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Cellular Communications

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Presentation on theme: "Cellular Communications"— Presentation transcript:

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)

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