Editing multiplexer

{{title|Multiplexer (MUX)}}{{logic device
|title            = Multiplexer
|symbol title     = Typical Symbol (2:1 MUX)
|symbol body      = [[File:Mux 2 1.svg|150px|center]]
|functional body  = [[File:mux functional.gif|center]]
|truth table      = {{truth table/mux}}
}}
A '''multiplexer''' ('''mux''') or a '''data selector''' or '''input selector''' is a [[combinational circuit]] device that selects one of ''N'' inputs and provides it on its output. A set of inputs called select lines determine which input should be passed to the output. For a 2:1 (two-to-one) MUX, when sel is 0, q = a and when sel is 1, q = b. A multiplexer with 2<sup>N</sup> input lines requires ''N'' select lines.

Multiplexers are useful in any application in which data must be chosen from multiple sources to a single destination. Multiplexers are also heavily used in I/O operations, data buses, and register files. Additionally multiplexers have also found their way to various other circuits such as adders.

== Description ==
<span style="float:right; display: inline-block;">[[File:3reg mux31.svg|200px|A simple selection of 3 [[register]]s.]]</span>
A multiplxer is a device that receives multiple inputs from usually different sources. A set of select lines are then used to choose which of those inputs gets produced as output. Signals to the select lines usually come from a control unit that determins which, if any, of the signals should be routed to some destination. MUXes are core components in most digital systems as they can be used to pass the correct signal based on some conditional logic. For example, consider a [[data bus]] that is connected to [[register file|multiple]] memory [[register|storage units]]. One can use a multiplexer to select which of those lines should be going to the shared data bus.

=== Enable ===
It's often desirable to add an [[enable]] (or strobe) input ''EN'' to a multiplexer. An enable input makes the multiplexer operate. When ''EN = 0'', the output is either [[LOW]] or [[High-Z]] (depending on the specific device). When ''EN = 1'', the multiplexer performs its operation depending on the selection line.

== Design ==
[[File:multiplxer general line count.svg|left|150px|2<sup>n</sup>:1 MUX]]
For a multiplexer with <math>N</math> inputs, you also need <math>\lceil\log_2(N)\rceil</math> selection lines. It does mean that for multiplexers with odd number of inputs, some selection line combinations are not allowed (e.g. in a 3:1 MUX, the SEL=11 combination is not allowed). MUXes in the form <math>2^N</math>:1 can be expressed by
:<math>\text{MUX}(I_0, ..., I_k) = \sum^{2^n-1}_{k=0}{m_k I_k}</math>
Where <math>m_k</math> is the kth [[minterm]] of the variable.

For 2, 4, and 8-input multiplexers the equations are thus:

:<math>\text{MUX21}(a, b) = (A\overline{S_0})+(BS_0)</math>
:<math>\text{MUX41}(a, b, c, d) = (A\overline{S_0 S_1})+(B\overline{S_0}S_1)+(C S_0\overline{S_1})+(D S_0 S_1)</math>
:<math>\text{MUX81}(a, b, c, d, e, f, g) = (A\overline{S_0 S_1 S_2})+(B\overline{S_0 S_1}S_2)+(C \overline{S_0} S_1 \overline{S_2})+(D \overline{S_0} S_1 S_2)+(E S_0 \overline{S_1 S_2})+(F S_0 S_1 \overline{S_2})+(G S_0 S_1 S_2)</math>
{{clear}}
== Implementations ==
Many different variations of multiplexers exist. Typically larger multiplxers (over 8 or 16 inputs) are built using smaller multiplxers using a [[/tree|multiplexer tree]].

=== 2:1 MUX (MUX21) ===
{| class="wikitable" style="float: right;"
! style="width:20em;" colspan="4" | 2:1 Mux
|-
! Sel !! A !! B !! Q
|-
| 0 || 0 || {{X}} || 0
|-
| 0 || 1 || {{X}} || 1
|-
| 1 || {{X}} || 0 || 0
|-
| 1 || {{X}} || 1 || 1
|}
[[File:Mux_2_1_equivalence.svg|left|200px]]
The simplest multiplexer is the '''2:1 MUX''' (or '''MUX21''') which simply selects its output from just two possible inputs. Its selection lines is therefore made of a [[bit|single bit]]. A [[truth table]] is provided on the right. The logic function of a 2:1 Mux is:

:<math>Q = (A \land \overline{S_0}) \lor (B \land S_0)</math>

====CMOS====
A very fast and compact, [[CMOS]]-based [[pass-transistor logic|PTL logic]], 2:1 MUX device can be built using two [[transmission gate]]s as shown below. While smaller overall, this multiplexer is also [[nonrestoring]].

