Receiver Design for Vehicular Communications

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Receiver Design for Vehicular Communications

OLIVIER GOUBET

Master’s Degree Project Stockholm, Sweden 2012

XR-EE-KT 2013:002

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Abstract

As the mobility of people increases, so does the density of vehicles on road networks. This comes at a cost, causing traffic congestions and raising the number of casualties. Wire- less communications between vehicles will enable the development of Intelligent Transport Systems (ITS), which are expected to assist and manage road transportation. The aim of ITS is to enhance safety, road network management, but also personal comfort for drivers and passengers. IEEE 802.11p has been chosen as the standard for the Physi- cal Layer (PHY) design for wireless vehicular communications. Unfortunately, vehicular channels are challenging for communications. While the dispersive nature of the channel has advantages, such as the possibility to communicate in the absence of line-of-sight, time variations present in high mobility scenarios lead to doubly selective channels. The systems are expected to allow reliable communications despite those conditions.

In this thesis we focus on PHY design for the receiver. We aim at implementing receivers able to perform channel estimation in the case of doubly selective channels, with minimal information from training sequences. The solutions considered all involve joint channel estimation and decoding, characterized by the use of an iterative structure.

As limited knowledge of the channel is available at the receiver, only a coarse estimate can be performed in the first place. Iterative structures allow the channel estimator to benefit from feedback produced by the decoder, used to refine the channel estimation and ultimately resulting in a smaller error rate. Different algorithms are considered, either based on Minimum Mean Square Estimation (MMSE), or Maximum A Posteriori (MAP) estimation. The latter requires a recursive description of the channel introduced as a Markov chain. Using these powerful methods to perform channel estimation, we decrease the error rate, despite the varying nature of the channel. However, improving the channel estimation comes at the cost of a higher complexity. An analysis on the trade-off between performance and complexity is also provided.

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Acknowledgments

First I would like to express my very great appreciation to Tobias Oechtering, Asst. Prof.

at the Communication Theory department at KTH Royal Institute of Technology, for giving me the opportunity to work on this project, as well as co-supervising it. I would also like to offer my special thanks to Fr´ed´eric Gabry, Ph.D. at KTH, for taking some of his time to supervise my thesis. His guidance throughout this project has been valuable and his thorough proofreading has greatly improved this report. I also wish to acknowledge the help provided by Asst. Prof. Ragnar Thobaben at a crucial point of this thesis.

I want to thank my fellow master thesis students and friends for making this period enjoyable. Our lively discussions and laughs around lunch were much appreciated.

Finally I would like to thank my parents for their constant support during my studies.

