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**Cellular Communications**

6. Channel Coding

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**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

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**Coded Communication System**

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

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**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)

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**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

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**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”

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**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

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**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

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**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

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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

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**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

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**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

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**Shannon Theorem p is probability bit error**

Probability of an error (incorrect decoding) can be made arbitrary small if

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**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:

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**Linear Codes: Decoding**

Parity check matrix Gives zero when multiplied by valid codeword When error is present, produce “syndrome”

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**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

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**Decoding using Syndrome**

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

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**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

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**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)

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Convolution Codes

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Trellis Diagram

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**States During the Encoding**

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**Decoding: Viterbi Algorithm**

Errors on the channel Find path with minimal total errors

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**Trellis Coded Modulation (TCM)**

Combined coding and modulation scheme Make most similar signals (phases) represent most different/distance codewords

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**Turbo Codes Use 2 convolutional codes on the same data**

Feed data in different order to the encoders

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**Turbo Codes Iterative Decoding Used in**

Each decoder takes into account “guess” from other Continue till produce same “guesses” Used in 3G/4G WiMAX

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**How Do they Work (© IEEE spectrum)**

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**How Do they Work (© IEEE spectrum)**

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