L_{2}} L_{1}$.

• $Y_{1} = 0 + 1 =$ 1
• $Y_{0} = 0 + \left(0 \cdot 1\right) =$ 0 ($L_{2} = 1$, so $\overline{L_{2}} = 0$)
• Output = 10 (input 2 wins, as it’s the highest one that is on) ✔

Everyday example: an ambulance, a bus and a car all reach a junction at the same time. The traffic police let the ambulance go first. It has the highest priority.

In one line: An encoder turns “one line on” into its binary number $\left(2^{n} \to n\right)$; a priority encoder lets the highest active input win when several are on. Exam words: encoder, 4-to-2 encoder, priority encoder, highest priority, don’t care (X).