Multiplexer

Multiplexer

• In digital systems, many times it is necessary to select single data line from several data–input lines, and the data from the selected data line should be available on the output. The digital circuit which does this task is a multiplexer.

• It is a digital switch. It allows digital information from several sources to be routed onto a single output line, as shown in Fig. 4.8.1. The basic multiplexer has several data–input lines and a single output line. The selection of a particular input line is controlled by a set of selection lines. Since multiplexer selects one of the input and routes it to output, it is also known as data selector. Normally, there are 2ⁿ input lines and n selection lines whose bit combinations determine which input is selected. Therefore, multiplexer is 'many into one' and it provides the digital equivalent of an analog selector switch.

 (a) Block diagram of 2ⁿ : 1 multiplexer (b) Equivalent circuit

1. 2 : 1 Multiplexer

Fig. 4.8.2 (a) shows 2 : 1 multiplexer. D₀ is applied as an input to one AND gate and D₁ is applied as an input to another AND gate. Enable input is applied to both gates as one input. Selection line S is connected as second input to second AND gate. An inverted S is applied as second input to first AND gate. Outputs of both AND gates are applied as inputs to OR gate.

(a) Logic diagram (b) Function table (c) Logic symbol

Working

• When E = 0, output is 0, i.e. Y = 0 irrespective of any input condition. When E = 1 the circuit works as follows :

• When S = 0, the inverted S, that is 1 gets applied as second input to first AND gate. Since S is applied directly as input to second AND gate; its output goes zero irrespective of first input. Since the second input of first AND gate is 1, its output is equal to its first input, that is D₀. Hence Y = D₀.

• Exactly opposite is the case when S = 1. In this case, second AND gate output is equal to its first input D₁ and first AND gate output is 0. Hence Y = D₁.these cases are summarized in truth table shown in Fig. 4.8.2 (b).

Deriving realization expression

• Table 4.8.1 shows the truth table for 2 : 1 multiplexer. From the truth table it is clear that Y = 1 when ED₀ = 1 or ESD₁ =1 as indicated by shaded rows.

2. 4 : 1 Multiplexer

• Fig. 4.8.3 (a) shows 4–to–1 line multiplexer. Each of the four lines, D₀ to D₃, is applied to one input of an AND gate. Selection lines are decoded to select a particular AND gate.

 (a) Logic diagram (b) Function table (c) Logic symbol

• For example, when S₁S₀ = 0 1, the AND gate associated with data input D₁ has two of its inputs equal to 1 and the third input connected to D₁. The other three AND gates have at least one input equal to 0, which makes their outputs equal to 0. The OR gate output is now equal to the value of D₁, thus we can say data bit D₁ is routed to the output when S₁ S₀ = 0 1.

 

3. 8 : 1 Multiplexer

Fig. 4.8.4 shows 8 : 1 multiplexer.

 (a) Logic diagram (b) Function table (c) Logic symbol

• There are eight input lines one output line and three select lines. As shown in the function table, the selection of a particular input line is controlled by three and selection lines.

 

4. Quadruple 2 to 1 Multiplexer

• In some cases, two or more multiplexers are enclosed within one IC package, as shown in Fig. 4.8.5. Fig. 4.8.5 shows quadruple 2–to–1 line multiplexer, i.e. four multiplexers, each capable of selecting one of two input lines. Output Y₁ can be selected to be equal to either A₁ or B₁. Similarly output Y₂ may have the value of A₂ or B₂, and so on. The selection line S selects one of two lines in all four multiplexers. The control input E enables the multiplexers in the 0 state and disables them in the 1 state. When E= 1, outputs have all 0's, regardless of the value of S.

Function table

 

5. Expanding Multiplexers

• It is possible to expand range of inputs for multiplexer beyond the available range by interconnecting several multiplexers in cascade. The circuit with two or more multiplexers connected to obtain the multiplexer with more number of inputs is known as multiplexer tree.

Examples for Understanding

Example: 1

Design 16 : 1 multiplexer using 8 : 1 multiplexer.

Solution :

Step 1 : Connect the select lines (S₂, S₁ and S₀) of two multiplexers in parallel.

Step 2 : Connect most significant select line (S₃) such that when S₃ = 0 MUX1 is enabled and when S₃ = 1, MUX2 is enabled.

Step 3 : Logically OR the outputs of two multiplexers to obtain the final output Y. Refer Fig. 4.8.6.

