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Computer Network - Essay Example

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Each codeword contains three copies of the string. If the bit-string is 01, we send the 6-bit string 010101. What is the Hamming distance of this code? How many errors is it possible to detect? How many errors is it possible to…
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A wireless protocol transmits over a channel 20 MHz wide over a noiseless channel. How many bits/sec can be sent if 64 level digital signals are used?
Channel bandwidth = 20 MHz = 20 * 106
Using the Nyquist theorem;
C = 2Blog2M
C = 2 * 20 *106 * log 264
=240Mbps
2) If a binary signal is sent over a 20 3KHz MHz channel whose signal-to-noise ratio is 40 20 dB, what is the maximum achievable data rate?
We know that 10log10S/N = 40dB
It follows that S/N = 100
Now from the Shannon theorem;
The maximum number of bits per second = H* log2 (1 + S/N) = 20KHz * log2(1+100)
Log2(101)= 6,658
= 20 * 6658
=133160
Now from the Nyquist theorem;
The maximum number of bits per second = 2 * H * log2V = 2 * 20 * log22
=40
The Nyquist theorem gives us a lower data rate hence it forms is the maximum digital signals used.
3) Radio antennas often work best when the diameter of the antenna is equal to the wavelength of the radio wave. An antenna has a diameter of 10 cm. What frequency range does it work best for?
From the formulae λf = c
We get f = c/λ.
Now c = 3×108 m/sec and λ = 3×10-2 m/cycle
We obtain (3×108 m/sec)/(3×10-2 m/cycle) = 3 × 1010 m/cycle = 30 GHz.
When λ = 5 m/cycle we obtain (3×108 m/sec)/(5 m/cycle) = 60×106 m/cycle = 60 MHz.
Hence the frequency range is 60 MHz to 30 GHz.
4) Suppose we want to send 2-bit strings. Each codeword contains three copies of the string. If the bit-string is 01, we send the 6-bit string 010101. What is the Hamming distance of this code? How many errors is it possible to detect? How many errors is it possible to correct?
The length of the data sequence is 6
The number of parity bits is 5
The length of the code word is 11
The parity scheme is represented as
P0 = D0 + D1 + D3 + D4
P1 = D0 + D2 + D3 + D5
P2 = D1 + D2 + D3
P3 = D4 + D5
P4 = D0 + D1 + D2 + D3 + D4 + D5 + P0 + P1 + P2 + P3
The syndrome matrix is
Top of Form
D5
D4
D3
D2
D1
D0
P4
P3
P2
P1
P0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
1
1
1
0
0
0
1
0
0
1
0
1
1
0
1
0
0
0
1
0
0
1
1
0
1
1
0
0
0
0
1
Bottom of Form
5) Ten-bit messages are transmitted using a Hamming code. How many check bits are needed to ensure that the receiver can detect and correct single-bit errors? Show the bit pattern transmitted for the message 1101001100. Assume that even parity is used in the Hamming code.
The parity scheme is
P0 = D0 + D1 + D3 + D4
P1 = D0 + D2 + D3 + D5
P2 = D1 + D2 + D3
P3 = D4 + D5
P4 = D0 + D1 + D2 + D3 + D4 + D5 + P0 + P1 + P2 + P3
The Syndrome Matrix isTop of Form
D5
D4
D3
D2
D1
D0
P4
P3
P2
P1
P0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
1
1
1
0
0
0
1
0
0
1
0
1
1
0
1
0
0
0
1
0
0
1
1
0
1
1
0
0
0
0
1
Bottom of Form
6) A 7-bit Hamming code whose binary value is 1100010 arrives at a receiver. What was the original value in hexadecimal? Assume that not more than 1 bit is in error. Assume the value that arrived was 0110011? Answer the same questions.
The length of the data sequence is 7, the number of parity bits is 5 and the length of the code word is 12
Its parity scheme is
P0 = D0 + D1 + D3 + D4 + D6
P1 = D0 + D2 + D3 + D5 + D6
P2 = D1 + D2 + D3
P3 = D4 + D5 + D6
P4 = D0 + D1 + D2 + D3 + D4 + D5 + D6 + P0 + P1 + P2 + P3
Its syndrome matric is
Top of Form
D6
D5
D4
D3
D2
D1
D0
P4
P3
P2
P1
P0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
1
1
0
0
0
1
0
0
1
1
0
1
1
0
1
0
0
0
1
0
1
0
1
1
0
1
1
0
0
0
0
1
Bottom of Form
Length of Data Sequence
Number of Parity Bits
Length of CodeWord
7
5
12
Parity Scheme
P0 = D0 + D1 + D3 + D4 + D6
P1 = D0 + D2 + D3 + D5 + D6
P2 = D1 + D2 + D3
P3 = D4 + D5 + D6
P4 = D0 + D1 + D2 + D3 + D4 + D5 + D6 + P0 + P1 + P2 + P3
Syndrome Matrix
Top of Form
D6
D5
D4
D3
D2
D1
D0
P4
P3
P2
P1
P0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
1
1
0
0
0
1
0
0
1
1
0
1
1
0
1
0
0
0
1
0
1
0
1
1
0
1
1
0
0
0
0
1
Bottom of Form Read More
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