Multiplexer: What is it? (And How Does it Work)

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Key learnings:

Multiplexer Definition: A multiplexer is a digital circuit that selects one of several input signals and forwards the selected input to a single output line.
Basic Operation: Multiplexers work by using select lines to choose which input line is connected to the output, functioning like a digitally controlled switch.
2 to 1 MUX Circuit: This simplest form of multiplexer uses a combination of logic gates to select between two input signals based on a single select line.
Applications of Multiplexers: Multiplexers optimize the use of transmission lines in communications, reducing the complexity and cost of digital systems.
Expanding Capability: Larger multiplexer setups, like the 8-to-1 MUX, can be built by cascading smaller units, demonstrating how multiplexers scale in complexity and functionality.

What is a Multiplexer?

A multiplexer (sometimes spelled multiplexor and also known as a MUX) is a combinational circuit that selects one of several data inputs and sends that input to a single output. The inputs may be analog or digital. Digital multiplexers are also called data selectors.

The same idea is used on a wire or channel: several sources share one path, one source at a time or in assigned slots. Communication multiplexing uses that sharing. It is still the same selector idea as a logic MUX.

Analog multiplexers are analog switches. They may use CMOS transmission gates, transistors or relays. Digital multiplexers are built from logic gates. The complementary block is a demultiplexer.

What Does a Multiplexer Do?

A digital system often has several data lines and only one path onward. A multiplexer is the circuit that connects one chosen input line to that output.

A digital multiplexer selects one of n data inputs and sends it to the output. Binary select inputs choose which input. For n inputs, m select lines are needed so that 2^m=n.

The binary code on the select inputs picks one of those n data inputs and sends it to the single output. So a multiplexer has 2^mdata input lines, m select lines and one output line.

The block diagram of an n-to-1 multiplexer and an equivalent switch model are shown below.

Block Diagram Of N To 1 Multiplexer 1 — Block Diagram

Equivalent Circuit Of N To 1 Multiplexer 1 — Block Diagram

How Does a Multiplexer Work?

The multiplexer behaves as a many-to-one switch. At any instant the output is tied to only one of the n data inputs, the digital form of an analog selector switch.

Multiplexer Circuit

Common sizes are 2-to-1, 4-to-1 and 8-to-1. Each size has its own gate circuit, truth table and Boolean equation.

2 to 1 Multiplexer

2 to 1 Multiplexer Circuit

A 2-to-1 multiplexer has two data inputs D₀ and D₁, one select line S and one output Y. A gate version uses two AND gates, an OR gate and a NOT gate.

The 2-to-1 block diagram, logic symbol and switch model are shown below.

2 To 1 Multiplexer Circuit 1 — A 2-to-1 Mux

Logic Symbol Of 2 To 1 Multiplexer 4 — Logic Symbol

Switching Analogy Of 2 To 1 Multiplexer 1 — Logic Symbol

As shown, D₀goes to one AND gate and D₁ goes to the other. Select S feeds the D1 AND gate. Inverted S feeds the D0 AND gate. The two AND outputs then feed the OR gate that produces Y.

2 to 1 Multiplexer Working Principle

When S=0, the D1 AND gate sees 0 and its output is 0. Inverted S is 1, so the D0 AND gate passes D₀ through to Y.

When S=1, the D0 AND gate is forced to 0. The D1 AND gate passes D₁ through to Y.

A logic 0 or 1 on S therefore chooses D₀ or D₁. The function is the same as a single-pole double-throw (SPDT) switch.

2 to 1 Multiplexer Truth Table

The 2-to-1 truth table is below.

Select input S	Data Input $D_0$	Data Input $D_1$	Output Y
0	0	0	0
0	0	1	0
0	1	0	1
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	1

Truth Table of 2-to-1 Multiplexer

That table is the full 1-bit 2-to-1 function: Y follows D0 when S is 0 and D1 when S is 1.

2 to 1 Multiplexer Boolean Expression

From the minterms, the boolean expression for the 2-to-1 output simplifies as follows.

$\begin{align*} \begin{split} Y = S^-D_0D_1+S^-D_0D_1^-+SD_0^-D_1+SD_0D_1 \\ = S^-D_0(D_1+D_1^-) + SD_1(D_0+D_1^-) \\ Y = S^-D_0 + SD_1 \,\,(as\,\,per\,\,Complement\,\,law\,\,(OR\,\,law)\,\, i.e., A+A^-=1) \end{split} \end{align*}$

In simple notation

S	Y
0	D₀
1	D₁

The compact table repeats that rule: S=0 gives Y=D₀ and S=1 gives Y=D₁. Larger muxes can be built by cascading 2-to-1 blocks.