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List of Figures

1.1 Examples of vehicular communication safety-related scenarios. . . 2

1.2 Relaying in vehicular communications . . . 2

1.3 Strucure of an OFDM frame. . . 3

1.4 Mapping of the concepts related to this thesis. . . 6

2.1 Block diagram of the transmitter. . . 7

2.2 Convolutional encoder. . . 9

2.3 PSK signal constellations with Gray coding. . . 12

2.4 QAM signal constellations with Gray coding. . . 13

2.5 Implementation of OFDM modulation. . . 16

2.6 Multipath delay profile. . . 19

2.7 Distribution of the channel coefficients of a single subcarrier. . . 22

2.8 Autocorrelation function of a Rayleigh fading channel. . . 23

3.1 Receiver structures. . . 28

3.2 Structure of the iterative receiver performing MMSE estimation. . . 29

3.3 Decoder. . . 36

3.4 Channel estimator. . . 37

3.5 SISO demapper. . . 39

3.6 SISO mapper. . . 39

3.7 Quantized channel coefficients. . . 42

3.8 Structure of the receiver in the case of a single subcarrier. . . 45

3.9 Complete structure of the receiver. . . 45

4.1 Performance of MMSE estimation for various frame length. . . 48

4.2 Minimum Mean Square Error (MMSE) estimation with a IEEE 802.11p compliant frame . . . 49

4.3 MMSE estimation with an additional pilot at the end of the frame . . . 50

4.4 MMSE estimation with one pilot every 20 symbols . . . 50

4.5 Influence of the frame length on the BER performance. . . 52

4.6 Influence of the speed on the BER performance. . . 53

4.7 BER performance depending on the constellation. . . 53

4.8 MAP estimation when strong fading occurs. . . 54

4.9 MAP estimation when no fading occurs. . . 55

4.10 MAP estimation when fading occurs and BPSK signaling is employed . . . 56

4.11 MAP estimation when strong fading occurs, with a postamble. . . 56

4.12 Performance of the complete system in the case of Maximum A Posteriori (MAP) estimation. . . 57

4.13 Comparison of the performance of the different receivers. . . 58 vii

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viii LIST OF FIGURES

4.14 Comparison of the performance of the different receivers. . . 59 4.15 Complexity of three receivers as a function of the frame length. . . 60 4.16 Complexity of the MAP receiver as a function of the frame length. . . 61 4.17 Complexity of the MAP receiver as a function of the size of the constellation 62 4.18 Complexity of the MAP receiver as a function of the number of quantized

channel coefficients. . . 62 4.19 Repartition of the execution time. . . 63

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List of Tables

1.1 Achievable theoretical rates in IEEE 802.11p . . . 4 2.1 Normalization factors for the different constellations . . . 14 2.2 ITU channel model for vehicular test environment . . . 20

ix

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Notation

We will use the following notation throughout this thesis:

Y random variable

Y alphabet or set

y realization of Y

Y ∼ pY random variable X with probability distribution pY

E{Y } expected value over random variable Y Pr(Y ) probability of event Y

CN (µ, σ²) complex normal distribution with mean µ and variance σ²

y vector

y^T transpose of y

y^H hermitian transpose of y

|y| absolute value of a complex number y byc largest integer less than or equal to y log natural logarithm

log₂ logarithm to the base 2 tanh hyperbolic tangent

arg max argument of the maximum value

J₀(·) zeroth order Bessel function of the first kind 1I indicator function of the interval I

max∗ Jacobian logarithm max∗

i concatenation of i Jacobian logarithm Nsc number of subcarriers

N_dsc number of data subcarriers N_s number of OFDM symbols Nb number of information bits

N_cbps number of coded bits per subcarrier

R_s symbol rate

Rb bit rate

Fs sample rate

T_s sampling time

Eb energy per information bit Es energy per symbol

xi

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xii NOTATION

R coding rate

L encoder’s constraint length

Nr number of shift registers in the encoder gi generator polynomial

p_i puncturing vector

c coded bit

b information bit

X symbol alphabet

X_n symbol at time instant n x realization of Xn

Cn⁽ⁱ⁾ i^th bit mapped to X_n c⁽ⁱ⁾x i^th bit mapped to x Cx set of bits mapped to x

M number of points in a constellation

m number of bits mapped to a constellation point

s[n] transmitted symbol at time index n r[n] received sample at time index n z[n] complex noise sample at time index n

h[n, m] m^th channel’s tap at time index n (multipath) h[n] channel coefficient at time index n (single path) a[n] absolute value of h[n]

φ[n] absolute value of φ[n]

Lch number of paths

σ² variance of the additive noise N₀ power spectral density of the noise f_d maximum Doppler shift

v relative speed between the vehicles

λ carrier wavelength

f_c carrier frequency

R(τ ) channel autocorrelation function ρ parameter of the Rayleigh distribution

M_n quantized amplitude at time index n mi realization of Mn

M set of realizations mi

α_i limit of quantized region of the amplitude Na number of quantized amplitudes

Tn quantized phase at time index n t_i realization of T_n

T set of realizations ti

Np number of quantized phases

Q_n quantized channel coefficient at time index n qi realization of Qn

Q set of realizations qi

N_h number of quantized channel coefficients

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xiii

w tap vector of Wiener filter p_AP(Y ) a priori information on Y pEXT(Y ) extrinsic information on Y p(r|Y ) likelihood regarding Y p(Y|r) a posteriori distribution of Y L(r) Log-Likelihood Ratio

