CELLULAR COMMUNICATIONS 6. Channel Coding. Motivation Wireless channel introduces errors due to ...
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Transcript of CELLULAR COMMUNICATIONS 6. Channel Coding. Motivation Wireless channel introduces errors due to ...
Motivation
Wireless channel introduces errors due to 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
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)
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
Error correction vs. Error Detection Error-detection
Detect that received sequence contains an error Request retransmission ARQ: Automatic Repeat Request/Query (HSDPA)
Error-correction Detect that received sequence contains an error Correct the error Forward Error Correction
“A Code allows correction of up to p errors and detection up to q (q>p) errors”
Block Codes vs. Convolution Codes Block 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
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
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
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
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
Sphere Packing Bound
Assume
Example 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
Shannon Theorem
p is probability bit error Probability of an error (incorrect
decoding) can be made arbitrary small if
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:
Linear Codes: Decoding Parity check matrix
Gives zero when multiplied by valid codeword
When error is present, produce “syndrome”
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
Decoding using Syndrome
Evaluate the syndrome s from r Lookup corresponding e in a pre-
computed table Correct codeword c=r+e
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
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)
Trellis Coded Modulation (TCM) Combined coding and modulation scheme Make most similar signals (phases) represent most different/distance codewords
Turbo Codes
Iterative Decoding Each decoder takes into account “guess”
from other Continue till produce same “guesses”
Used in 3G/4G WiMAX