Multiplexers | Sahithyan's Notes

A multiplexer (MUX) is a combinational circuit that routes 1 of $2^n$ input lines to a single output. $n$ select lines determine which input is forwarded. Available as ICs.

Select-Line Logic

For a $2^n$ -to- $1$ MUX with inputs $I_0, \ldots, I_{2^n-1}$ and select lines $S_{n-1}, \ldots, S_0$ :

Y = \sum_{k=0}^{2^n-1} m_k \cdot I_k

$m_k$ : minterm corresponding to binary value $k$ of the select lines.

For a 4-to-1 MUX:

Y = \overline{S_1}\,\overline{S_0}\,I_0 + \overline{S_1}\,S_0\,I_1 + S_1\,\overline{S_0}\,I_2 + S_1\,S_0\,I_3

$S_1$	$S_0$	$Y$
0	0	$I_0$
0	1	$I_1$
1	0	$I_2$
1	1	$I_3$

Implementing Boolean Functions

Any $n$ -variable boolean function can be implemented with a $2^n$ -to- $1$ MUX:

Use the $n$ input variables as select lines.
Connect each data input $I_k$ to $1$ if the function is $1$ for minterm $k$ , else to $0$ .

With a $2^{n-1}$ -to- $1$ MUX, 1 variable becomes a data input. Each $I_k$ is connected to $0$ , $1$ , the variable, or its complement based on the last truth table column.

Advantages

Reduction of wires
Reduction of circuit complexity
Reduction of cost
Any combinational function can be implemented using MUX

Demultiplexer

A demultiplexer (DEMUX) routes 1 input to 1 of $2^n$ outputs based on $n$ select lines. The inverse operation of MUX.

For a 1-to-4 DEMUX with input $D$ :

$S_1$	$S_0$	$Y_0$	$Y_1$	$Y_2$	$Y_3$
0	0	$D$	$0$	$0$	$0$
0	1	$0$	$D$	$0$	$0$
1	0	$0$	$0$	$D$	$0$
1	1	$0$	$0$	$0$	$D$