A convolutional code is a type of error-correcting code that is used in digital communication systems to transmit data over a noisy channel. These codes work by adding redundant bits, known as parity bits, to the transmitted data in order to allow the receiver to detect and correct errors that may occur during transmission.

One example of a convolutional code is the binary convolutional code, which uses a fixed set of parity bits to encode each input bit. These parity bits are generated using a linear shift register, which is a type of finite state machine that consists of a series of flip-flops that are connected in a feedback configuration.

The shift register is initialized with a known set of values, called the generator polynomial, which determines the pattern of parity bits that will be added to the transmitted data. As the input bits are transmitted, the shift register shifts its internal state based on the values of the input bits and the generator polynomial.

For example, consider a binary convolutional code with a generator polynomial of (1,1,0). This code would add two parity bits to each input bit, resulting in a code rate of 1/3 (that is, for every three bits transmitted, one of them is an input bit and two are parity bits).

If we want to transmit the input sequence 011011, the encoder would first shift the shift register based on the first input bit (0), resulting in a parity bit sequence of 00. The second input bit (1) would then be transmitted along with the first two parity bits (01), resulting in a transmission of 0101. This process would continue until all of the input bits have been transmitted.

At the receiver, the decoder uses the same generator polynomial to reconstruct the original input sequence. If any errors are detected during transmission, the decoder can use the parity bits to correct them.

Convolutional codes are widely used in a variety of communication systems, including satellite communication, cellular networks, and digital audio broadcasting. They are particularly useful in situations where the channel conditions may vary over time, as they can adapt to these changes and maintain a high level of error correction.

Overall, convolutional codes are a powerful tool for ensuring the reliability of data transmission in noisy communication channels.