Ch5-Part2-ChannelCoding

7/27/2019 Ch5-Part2-ChannelCoding

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by Assoc. Prof. Thuong Le-Tien 1

Channel Coding Part II

DIGITAL COMMUNICATIONS

Lectured by Assoc. Prof. Dr. Thuong Le-Tien

October 2011


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3. Convolutional Codes


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Convolutional encoder with n=2, k=1, and L=2

Figure 3-2


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Code tree for 2,1,2) encoder


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(a) Code trellis (b) state diagram for (2, 1, 2) encoder (c)

illustrative sequence

Figure 13.3-4


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Termination of (2, 1, 2) code trellis

Figure 3-7

Each branch has been labeled with the number of 1s in the encoded bits

Free Distance and Coding Gain


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(7)


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where


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bcRbc

eR

2/1

4


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DECODING METHOD


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Illustration of the Viterbi algorithm for maximum-likelihood decoding

Figure 3-11


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Turbo code

Turbo codes, or parallel concatenated codes (PCC) are a

relatively new class of convolutional codes first introduced

in 1993 by Berrou et al., Berrou (1996), Hagenauer et al.(1996), and Johannesson and Zigangirov (1999).

They have enabled channel capacities to near

reach the Shannon limit.

Shannons theorem for channel capacity assumes random

coding with the BER approaching zero as the codes block

or constraint length approaches infinity


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Turbo encoder

The RSC is Recursive Systematic Convolutional encoder with rate. Both RSC produce parity check bits then overall rate is 1/3.

However it can be reduced to using the process of puncturing by

eliminating the odd parity check bits of the first RSC and the even

parity check bits of the second RSC


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For the particular encoder in the figure, the polynomial

describing the feedback connections is 1+D3+D4=10011=238and the polynomial for the output is

1+D+D2+D4=11101=358. Hence, the literature often refers

this to as G1=23, G2=35 or simply a (23,35) encoder.

RSC encoder with R=1/2, G1=23, G2=35, L=2


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Turbo decoder: Consist of two Maximum a Posterior

(MAP) decoders and feedback path. The first decodertakes the information from the received signal and

calculate the A Posterior Probability (APP) value. This

value is then used as the APP value for the second decoder

Turbo decoder


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Instead of using the Viterbi algorithm, the MAP decoder usesa modified form of the BCJR (Bahl, Cocke, Jelinek, and

Raviv, 1972) algorithm that take into account the recursive

character of the RSV codes and computes a log-likelihood

ratio to estimate the APP for each bit.

The results by Berrou et al. are impressive. When encodingusing rate R=1/2, G1=37 and G2=21, 65,537 interleaving,

and 18 iterations, they were able to achieve a BER of 1/100000

and Eb/N0=0.7dB.

The main disadvantage of turbo codes with their relatively

large code words and iterative decoding process is their longlatency. A system with 65,537 interleaving and 18 iterations

may have too long a latency for voice telephony


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Reed Solomon Code (RS Code)

* RS codes are nonbinary cyclic codes with code symbols from

A Galois field. They were discovered in 1960 by I. Reed andG. Solomon. The work was done when they were at MIT laboratory.

* In the decades since their discovery, RS codes have enjoyed

Countless applications from compact discs an digital TV in living

Room to spaccraft and satellite in outer space.

* The most important RS codes are codes with symbols from GF(2m)The minimum distance of an (n,k) RS code is n-k+1. Codes of this

kind are called maximum-distance-seperable codes


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A systematic RS code word and

some RS code parameters


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Example. The following code is a (255,233) RS code.

It is NASA standard code for satellite and space communication


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Encoding of RS code


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The encoding circuit is shown below (Lin/Costello)


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Ch5-Part2-ChannelCoding