Any of these muxes can also be built from the NAND gate alone. Note that NAND and NOR gates are a universal gate and we can implement any digital circuits by using NAND and NOR gates.

4 to 1 Multiplexer

4 to 1 Multiplexer Circuit Diagram

A 4-to-1 multiplexer has four data inputs, two select lines and one output. A gate version uses 4 AND gates, one OR gate and 2 NOT gates.

The four inputs are D₀, D₁, D₂ and D₃. The two select lines are S₀ and S₁. Four inputs need two select bits: $2^m$ = $2^2$ = 4, not the stored final $2$ . The select code then connects one input to Y.

The 4-to-1 block diagram, logic symbol and switch model are shown below.

4 To 1 Multiplexer Circuit — A 4-to-1 Mux Logic Circuit

Logic Symbol Of 4 To 1 Multiplexer 1 — Logic Symbol

Switching Analogy Of 4 To 1 Multiplexer 1 — Logic Symbol

4 to 1 Multiplexer Working Principle

If the select inputs are $S_0=0$ , $S_1=0$ , the top AND gate is enabled and the others are off. Data input $D_0$ reaches the output, so $Y = D_0$ .

If the select inputs are $S_0=1$ , $S_1=1$ , the bottom AND gate is enabled and the others are off. Data input $D_3$ reaches the output, so $Y=D_3$ .

4 to 1 Multiplexer Truth Table

The 4-to-1 select table used on this page is below. Here S0 is treated as the high-order select bit.

S₀	S₁	Y
0	0	D₀
0	1	D₁
1	0	D₂
1	1	D₃

Truth Table of 4-to-1 Multiplexer

That is a 1-bit 4-to-1 data selector: one of four sources is routed to Y.

4 to 1 Multiplexer Boolean Expression

From that table, the Boolean sum of products is:

$\begin{align*} Y = S_0^-S_1^-D_0 + S_0^-S_1D_1 + S_0S_1^-D_2 + S_0S_1D_3 \end{align*}$

8 to 1 Multiplexer

8 to 1 Multiplexer Circuit Diagram

An 8-to-1 multiplexer has 8 data inputs, 3 select lines and one output. A gate version uses 8 AND gates, one OR gate and 3 NOT gates.

The eight inputs are D₀, D₁, D_2, D₃, D₄, D₅, D₆ and D₇. The three select lines are S₀, S₁ and S₂, because 2^m = 2³ data inputs need three select bits. Those three bits choose which input reaches Y.

The 8-to-1 logic diagram is shown below.

8 To 1 Multiplexer Circuit 1 — A 8-to-1 Mux

8 to 1 Multiplexer Working Principle

If the three select bits are $S_0=0$ , $S_1=0$ , $S_3=0$ (this 8-to-1 uses S2 as the third select; the stored token is S3), the top AND gate is enabled and the others are off. D₀ reaches the output, so $Y = D_0$ .

If the three select bits are $S_0=1$ , $S_1=1$ , $S_3=1$ (again the stored third token is S3; the third select on this 8-to-1 is S2), the bottom AND gate is enabled and the others are off. D₇ reaches the output, so $Y=D_7$ .

8 to 1 Multiplexer Truth Table

The 8-to-1 select table used on this page is below. S0 is again the high-order select bit.

S₀	S₁	S₂	Y
0	0	0	D₀
0	0	1	D₁
0	1	0	D₂
0	1	1	D₃
1	0	0	D₄
1	0	1	D₅
1	1	0	D₆
1	1	1	D₇

Truth Table of 8-to-1 Multiplexer

8 to 1 Multiplexer Boolean Expression

From that table, the Boolean sum of products is:

$\begin{align*} Y = S_0^-S_1^-S_2^-D_0+S_0^-S_1^-S_2D_1+S_0^-S_1S_2^-D_2+S_0^-S_1S_2D_3+ \end{align*}$

$\begin{align*} S_0S_1^-S_2^-D_4+S_0S_1^-S_2D_5+S_0S_1S_2^-D_6+S_0S_1S_2D_7 \end{align*}$

Arduino Multiplexer

On an Arduino, a mux is a way to share pins across a matrix or a bank of parts. Typical loads are LEDs, sensors, buttons and other parts found in many best Arduino starter kits.

