JNTU B.Tech III SWITCHING THEORY AND LOGIC DESIGN Previous Question Paper (September - 2021 - R13)

i) To find the base x for (211)_x = (152)₈, convert both sides to decimal:

Using the quadratic formula, x = [-1 ± sqrt(1 - 4(2)(-105))] / 4 = [-1 ± sqrt(841)] / 4 = (-1 ± 29) / 4. Taking the positive root, x = 28/4 = 7. Thus, the base is 7.

ii) For (193)_x = (623)₈:

Using the quadratic formula, x = [-9 ± sqrt(81 - 4(1)(-400))] / 2 = [-9 ± sqrt(1681)] / 2 = (-9 ± 41) / 2. Taking the positive root, x = 32/2 = 16. Thus, the base is 16.

Hamming code is an error-correcting code that adds parity bits to data bits to detect single-bit errors and locate their position for correction. For a data word of length m, the number of parity bits k must satisfy the condition 2^k ≥ m + k + 1.

The process involves:

1. Assigning parity bits at positions that are powers of two (1, 2, 4, 8, ...).
2. Calculating each parity bit such that the total number of 1s in its assigned set of positions is even (for even parity).
3. During reception, calculating the syndrome by checking the parity of the same sets. If the syndrome is 0, no error occurred. If non-zero, the value indicates the bit position of the error.

Example: For 4-bit data 1011, we need 3 parity bits (k=3). Positions: P1, P2, D3, P4, D5, D6, D7. Data bits at 3, 5, 6, 7 are 1, 0, 1, 1. P1 checks 1, 3, 5, 7; P2 checks 2, 3, 6, 7; P4 checks 4, 5, 6, 7. Calculating these yields P1=0, P2=1, P4=0. The transmitted word is 0110011. If bit 3 flips to 0, the syndrome calculation will identify position 3 as the error, allowing the receiver to flip it back to 1.