Question

Assume that you are sending a message consisting of 32 bits over a noisy communication channel using CRC-16 with the polynomial x¹⁶ + x¹⁵ + x² + 1. If the message is received with errors in bits 10, 18, and 27, Can the receiver detect the errors using CRC? If yes, explain how. If not, explain why not. Show the process as well.

Answer 1

CRC-16 is used to encode the message for transmission over a noisy channel. The receiver can detect errors by performing division using the received message and the polynomial. Errors in bits 10, 18, and 27 are detected in this case.

Step-by-step explanation:

The message is encoded in the signal using CRC-16 (Cyclic Redundancy Check) with a polynomial of x¹⁶ + x¹⁵ + x² + 1.

In this encoding scheme, the message bits are divided by the polynomial and the remainder is added as an extra set of bits called the CRC (Checksum).

To check for errors, the receiver performs the same division operation using the received message and the polynomial. If the remainder is zero, it means that no errors were detected.

However, if the remainder is non-zero, it indicates that errors were present during transmission.

In the given scenario, the receiver can detect the errors in bits 10, 18, and 27 because when the received message is divided by the polynomial, a non-zero remainder is obtained, indicating the presence of errors.

Answer 2

The message is encoded in the signal using CRC-16, and the receiver can detect errors using CRC. The receiver performs polynomial division to check the remainder, and if it's non-zero, errors are present.

Step-by-step explanation:

The message is encoded in the signal using CRC-16. CRC stands for Cyclic Redundancy Check, which is an error-detection algorithm. To encode the message, the sender first appends a CRC checksum to the original message.

The checksum is generated by performing a polynomial division using the CRC polynomial. The resulting message, including the appended checksum, is then sent over the communication channel.

In this case, the polynomial used is x¹⁶ + x¹⁵ + x² + 1. So, the sender appends a 16-to the 32-bit message. When the receiver receives the message, it performs the same polynomial division using the CRC polynomial.

If there are no errors in the message, the remainder of the division will be 0. However, if errors are present, the remainder will be non-zero, indicating the presence of errors.

To detect errors, the receiver must perform the polynomial division and check the remainder. In this case, if errors are present in bits 10, 18, and 27, the receiver will detect the errors because the remainder will be non-zero. The receiver can then request retransmission of the message to ensure data integrity.