Hi hypothesis on a bit value S_n encoder state at time index n si realization of Sn

αn(·) alpha metric in BCJR algorithm β_n(·) beta metric in BCJR algorithm γ_n(·, ·) gamma metric in BCJR algorithm mn(·, ·) transition probability in BCJR algorithm A_n(·) alpha metric in log BCJR algorithm B_n(·) beta metric in log BCJR algorithm Γn(·, ·) gamma metric in log BCJR algorithm M_n(·, ·) transition probability in log BCJR algorithm

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List of Acronyms

16-QAM 16 Quadrature Amplitude Modulation 64-QAM 64 Quadrature Amplitude Modulation acf autocorrelation function

ADC Analog-to-Digital Converter AP A Priori Probability

APP A Posteriori Probability

AWGN Additive White Gaussian Noise

BER Bit Error Rate

BPSK Binary Phase-Shift Keying

CP Cyclic Prefix

DAC Digital-to-Analog Converter DFT Discrete Fourier Transform

DS Delay Spread

FDMA Frequency Division Multiple Access

FER Frame Error Rate

FFT Fast Fourier Transform FIR Finite Impulse Response ICI InterCarrier Interference IFFT Inverse Fast Fourier Transform ISI InterSymbol Interference

ITS Intelligent Transportation Systems ITU International Telecommunication Union JCED Joint Channel Estimation and Decoding LCR Level Crossing Rate

LLR Log-Likelihood Ratio

LS Least Squares

LTI Linear Time Invariant LTV Linear Time Varying

MAP Maximum A Posteriori

MMSE Minimum Mean Square Error MPE Minimum Error Probability

OFDM Orthogonal Frequency-Division Multiplexing OSI Open Systems Interconnection

PAPR Peak-to-Average Power Ratio pdf probability density function PDP Power Delay Profile

PHY Physical Layer

pmf probability mass functions PSK Phase-Shift Keying

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xvi LIST OF ACRONYMS

QAM Quadrature Amplitude Modulation QPSK Quadrature Phase-Shift Keying

RMS Root Mean Square

SISO Soft-Input Soft-Output SNR Signal-to-Noise Ratio V2I Vehicle to Infrastructure V2V Vehicle to Vehicle

WLAN Wireless Local Area Networks

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Chapter 1

Introduction

Nowadays vehicles are equipped with a growing number of safety systems aiming at preventing accidents. Unfortunately, these are based on sensors that operate only in the car’s own vicinity. Besides, global traffic management of vehicles is not well developed;

information conveyed over long distances is communicated to the driver through road sign panels and FM radio. Intelligent Transportation Systems (ITS) are expected to dramatically improve safety and global management, by allowing different vehicles to communicate with each other and receive relevant information about the state of the road network.

Communicating over the vehicular channel is challenging and researchers are looking to make communications both faster and more reliable. This thesis focuses on receiver design and aims at improving the reliability of the communications.

1.1 Applications of Vehicular Communications

Safety is the main focus in the development of ITS. The communication systems are expected to provide notifications to the driver in dangerous situations. Different scenarios are considered, such as emergency breaking, lane changing, nearing vehicles, driving in tun- nels, approaching an intersection, heavy traffic conditions, etc. In most of these situations, sensing the close vicinity of the car or relying on sight is not enough to guarantee the safety of the passengers. Three examples, namely approaching an intersection, changing lanes and approaching vehicles slowed down in traffic congestion, are illustrated in Figure 1.1.

When approaching an intersection, large objects such as buildings or trees usually hide potential vehicles coming on the intersecting road. Changing lanes is a typical example where blind spots affect the visibility of the driver; moreover, evaluating the speed of the incoming vehicles might be troublesome. The vehicles at the end of a traffic jam should have their hazard warning lights turned on to alert incoming vehicles; however, if it is not the case, there is a risk of accident due to the lack of visual indications. Communication systems operate beyond the line-of-sight constraint, and allow a wide and fast spreading of the information, through relaying for example. In a relaying scenario, information is passed along from vehicle to vehicle, without going through a centralized infrastructure.

