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A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (mi ; ci), where ci is the codeword corresponding to the ith message mi, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?
- a)1 0 1 1 0 1 0
- b)0 1 1 0 1 0 0
- c)0 0 0 1 0 1 1
- d)1 1 0 1 0 0 1
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A digital transmission system uses a (7, 4) systematic linear Hamming ...
Let d = (d1, d2 …. dk)
Codeword C can be written as:
Codeword C can be written as:
Cd = d1
C2 = d2
C3 = d3
⋮
Ck = dk
Ck = dk
Note: The parity bits are generated by a linear combination of message bits.
Application:
Given (7, 4) linear Hamming code.
(mi, ci) given are:
(1100 ; 0101100)
(1110 ; 0011110)
(0110 ; 1000110)
Clearly, we can observe that the first 3 bits are parity bits and the last 4 bits are message bits.
Clearly, we can observe that the first 3 bits are parity bits and the last 4 bits are message bits.
Observe when P0, P1, and P2 are one, and write their combination, i.e.
P0 = d0 ⊕ d1 ⊕ d3
P1 = d1 ⊕ d2 ⊕ d3
P2 = d0 ⊕ d1 ⊕ d2
Now checking each of the options:
Option (1): 1011010
Message bits are:
Message bits are:
P0 = d0 ⊕ d1 ⊕ d3 = 1
P1 = d1 ⊕ d2 ⊕ d3 = 1
P2 = d0 ⊕ d1 ⊕ d2 = 0
So, the codeword is 1101010.
Hence option (1) is invalid codeword.
Option (2): 0110100
Message bits are:
Message bits are:
P0 = d0 ⊕ d1 ⊕ d3 = 1
P1 = d1 ⊕ d2 ⊕ d3 = 1
P2 = d0 ⊕ d1 ⊕ d2 = 1
So, the valid codeword is 1110100
Hence option (2) is an invalid codeword.
Option (3): 0001011
Message bits are:
Message bits are:
P0 = d0 ⊕ d1 ⊕ d3 = 0
P1 = d1 ⊕ d2 ⊕ d3 = 0
P2 = d0 ⊕ d1 ⊕ d2 = 0
So the valid codeword is 0001011
Hence option (3) is a valid codeword.
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Community Answer
A digital transmission system uses a (7, 4) systematic linear Hamming ...
Given information:
- We have a (7, 4) systematic linear Hamming code.
- We are given three message-codeword pairs: (1 1 0 0; 0 1 0 1 1 0 0), (1 1 1 0; 0 0 1 1 1 1 0), and (0 1 1 0; 1 0 0 0 1 1 0).
To find:
- We need to determine which of the given options is a valid codeword in this code.
Explanation:
A (7, 4) Hamming code has 7 bits in each codeword, with 4 message bits and 3 parity bits. The parity bits are calculated using the message bits according to the Hamming code algorithm.
Hamming Code Algorithm:
1. The 4 message bits (m1, m2, m3, m4) are placed in the codeword at positions 3, 5, 6, and 7.
2. The remaining positions (1, 2, 4) are filled with parity bits (p1, p2, p3) calculated based on the message bits.
Calculating Parity Bits:
To calculate the parity bits, we need to find the values of p1, p2, and p3.
For the first pair (1 1 0 0; 0 1 0 1 1 0 0):
- p1 = m1 XOR m2 XOR m4 = 1 XOR 1 XOR 0 = 0
- p2 = m1 XOR m3 XOR m4 = 1 XOR 0 XOR 0 = 1
- p3 = m2 XOR m3 XOR m4 = 1 XOR 0 XOR 0 = 1
For the second pair (1 1 1 0; 0 0 1 1 1 1 0):
- p1 = m1 XOR m2 XOR m4 = 1 XOR 1 XOR 0 = 0
- p2 = m1 XOR m3 XOR m4 = 1 XOR 1 XOR 0 = 0
- p3 = m2 XOR m3 XOR m4 = 1 XOR 1 XOR 0 = 0
For the third pair (0 1 1 0; 1 0 0 0 1 1 0):
- p1 = m1 XOR m2 XOR m4 = 0 XOR 1 XOR 0 = 1
- p2 = m1 XOR m3 XOR m4 = 0 XOR 1 XOR 0 = 1
- p3 = m2 XOR m3 XOR m4 = 1 XOR 1 XOR 0 = 0
Valid Codewords:
Now, let's check which of the given options satisfy the calculated parity bits.
