Cyclic Redundancy Check Part-1
Duration: 8 min
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The video lecture introduces Cyclic Codes, defining them as linear block codes with the property that cyclic shifts of a codeword result in another valid codeword. It then transitions to Cyclic Redundancy Check (CRC), explaining the encoding process where a dataword is augmented with zeros and divided by a generator using modulo-2 arithmetic. The remainder is appended to the dataword to form the codeword. The instructor demonstrates these concepts with whiteboard examples of division and codeword construction.
Chapters
0:00 – 2:00 00:00-02:00
The video begins with a slide titled CYCLIC CODES. The instructor defines cyclic codes as special linear block codes with an extra property: if a codeword is cyclically shifted or rotated, the result is another codeword. He provides an example on the slide: if 1011000 is a codeword, a cyclic left shift results in 0110001, which is also a codeword. The instructor then moves to the whiteboard to visually demonstrate this rotation. He writes the sequence 1011000 and draws an arrow indicating the left shift, showing how the leading 1 moves to the end to form 0110001. This visual aid reinforces the concept of rotational invariance inherent in cyclic codes, setting the stage for their application in error detection.
2:00 – 5:00 02:00-05:00
The lecture transitions to the practical application of cyclic codes in Cyclic Redundancy Check (CRC). A detailed slide outlines the encoding process. It specifies that in the encoder, a dataword of k bits (4 bits in the example) is transformed into a codeword of n bits (7 bits in the example). The process involves augmenting the dataword by appending n - k zeros (3 zeros) to the right-hand side. This augmented n-bit result is then fed into a generator. The generator utilizes a predefined divisor of size n - k + 1 (4 bits). The core operation is modulo-2 division, where the generator divides the augmented dataword by the divisor. The quotient is discarded, and the remainder (r2 r1 r0) is appended to the original dataword to create the final codeword. A block diagram illustrates the sender and receiver sides, showing the flow of datawords, remainders, and codewords through the generator and checker units.
5:00 – 8:08 05:00-08:08
The instructor demonstrates the modulo-2 division process on the whiteboard to solidify the theoretical explanation. He sets up a long division problem with the divisor 1011 and the augmented dataword 1001000. He performs the XOR-based division step-by-step. In the first step, he aligns the divisor with the dividend and performs the XOR operation, bringing down subsequent bits. He continues this process until the remainder is smaller than the divisor. The final remainder calculated is 110. He then constructs the final codeword by appending this remainder to the original dataword 1001, resulting in 1001110. He proceeds to a second example on the right side of the board, using the same divisor 1011 but a different dividend 1001110, likely to demonstrate the checking process at the receiver end where a zero remainder indicates no errors.
The video provides a comprehensive introduction to Cyclic Codes and their implementation via Cyclic Redundancy Check (CRC). It starts by defining the mathematical property of cyclic codes—rotational invariance—where shifting a codeword yields another valid codeword. It then details the CRC encoding mechanism, explaining how datawords are augmented with zeros and divided by a generator polynomial using modulo-2 arithmetic to produce a remainder. This remainder is appended to the dataword to form the codeword. The instructor reinforces these concepts with clear whiteboard demonstrations of the division algorithm, showing how to calculate the remainder and construct the final codeword, as well as hinting at the error detection process at the receiver.