Large scale relaying will be performed by allowing communication with road side units;

this would ensure a spreading of traffic information over large geographical areas. On the other hand, small scale relaying will be performed through car to car communication;

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2 CHAPTER 1: INTRODUCTION

(a) Approaching an intersection (b) Changing lanes (c) Approaching a traffic jam

Figure 1.1: Examples of scenarios where safety is not guaranteed by only sensing the vicinity of the vehicle or relying on sight.

Figure 1.2: Relaying in vehicular communications

warning signals on an imminent danger would be spread promptly to the vehicles involved.

This particular situation is depicted in Figure 1.2. Consequently, both Vehicle to Infras- tructure (V2I) and Vehicle to Vehicle (V2V) communications are considered. Combining both will also allow to have a wide network without the need of a large infrastructure, which would be costly and would require substantial maintenance.

The applications are not all safety related. Vehicular communications are expected to have an important role in traffic management, by guiding drivers according to the state of the road network. Access to the Internet, or television channels could improve the comfort of the passengers. This wide range of applications introduces strong requirements for the development of lower layers, defined in the Open Systems Interconnection (OSI) model.

In the case of safety related applications, the systems must show both robustness and low latencies. Indeed, receiving a crucial piece of information either too late or containing errors could lead to an accident. On the other hand, in-vehicle television will require high data rates. Thus, challenges appear on different levels, from a networking point of view, to the physical layer design. The biggest obstacle for the physical layer is to offer a reliable system, while meeting the requirements on transmission rates. This is problematic considering the dispersive, fast varying nature of the channel.

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1.2. SPECIFICATIONS OF THE PHYSICAL LAYER 3

pilot data null

time

frequency

Figure 1.3: Strucure of an OFDM frame.

1.2 Specifications of the Physical Layer

Frequency Band Allocation The ITS applications have been allocated a 70 MHz spec- trum at 5.9 GHz¹. The total bandwidth is divided into seven 10 MHz-wide channels. Six of these channels are used for services and the last one is a control channel, used for the transmission of high priority messages and management information.

Physical Layer Specifications The IEEE 1906 Family of Standards applies to vehicular communications and defines a standardized architecture for the systems. The Physical Layer (PHY) specifications are given by the IEEE 802.11p standard. It is an amendment to the 802.11 standard, which is used in the implementation of Wireless Local Area Net- works (WLAN). IEEE 802.11p improves 802.11 in order to enable the development of ITS applications, such as data exchange among high speed vehicles and road side infrastructure.

Orthogonal Frequency-Division Multiplexing (OFDM), a multi-carrier modulation technique, was chosen as modulation scheme. The structure of an OFDM frame is depicted on Figure 1.3. In IEEE 802.11p, the system communicates over 64 subcarriers. Among these 64 subcarriers, only 52 are employed. The DC carrier is not used in order to avoid the undesirable DC offset introduced in the transceiver [1]. The remaining 11 are set to null to avoid leakage to adjacent frequency bands. 48 subcarriers are actually used to carry data, and the remaining four contain pilot sequences, called comb pilots, used in carrier synchronization. Pilots are not only spread in frequency, but also in time; the first two OFDM symbols of each of the 48 data subcarriers are dedicated to pilot sequences, called block pilots.

The 48 subcarriers containing data are modulated using Binary Phase-Shift Key- ing (BPSK), Quadrature Phase-Shift Keying (QPSK), 16 Quadrature Amplitude Mod- ulation (16-QAM) or 64 Quadrature Amplitude Modulation (64-QAM). Before that, the information bits are encoded with a convolutional code of rate 1/2. This rate is adjustable

1In Europe, the frequency band 5850-5905 MHz has been allocated. The range 5905-5925 MHz may be used in the future.