Option a) 1 0 1 1 0 1 0:
- p1 = 1 XOR 0 XOR 1 XOR 1 = 1
- p2 = 1 XOR 0 XOR 1 XOR 1 = 1
- p
- We have a (7, 4) systematic linear Hamming code.
- We are given three message-codeword pairs: (1 1 0 0; 0 1 0 1 1 0 0), (1 1 1 0; 0 0 1 1 1 1 0), and (0 1 1 0; 1 0 0 0 1 1 0).
To find:
- We need to determine which of the given options is a valid codeword in this code.
Explanation:
A (7, 4) Hamming code has 7 bits in each codeword, with 4 message bits and 3 parity bits. The parity bits are calculated using the message bits according to the Hamming code algorithm.
Hamming Code Algorithm:
1. The 4 message bits (m1, m2, m3, m4) are placed in the codeword at positions 3, 5, 6, and 7.
2. The remaining positions (1, 2, 4) are filled with parity bits (p1, p2, p3) calculated based on the message bits.
Calculating Parity Bits:
To calculate the parity bits, we need to find the values of p1, p2, and p3.
For the first pair (1 1 0 0; 0 1 0 1 1 0 0):
- p1 = m1 XOR m2 XOR m4 = 1 XOR 1 XOR 0 = 0
- p2 = m1 XOR m3 XOR m4 = 1 XOR 0 XOR 0 = 1
- p3 = m2 XOR m3 XOR m4 = 1 XOR 0 XOR 0 = 1
For the second pair (1 1 1 0; 0 0 1 1 1 1 0):
- p1 = m1 XOR m2 XOR m4 = 1 XOR 1 XOR 0 = 0
- p2 = m1 XOR m3 XOR m4 = 1 XOR 1 XOR 0 = 0
- p3 = m2 XOR m3 XOR m4 = 1 XOR 1 XOR 0 = 0
For the third pair (0 1 1 0; 1 0 0 0 1 1 0):
- p1 = m1 XOR m2 XOR m4 = 0 XOR 1 XOR 0 = 1
- p2 = m1 XOR m3 XOR m4 = 0 XOR 1 XOR 0 = 1
- p3 = m2 XOR m3 XOR m4 = 1 XOR 1 XOR 0 = 0
Valid Codewords:
Now, let's check which of the given options satisfy the calculated parity bits.
Option a) 1 0 1 1 0 1 0:
- p1 = 1 XOR 0 XOR 1 XOR 1 = 1
- p2 = 1 XOR 0 XOR 1 XOR 1 = 1
- p
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A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (mi; ci), where ciis the codeword corresponding to the ithmessage mi, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?a)1 0 1 1 0 1 0b)0 1 1 0 1 0 0c)0 0 0 1 0 1 1d)1 1 0 1 0 0 1Correct answer is option 'C'. Can you explain this answer?
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A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (mi; ci), where ciis the codeword corresponding to the ithmessage mi, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?a)1 0 1 1 0 1 0b)0 1 1 0 1 0 0c)0 0 0 1 0 1 1d)1 1 0 1 0 0 1Correct answer is option 'C'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (mi; ci), where ciis the codeword corresponding to the ithmessage mi, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?a)1 0 1 1 0 1 0b)0 1 1 0 1 0 0c)0 0 0 1 0 1 1d)1 1 0 1 0 0 1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (mi; ci), where ciis the codeword corresponding to the ithmessage mi, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?a)1 0 1 1 0 1 0b)0 1 1 0 1 0 0c)0 0 0 1 0 1 1d)1 1 0 1 0 0 1Correct answer is option 'C'. Can you explain this answer?.
A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (mi; ci), where ciis the codeword corresponding to the ithmessage mi, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?a)1 0 1 1 0 1 0b)0 1 1 0 1 0 0c)0 0 0 1 0 1 1d)1 1 0 1 0 0 1Correct answer is option 'C'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (mi; ci), where ciis the codeword corresponding to the ithmessage mi, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?a)1 0 1 1 0 1 0b)0 1 1 0 1 0 0c)0 0 0 1 0 1 1d)1 1 0 1 0 0 1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A digital transmission system uses a (7, 4) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code (mi; ci), where ciis the codeword corresponding to the ithmessage mi, are known to be (1 1 0 0 ; 0 1 0 1 1 0 0), (1 1 1 0 ; 0 0 1 1 1 1 0) and (0 1 1 0 ; 1 0 0 0 1 1 0), then which of the following is a valid codeword in this code?a)1 0 1 1 0 1 0b)0 1 1 0 1 0 0c)0 0 0 1 0 1 1d)1 1 0 1 0 0 1Correct answer is option 'C'. Can you explain this answer?.
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