Today, well study multiplexers, which are just as commonly used as the decoders we presented last ti

1
Multiplexers

• Today, well study multiplexers, which are just
  as commonly used as the decoders we presented
  last time. Again,
• These serve as examples for circuit analysis and
  modular design.
• Multiplexers can implement arbitrary functions.
• We will actually put these circuits to good use
  in later weeks, as building blocks for more
  complex designs.

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Multiplexers

• A 2n-to-1 multiplexer sends one of 2n input lines
  to a single output line.
• A multiplexer has two sets of inputs
• 2n data input lines
• n select lines, to pick one of the 2n data inputs
• The mux output is a single bit, which is one of
  the 2n data inputs.
• The simplest example is a 2-to-1 mux
• The select bit S controls which of the data bits
  D0-D1 is chosen
• If S0, then D0 is the output (QD0).
• If S1, then D1 is the output (QD1).

Q S D0 S D1
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More truth table abbreviations

• Here is a full truth table for this 2-to-1 mux,
  based on the equation
• Here is another kind of abbreviated truth table.
• Input variables appear in the output column.
• This table implies that when S0, the output
  QD0, and when S1 the output QD1.
• This is a pretty close match to the equation.

Q S D0 S D1
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A 4-to-1 multiplexer

• Here is a block diagram and abbreviated truth
  table for a 4-to-1 mux.
• Be careful! In LogicWorks the multiplexer has an
  active-low EN input signal. When EN 1, the mux
  always outputs 1.

Q S1 S0 D0 S1 S0 D1 S1 S0 D2 S1 S0 D3
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Implementing functions with multiplexers

• Muxes can be used to implement arbitrary
  functions.
• One way to implement a function of n variables is
  to use an n-to-1 mux
• For each minterm mi of the function, connect 1 to
  mux data input Di. Each data input corresponds to
  one row of the truth table.
• Connect the functions input variables to the mux
  select inputs. These are used to indicate a
  particular input combination.
• For example, lets look at f(x,y,z) ?m(1,2,6,7).

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A more efficient way

• We can actually implement f(x,y,z) ?m(1,2,6,7)
  with just a 4-to-1 mux, instead of an 8-to-1.
• Step 1 Find the truth table for the function,
  and group the rows into pairs. Within each pair
  of rows, x and y are the same, so f is a function
  of z only.
• When xy00, fz
• When xy01, fz
• When xy10, f0
• When xy11, f1
• Step 2 Connect the first two input variables of
  the truth table (here, x and y) to the select
  bits S1 S0 of the 4-to-1 mux.
• Step 3 Connect the equations above for f(z) to
  the data inputs D0-D3.

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Example multiplexer-based adder

• Lets implement the adder carry function,
  C(X,Y,Z), with muxes.
• There are three inputs, so well need a 4-to-1
  mux.
• The basic setup is to connect two of the input
  variables (usually the first two in the truth
  table) to the mux select inputs.

With S1X and S0Y, then QXYD0 XYD1 XYD2
XYD3
Equation for the multiplexer
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Multiplexer-based carry

• We can set the multiplexer data inputs D0-D3, by
  fixing X and Y and finding equations for C in
  terms of just Z.

When XY00, C0 When XY01, CZ When XY10,
CZ When XY11, C1
C X Y D0 X Y D1 X Y D2 X Y D3 X
Y 0 X Y Z X Y Z X Y 1 X Y Z X Y
Z XY ?m(3,5,6,7)
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Multiplexer-based sum

• Heres the same thing, but for the sum function
  S(X,Y,Z).

When XY00, SZ When XY01, SZ When XY10,
SZ When XY11, SZ
S X Y D0 X Y D1 X Y D2 X Y D3 X
Y Z X Y Z X Y Z X Y Z ?m(1,2,4,7)
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Dual multiplexer-based full adder

• We need two separate 4-to-1 muxes one for C and
  one for S.
• But sometimes its convenient to think about the
  adder output as being a single 2-bit number,
  instead of as two separate functions.
• A dual 4-to-1 mux gives the illusion of 2-bit
  data inputs and outputs.
• Its really just two 4-to-1 muxes connected
  together.
• In LogicWorks, its called a Mux-4x2 T.S.

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Dual muxes in more detail

• You can make a dual 4-to-1 mux by connecting two
  4-to-1 muxes. (Dual means two-bit values.)
• LogicWorks labels input bits xDy, which means
  the xth bit of data input y.
• In the diagram on the right, were using S1-S0 to
  choose one of the following pairs of inputs
• 2D3 1D3, when S1 S0 11
• 2D2 1D2, when S1 S0 10
• 2D1 1D1, when S1 S0 01
• 2D0 1D0, when S1 S0 00

You can see how 8-way multiplexer (k-to-1) can be
used to select from a set of (k) 8-bit numbers
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Summary

• A 2n-to-1 multiplexer routes one of 2n input
  lines to a single output line.
• Just like decoders,
• Muxes are common enough to be supplied as
  stand-alone devices for use in modular designs.
• Muxes can implement arbitrary functions.
• We saw some variations of the standard
  multiplexer
• Smaller muxes can be combined to produce larger
  ones.
• We can add active-low or active-high enable
  inputs.
• As always, we use truth tables and Boolean
  algebra to analyze things.
• Wednesday, we start to discuss how to build
  circuits to do arithmetic.

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Practice

• Implement F(x,y,z) ?m(1,3,4,5) using a decoder.

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Practice

• Implement F(x,y,z) ?m(1,3,4,5) using a
  multiplexer.