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Modulation Coding rate Data rate (Mbps)

BPSK 1/2 3

BPSK 3/4 4.5

QPSK 1/2 6

QPSK 3/4 9

16-QAM 1/2 12

16-QAM 3/4 18

64-QAM 2/3 24

64-QAM 1/2 27

Table 1.1: Achievable theoretical rates in IEEE 802.11p

using two different puncturing patterns, which leads to coding rates 1/2, 2/3 and 3/4. The IEEE 802.11p standard is thus similar to 802.11a, besides one exception. While systems based on 802.11a communicate over 20 MHz bands, with a symbol duration of 4µs, those based on 802.11p communicate using 10 MHz bands, with a symbol duration of 8µs. Using a larger symbol duration makes 802.11p more robust against fading. On the downside, the achievable transmission rates are divided by a factor of two. The different theoretical rates available are in the range 3-27 Mbps, and the corresponding transmission settings are described in Table 1.1. A more thorough description of the standards used in ITS in Europe can be found in [2].

1.3 Limitations of the Current Systems

As mentioned in Section 1.1, the systems that will enable communications between vehicles will be used in environments characterized by time-varying fast fading channels. It is important to choose a standard that is adapted for transmission through such unfavorable environments. In [3], characterization of the vehicular channel is discussed.

First, the dispersive property of the channel, which introduces InterSymbol Interference (ISI), must be alleviated. It was noted in Section 1.2 that the standard uses OFDM as modulation scheme. This will allow the systems to transmit data on parallel carriers. Each carrier is sufficiently narrow to consider the channel constant over its bandwidth.

Secondly, the fast variations of the channel must be dealt with. In the case of V2I, the relative movement between the moving car and the fixed antenna leads to distortions caused by the Doppler effect. The strength of these distortions, which is characterized by the maximum Doppler shift, increases with speed. In the case of V2V, the two vehicles are in motion, and the relative speed can be doubled. The fact that the reflectors will be in motion as well must be considered, increasing the maximum Doppler shift. Tackling this issue requires either a high density of pilot sequences or a powerful channel estimators.

The standard reserves two OFDM symbols at the beginning of each frame to place pilot symbols. Using classical channel estimation methods, such as the average Least Squares (LS) estimation [4], the channel will be estimated using these two pilot symbols and then be considered constant over a frame. While the approximation might hold in certain cases, typically at low speeds and using short frames, the performance drops quickly when the frame’s length and the speed increase [4]. This loss of performance is due to the low density of pilot symbols.

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1.4. METHODOLOGY 5

In order to ensure acceptable performance regardless of frame length or speed, we will opt for high performance channel estimation, at the cost of a higher complexity. This thesis focuses on solutions to perform channel estimation under those circumstances. The solutions that are examined employ Joint Channel Estimation and Decoding (JCED), which involves an iterative structure in the receiver, allowing the channel estimator and the decoder to exchange information and improve the performance of the system.

1.4 Methodology

Work Organization The design of the receiver is done in several steps. First, an IEEE 802.11p compliant transmitter is implemented. Then the channel is simulated using models for the different distortions. Before designing an iterative receiver, a classical “linear”

receiver is implemented. It can be used as a reference to evaluate the performance of the iterative receivers.

Two different types of receivers are considered, both relying on JCED. The difference between the two types lies in the channel estimation. While the first produces a Minimum Mean Square Error (MMSE) estimate of the channel, the second offers a Maximum A Posteriori (MAP) estimation. The communication devices are multi-carrier systems, but the receivers are first designed to work on a single OFDM subcarrier. This subsystem is then incorporated in the complete, 802.11p compliant system. The diagram in Figure 1.4 describes the different concepts that were considered throughout this thesis as well as how they relate to each other.

Tools The algorithms were implemented in MATLAB. Note that they have been opti- mized for this environment, and are not directly applicable to other setups (for instance for an implementation on a digital signal processor using C code). All the simulations were also conducted using MATLAB.

1.5 Thesis Outline

This report is organized as follows. In Chapter 2 we introduce the models used for the transmitted signals and the channel. In order to understand the signal model, the structure of the transmitter is fully described. Details are given on its modules, namely the bit source, the encoder, the interleaver, the mapper and the OFDM modulator. The different distortions characterizing the channel are then modeled. We then focus on modeling a single OFDM subcarrier. Finally, we derive a Markov model for the channel. In Chap- ter 3 the receiver designs that have been developed throughout this thesis are presented.

