Digital Logic Problems II

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Digital Logic Problems (II)
Chapter 1

• Prof. Sin-Min Lee
• Department of Mathematics and Computer Science

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Multiplexers

• A combinational circuit that selects info from
  one of many input lines and directs it to the
  output line.
• The selection of the input line is controlled by
  input variables called selection inputs.
• They are commonly abbreviated as MUX.

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Implementing Boolean functions with multiplexers

• Any Boolean function of n variables can be
  implemented with 2n-1-to-1 multiplexer. The
  procedure for implementing a Boolean function
  with a multiplexer is
• 1.Express the function in its sum of minterms
  form.
• 2. Order the sequence of variables chosen for the
  minterms. Suppose the sequence is , where A is
  the leftmost variable, and are the remaining
  n-1 variables.

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• 3.Connect the n-1 variables to the selection
  lines of the 2n-1-to-1 multiplexer, with B
  connected to the highest order selection line,
  and so on.
• 4. Construct the implementation table List all
  the minterms in two rows.
• The first row consists of minterms 0 to 2n-1 -1
  (in all of which A is complemented).
• The second row consists of minterms 2n-1 to 2n-1
  (in all of which A is uncomplemented). .

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• 5. Circle all the minterms of the function and
  inspect each column in the implementation table
  separately
• If the two minterms in a column are not circled,
  apply 0 to the corresponding multiplexer input.
• If the two minterms are circled, apply 1 to the
  corresponding multiplexer input.
• If the bottom minterm is circled, and the top is
  not circled, apply A to the corresponding
  multiplexer input.
• If the top minterm is circled but not the bottom,
  apply A

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• Multiplexers and decoders are used when many
  lines of information are being gated and passed
  from one part of a circuit to another.
• Multiplexing is when multiple data signals share
  a common propagation path. Time multiplexing is
  when different signals travel along the same wire
  but at different times. These devices have data
  and address lines, and usually include an
  enable/ disable input. When the device is
  disabled the output is locked into some
  particular state and is not effected by the
  inputs.

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Consider the function of 3 variables
1. Input variables B and C are applied to the
selection lines s1 and s0, respectively. 2.
Construct the implementation table, and circle
all the minterms of the function in the
implementation table
3. Apply 0, 1, A, and A   to the inputs I0
through I3.
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Multiplexers (continued)

• S0 and S1 are the selection inputs.
• D0, D1, D2, D3 are the input lines.

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Multiplexers (continued)

• MUX blocks can be combined in parallel with
  common selection and enable inputs to perform
  selection on multiple bit quantities.

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Combinational circuit implementation using MUX

• We can use Multiplexers to express Boolean
  functions also.
• Expressing Boolean functions as MUXs is more
  efficient than as decoders.
• First n-1 variables of the function used as
  selection inputs last variable used as data
  inputs.
• If last variable is called Z, then each data
  input has to be Z, Z, 0, or 1.

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Karnaugh Map Method of Multiplexer Implementation
Consider the function
A is taken to be the data variable and B,C to be
the select variables.
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Example of MUX combo circuit

• F(X,Y,Z) Sm(1,2,6,7)

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Implement g(w,x,y)wxxywy using a 4-1
multiplexer.
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