Multiplexers

Multiplexers

AU: Dec.-08,11,14, 16,19, May-04,06,07,11,12,14,17

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 the Fig. 1.17.1. 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  input linesand 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.

2 : 1 Multiplexer

• Fig. 1.17.2 (a) shows 2: 1 multiplexer.

• D₀ to 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.

Working

• When E = 0, output is 0, i.e. Y = 0 irrespective of any input condition.

• When E = 1, the circuit works as follows:

a. 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.

b. Exactly opposite is the case when S = 1. In this case, second AND gate output is equal to its first input D₀ and first AND gate output is 0. Hence Y = D₀. Both these cases are summarized in truth table shown in Fig. 1.17.2 (b).

Deriving realization expression

The Table 1.17.1 shows the truth table for 2 : 1 multiplexer. From the truth table it is clear that Y=1 when EŜD₀=1 or ESD₁=1 as indicated by shaded rows.

4 :1 Multiplexer

• Fig. 1.17. (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.

• For example, when S₁ S₀ = 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.

8: 1 Multiplexer

• Fig. 1.17.4 shows 8 : 1 multiplexer.

• 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 selection lines.

Quadruple 2 to 1 Multiplexer

• In some cases, two or more multiplexers are enclosed within one IC package, as shown in Fig. 1.17.5.

• The Fig. 1.17.5 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 A₂ or B₂.

• Output Y₂ may have the value of A₂ or B₂, and soon.

•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

Illustrative Examples

Example 1.17.1 Construct and explain the working of 2 to 1 line multiplexer using

three-state gates.

Solution : Three state/tri-state gates exibit three states. Two states: Logic 1 and Logic 0 as in a conventional gate. The third state is a high-impedance state in which logic behaves like an open circuit. Fig. 1.17.6 (a) shows 2 to 1 line multiplexer using 2 three-state buffers and an inverter. Since buffers are tri-state buffer their outputs can be connected together to form a single output line. The circuit behaves as per the truth table shown in Fig. 1.17.6 (b).

 

Example 1.17.2 Construct 4 to 1 line multiplexer using three-state gates and decoder.

Solution: Here, the output of 4 three-state buffers are connected together to form a single output line. The control inputs for the four buffer are determined using 2: 4 decoder as shown in the Fig. 1.17.7. When S₁ S₀ = 00, the upper control buffer is selected and I₀ signal is routed to the output. Similarly, according to the select inputs of the decoder one of input is routed to the output.

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 1.17.3 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 MUX 1 is enabled and when S₃ = 1, MUX 2 is enabled.

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

Example 1.17.4 Design 16 : 1 multiplexer using 4 : 1 multiplexers.

Solution : Since there are 16-inputs for the multiplexers we 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 MUX 5.

Step 3 : Connect the outputs Y₀ ,Y₁ ,Y₂ and Y₄ of four multiplexers as data inputs for the MUX 5, as shown in the Fig. 1.17.9.

Examples for Practice

Example 1.17.5 Design 32: 1 MUX using 8: 1 MUX.

Example 1.17.6 Show how two 2: 1 multiplexers(with no added gates) could be connected to form a 3: 1 multiplexer. Input selection should be as follows:

If AB = 00, Select I₀.

AB = 01, Select I₁.

AB = 1 × (B is don't care), Select I₂.

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.

Illustrative Examples

Example 1.17.7 Implement the given function using 8 : 1 multiplexer.

F(A,B,C) = ∑m(1, 3, 5, 6).   AU: Dec.-14, Marks 2

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 1.17.8 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 ∑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 nvariables 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.

Example 1.17.9 Implement the following Boolean function using 4: 1 multiplexer. F (A, B, C) = ∑m (1, 3, 5, 6)

Solution :

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

Step 2: Derive inputs for multiplexer using implementation table.

As shown in the Fig. 1.17.12 (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 the 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).

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 1.17.11 Implement the following Boolean function with 8: 1 multiplexer F (A, B, C, D) = л M (0, 3, 5, 6, 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. 1.17.14 shows the implementation of Boolean function with 8 : 1 multiplexer.