The two types of receivers that were implemented are described here, starting with the receiver performing MMSE channel estimation. Then, the second receiver, performing MAP channel estimation, is explained. In Chapter 4, we define the criteria for our system analysis. The results obtained during the numerical simulations of the systems described previously are then presented and analyzed. Chapter 5 concludes this thesis. We discuss the results obtained when simulating the receivers, and propose directions for future work on the subject.

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Receiver design for vehicular communications

Transmitter

encoder interleaver

mapper

ofdm

channel additive noise

multipath

Doppler effect

Receiver

Classical receiver

iterative receiver

JCED

MMSE MPA

Markov modeling

of the channel

Inference problem solving IEEE

802.11p standard

ITU channel models

factor graphs

Figure 1.4: Mapping of the concepts related to this thesis.

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Chapter 2

Channel and Signal Model

In this chapter, we discuss the models for the transmitted signals and the channel. In Section 2.1, we describe the different components of the receiver. In order to properly test the receivers that are implemented, it is important that the transmitted signals are shaped as the ones used in vehicular communications, which are defined by the IEEE 802.11p standard. Therefore, we must generate the transmitted signals using an 802.11p compliant transmitter.

Then, in Section 2.2, we introduce and model the distortions introduced by the channel through which the transmitted signals propagate. These distortions can then be accounted for in the design of the receiver.

2.1 Transmitter

In this section we describe the different components of the transmitter. The requirements imposed by the standard were already briefly discussed in section 1.2. Let us examine the structure of the transmitter in more details. The information bits go first through a scrambler, whose goal is to randomize them. The scrambled data then goes through a convolutional encoder of rate 1/2, adaptable through puncturing. The coded bits are then interleaved to ensure that the additive noise samples introduced in the channel are uncorrelated. The obtained sequence is mapped to predefined constellations, namely BPSK, QPSK, 16-QAM and 64-QAM. Then the OFDM symbols are assembled and the pilot symbols are added. Finally, the Inverse Fast Fourier Transform (IFFT) is applied to perform pulse shaping. The block diagram of the transmitter is shown in Figure 2.1.

A complete description of an 802.11p compliant receiver can also be found in [4]. The receiver will have to be designed according to these requirements. The received data will be demodulated, demapped, deinterleaved, decoded and descrambled accordingly.

Scrambler Encoder Interleaver Mapper OFDM

Pilots

Figure 2.1: Block diagram of the transmitter.

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8 CHAPTER 2: CHANNEL AND SIGNAL MODEL

Note that the scrambler will not be considered since it is used to randomize the data sequence. In our case, that sequence is not provided by the upper layers, but instead generated randomly.

2.1.1 Source

In real transmission systems, the upper layers feed the information bits that are to be transmitted. These bits contain the actual information that the application needs to send, as well as various overhead added by the different layers. In this thesis, the transmission system is only simulated, and the content is irrelevant. The bits that must be transmitted can thus be generated randomly by a source. First the number of transmitted OFDM symbols is chosen. Choosing different frame lengths shows the effect of the message length on the performance of the system. A sufficient number of bits must be generated to match the desired number of symbols. Let us denote:

• R, the coding rate (after puncturing),

• m, the number of bits mapped to a constellation point. This represents the number of bits per subcarrier in an OFDM symbol,

• Ndsc, the number of data subcarriers,

• Ns, the number of OFDM symbols,

• L, the constraint length of the encoder (see Section 2.1.2 for more details),

• Nb, the number of information bits that must be generated in the source.

We separate two cases to compute Nb, whether the code is terminated or truncated.

Termination and truncation are explained in Section 2.1.2. The formulas are given respectively in Equation (2.1) and Equation (2.2).

N_b = RmN_dscN_s− (L − 1) (2.1)

N_b = RmN_dscN_s (2.2)

The bits can take the values “1” and “0”. In order to generate N_b bits, a sequence of N_b values are drawn from a uniform distribution on [0, 1]. Values larger than 0.5 are mapped to “1”, while values lower than 0.5 are mapped to “0”.

