Reliability and Channel Coding. How fast can we send information over a link? Key channel...
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Transcript of Reliability and Channel Coding. How fast can we send information over a link? Key channel...
![Page 1: Reliability and Channel Coding. How fast can we send information over a link? Key channel properties: The bandwidth (B), single strength (S), and noise.](https://reader036.fdocuments.in/reader036/viewer/2022062715/56649d6e5503460f94a500ae/html5/thumbnails/1.jpg)
Reliability and Channel Coding
![Page 2: Reliability and Channel Coding. How fast can we send information over a link? Key channel properties: The bandwidth (B), single strength (S), and noise.](https://reader036.fdocuments.in/reader036/viewer/2022062715/56649d6e5503460f94a500ae/html5/thumbnails/2.jpg)
How fast can we send information over a link?
• Key channel properties: The bandwidth (B), single strength (S), and noise strength (N)– B limits the rate of transmissions– S and N limit how many signal levels we can
distinguish
Bandwidth B Signal S,Noise N
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Nyquist Theorem
• The maximum symbol rate is 2B
• Thus, if there are K signal levels, ignoring noise, the maximum bit rate is:
0 1 0 1 0 1 0 1
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The Effect of Noise on Communication
• In practice, the signal levels we can distinguish depends on S/N– Or SNR, the Signal-to-Noise Ratio– SNR often given on a log-scale in deciBels:• dB = 10log10(S/N)
• Shannon’s Theorem
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Calculate Channel Capacity with S/N in dB
• The voice telephone system:– Signal-to-noise ratio: about 30 dB– An analog bandwidth: about 3 kHz
• Calculation– Step 1: Converting the S/N in dB into a simple
fraction: S/N = 10(dB/10)
• 30dB 1000
– Step 2: Applying Shannon's Theorem
about 30,000 bps
dB = 10log10(S/N)
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Exercise• If two signal levels are used, what is
the data rate that can be sent over a coaxial cable that has an analog bandwidth of 6.2 MHz?
• If a system has an average power level of 100, an average noise level of 33.33, and a bandwidth of 100 MHz, what is the effective limit on channel capacity?
• If a telephone system can be created with a signal-to-noise ratio of 40 dB and an analog bandwidth of 3000 Hz, how many bits per second could be transmitted?
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The Significance of Channel Capacity• Nyquist's work has provided an incentive to
explore complex ways to encode bits on signals.
• Shannon's Theorem is more fundamental because it represents an absolute limit derived from the laws of physics.
• Engineers recognize that faster transmission speeds will only be possible if the signal-to-noise ratio can be improved.
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Errors Happen!
Illustration of a burst error that results from interference
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Channel Coding: Strategies to Handle Channel Errors
• Forward Error Correction (FEC)– Adding informaion on the sender side– Detecting and correcting errors on the receiver side
• Automatic Repeat reQuest (ARQ)– Requiring cooperation and further message
exchanges between the sender and receiver.
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Forward Error Correction
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Using Error Codes
• Codeword consists of dataword D plus check bits R
D R=fn(D)
• Sender:– Compute check bits R based on the dataword D;
send the codeword D+R
Dataword Check bits
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Using Error Codes
• Receiver:– Receive codeword D’+R’ with unknown errors
• The check bits could contain errors, too!
– Recompute R check bits based on the dataword D’; error if R does not match R’
D’ R’Dataword Check bits
R=fn(D’)=?
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Example: Single Parity Checking
• Adding an extra bit to each byte to make the number of 1 bits even (Even Parity) or odd (Odd Parity)
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How well does parity work?
• If one of the nine bits is changed, will the error be detected?
• How about if two bits are changed?
• What about larger errors (3, 4, 5,… bits are changed)?– Can detect if odd number of bits are changed
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How to Choose Coding Schemes?
• An ideal coding scheme:– Any change of bits in a valid data leads to an invalid
result.
• In practice:– What is the minimum number of bits of a valid
codeword that must be changed to produce another valid codeword?
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Hamming distance
• Distance between two codewords D and D’ is the number of bit flips needed the change D to D’
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Hamming distance
• Hamming distance of a code is the minimum distance between any pair of codewords– Example: Single parity checking:
Hamming distance = 2
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Hamming Distance
• Error detection:– For a code of distance d+1, up to d errors will
always be detected
• Error correction:– For a code distance 2d+1, up to d errors can
always corrected by mapping to the closest codeword
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.
Error Correction with Row and Column (RAC) Parity
• If an error is detected, how do we know which bit went wrong?
• Adding bits for both rows and columns
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Correcting a Single-Bit Error
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Row and Column (RAC) Parity
• If one of the nine bits is changed, will the error be detected?
• How about if two bits are changed?
• What about larger errors (3, 4, 5,… bits are changed)?– Can detect if odd number of bits are changed
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The 16-Bit Checksum Used in the Internet
• The Internet checksum treats data in a message as a series of 16-bit integers.– The size of a dataword can vary.– Adding 0s if the data is not an exact multiple of 16
bits.
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The 16-Bit Checksum Used in the Internet
0100 1000 0110 01010110 1100 0110 1100
+ 0110 1111 0010 0001 1 0010 0011 1111 0010
0010 0011 1111 0010 + 1
0010 0011 1111 0011
1101 1100 0000 1100
Step 1: add 16-bit values
Step 2: add overflow
Step 3: invert result
Overflow
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The 16-Bit Checksum Used in the Internet
• How well does the checksum work?– What is the Hamming distance of the code?– How many errors will it detect/correct?
• What about larger errors?– Correct all burst errors up to 16 bits!