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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:

– 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 ⋅

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http://www.ii.uib.no/~paale/oblig.xhtmlHow 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.

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Design of turbo codes• Turbo codes can be designed for performance at

• Low SNR

• High SNR

• Choices: Constituent codes, interleaver

Errors that an ML decoder would

also make

Errors due to imperfectness of

decoding algorithm

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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

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Weight-two inputs• Assume primitive feedback polynomial

• 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 + 2 zmin

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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.

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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

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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

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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

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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

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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

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

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EXIT charts• Next, plot the EXIT curve for one SNR, together with its mirror image. These

curves represent the EXIT curves of the two constituent decoders

Open tunnel:

Decoding will

proceed

to convergence

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EXIT charts• EXIT chart for another SNR:

Closed tunnel:

Decoding will

get stuck

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SNR Threshold• SNR Threshold: The smallest SNR with an open EXIT chart tunnel

• Defines the start of the waterfall region

Tunnel opens

Non-convergence

becomes a small

problem

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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

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Iterative decoding• Decoding examples• Some observations

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Decoding example

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Decoding example: K=4

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The effect of many iterations

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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

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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

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Suggested exercises

• 16.16-16.30