2.1.2 Encoding

The data bits generated in the source are passed through a convolutional encoder. The encoder will add redundancy to the sequence. This additional information is then used at the receiver to detect and correct errors caused by distortions applied to the transmitted signal by the channel. We will see in Chapter 3 that in the case of iterative receivers, the additional information brought by the encoder is not only used to detect and correct errors, but also to refine the estimation of the channel. The code is thus of utmost importance.

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2.1. TRANSMITTER 9

b[n] D D D D D D

+

c1[n]

c₂[n]

Figure 2.2: Convolutional encoder.

A convolutional encoder can be represented with shift registers connected in series. Each shift register corresponds to a time delay. Thus, the encoder stores the last input bits, which define the state in which the system is. Hence a convolutional encoder works as a finite-state machine [5]. At each time index, the encoder takes n data bits as an input and generates k coded bits at the output, leading to a coding rate of n/k. The output bits are computed by modulo-2 adders connected to a specific set of shift registers. The constraint length of the code is L = Nr + 1, where Nr is the number of shift registers.

This quantity represents the number of data bits taken into account to compute a coded bit. A high constraint length will increase the performance of the code, at the cost of a higher complexity.

The convolutional code specified in the standard has a rate 1/2. At each time index, it takes one data bit as an input and produces two coded bits. The generator polynomials for the two coded bits are g0= 1338 = [01011011]2 and g1 = 1718 = [01111001]2.

This encoder can be represented with Nr = 6 shift registers. The binary representation of the generator polynomials shows which shift registers are connected to the modulo-2 adders; a “1” signifies a connection between the shift register and the modulo-2 adder, while a “0” signifies the absence of connection. The encoder is shown in Figure 2.2.

The encoder can be implemented using convolutions. Indeed, the output streams of coded bits c₁[n] and c₂[n] can be computed as follows:

(c₁[n] = (g₁∗ b)[n]

c₂[n] = (g₂∗ b)[n],

where∗ denotes the convolution operator. The convolution of two discrete signals x[n]

and y[n] is defined as:

(x∗ y)[n] ,

∞

X

k=−∞

x[k]y[n− k]

Note that the sequences we work with are time limited and the input stream b[n] must be padded with N_r zeros at the beginning (corresponding to the initial state of the encoder).

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The communication systems specified by the standard transmit and receive frames containing a certain number of bits. Consequently, we want to input blocks of data in the encoder. However, convolutional codes are not block codes, since the encoder takes a con- tinuous stream of bits as an input. It is however possible to convert a convolutional code into a block code using two different methods: truncation or termination.

Truncation consists of separating the sequence of data bits coming from the source into blocks of length N . These blocks are then fed to the receiver, which is reset to the all-zero state before encoding each block. The main drawback of this method is that the last bits are less protected, since they will influence less output bits (or none).

Termination solves this issue by appending a sequence of Nr bits to the block of N data bits. These N_r bits are chosen so that the encoder is back to the all-zero state after encoding the sequence. In the case of non-recursive codes, sending N_r zeros will always set the encoder in the all-zero state. In the following, termination is used due to better since it leads to a more reliable decoding.

The IEEE 802.11p standard defines a base code of rate 1/2, which is described above.

It also includes the possibility to encode data at rates 2/3 and 3/4. This is done by applying puncturing after the encoding. Puncturing consists of transmitting only certain bits according to a pattern. This method relies on the ability of the decoder to correct errors in order to reconstruct properly the whole sequence.

The puncturing vectors specified in the standard are p₀ = [1110] and p₁ = [110110].

Here, a “0” signifies that the bit will not be transmitted. With the first pattern, every fourth bit is not transmitted and with the second pattern, every third bit is not sent. This leads to code rates 2/3 and 3/4 respectively. A more detailed description of puncturing can be found in [5].

