Don’t Care (X) Conditions in K-Maps

The “Don’t Care” conditions allow us to replace the empty cell of a K-Map to form a grouping of the variables. While forming groups of cells, we can consider a “Don’t Care” cell as either 1 or 0 or we can simply ignore that cell. Therefore, “Don’t Care” condition can help us to form a larger group of cells.

A Don’t Care cell can be represented by a cross(X) in K-Maps representing a invalid combination. For example, in Excess-3 code system, the states 0000, 0001, 0010, 1101, 1110 and 1111 are invalid or unspecified. These are called don’t cares. Also, in design of 4-bit BCD-to-XS-3 code converter, the input combinations 1010, 1011, 1100, 1101, 1110, and 1111 are don’t cares.

A standard SOP function having don’t cares can be converted into a POS expression by keeping don’t cares as they are, and writing the missing minterms of the SOP form as the maxterm of POS form. Similarly, a POS function having don’t cares can be converted to SOP form keeping the don’t cares as they are and write the missing maxterms of the POS expression as the minterms of SOP expression.



Example-1:
Minimise the following function in SOP minimal form using K-Maps:

f = m(1, 5, 6, 12, 13, 14) + d(4) 

Explanation:
The SOP K-map for the given expression is:

Therefore, SOP minimal is,

f = BC' + BD' + A'C'D 

Example-2:
Minimise the following function in SOP minimal form using K-Maps:

F(A, B, C, D) = m(0, 1, 2, 3, 4, 5) + d(10, 11, 12, 13, 14, 15) 

Explanation:
Writing the given expression in POS form:

F(A, B, C, D) = M(6, 7, 8, 9) + d(10, 11, 12, 13, 14, 15) 

The POS K-map for the given expression is:

Therefore, POS minimal is,


F = A'(B' + C') 

Example-3:
Minimise the following function in SOP minimal form using K-Maps: F(A, B, C, D) = m(1, 2, 6, 7, 8, 13, 14, 15) + d(3, 5, 12)

Explanation:
The SOP K-map for the given expression is:

Therefore,

f = AC'D' + A'D + A'C + AB 

Significance of “Don’t Care” Conditions:
Don’t Care conditions has the following significance with respect to the digital circuit design:

  1. Simplification:
    These conditions denotes the set of inputs which never occurs for a given digital circuits. Thus, they are being used to further simplify the boolean output expression.
  2. Lesser number of gates:
    Simplification reduces the number of gates to be used for implementing the given expression. Therefore, don’t cares make the digital circuit design more economical.
  3. Reduced Power Consumption:
    While grouping the terms long with don’t cares reduces switching of the states. This decreases the required memory space which in turn results in less power consumption.
  4. States in Code Converters:
    These are used in code converters. For example- In design of 4-bit BCD-to-XS-3 code converter, the input combinations 1010, 1011, 1100, 1101, 1110, and 1111 are don’t cares.
  5. Prevention of Hazards:
    Don’t cares also prevents hazards in digital systems.



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