(ii) Explain Cyclic Redundancy Check (CRC). Given a data message D(x)= 1100001 a Generator polynomial G(x) = x³ +1. Calculate the codeword at the sender side. Als do the calculations on the receiver side.
(ii) Explain Cyclic Redundancy Check (CRC). Given a data message D(x)= 1100001 a Generator polynomial G(x) = x³ +1. Calculate the codeword at the sender side. Als do the calculations on the receiver side.
Computer Networking: A Top-Down Approach (7th Edition)
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![### Cyclic Redundancy Check (CRC)
**Problem Statement:**
Given a data message \( D(x) = 1100001 \) and Generator polynomial \( G(x) = x^3 + 1 \). Calculate the codeword at the sender side. Also do the calculations on the receiver side.
**Solution:**
#### 1. Introduction to CRC
Cyclic Redundancy Check (CRC) is a method used in digital networks and storage devices to detect accidental changes to raw data. The CRC employs a generator polynomial to perform binary division of the input data and appends the remainder of this operation to the input data to form the codeword.
#### 2. Given:
- Data message \( D(x) \): 1100001
- Generator polynomial \( G(x) \): \( x^3 + 1 \)
#### 3. Steps at the Sender Side:
1. **Append Zeros:**
- Append \( n \) zeros to the end of \( D(x) \), where \( n \) is the degree of \( G(x) \).
- Degree of \( G(x) = 3 \)
- Modified data message: \( 1100001000 \)
2. **Binary Division:**
- Perform binary division of the modified data message by the generator polynomial.
- The generator polynomial \( G(x) = x^3 + 1 \) corresponds to 1001 in binary.
- Use XOR division (each division step will be modulo 2).
3. **Calculate Remainder:**
- The remainder after the binary division will be the CRC.
4. **Form Codeword:**
- Append the CRC remainder to the original data message to form the codeword.
#### 4. Example Calculation:
Let's break down the binary division:
1. **Initial Data Message**: \( 1100001000 \)
2. **Generator Polynomial**: \( 1001 \)
#### Binary Division Steps (simplified overview):
- Perform XOR starting from the leftmost bit where we encounter a '1':
\[
\begin{aligned}
1100 001000 \quad \text{(Initial Step)} \\
1001 & \quad \text{XOR} \\
\hline
0101000 \\
\end{aligned}
\]
- The process continues, repeatedly applying the XOR and bringing down the next bit until all](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F039f7f42-bea1-420c-8e74-506e9aacc740%2F238a4844-3dbe-48ad-946a-52367bf8ef51%2F2n2nn5_processed.png&w=3840&q=75)
Transcribed Image Text:### Cyclic Redundancy Check (CRC)
**Problem Statement:**
Given a data message \( D(x) = 1100001 \) and Generator polynomial \( G(x) = x^3 + 1 \). Calculate the codeword at the sender side. Also do the calculations on the receiver side.
**Solution:**
#### 1. Introduction to CRC
Cyclic Redundancy Check (CRC) is a method used in digital networks and storage devices to detect accidental changes to raw data. The CRC employs a generator polynomial to perform binary division of the input data and appends the remainder of this operation to the input data to form the codeword.
#### 2. Given:
- Data message \( D(x) \): 1100001
- Generator polynomial \( G(x) \): \( x^3 + 1 \)
#### 3. Steps at the Sender Side:
1. **Append Zeros:**
- Append \( n \) zeros to the end of \( D(x) \), where \( n \) is the degree of \( G(x) \).
- Degree of \( G(x) = 3 \)
- Modified data message: \( 1100001000 \)
2. **Binary Division:**
- Perform binary division of the modified data message by the generator polynomial.
- The generator polynomial \( G(x) = x^3 + 1 \) corresponds to 1001 in binary.
- Use XOR division (each division step will be modulo 2).
3. **Calculate Remainder:**
- The remainder after the binary division will be the CRC.
4. **Form Codeword:**
- Append the CRC remainder to the original data message to form the codeword.
#### 4. Example Calculation:
Let's break down the binary division:
1. **Initial Data Message**: \( 1100001000 \)
2. **Generator Polynomial**: \( 1001 \)
#### Binary Division Steps (simplified overview):
- Perform XOR starting from the leftmost bit where we encounter a '1':
\[
\begin{aligned}
1100 001000 \quad \text{(Initial Step)} \\
1001 & \quad \text{XOR} \\
\hline
0101000 \\
\end{aligned}
\]
- The process continues, repeatedly applying the XOR and bringing down the next bit until all
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