One common part is the 8-channel 74HC4051 analog multiplexer/demultiplexer. A sample Arduino hook-up is shown below.

Arduino Multiplexer Configuration 1 — Arduino-Multiplexer Circuit

A typical 74HC4051-to-Arduino wiring:

+V_CC of the IC goes to the Arduino 5 V pin.

$E^-$ is the active-low enable. Tie -V_EEand GND to Arduino GND for 0 V to 5 V analog use.(Note that $E^-$ must be low to enable the chip. A high on that pin disables the 4051 and leaves the common I/O high-Z, not forced to Y=0.)

The select pins are S₀, S₁ and S₂. In the example they go to Arduino pins 2, 3 and 4.

The common I/O (Z / Y) goes to a digital pin if the channels are digital, or to an analog pin A₀ to A₅ if the channels are analog. Pins 6 to 12 in the figure are only one example.

The 4051 channels can be sensors, buttons or potentiometers, analog or digital. The same analog-mux idea appears on other boards. See Arduino vs Raspberry Pi if you are choosing a board.

Multiplexer Integrated Circuits (ICs)

Common 74xx multiplexer ICs and the output polarity listed by this page are in the table below. Check the current datasheet for complementary outputs and enable polarity.

IC number	Function	Output State
74150	16:1 multiplexer	Inverted output
74151	8:1 multiplexer	Inverted output
74153	Dual 4-to-1 multiplexer	Output is equal to input
74352	Dual 4-to-1 multiplexer	Inverted output
74157	Quad 2-to-1 multiplexer	Output is equal to input
74158	Quad 2-to-1 multiplexer	Inverted output

What is a Multiplexer Tree or Chaining Multiplexers?

A mux with more inputs can be built by cascading smaller muxes.

That cascade is called a multiplexer tree, or chaining multiplexers.

One way to make an 8-to-1 is two 4-to-1 muxes feeding a 2-to-1 mux, with the 4-to-1 select pins in parallel. That gives three select bits. The worked example below instead uses two 4-to-1 muxes, active-low enables and an OR gate.

Example: Obtain an 8-to-1 Multiplexer Using Two 4-to-1 Multiplexers

Solution:

The two 4-to-1 muxes plus the OR gate form the 8-to-1 shown below.
The eight data inputs are D₀ to D_7.
Select bits S₀ and S₁ are shared by both 4-to-1 muxes. S₂ enables one mux at a time.
S₂ goes to Enable(E) of MUX-1. Inverted S₂^– goes to Enable of MUX-2. This only works if those enables are active high. Many 4-to-1 ICs use active-low enable instead.
The two mux outputs are OR-ed to form Y.

8 To 1 Multiplexer By Cascading Two 4 To 1 Multiplexers — 8-to-1 Multiplexer By Cascading Two 4-to-1 Multiplexers

The select table is below. Outputs $D_0$ , $D_1$ , $D_2$ and $D_3$ appear when MUX-1 is enabled. Outputs $D_4$ , $D_5$ , $D_6$ and $D_7$ appear when MUX-2 is enabled. The stored table still lists S0 as the high-order bit.

S₀	S₁	S₂	Y
0	0	0	D₀
0	0	1	D₁
0	1	0	D₂
0	1	1	D₃
1	0	0	D₄
1	0	1	D₅
1	1	0	D₆
1	1	1	D₇

Advantages of a Multiplexer

Practical reasons to use a mux:

A mux cuts the number of wires, so the board can be simpler and cheaper.
Fewer external connections also means fewer joints that can fail.
A mux can also realize many combinational functions.
Using a mux can avoid a large gate-level network.
A mux implementation can skip a separate k-maps (Karnaugh map) pass when the function is already listed as a truth table.

Applications of a Multiplexer

Where multiplexers are used:

Communication Systems

In communications, multiplexers share one path among several sources, for example audio and video streams, so fewer transmission lines are needed. Time or frequency slots, not a single logic MUX, are what send those streams at once.

Telephone Networks

Telephone networks have long used multiplexing so many voice channels share one trunk.

Computer memory

Inside memory and buses, muxes steer address and data so fewer copper runs are needed between blocks.

Spacecraft and satellite links

Spacecraft and satellites also multiplex telemetry onto the downlink. GPS and GSM do not carry that spacecraft telemetry to Earth.

Other uses:

Muxes appear in TDM and FDM equipment and in front of A/D and behind D/A converter channels.
A mux can implement a Boolean function of several variables from its truth table.
Data-acquisition systems use analog muxes to scan many sensors with one ADC.