Example: 2

Design 16 : 1 multiplexer using 4 : 1 multiplexers.

Solution :

Since there are 16–inputs for the multiplexers we require require four 4 : 1 multiplexers to satisfy input needs. The four outputs of 4: 1 multiplexers are again multiplexed by 4: 1 multiplexer to generate final output.

Step 1: Connect the select lines (S₁ and S₀) of four multiplexers in parallel.

Step 2: Connect the most significant select lines (S₃ and S₂) to the MUX5.

Step 3: Connect the outputs Y₀, Y₁, Y₂ and Y₄ of four multiplexers as data inputs for the MUX 5, as shown in Fig. 4.8.7.

Examples with Solutions

Example: 3

Draw 32 : 1 multiplexer using two 16: 1 multiplexers and one 2 : 1 multiplexer.

Solution :

Fig. 4.8.8 shows the connection diagram for 32 : 1 multiplexer using two 16 : 1 multiplexer and one 2 : 1 multiplexer. Here, select lines S₀, S₁, S₂ and S₃ are connected in parallel to 16 : 1 multiplexers and select line S₄ is used to select either Y₁ or Y₂ using 2: 1 multiplexer.

Example: 4

Draw 64 : 1 multiplexer tree using 16 : 1 multiplexer.

Solution :

Fig. 4.8.9 shows 64 : 1 multiplexer using four 16 : 1 multiplexers and one 4 : 1 multiplexer.

Examples for Practice

Example: 5 Draw the block diagram of a 4 : 1 multiplexer using 2 : 1 MUX

Example: 6 Design 32 : 1 MUX using 8 : 1 MUX.

 

6. Implementation of Combinational Logic using MUX

• A multiplexer consists of a set of AND gates whose outputs are connected to single OR gate. Because of this construction any Boolean function in a SOP form can be easily realized using multiplexer. Each AND gate in the multiplexer represents a minterm. In 8 to 1 multiplexer, there are 3 select inputs and 23 minterms. By connecting the function variables directly to the select inputs, a multiplexer can be made to select the AND gate that corresponds to the minterm in the function. If a minterm exists in a function, we have to connect the AND gate data input to logic 1; otherwise we have to connect it to logic 0. This is illustrated in the following example.

Examples for Understanding

Example: 7

Implement the given function using multiplexer. F(x,y,z) = (0,2,6,7).

Solution :

Step 1: Select the multiplexer. Here, Boolean expression has 3 variables, thus we require 2³ = 8: 1 multiplexer.

Step 2 : Connect inputs corresponds to the present minterms to logic 1.

Step 3 : Connect remaining inputs to logic 0.

Step 4 : Connect input variables to select lines of MUX.

Example: 8

Implement the Boolean function represented by the given truth table using multiplexer.

Solution :

Step 1: Select the multiplexer. Here, there are three input variables, thus we require 2³ = 8 : 1 multiplexer.

Step 2 : Find the minterm expression.

Minterm expression for given truth table is Σ m (1, 2, 5, 7).

Step 3 : Connect inputs corresponds to the present minterms to logic 1.

Step 4 : Connect remaining inputs to logic 0.

Step 5 : Connect input variables to select lines of MUX.

• In the above example, we have seen the method for implementing Boolean function of 3 variables with 2³(8) – to –1 multiplexer. Similarly, we can implement any Boolean function of n variables with 2ⁿ–to–1 multiplexer. However, it is possible to do better than this. If we have Boolean function of n + 1 variables, we take n of these variables and connect them to the selection lines of a multiplexer. The remaining single variable of the function is used for the inputs of the multiplexer. In this way we can implement any Boolean function of n variables with 2^n–1–to–1 multiplexer. Let us see some example.

Example: 9

Implement the following Boolean function using 4: 1 multiplexer. F (A, B, C) = Σ m (1, 3, 5, 6).

Solution :

Step 1: Connect least significant variables as a variables as a select inputs of multiplexer. Here,  connect C to S₀ and B to S₁.brobable

Step 2 : Derive inputs for multiplexer using implementation table.

• As shown in Fig. 4.8.12 (a) the implementation table is nothing but the list of the inputs of the multiplexer and under them list of all the minterms in two rows. The first row lists all those minterms where A is complemented, and the second row lists all the minterms with A uncomplemented. The minterms given in th function are circled and then each column is inspected separately as follows :

■ If the two minterms in a column are not circled, 0 is applied to the corresponding multiplexer input (see column 0).

