# Bounds on the Frame Error Probability for Turbo Codes in Block Fading Channels

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## Wireless Channel Impairments

In this section, we discuss the properties of the wireless channel, which motivates the design decisions made in this thesis. In a wireless network, the data which is transferred from a transmitter to a receiver has to propagate through a physical medium, or wireless channel. During propagation several phenomena will distort the signal. In the following we briefly describe the channel impairments (noise, path loss propagation and fading) which can be modeled stochastically, as presented in the last sub-section.

Noise

For a typical communication system, the noise is generally modeled as an additive distortion. The main sources of noise in a wireless network are the combination of a wide variety of sources, ambient heat in the transmitter/receiver, co-channel and adjacent channel interference from other communications systems, climatic phenomena and even cosmic background radiation [Wic95]. This noise is stochastically modeled using a zero-mean Gaussian probability density with variance σ2. The total noise power will be N0 = 2σ2. This noise model is known as additive white Gaussian noise (AWGN). The signal-to-noise ratio (SNR) is a widely used value to indicate the signal quality at the receiver. It is an important parameter used to quantify the noise with respect to the signal.

### Path Loss and Fading

The signal is attenuated mainly by the eﬀects of path loss and fading, both repre-senting multiplicative distortions. The path loss eﬀect is defined as the signal power attenuation resulting from propagation over long distances. As long as the distance between the transmitter and the receiver does not change too much, it can be assumed to be constant for the whole transmission. When the receiver and the transmitter are separated by a distance d, it is generally considered that the power of the received signal is attenuated proportionally to d−β , where β denotes the path-loss exponent and it is often assumed to be 2 ≤ β ≤ 6 [Ber00]. For instance, the path loss exponent β is assumed to be 2 in the case of propagation in free space. For lossy environments, β can be in the range of 4.
Furthermore, in wireless channels, the presence of obstacles cause diﬀerent eﬀects such as reflections, diﬀractions or shadowing eﬀects, resulting in multi-path propagation which induces fadings. The so-called multi-path propagation causes that diﬀerent parts of the transmitted signal spectrum to be attenuated diﬀerently, which is known as frequency-selective fading. In addition to this, due to the mobility of transmitter and/or receiver or some other time-varying characteristics of the transmission environment, the principal characteristics of the wireless channel change in time which results in time-varying fading of the received signal.
A lot of statistical models exist to describe the time-varying nature of the received signals. The most known are the Rayleigh distribution in the no line of sight (NLOS) case and the Rice distribution in the line of sight (LOS) case. In the course of the thesis, we will consider the Rayleigh distribution to model fading channels. The coherence bandwidth measures the separation in frequency after which two sig-nals will experience uncorrelated fading.
• In flat fading, the coherence bandwidth of the channel is larger than the bandwidth of the signal. Therefore, all frequency components of the signal will experience the same magnitude of fading.
• In frequency-selective fading, the coherence bandwidth of the channel is smaller than the bandwidth of the signal. Diﬀerent frequency components of the signal therefore experience decorrelated fading.

The DMT was formally defined and studied in the context of point-to-point coherent multiple antenna communications in [ZT03]. It is defined in order to understand the interplay between rate and reliability in the high SNR regime. In [ZT03], it was found that the design of channel codes aim to achieve either the full diversity or the full multiplexing gain oﬀered from the channel. Furthermore, the DMT framework is usually adopted as a helpful performance metric to stress the diﬀerence between several systems achieving the same diversity order. Next, we summarize several important definitions that will be used for the DMT analysis:
1. A function f (ρ) is said to be exponentially equal to ρb when lim log (f (ρ)) = b. (2.16) log (ρ) ρ→∞
Exponential equality is denoted by =, i.e., f (ρ) = ρb. b is called the exponential order of f (ρ).
2. We consider a family of codes {C(ρ)} such that the code C(ρ) has a rate R(ρ), a throughput eﬃciency η(ρ) and a maximum likelihood error probability Pe(ρ). For this family, the multiplexing gain, the eﬀective multiplexing gain (also known as the eﬀective rate), and the diversity gain are defined, respectively, as: r = lim R(ρ) (2.17) ρ→∞ log (ρ).

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Background on Forward Error Correcting

The diversity techniques were introduced to combat wireless channel impairments. However, these techniques remain mostly insuﬃcient to guarantee a reliable transmis-sion. Therefore, we have to further use the FEC scheme, a system of error control for data transmission. Forward refers to the non feedback aspect of error control where a code automatically corrects errors detected at the receiver. The main idea behind channel coding is to introduce a carefully designed redundancy to the information data in order to make the transmission more reliable. However, this operation causes a re-duction in data rate or an expansion in bandwidth. In this section, we describe some well-known channel codes considered in the course of the thesis.

