A bit stream 1101011011 is transmitted using the standard CRC method. The generator polynomial is x+x+1. What is the actual bit string transmitted?
Question
A bit stream 1101011011 is transmitted using the standard CRC method. The generator polynomial is x+x+1. What is the actual bit string transmitted?
Solution
To solve this problem, we need to follow these steps:
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First, we need to understand the generator polynomial. The given polynomial is x^2 + x + 1. In binary form, this polynomial is represented as 111.
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Next, we need to append zeros to the bit stream. The number of zeros to append is one less than the degree of the polynomial. In this case, the degree of the polynomial is 2 (from x^2), so we append two zeros. The bit stream becomes 110101101100.
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Now, we perform binary division of the bit stream (including the appended zeros) by the generator polynomial. The remainder of this division will be our CRC checksum, which is added to the end of the original bit stream.
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The binary division is performed as follows:
110101101100 divided by 111
The first three bits (110) are divided by 111. Since 110 is less than 111, we write a 0 in the quotient and XOR the divisor with the next three bits from the dividend.
This process is repeated until we've gone through all the bits in the dividend.
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The remainder from this division is 10.
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Finally, we append this remainder to the original bit stream (before we appended the zeros). So, the actual bit string transmitted is 110101101110.
Please note that the process of binary division is similar to long division, but uses XOR instead of subtraction.
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