■ If the two minterms in a column are circled, 1 is applied to the corresponding multiplexer input (see column 1).

■ If the minterm in the second row is circled and minterm in the first row is not circled, A is applied to the corresponding multiplexer input (see column 2).

■ If the minterm in the first row is circled and minterm in the second row is not circled, Ā is applied to the corresponding multiplexer input (see column 3).

Example: 10

Implement the following Boolean function using 8 : 1 multiplexer 

Solution :

Step 1: Express Boolean function in the minterm form.

The given Boolean expression is not in standard SOP form. Let us first convert this in standard SOP form

= Σ m (6, 4, 15, 11, 3, 5, 1) = Σ m (1, 3, 4, 5, 6, 11, 15)

Step 2 : Implement it using implementation table.

From the Boolean function in the minterm form can be implemented using 8 :1 multiplexer as follows :

Example: 11

Implement the following Boolean function with 8: 1 multiplexer

F(A, B, C, D) = π M (0, 3, 5, 8, 9, 10, 12, 14).

Solution :

Here, instead of minterms, maxterms are specified. Thus, we have to circle maxterms which are not included in the Boolean function. Fig. 4.8.14 shows the implementation of Boolean function with 8:1 multiplexer.

Example: 12

Implement the following Boolean function with 8 : 1 multiplexer.

F(A, B, C, D) = Σ m (0, 2, 6, 10, 11, 12, 13) + d (3, 8, 14).

Solution :

In the given Boolean function three don't care conditions are also specified. We know that don't care conditions can be treated as either 0s or 1s. Fig. 4.8.15 shows the implementation of given Boolean function using 8 : 1 multiplexer.

In this example, by taking don't care conditions 8 and 14 as 1s we have eliminated Ā term and hence the inverter.

Examples with Solutions

Example: 13

Implement full adder circuit using 8 : 1 multiplexer.

Solution :

Example: 14

Implement full adder circuit using dual 4 to 1 multiplexer.

Solution : Implementation tables :

Example: 15

Realize F(w, x, y, z) = Σ (1, 4, 6, 7, 8, 9, 10, 11, 15) using 4 to 1 MUX.

Solution :

Example: 16

Implement the function with a multiplexer.

F(A, B, C, D) = Σ (0,1,3,4,8,9,15)

Solution : Implementation table :

Implementation

Example: 17

Implement the following function using suitable multiplexer. F(x, y, z) = Σ m (0, 2, 5, 7).

Solution :

Example: 18

Implement the given Boolean function using 8 : 1 multiplexer.

F(A, B, C) = Σ (1, 3, 5, 6).

Solution :

Example: 19

Implement the following function using suitable multiplexer

F(A, B, C, D) = Σ (1, 3, 4, 11, 12, 13, 14, 15).

Solution :

Example: 20

Realize the Boolean function using appropriate multplexer

 F(A, B, C) = Σ (0, 1, 3, 7).

Solution :

Examples with practice

Example: 21

Implement the following function using 8:1 multiplexer f(a, b, c, d) = Σ m (0, 1, 5, 6, 8, 10, 12, 15)  

Example: 22

Implement Boolean function f = AB + D + AC using 8 : 1 multiplexer.

7. Applications of Multiplexer

1. They are used as a data selector to select one out of many data inputs.

2. They can be used to implement combinational logic circuit.

3. They are used in time multiplexing systems.

4. They are used in frequency multiplexing systems.

5. They are used in A/D and D/A converter.

6. They are used in data acquisition systems.

8. Multiplexer ICs

IC number          –       Function

74150                    –      16: 1 multiplexer

74151                     –     8: 1 multiplexer

74153                     –    Dual 4: 1 multiplexer

74157                     –   Quad 2–input multiplexer

 

Review Questions

1. Define multiplexer.

2. What is data selector?

3. Draw and explain the working of 2 : 1 multiplexer and realize it using basic gates.

4. Explain 4 : 1 multiplexer with the help of logic circuit and truth table.

5. Write the truth table of a 4: 1 multiplexer.

6. Explain 8 : 1 multiplexer with the help of logic circuit and truth table.

7. Explain the concept and working of quadruple 2 to 1 line multiplexer.

8. State the applications of multiplexers.

9. Illustrate the concept of basic 4–input mutiplexer.