Example 1.17.12 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. 1.17.15 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.

Example 1.17.13 Implement full adder circuit using 8: 1 multiplexer.    AU: May-11, Marks 5

Solution :

Example 1.17.14 Implement full adder circuit using dual 4 to 1 multiplexer.  AU: May-12, Dec.-14, Marks 8

Solution: Implementation tables :

Example 1.17.15 Realize F(w, x, y, z) = Σ (1, 4, 6, 7, 8, 9, 10, 11, 15) using 4 to 1 MUX.  AU May-04, Marks 8

Solution :

Example 1.17.16 Design even parity generator circuit for 4 bit input using multiplexer.  AU: May-07

Solution : The Table 1.17.3 shows the truth table for even parity generator circuit.

Implementation Table :

Implementation using 8 : 1 multiplexer

Example 1.17.17 A 8 x 1 multiplexer has inputs A, B, and C connected to the selection inputs S₂, S₁ and S₀ respectively. The data inputs I₀ to I₇ are as follows I₀ = I₂ = I₇ = 0; I₃ = I₅ = 1; I₀ = I₄ = D and I₆= D'. Determine the Boolean function that the multiplexer implements.  AU: Dec.-08, Marks 10.

Solution: Since input A is the MSB, The D input is LSB. Let us determine the implementation table for multiplexer from given information.

If I is connected to 0 don't encircle any term. If I is connected to 1, encircle both the terms in a row. If I is connected to Ď, encircle the corresponding term in the column Ď. If I is connected to D, encircle the corresponding term in the column D. This is illustrated in the implementation table of Fig. 1.17.20. Now write down the encircled terms to get the boolean function. It is given as,

F(A, B, C, D) = ∑m (1, 6, 7, 9, 10, 11, 12)

Example 1.17.18 Design a combinational logic using a suitable multiplexer to realize the following Boolean expression.

Y = ADʹ + B'C + BC' D'. AU May-11, Marks 10

Solution:

Step 1: Convert the given Boolean expression in standard form

Step 2: Implementation

Example 1.17.19 Implement the function with a multiplexer.

F(A, B, C, D) = ∑ (0, 1, 3, 4, 8, 9, 15)  AU: Dec.-08, 11, Marks 6

Solution: Implementation table :

Implementation

Example 1.17.20 How can a multiplexer be used to convert 8-bit parallel data into serial form? Draw the circuit and briefly explain. AU May-06, Marks 2

Solution: Multiplexer can be used to convert 8-bit parallel data into serial form as shown in the Fig. 1.17.23.

Here, binary counter is used to drive the select inputs of the multiplexer so that as binary counter increments its count, the next bit is available at the output of the multiplexer. The binary counter counts from 000 to 111, therefore, D₀ through D₇, bits are available at the output of multiplexer as serial output.

Example 1.17.21 Compare and contrast between encoders and multiplexers. AU May-11, Marks 6

Solution :

Examples for Practice

Example 1.17.22 Implement F (A, B, C, D) =∑ m (0, 1, 3, 4, 8, 9, 15) using 8: 1

multiplexer.

Example 1.17.23 Implement the following Boolean function with 8: 1 mux. F(A, B, C, D) = πM (0, 3, 5, 8, 9, 10, 12, 14).

Example 1.17.24 Implement following function using 16: 1 multiplexer. f(A, B, C, D) = ∑m (1, 9, 10, 14, 18).

Example 1.17.25 Design a 4 to 1 MUX using a 2 to 4 decoder and basic gates.

Example 1.17.26 Realize a full adder using 4 x 1 multiplexers.

Example 1.17.27 Realize the function f(A, B, C, D) = π(1,4,6,10,14). d (0,8,11,15) using i) 16:1 MUX ii) 8:1 MUX.

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.

Multiplexer ICs

Review Questions

1. Draw the circuits for 2-to-1 line multiplexer.   AU: Dec.-16, May-17, Marks 2

2. What is multiplexer ? Show the block diagram of 4 x 1 multiplexer.  AU: Dec.-19, Marks 2