1. Cellular Communications
6. Channel Coding

2. Motivation Wireless channel introduces errors due to Option A Option B
Noise and Interference
Multipath Effect resulting in fast fading
Option A
Increase power of transmission
Waste of energy and interference
Option B
Send redundant information
Errors can be detected and re-transmission requested
Errors can be corrected
Forward Error Correction(FEC) or Channel Coding

3. Coded Communication System

4. Coding advantages Pn
10-3
uncoded
Coding gain
coded
10-8 8 19
Eb/N0 dB

5. Binary Symmetric Channel
Transmission medium introduce errors
Demodulator produces errors
Model as a channel
Memoryless: probability of error is independent from one symbol to the next
Symmetric: any error is equally probable
Binary Symmetric Channel (BSC)

6. Error Correcting Codes (ECC)
Redundancy added to information
Encode message of k bits with n (n>k) bits
Example: Systematic Encoding
Redundant symbols are appending to information symbols to obtain a coded sequence
Codeword

7. Error correction vs. Error Detection
Detect that received sequence contains an error
Request retransmission
ARQ: Automatic Repeat Request/Query (HSDPA)
Error-correction
Correct the error
Forward Error Correction
“A Code allows correction of up to p errors and detection up to q (q>p) errors”

8. Block Codes vs. Convolution Codes
Encode information block by block
Each block encoded independently
Encoding/Decoding is a memoryless operation
Convolutional Codes
Next symbol depend on a history of inputs/outputs

9. Example Single-bit message 0 or 1 Extend to 3 bit messages (codewords)
010
101
Only 2 valid codeword out of 8
Due to the error can receive any sequence
Can associate invalid code with valid
Invalid codes differs from valid only by 1 bit

10. Hamming distance and Block Code
Two vectors of size n have a Hamming Distance of d if they differ in d bits
Block code is a subset of 2^n bit sequences
Code distance is minimum hamming distance between any two members of the code
Assume code distance d=2t+1
Can detect up to 2t errors
Can correct up to t-1 errors

11. Good Code Design
Select a subset of 2^n (e.g. n=1024) vectors such that
Distance between codewords is large as possible
Can find correct message without comparing with all possible codewords

12. Sphere Packing Bound Assume have to encode M messages
Want being able to correct up to t errors
Each codeword have a ‘sphere” around it with codes of distance up to t
Spheres around different codewords are not overlapping

13. Sphere Packing Bound Assume Example Code rate n=8, t=2=>M<6.9
Found only code of size 4
Code rate
Example (8,2) systematic code, 4 message, 2 information bits
Code rate 2/8=1/4

14. Shannon Theorem p is probability bit error
Probability of an error (incorrect decoding) can be made arbitrary small if

15. Linear Codes Linear combination of valid codewords is also a codeword
Code distance is a minimum among all nonzero codeword weights (number of 1s)
Linear space spanned by basis:

16. Linear Codes: Decoding
Parity check matrix
Gives zero when multiplied by valid codeword
When error is present, produce “syndrome”

17. Syndrome Syndrome depends only on error pattern
Different errors=>different syndromes except for the addition of codeword
Can identify error patterns of weight w<=t by looking at the syndrome
One-to-one between syndromes and errors w<=t

18. Decoding using Syndrome
Evaluate the syndrome s from r
Lookup corresponding e in a pre-computed table
Correct codeword c=r+e

19. Non-binary Codes Alphabet of more then two symbols
q=p^n where p is prime, GF(p^n)
Each element of the alphabet is a vector
Element-wise: operation modulo p
Vector-wise: view as a polynomial

20. Reed-Solomon Code a block code
a cyclic code (some additional constrains)
Used in
CD/DVD
Bar Codes
Deep Space Communications
Cellular Digit Packet Data CPDP RS(63,47)

21. Convolution Codes

22. Trellis Diagram

23. States During the Encoding

24. Decoding: Viterbi Algorithm
Errors on the channel
Find path with minimal total errors

25. Trellis Coded Modulation (TCM)
Combined coding and modulation scheme
Make most similar signals (phases) represent most different/distance codewords

26. Turbo Codes Use 2 convolutional codes on the same data
Feed data in different order to the encoders

27. Turbo Codes Iterative Decoding Used in
Each decoder takes into account “guess” from other
Continue till produce same “guesses”
Used in
3G/4G
WiMAX

28. How Do they Work (© IEEE spectrum)

29. How Do they Work (© IEEE spectrum)