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http://www.ii.uib.no/~paale/oblig.xhtml. Mandatory exercise for Inf 244 Deadline: October 29th The assignment is to implement an encoder/decoder system in Matlab using the Communication Blockset. The system must simulate communication over an AWGN channel using either of these codes:

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  1. http://www.ii.uib.no/~paale/oblig.xhtml • Mandatory exercise for Inf 244 • Deadline: October 29th • The assignment is to implement an encoder/decoder system in Matlab using the Communication Blockset. The system must simulate communication over an AWGN channel using either of these codes: • Block code • Convolutional code • PCCC • LDPC • You are free to implement any of these coding techniques, as long as the requirements below are fullfilled: • Information length k = 1024 Block length n = 3072 E b / N o = 1.25 • We will test your answers on our own computer and evaluate them based on bit error rate versus CPU time usage according to the following formula: p = T ⋅ BER

  2. http://www.ii.uib.no/~paale/oblig.xhtml How to create and run simulations in MATLAB from scratch • Run matlab from a command window. • Type in simulink in MATLAB's command window. • Choose File -> New -> Model in Simulink's main window. • Create the model by dragging and dropping elements into it. How to finish this exercise starting from a demo • Run matlab from a command window. • Type in sccc_sim in MATLAB's command window. A ready-made demo of an SCCC opens. • Study the demo closely and modify it to your needs.

  3. Design of turbo codes Errors that an ML decoder would also make Errors due to imperfectness of decoding algorithm • Turbo codes can be designed for performance at • Low SNR • High SNR • Choices: Constituent codes, interleaver

  4. Design of turbo codes for high SNR • Goal: • Maximize minimum distance d • Minimize the number of codewords Ad of weight d • In general, design code for a thin weight spectrum • Use recursive component encoders! • Simple approach: • Concentrate on the weight two-inputs • Simple (but flawed!) approach: This applies if the interleavers are chosen at random. But it is possible (and even easy) to avoid the problem

  5. Weight-two inputs • Two weight input vectors that will take the encoder out of and back to the initial state: • (1, 0,0, 1)  corresponds to parity weight zmin • (1, 0,0, 0,0,0, 1) • In general, 1 followed by 3m-1 0s followed by a 1 • Even more general, 1 followed by (2-1)m-1 0s followed by a 1 • deff= 2 + 2zmin • Assume primitive feedback polynomial

  6. Theorem on zmin • Theorem: Let G(D) = [ 1, e(D)/d(D) ], where the denominator polynomial d(D) is primitive of degree . Then zmin=2-1 + 2s, s=1 if e(D) has constant coefficient 1 and degree , s=0 otherwise. • Proof: • d(D) is the generator polynomial of a cyclic (2-1, 2-1-)Hamming code • q(D)=(D2-1+1) /d(D) is the generator polynomial of a cyclic (2-1, )maximum-length code of minimum distance 2-1 • deg e(D) < : e(D) q(D) is a codeword in the maximum-length code and so has weight 2-1 • deg e(D) = : e(D)=DD-1 + e(2)(D). c1(D) =D-1q(D) and c2(D) = e(2)(D)q(D) are both codewords in the maximum-length code and so have weight 2-1. Dc1(D) = [cycl. shift of c1(D)]+ D2-1+1, so e(D) q(D) is [a codeword with const.coeff=0] + 1 + D2-1.

  7. Convolutional Recursive Encoders for PCCC codes Max 6 10 13 16 18 5 8 10 12 13 15 3 5 6 7 8 10

  8. Convolutional Recursive Encoders for PCCC codes Max 2 3 4 5 6 7 2 3 4 4 5

  9. Choice of component codes • The listed codes may not have the best free distance, but have a better mapping (compared to ”optimum” CCs) of input weights to output weights • The overall turbo code performance depends also on the actual interleaver used

  10. Choice of interleaver • Pseudorandom interleavers with enhanced requirements: • Interleavers that avoid problem with weight-2 inputs:* • If | i-j | = (2-1)m, then | (i)-(j) |  (2-1)n (for n+m small) • S-random interleaver: • If | i-j |  S, then | (i)-(j) |  S • Interleavers specialized to accommodate the actual encoders* • Maintains a list of ”critical” sets of positions, which are the information symbols of low weight words • Do not map one critical set into another

  11. Design of turbo codes for low SNR • The foregoing discussion assumes that the decoding is close to maximum likelihood. This is not the case for very low SNRs • Goal for low SNR: • Optimize interchange of information between the constituent decoders • Analyze this interchange by using density evolution or EXIT charts

  12. EXtrinsic Information Transfer charts • Approach: A SISO block produces a more accurate information about the transmitted information at its output than what is available at the input • The amount of information can be precisely quantified using information theory • The entropy H(X) of a stochastic variable X is given as H(X) = - xP(X=x)log(P(X=x)). It is a measure of uncertainty • The mutual information I(X;Y) = H(X)-H(X|Y) • For a specified SNR (and thus a known information about the ul due to the channel values): • Ia(ul,La(ul)) • Ie(ul,Le(ul)) • EXIT chart: Ie(ul,Le(ul)) as a function of Ia(ul,La(ul)) by log-MAP

  13. EXIT curves • Obtained by simulations (But much simpler than turbo code simulations)

  14. EXIT charts Open tunnel: Decoding will proceed to convergence • Next, plot the EXIT curve for one SNR, together with its mirror image. These curves represent the EXIT curves of the two constituent decoders

  15. EXIT charts Closed tunnel: Decoding will get stuck • EXIT chart for another SNR:

  16. SNR Threshold Tunnel opens Non-convergence becomes a small problem • SNR Threshold: The smallest SNR with an open EXIT chart tunnel • Defines the start of the waterfall region

  17. EXIT chart • A property of the constituent encoder • Can be used to find good constituent encoders for low SNRs • In general, simple is good (flatter EXIT curve) • Can be used for codes with different constituent encoders too. The constituent coders can in this case be fitted to each other’s EXIT curve, providing a lower SNR threshold • It is assumed that the interleavers are very long, so that a Gaussian Approximation applies: Errors in the extrinsic values occur according to a Gaussian distribution

  18. Iterative decoding • Decoding examples • Some observations

  19. Decoding example

  20. Decoding example: K=4

  21. The effect of many iterations

  22. Iterative decoding: Stopping criteria • Fixed number of iterations • Hard-decisions • If the hard decisions of the two extrinsic value vectors coincide; assume that convergence has been achieved • Cross-entropy • Outer error-detecting codes

  23. Iterative decoding: Some observations • Parallel implementations: The constituent decoders can work in parallel • Final decision can be taken from a posteriori values of either constituent decoder; their average; or the extrinsic values • The decoder may sometimes, depending on the SNR and on the occurence of structural faults in the interleaver, oscillate between correct and incorrect decisions • Max-log-MAP can be shown to be equivalent to SOVA • Max-log-MAP is (a little!) simpler to implement than log-MAP, but suffers a penalty of about 0.5 dB

  24. Suggested exercises • 16.16-16.30

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