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.