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 2
^{n }possible output lines. - The selection of specific output line is controlled by the bit values of ‘n’ selection lines.

- 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 2
^{n}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 2

^{n }:1 MUX by using 2:1 MUX, the total number or 2:1 MUX required is 2^{n}-1Given 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 2

^{n}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.