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K-map

Before seeing the chapter please follow the previous chapter: 

Gray Code (Un-weighted BCD)

K – map: (Karnaugh Map)

# K-map is one of the minimization techniques used to simplify logical expressions.

# Unit-distance code or grey code representation is used in K – map.

# In K-map, the grouping of bits is done in powers of 2, 

(depending on the number of variables used).

# K-map will provide minimum expression, but not necessarily unique.

# In order to group minterms group of 1’s are used, where ‘d’ or ‘X’ (don’t care, can be either 0 or 1) can also be a part of each group. In a particular group, there is no restriction in the numbers of 1’s and the number of don’t care terms. Minimizing expression with a set of minterms produces the SOP form of the simplified expression.

# In order to group maxterms, group of 0’s are used, where ‘d’ or ‘X’ (don’t care, can be either 0 or 1) can also be a part of each group. In a particular group, there is no restriction in the number of 0’s and the number of don’t care terms. Minimizing expression with a set of maxterms produces a POS form of the simplified expression.

Now why we are using Gray Code in K-Map:

We are not going to minimize it because it is already minimized because only one bit changed here.

There is no common bit.

But if we have, 

So, in that case, we do minimize.

So, if there is only one change of a bit in that case we are going to minimizing change of one bit is generally gray code that’s why we use gray code in k-map.

[In gray code the successive terms, there is only one bit of change at a time]

Now see how we are using gray code in K-MAP:

Now the number of cells possible in K-Map:

Now how to represent SOP and POS at K-Map:

Sum of Product (SUM) representation at K-Map: 

Product of Sum (POS) representation at K-Map:

Now we change:

We can change the K-Map POS to SOP or vice versa by we can change the position of LSB or MSB.

Change it to POS:

Note: First we see it is SOP or POS then check the position of LSB and MSB of K-MAP. After that, we are going to solve it.

Three variables SOP K-MAP

Three variables POS K-MAP

Here in the case of three variables K-MAP we can change MSB and LSB.

Four variable SOP and POS K-MAP

We can represent four variables K-MAP as: