Higher Mux by Lower Mux

Constructing Higher MUX using Lower MUX derivation of formula

To construct a ${\color{Red} M\times 1}$ using ${\color{Red} N\times 1}$

no. of levels (K) = $\left \lceil log _{n}^{m}\right \rceil$

and Total number of MUX needed(n)= ${\color{Red} \sum_{p=1}^{k}\left \lceil M/N^{p} \right \rceil}$

Prove:-

Example:

1. Construct $32\times 1$ using $2\times 1$

Solution:

Total number of levels= $\left \lceil log_{2}^{32} \right \rceil$ = 5

Total number of MUX= $\left \lceil 32/2^{1} \right \rceil +\left \lceil 32/2^2 \right \rceil+\left \lceil 32/2^3 \right \rceil +\left \lceil 32/2^4 \right \rceil+\left \lceil 32/2^5 \right \rceil$

= $16+8+4+2+1=31$

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Constructing Higher MUX using lower MUX

let's talk informally :

We know that $4\ast 1$ means there are 4 input lines, 2 select lines, and 1 output line. In other words, we can say that

To map 4 input lines we need 1 MUX. Why?? because there are 4 input line and only one $4\ast 1$ MUX is enough to map those lines.

So this is the idea to implement higher MUX using lower MUX.

Example:

1. Implement $64\times 1$ using $4\times 1$ .

Solution:-

let's talk about $4\times 1$ first

To implement 4 input lines we need ................... 1 MUX

" " 1 input line we need ......................1/4 MUX

In $64\times 1$ there are 64 input lines

To implement 64 input lines we need ..................64*(1/4)=16 MUX //level 1

But 16 MUX will have 16 output lines, we have to map until the total output line is 1 or less than 1, again consider these 16 output lines as input lines for next level MUX

To implement 16 input lines we need ..................16*(1/4)=4 MUX //level 2

Again 4 MUX will have 4 output lines, we have to map until the total output line is 1 or less than 1, again consider these 4 output lines as input lines for next level MUX.

To implement 4 input lines we need .................. 4*(1/4)=1 MUX //level3

Now we can stop. Think Why??

The total number of $4\times 1$ MUX required to construct $64\times 1$ = $16+4+1=21$

Now we will see a formula to directly calculate the number of level and MUX.

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