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Wednesday, December 25, 2013

DIGITAL BASIC - 1.5 : Multiplexer (MUX)



Index Chapter1 Chapter2 Chapter3 Chapter4
Digital
Background
Semiconductor Background CMOS
Processing


1.1 1.2 1.3a 1.3b 1.4 1.5 1.6
Number
System
Digital
Arithmetic
Logic
Gates
Logic
Gates
Combinational
Circuits
Multiplex
(MUX)


De-Multiplexer:
  • Receives information on a single line and transmits that information on one of 2n possible output lines.
  • The selection of specific output line is controlled by the bit values of ‘n’ selection lines.
Multiplexer:
  • Multiplexing means transmitting a large number of information units over a smaller number of channels lines.
  • A digital multiplexer is a combinational circuit that selects binary information from one of many inputs lines and directs it in a signal output line.
  • The selection of a particular line is controlled by a set of selection lines.
  • Normally, there are 2n input lines and ‘n’ selection lines whose bit combinations determine which input is selected.”
  • Multiplexers can be used for the implementation of Boolean functions, combinational circuits. They can also used for parallel to serial conversion.
  • Multiplexer is also called data selector or universal circuit.
  • It is used for connection two or more sources to a single destination among computer units and it is useful for constructing a common bus system







Important Points:
To implement 2n :1 MUX by using 2:1 MUX, the total number or 2:1 MUX required is 2n-1

Given MUX
To be implemented
MUX
Required
No of MUX
4 : 1
16 : 1
4+1=5
4 : 1
64 : 1
16+4+1=21
8 : 1
64 : 1
8+1 =9
8 : 1
256 : 1
32+4+1=37

Implementation of Higher Order MUX using Lower Order MUX:
4:1 MUX by 2:1 MUX

Total number of 2: 1 MUX = 3

MUX as a universal logic gate

Gate Type
Implemented by MUX  + Equation
Buffer
Y=output = A

NOT/Inverter
Y=A’

AND
Y=A.B

OR
Y=A+B

NOR
Y=(A+B)’

NAND
Y=(A.B)’

XOR


XNOR



Implementation of Boolean function using Multiplexer:
The Boolean function may be implemented in 2n to 1 multiplexer.
  • If we have a Boolean function of n variables, we take n-1 of these variables and connect them to the selection lines of a multiplexer (let’s say these are “select variables”).
  • The remaining single variable (MSB variable) of the function is used for the inputs of the multiplexer (let’s say these are “input variable”).
  • Now form the implementation table
    • First row lists all those minterms where “input variable” is complemented (say 0).
    • Second row lists all those minterms where “input variable” is in its normal form (say 1).
  • The minterms are circled as per the given Boolean function. Now use the following steps to find out final multiplexer inputs.
    • If the 2 minterms in a column are not circled, 0 is placed to the corresponding multiplexer inputs.
    • If the 2 minterms in a column are circled, 1 is placed to the corresponding multiplexer inputs.
    • If the minterms in the second row is circled and the first row is not circled, apply second row of variable to the corresponding multiplexer inputs.
    • If the minterms in the first row is circled and not the second row, apply first row of the variable to the corresponding multiplexer inputs.
Example:  Implementation of given function using 8 to 1 multiplexer
F(A,B,C,D) = Ʃ (1,3,4,11,12,13,14,15)
Solution.
  • Total number of variable n = 4 (A,B,C,D)
  • Number of select lines:  n-1= 3 (B, C, D)
  • The given function has 4 variable, so 16 possible minterms (0 – 15) are entered in the implementation table.
  • All the minterms are divided into 2 groups
    • The first group (0-7) minterms  are entered in the first row  (Variable A =0)
    • The second group (8–15) minterms are entered in the second row (Variable A= 1)
  • Circle the minterm number as per function, which you have to implement (in this case it’s 1,3,4,11,12,13,14,15)
  • Find out the multiplexer input as per above given steps.








Implementation Table

Given multiplexer is 8:1
Logic diagram

Example
Implement the following Boolean function using 8 : 1 MUX
F(A,B,C,D) = Ʃ m(0,1,2,4,6,9,12,14)
Solution.
Select lines are B, C and D

Follow all the steps as per above points.


Example
Implement the following Boolean function with 8 : 1 multiplexer
F(A,B,C,D) = ∏M (0,3,5,6,8,9,10,12,14)

Solution
The given maxterms are inverted to obtain minterms. From the minterms, we can implement the above Boolean function by using 8 : 1 multiplexer. Select lines are B, C and D, the input variable is A.
F(A,B,C,D) = Ʃ m(1,2,4,7,11,13,15)


Example  
Implement the following Boolean function with 8 : 1 multiplexer
F(A,B,C,D) = Ʃ m (0,2,6,10,11,12,13) + Ʃ d(3,8,14)
Solution.
The Boolean function has three don’t care conditions which can be treated as either 0’s or 1’s. In this example don’t care condition is consider as 1.




7 comments:

  1. Very good explaination

    ReplyDelete
  2. very helpful..are you going to post remaining concepts of digital designing

    ReplyDelete
    Replies
    1. Yes I will do. but it will take some time.

      Delete
  3. How many 16:1mux required for 64:1 mux

    ReplyDelete
  4. Someone gave me answer as 4 -64/16= 4
    And some as 5- 64/16=4+1(for next one)=5
    So which one is correct 5 or 4

    ReplyDelete
    Replies
    1. if you can't use anyother gate.. like AND, OR, XOR .. then 5 is correct. but in case you can use any such basic gates - then 4 is correct answer.

      Delete
  5. Is any mathematical formula to make 64:1 mux by using 4:1 mux???

    ReplyDelete

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