Question:
The information bit sequence \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1\}\) is to be transmitted by encoding with Cyclic Redundancy Check 4 (CRC-4) code, for which the generator polynomial is \(C(x) = x^4 + x + 1\). The encoded sequence of bits is \(\_\_\_\_\).
The information bit sequence \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1\}\) is to be transmitted by encoding with Cyclic Redundancy Check 4 (CRC-4) code, for which the generator polynomial is \(C(x) = x^4 + x + 1\). The encoded sequence of bits is \(\_\_\_\_\).
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In cyclic redundancy check (CRC) encoding, ensure you append zeros equal to the degree of the generator polynomial before performing division. This step is crucial for determining the correct remainder to form the encoded sequence.
Updated On: Jan 31, 2025
- \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 0\}\)
- \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 1\}\)
- \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 1 \, 0\}\)
- \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 0\}\)
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The Correct Option is A
Solution and Explanation
Step 1: Append 4 zeros to the information bits.
The given information bit sequence is \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1\}\). Add 4 zeros to the end of the sequence, resulting in: \[ \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 0 \, 0 \, 0\}. \] Step 2: Perform modulo-2 polynomial division.
Divide the appended sequence by the generator polynomial \(C(x) = x^4 + x + 1\) using modulo-2 arithmetic. The resulting remainder is: \[ \{1 \, 1 \, 0 \, 0\}. \] Step 3: Create the encoded sequence.
Add the remainder to the appended sequence, replacing the appended zeros. The encoded sequence becomes: \[ \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 0\}. \] Final Answer: \[ \boxed{{(1) } \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 0\}} \]
The given information bit sequence is \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1\}\). Add 4 zeros to the end of the sequence, resulting in: \[ \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 0 \, 0 \, 0\}. \] Step 2: Perform modulo-2 polynomial division.
Divide the appended sequence by the generator polynomial \(C(x) = x^4 + x + 1\) using modulo-2 arithmetic. The resulting remainder is: \[ \{1 \, 1 \, 0 \, 0\}. \] Step 3: Create the encoded sequence.
Add the remainder to the appended sequence, replacing the appended zeros. The encoded sequence becomes: \[ \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 0\}. \] Final Answer: \[ \boxed{{(1) } \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 0\}} \]
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