{| class="center"
| [[File:MUX21 using transmission gates.svg|200px]]
|[[File:Transmission gate 2 1 mux.png|200px]]
|}
{{clear}}

=== 4:1 Mux ===
{| class="wikitable" style="float: right;"
|+ 4:1 Mux 
|-
! Sel<0> !! Sel<1> !! I<0> !! I<1> !! I<2> !! I<3> !! Q
|-
| 0 || 0 || 0 || X || X || X || 0
|-
| 0 || 0 || 1 || X || X || X || 1
|-
| 0 || 1 || X || 0 || X || X || 0
|-
| 0 || 1 || X || 1 || X || X || 1
|-
| 1 || 0 || X || X || 0 || X || 0
|-
| 1 || 0 || X || X || 1 || X || 1
|-
| 1 || 1 || X || X || X || 1 || 1
|-
| 1 || 1 || X || X || X || 0 || 0
|}
A '''4:1''' Multiplexer is a common multiplexer that takes selects one input among 4 and connects it to its output based on a 2-bit select line. There are many way to construct a 4:1 Mux, one possibility is using 2:1 Mux as shown below:

[[File:Mux 4 1 using mux 2 1.svg|200px|center]]

Alternatively, a 4:1 Mux can be built out of basic gates. Its function is shown below:

Q = <math>(A \land \overline S_0 \land \overline S_1) \lor (B \land \overline S_0 \land S_1) \lor (C \land S_0 \land \overline S_1) \lor (D  \land S_0 \land S_1)</math>

Where A, B, C, and D are the four inputs. Q is the output.

=== Larger Multiplexers ===
Multiplexers generally only come in a few common sizes. Even in [[ASIC]] design, arbitrary sized multiplexers are not always offered. Large multiplexers can always be built from a collection of smaller ones. Consider a [[register file]] with 32 registers where we only want to select a single register at any given time. Such multiplexer can be design from four 8:1 Mux.

==== 74151 - 8:1 Mux ====
[[File:Mux 8 1 equivalence.svg|500px|right|thumb|74151 IC Chip]]
A common multiplexer is the 8:1 Mux which selects one of 8 bits of input. The [[74151]] is a popular 16-pin [[DIP]] IC that implements an 8:1 mux. Note that the implementation below is an active-low.

To the right is the typical schematic of the 74151, 16-pin DIP IC. Vcc is on pin 16 and GND is on pin 8. Pins 5 and 6 are the outputs, the output on pin 6 is the inverted version of the output on pin 5. The enable is on pin 7.

{| class="wikitable"
|-
! colspan="6" | 8:1 Mux
|-
! colspan="4" | Inputs !! rowspan="2" colspan="2" | Output
|-
! colspan="3" | Select !! Enable
|-
! Sel<0> !! Sel<1> !! Sel<2> !! EN !! Q !! <span style="text-decoration:overline;">Q</span>
|-
| X || X || X || 1 || 1 || 0
|-
|0 || 0 || 0 || 0 || I<0> || <span style="text-decoration:overline;">I<0></span>
|-
|0 || 0 || 1 || 0 || I<1> || <span style="text-decoration:overline;">I<1></span>
|-
|0 || 1 || 0 || 0 || I<2> || <span style="text-decoration:overline;">I<2></span>
|-
|0 || 1 || 1 || 0 || I<3> || <span style="text-decoration:overline;">I<3></span>
|-
|1 || 0 || 0 || 0 || I<4> || <span style="text-decoration:overline;">I<4></span>
|-
|1 || 0 || 1 || 0 || I<5> || <span style="text-decoration:overline;">I<5></span>
|-
|1 || 1 || 0 || 0 || I<6> || <span style="text-decoration:overline;">I<6></span>
|-
|1 || 1 || 1 || 0 || I<7> || <span style="text-decoration:overline;">I<7></span>
|}

[[File:74151.svg|left|thumb|300px|74151 IC 8:1 MUX]]
{{clear}}

== Tri-State Outputs ==
Some commercial multiplexers have tri-state outputs. When the EN input is LOW, instead of the output being forced into 0, it gets forced into a Hi-Z state.

== See also ==
* [[choose function]]

[[Category:Logic gates]]