1. Reliability and Channel Coding

2. 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

3. Nyquist Theorem The maximum symbol rate is 2B
Thus, if there are K signal levels, ignoring noise, the maximum bit rate is:

4. 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

5. 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)

6. 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?

7. 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.

8. Errors Happen!
Illustration of a burst error that results from interference

9. 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.

10. Forward Error Correction

11. Using Error Codes Codeword consists of dataword D plus check bits R
R=fn(D)
Sender:
Compute check bits R based on the dataword D; send the codeword D+R

12. Using Error Codes Receiver: D’ R’ R=fn(D’)
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’
Dataword
Check bits D’ R’ =?
R=fn(D’)

13. 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)

14. 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

15. 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?

16. Hamming distance
Distance between two codewords D and D’ is the number of bit flips needed the change D to D’

17. Hamming distance
Hamming distance of a code is the minimum distance between any pair of codewords
Example: Single parity checking:
Hamming distance = 2

18. Hamming Distance Error detection: Error correction:
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

19. 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 .

20. Correcting a Single-Bit Error.

21. 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

22. 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.

23. The 16-Bit Checksum Used in the Internet
Step 1: add 16-bit values
Overflow
Step 2: add overflow
Step 3: invert result

24. 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!