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## Performance requirements for a TC interleaver

As introduced in section 1.1.1.5, TC performance in the error floor region is mainly related to dmin. Better performance in this region is obtained with higher values of dmin. In order to obtain such values, the interleaver has to avoid feeding the second constituent encoder with the same data sequence patterns that produce low Hamming weight codewords in the first constituent encoder of the TC. Independently of the interleaver choice, data sequences with a single 1 will produce the same codeword weights in both constituent encoders. This motivates the use of RSC codes as constituent codes for the TC, because the output weight due to a weight-1 data sequence is high and even infinite for infinite frame sizes.

A Return To Zero (RTZ) sequence is defined as any finite input sequence which makes a RSC code leave the zero state and return back to the same state. Short RTZ sequences have an important impact on the RSC distance spectrum. Indeed, low weight codewords, in the RSC code, correspond to short RTZ sequences. As far as the TC is concerned, the Hamming weight of codewords resulting from a single RTZ sequence is increased by maximizing the scattering of data through the interleaver. It was shown in [28] that the increase in scattering of data leads to a better TC performance in the error floor region. In the case of codewords made up of more than one RTZ sequence a disorder must be introduced in the permutation function [17]. In fact, multiple RTZ sequences evenly permuted can compose new RTZ sequences for the second constituent code. Furthermore, the disorder in the permutation makes the parity generation of both constituent encoders as diverse as possible, increasing the weight of such codewords [4]. Then, in terms of TC performance, the interleaver should allow a maximum scattering of data, and a maximum disorder in the interleaved data sequence.

### Implementation requirements for a TC interleaver

The principal requirements of a TC interleaver in terms of implementation are: a reduced hardware complexity, which corresponds to the computational complexity of the interleaved addresses and/or the memory needed to store them, and the adaptation to the throughput requirements of the transmission system.

In cutting-edge wireless communication systems, high turbo decoding throughputs are required. This is achieved by using internal parallelization in the turbo decoder. The internal parallelism method, also known as the method of parallel windows [50], involves dividing the trellis of the constituent convolutional codesinto W windows of the same length L. Then, in the turbo decoder, W processors working in parallel treat the W windows of the code trellis increasing the decoding throughput by a factor W. Thus, a degree of parallelism of W is achieved. For parallel processing of W windows of size L, extrinsic values must be stored in W memory banks. The access connections of the processors to the memory banks are defined by the TC interleaving/deinterleaving function. Memory access contentions appear when two or more processors try to access the same memory bank concurrently. These contentions are avoided if the TC interleaver is contention free for a window of size L. Then, in order to allow an efficient parallelization of the turbo decoding the TC interleaver has to be contention free for the desired degree of parallelism.

#### Interleaver design criteria

According to the performance requirements, some interleaver design criteria can be established. The first one is based on the total spatial distance or span between non-interleaved and interleaved data positions. In fact, the maximization of the span results in the maximization of the scattering of data. The second one is based on the correlation between the extrinsic information in the turbo decoder.

**Random permutations**

Random interleavers, as introduced by Divsalar and Pollara [56], are defined by a random permutation of K elements with no repetitions. The data sequence is written into the interleaver memory block and read out in a specific random order. The same interleaver is used for all subsequent data sequences. In [57], an analysis of the effect of random interleavers on low-weight data sequence, which may produce low codeword weights, was carried out. It was shown that the probability to have a low-weight data sequence interleaved to a similar sequence decreases with the data sequence weight. Then, it was concluded that the minimum distance of TCs is more likely to be governed by weight-2 data sequences.

An analysis on the effect of the interleaver on the distance spectrum of TCs in a short data sequence was also performed in [56]. It was shown that better TC distance spectra were obtained by using random permutations than by using structured permutations like row-column block. In fact, low-weight data sequences such as the one shown in Fig. 1.11 cannot be broken by a row-column block interleaver. A useful tool for the design of TCs was introduced by Benedetto and Montorsi [14]. The so called uniform interleaver is a probabilistic interleaver based on random permutations that generate all possible permutations for a given data sequence size with equal probability. It has been used to calculate error-rate bounds as shown in [14]. Furthermore, the uniform interleaver allows the evaluation of the average TC performance for all possible interleavers, providing a TC performance benchmark regardless of the interleaver selection.