Rate Compatible Punctured Convolutional Codes

Convolutional codes are extensively used in various applications in order to achieve reliable transmission. Figure 2.4.1 shows the block diagram of a convolutional encoder with memory m = 3. D represents the memory register, responsible of forwarding the bits with a delay of one time unit. Such a code is referred to as systematic, since the input being encoded is included in the output sequence. The code is recursive because the current memory bits are fed back to compute the new memory bits. The code rate is Rc = 1/2. It is defined as the ratio between the information bits and the total coded bits, which corresponds for this example to two coded bits for each information bit. The generator polynomials in octal form of this recursive systematic convolutional (RSC) code are (13, 15)8.

Abstract
R´esum´e
Remerciements
List of Figures
List of Tables
Abbreviations
Notations
1 Introduction
1.1 Overview and Motivations
1.2 Contributions and Outlines of the Dissertation
2 Point-to-Point Communications
2.1 Wireless Channel Impairments
2.1.1 Noise
2.1.2 Path Loss and Fading
2.1.3 Channel Model
2.2 Channel Capacity
2.2.1 Entropy and Mutual Information
2.2.2 Capacity and Outage Probability
2.3 Diversity Techniques
2.4 Background on Forward Error Correcting
2.4.1 Rate Compatible Punctured Convolutional Codes
2.4.2 Parallel Concatenated Convolutional Codes
2.4.2.1 Log-Likelihood Ratios
2.4.2.2 Iterative Decoding
2.4.3 Bounds on the Frame Error Probability for Turbo Codes in Block Fading Channels
2.4.3.1 Performance of Turbo Codes
2.4.3.2 Performance of Punctured Turbo Codes
2.5 Background on Retransmission Protocols
2.5.1 Hybrid Automatic Repeat Request Protocols
2.5.2 Incremental Redundancy Hybrid Automatic Repeat Request Protocols
2.5.3 Throughput Efficiency Performance
2.6 Multi-Source Transmission
2.6.1 Interleaved Division Multiple Access
2.6.1.1 IDMA Transmitter and Receiver Structures
2.6.1.2 The Elementary Signal Estimator Function
2.7 Transmission of Correlated Sources Over a Multiple Access Channel
2.8 Chapter Summary
3 Cooperative HARQ Protocols for Turbo Coded Cooperation
3.1 Background on Cooperative Communications
3.1.1 The Relay Channel
3.1.2 Cooperative Strategies
3.1.2.1 Amplify-and-Forward
3.1.2.2 Decode-and-Forward
3.1.2.3 Compress-and-Forward
3.1.3 Distributed Turbo Codes for the Relay Channel
3.1.4 Cooperative HARQ Protocols
3.2 The Coded Cooperation
3.2.1 System and Channel Models
3.2.2 Coded Cooperation Scheme
3.2.2.1 The Cyclic Redundancy Code
3.2.3 Turbo Coded Cooperation Scheme
3.2.4 Performance Evaluation
3.2.4.1 Rate Compatible Punctured Convolutional Codes
3.2.4.2 Turbo Codes
3.3 Cooperative HARQ Protocols for Turbo Coded Cooperation
3.3.1 Destination-Level HARQ Protocol
3.3.2 Relay-Level HARQ Protocol
3.3.3 Two-Level HARQ Protocol
3.4 Performance Evaluation
3.4.1 Bit Error Rate
3.4.2 Throughput Efficiency
3.5 Multi-Source Transmission with Interleave Division Multiple Access .
3.6 Chapter Summary
4 Performance Analysis of Coded Cooperation with HARQ
4.1 Outage Probability
4.1.1 Coded Cooperation Scheme
4.1.2 Destination-Level HARQ Protocol
4.1.3 Relay-Level HARQ Protocol
4.1.4 Two-Level HARQ Protocol
4.1.5 Numerical Results
4.2 Asymptotic Analysis and Diversity Order
4.3.1 Coded Cooperation Tradeoff Curve
4.3.2 Destination-Level HARQ Tradeoff Curve
4.3.3 Relay-level HARQ
4.3.4 Two-level HARQ
4.4 Bounds on the Frame Error Probability for Cooperative HARQ Protocols based on Turbo Coded Cooperation
4.5 Frame Error Probability Based Analysis
4.5.1 Frame Error Probability Retransmission Gain
4.5.2 Frame Error Probability Cooperation Gain
4.6 Throughput Efficiency Based Analysis
4.6.1 Throughput Efficiency Retransmission Gain
4.6.2 Throughput Efficiency Cooperation Gain
4.7 Geometrical Analysis
4.8 Chapter Summary
5 Decode-and-Forward Relaying with Correlated Sources
5.1 System Model
5.2 Error Bounds for Gaussian Channels in Single-Relay Case
5.2.1 Virtual Channel-Maximum a Priori Decoder
5.2.2 Maximum a Priori Decoder
5.3 Error Bounds for Rayleigh Channels for the Single-Relay Case
5.4 Numerical Results
5.5 Chapter Summary
6 Conclusions
A Proof of the Tradeoff Curve of the Coded Cooperation Scheme
B Proof of the Tradeoff Curve
Bibliography

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