 Namaste welcome to the session design of multiplexer tree at the end of this session students will be able to design a multiplexer tree in this session we are going to design a multiplexer tree now before proceeding further take a pause here and recall what do you mean by multiplexer so multiplexer is a combinational circuit which selects binary information from 2 raise to n number of input lines using n number of selection or also known as a control inputs and directs it to the single output line multiplexer is also known as data selector now let us see block diagram of n is to 1 multiplexer so where you can see there are 2 raise to n data inputs and where n number of select or control inputs are there and 1 output data line so from this block diagram we can observe that there are 2 raise to n data inputs n number of control inputs are also known as select inputs and 1 data output line now let us see advantages of multiplexer using multiplexer it is easy to design a combinational circuit because whenever you are designing a combinational circuit sometimes it is required to minimize the equation of the combinational circuit using k-map or other technique but whenever we are designing a combinational circuit using multiplexer there is no need to minimize the logical equation it is also easy to debug a circuit which is designed using multiplexer now disadvantage of multiplexer is that if number of inputs are more to a combinational circuit then circuit becomes very large if designed using multiplexer now let us design a multiplexer circuit for example we are going to design a 4 is to 1 multiplexer before drawing a block diagram so we need to know how many inputs are there from number of inputs we can find out number of select inputs required so according to the equation 2 raise to m is equal to n where m is number of select inputs and n is number of data inputs so if number of inputs are 4 then select inputs required that is m is 2 so for n is equal to 4 data inputs we require m is equal to 2 select inputs so let us draw the block diagram here so here the data inputs are represented by letter i so 4 inputs are i0 i1 i2 i3 and let us consider letter s for select inputs as s1 and s0 along with these inputs we have one more input that is enable input denoted as e which is used to start or stop the operation of multiplexer and then we have one output represented as f now the second step is to write a truth table so we know that for 4 is to 1 multiplexer we have 2 select inputs or control inputs as s1 and s0 and 4 data inputs as i0 i1 i2 i3 f as the output and e as a enable input so for this block diagram we have this truth table where we have indicated enable input as active i so whenever enable input is 0 none of the input is get selected or are disabled so we are getting output f as 0 so when enable input is 1 now let us consider 4 inputs i0 i1 i2 i3 maybe 0 or 1 as binary inputs are given and consider enable input as 1 so when select input s1 and s0 are 0 0 then i0 will be get selected so i0 may be 0 or 1 so accordingly output will also 0 or 1 so whenever s1 and s0 are 0 i0 will be get selected and that is connected to the output so output may be 0 or 1 when s1 is 0 and s0 is 1 then i1 should get selected and according to the i1 value whether it may be 0 or 1 we are getting output as 0 or 1 so here only i1 is get selected and that is connected to the output now similarly when s1 and s0 are 1 0 respectively then i2 is get selected and then it is connected to output line when s1 and s0 both are 1 1 then i3 is get selected and depending upon value of i3 output will be 0 or 1 so whenever both are 1 1 we are getting output selected from the i3 now after truth table we are writing logical equation for 4 is to 1 multiplexer so from this truth table we can observe that we are getting i0 selected when both s1 and s0 are 0 0 so here equation becomes i0 into s1 bar s0 bar plus i1 is get selected when s1 is 0 and s0 is 1 so here the equation becomes i1 s1 bar s0 plus i2 is get selected when s1 is 1 and s0 is 0 so i2 s1 s0 bar plus i3 will be get selected here when s1 and s0 both are 1 1 so equation becomes i3 s1 s2 and in a ball is 1 so into e so this is logical equation for 4 is to 1 multiplexer the next step we are going to draw logic circuit for 4 is to 1 multiplexer so here you can observe the circuit is drawn according to the equation now let us design multiplexer tree so what is the multiplexer tree a larger multiplexer can be implemented using a tree of smaller multipliers so when we are designing 4 is to 1 multiplexer using 2 is to 1 multiplexer we require 3 2 is to 1 multiplexer and we require 2 select inputs let us see how so here you can see in the first row we have 2 multiplexers which are having 2 inputs each and output of these first 2 2 multiplexers are given to third 2 is to 1 multiplexers as a input so here you can see the first multiplexer having i0 i1 input second 2 is to 1 multiplexer having i2 i3 input and third 2 is to 1 multiplexer having input which are the output from the first to 2 is to 1 multiplexers so first to 2 is to 1 multiplexer having common select input as s0 and third one having select input or control input as s1 now let us see how it works now let us consider s1 and s0 both are 0 0 where s1 is msb so when s0 is 0 so i0 and i2 will be get selected and as s1 is 0 first input of the third 2 is to 1 multiplexer it get selected so in this way i0 is get selected and that is connected to the output line now when s1 is 0 and s0 is 1 as s0 is 1 i1 and i3 are get selected and as s1 is 0 the first input of 2 is to 1 multiplexer is get selected so here i1 is get selected and that is connected to the output where i3 is not selected now when s1 is 1 and s0 is 0 then because of s0 is 0 i0 and i2 is get selected and as s1 is 1 the second input of 2 is to 1 multiplexer it get selected so here you can observe that i2 is get selected and that is connected to the output line now when s1 and s0 are both 1 1 because of s0 is 1 i1 and i3 will be get selected and as s1 is 1 the second input of third 2 is to 1 multiplexer is get selected so in this way i3 is get selected and that is connected to the output line so in this way multiplexer tree works so let us have one more example 16 is to 1 multiplexer using 4 is to 1 multiplexer so in this case we require 5 numbers of 4 is to 1 multiplexer and in 16 is to 1 multiplexer we require 4 select inputs as s1 s0 s2 s3 and 16 inputs as i0 to i15 and f is the output so here is the block diagram where you can see we have used 4 4 is to 1 multiplexer in the first row where it is having 16 inputs i0 to i15 and in the first row 4 is to 1 multiplexer are having 2 select inputs common as s1 and s0 fifth 4 is to 1 multiplexer where it is having select input as s1 and s2 so in this way you can design any multiplexer tree using smaller multiplexers these are references thank you