Multiplexers (MUX)

Start here A TV has many channels but one screen. The remote’s channel number selects which channel reaches the screen. A multiplexer does exactly this with signals: many inputs, one output, and a select code that picks which input gets through.

• N data inputs, 1 output.
• log₂N select lines choose which input is sent to the output.

Equation: $Y = \overline{S} \cdot I_{0} + S \cdot I_{1}$

The slowest path is S → NOT → AND → OR → Y.

delay = $\operatorname{NOT} + \operatorname{AND} + \operatorname{OR}$

The two select bits $S_{1} S_{0}$ form a number 0–3, and that input goes out.

$Y = \overline{S_{1}} \overline{S_{0}} \cdot I_{0} + \overline{S_{1}} S_{0} \cdot I_{1} + S_{1} \overline{S_{0}} \cdot I_{2} + S_{1} S_{0} \cdot I_{3}$

Method 1: all variables on the select lines. An n-variable function needs $a 2^{n} : 1 \operatorname{MUX}$. Put the truth-table output column (0s and 1s) on the data inputs.

Method 2: split on one variable (very useful for 2:1 MUX questions). Pick a variable, say A, for the select line:

$F = \overline{A}$·(F when A = 0) + A·(F when $A = 1$)

Example — $F \left(X , Y , Z\right) = \Pi M \left(0 , 4\right)$ (2026 T2 Quiz 1, answer 1) $F = 0$ only in rows 000 and 100, so $F =$ $Y + Z$. Put Y on the select line: $Y = 0 \to F = Z$; $Y = 1 \to F = 1$. One $2 : 1 \operatorname{MUX} : I_{0} = Z , I_{1} = 1 , S = Y$. ✔

NOT from a MUX: $\overline{A} = \operatorname{MUX} \left(S = A , I_{0} = 1 , I_{1} = 0\right)$. It costs one whole MUX.

In one line: A multiplexer sends one of N inputs to the output using log₂N select lines $\left(2 : 1 \operatorname{MUX} : Y = \overline{S} I_{0} + S I_{1}\right)$, and by splitting on one variable at a time a MUX can build any function. Exam words: multiplexer, MUX, select lines, data inputs, 2:1, 4:1, MUX delay, function implementation, Shannon expansion.