Parity bits and 3-bit odd parity generator

Importance of parity bits

A parity bit is one extra bit attached to a data word so that the total count of 1s is either even (even parity) or odd (odd parity). In communication systems it matters because:

• Error detection: noise or interference may flip a bit. The receiver checks the parity; a mismatch shows an error, and the data can be re-requested.
• Low cost: only one extra bit per word and a few XOR gates.
• Single-bit errors are always detected (in fact any odd number of errors).
• Limits: an even number of bit errors goes unnoticed, and the faulty bit cannot be located or corrected. Codes like Hamming code extend the parity idea to correct errors.
• Odd parity is often preferred because an all-zero word (e.g. a dead line) still carries a 1, so a stuck-at-0 line is detected.

3-bit odd parity generator

Inputs $A, B, C$; output $P$ is chosen so that $A, B, C, P$ together contain an odd number of 1s.

$$P = \Sigma m(0,3,5,6) = A'B'C' + A'BC + AB'C + ABC'$$

The K-map has no adjacent 1s, so it is simplified with XOR/XNOR:

$$\begin{aligned} P &= A'(B \odot C) + A(B \oplus C) \\ &= \overline{A \oplus B \oplus C} \end{aligned}$$

 A --+
     +--[XOR]--+
 B --+         +--[XNOR]--- P
 C ------------+

Working: if the data already has an odd number of 1s, $P = 0$; if even, $P = 1$. Example: data $110$ (two 1s) gives $P = \overline{1 \oplus 1 \oplus 0} = 1$, so the word $1101$ has three 1s (odd). The receiver's odd parity checker computes $\overline{A \oplus B \oplus C \oplus P}$, which is 0 when no error is present.