**Table of contents :**

Contents

List of Figures

List of Tables

Abbreviations

Notations

Résumé de la thèse

Introduction

**1 Channel coding and interleaving techniques **

1.1 Channel coding

1.1.1 Parallel convolutional turbo codes

1.1.1.1 Turbo encoder structure

1.1.1.2 Decoding algorithm for turbo codes

1.1.1.3 Error rate performance evaluation of turbo codes

1.1.1.4 Convergence analysis of TCs using EXtrinsic Information Transfer (EXIT) chart

1.1.1.5 Distance spectrum estimation methods

1.1.2 Precoded turbo codes

1.1.2.1 Precoded turbo encoder structure

1.1.2.2 Decoding algorithm for precoded turbo codes

1.2 Interleaving techniques for turbo codes

1.2.1 Performance requirements for a TC interleaver

1.2.2 Implementation requirements for a TC interleaver

1.2.3 Interleaver design criteria

1.2.3.1 Span properties

1.2.3.2 Correlation girth

1.2.4 Interleaver types

1.2.4.1 Random permutations

1.2.4.2 Partially random permutations

1.2.4.3 Regular Interleaver

1.2.4.4 Dithered Relative Prime

1.2.4.5 Quadratic Permutation Polynomial

1.2.4.6 Almost Regular Permutation

1.2.4.7 Comparison of algebraic interleavers

1.3 Channel interleaving techniques

1.3.1 Channel interleaver types

1.3.1.1 Adaptive interleaver based on Doppler shift

1.3.1.2 Adaptive interleaver based on Channel State Information

1.3.1.3 The channel interleaver of DVB-T2

1.3.1.4 Comparison of channel interleavers

1.4 Conclusion

**2 Design of efficient puncture-constrained interleavers for turbo codes **

2.1 Overview of the proposed puncture-constrained interleaver design method

2.2 Puncturing mask selection proposal

2.3 Puncturing constraints

2.3.1 State-of-the-art data puncture-constrained interleavers for TCs

2.3.2 Proposed parity puncturing constraint

2.3.2.1 Extrinsic information analysis

2.3.2.2 Connection strategy for the unpunctured parity positions

2.4 New representation of the correlation graph for TCs

2.5 Proposed interleaver design method

2.5.1 Choice of a suitable TC interleaver structure

2.5.1.1 DRP interleavers expressed as ARP interleavers .

2.5.1.2 QPP interleavers as special cases of ARP interleavers

2.5.1.3 Conclusion on the choice of the TC interleaver structure

2.5.2 Graph-based method to select the ARP interleaver parameters

2.6 Application examples

2.6.1 8-state CRSC(13, 15)8 TC

2.6.1.1 Puncturing mask selection

2.6.1.2 Parity puncturing constraint

2.6.1.3 Puncture-constrained ARP interleaver design

2.6.2 8-state CRSC(13, 15, 17)8 TC

2.6.2.1 Puncturing mask selection

2.6.2.2 Parity puncturing constraint

2.6.2.3 Puncture-constrained ARP interleaver design .

2.6.3 Simulated TC error rate performance

2.7 Conclusion

**3 Exploring precoding techniques for turbo codes **

3.1 Selection of a suitable precoding structure

3.1.1 Analysis of the reference precoding structure

3.1.1.1 Proposed design tool based on extrinsic information exchange

3.1.2 Hybrid precoding structure

3.1.3 Composite precoding structure

3.1.4 Conclusion on the choice of a suitable precoding structure .

3.2 Improving the hybrid precoding structure

3.2.1 Studying the precoding pattern

3.2.2 Analysis of different constituent codes for the precoder

3.3 Design criteria for PTC interleavers

3.3.1 Span properties in the three-dimensional case

3.3.2 Proposed correlation graph for the three-dimensional case

3.4 An example of hybrid PTC with optimized interleavers

3.5 Conclusion

**4 Channel interleavers for terrestrial broadcast: analysis and design **

4.1 Identification of relevant design criteria for channel interleavers .

4.1.1 Span properties

4.1.2 Joint representation of the time and frequency span properties in the L1 space

4.1.3 Mutual information distribution

4.2 Performance analysis of DVB-T2 channel interleavers

4.3 Guidelines for designing improved channel interleavers

4.3.1 Regular Interleaver

4.3.2 Double Regular Interleaver

4.3.3 Almost Regular Permutation Interleaver

4.3.4 Double Almost Regular Permutation Interleaver

4.4 Application of interleaver design guidelines and further analysis .

4.4.1 Application of guidelines to design channel interleavers for DVB-T2

4.4.2 Interaction between channel interleaver and BI

4.4.3 Impact on latency and complexity

4.4.3.1 Impact on latency

4.4.3.2 Impact on complexity

4.5 Performance evaluation of the proposed channel interleavers

4.6 Conclusion

Conclusions and future works

A Application examples of the equivalent expressions of DRP and QPP interleavers in the form of the ARP interleaver

B Additional application examples of the puncture-constrained interleaver

design method for 8-state CRSC(13,15)8 TC

B.1 Code rate 2/3 and data sequence size

B.2 Code rate 4/5 and data sequence size

B.3 Code rate 4/5 and data sequence size

C Application of guidelines to design 2ARP channel interleaver

D Effectiveness of the new BI mapping mask in improving BER performance

**Bibliography **

List of publications