2.1.3 Interleaving

It is common in wireless communications to consider the channels memoryless. However, this assumption is not rigorously correct; indeed, errors typically come in burst, rather than independently. Shuffling the coded bits allows creating a more uniform distribution of errors. In other words, the correlated noise samples appear uncorrelated if interleaving is used, which concurs with the assumption of a memoryless channel. In multi-carrier systems, such as OFDM, it is also interesting to perform interleaving across carriers. It mitigates the effect of excessive noise or fading on a few specific carriers.

IEEE 802.11p uses a block interleaver defined by two permutations. The first permutation ensures that adjacent bits are on non-adjacent subcarriers. The second permutation ensures that adjacent coded bits are mapped onto less and more significant bits (respectively the rightmost and leftmost bits) of the constellation symbols to avoid long runs of low reliability [6]. The block size is chosen to be the number of coded bits per OFDM symbols. It is calculated according to Equation (2.3).

Ncbps = mNdsc, (2.3)

where:

• m is the number of bits mapped to a constellation point,

• Ndsc is the number of data subcarriers,

• Ncbps is the number of coded bits per OFDM symbol.

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2.1. TRANSMITTER 11

Let us denote k the index before any permutation, i the index after the first permutation and before the second and j the index after the second permutation. The first permutations is given in Equation (2.4). It is equivalent to defining a matrix with 16 rows and a number of columns depending on the modulation scheme used; the matrix is filled row by row by the input bits and read column by column.







i = N_cbps

16 k⁰+ k 16

, k = 0, . . . , Ncbps

k⁰ ≡ k (mod 16),

(2.4)

wherebxc is the largest integer less than or equal to x. The second permutation is:











j = s· i s

+ l, i = 0, . . . , Ncbps

l≡ i + Ncbps− 16· i N_cbps

(mod s).

The parameter s depends on the constellation and is computed as:

s = maxm 2, 1

.

Let us note that for constellations with four symbols or less, the second permutation leaves the bits unchanged.

2.1.4 Mapping

The rate at which the information is sent is limited by the available channel bandwidth.

In order to achieve higher transmission rates, the coded bits are first mapped to symbols that are usually non-binary.

The symbols are chosen from a symbol alphabet X which contains M elements. It is possible to map m bits per symbol, with m = log₂(M ). Let us denote the symbol rate Rs

and the bit rate R_b. The relation between Rs and R_b is given by:

Rb = mRs.

As mentioned previously, R_sis limited by the available bandwidth. However, R_b is not limited, as long as we increase the size of the constellation, which corresponds to increasing m. In practical systems, Rb is actually bounded because of the presence of noise and the limited power available at the transmitter.

The complex symbols in the alphabet correspond to the baseband complex signal. If the passband carrier is considered, it is actually modulated in phase and amplitude. The set of points representing the complex symbols is called the signal constellation.

When designing a signal constellation, three criteria must be considered. First, the mean signal power must be kept as low as possible, in order to limit the power required at the transmitter. Then, the minimum distance between the symbols must be as high as possible in order to minimize the error rate. Finally, the geometry of the constellation must be kept relatively simple, so that it is not too complex to demodulate.

The mapping of the bits onto the symbols will also influence the performance of the system. We want to minimize the bit errors in the case where a symbol is improperly decoded. Using Gray coding allows doing this. In this case, the neighboring symbols differ

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I Q

-1 1

1 0

(a) BPSK constellation

I Q

-1

1

1 11

01

00 10

(b) QPSK constellation

Figure 2.3: PSK signal constellations with Gray coding.

by exactly one bit. Since choosing a neighboring symbol instead of the proper one is the most probable error, we ensure that, in most of the cases, an error in the symbol decoding will lead to a single error in the bit stream.

The IEEE 802.11p standard defines four possible constellations: BPSK, QPSK, 16-QAM and 64-QAM. With Phase-Shift Keying (PSK), it is possible to choose the argument of the complex symbols, which have a constant magnitude. In the case of Quadrature Amplitude Modulation (QAM), variations of the amplitude along the in-phase (I) and quadrature (Q) components generate the different symbols. The PSK constellations are shown in Figure 2.3 and the QAM constellations are shown in Figure 2.4.

These constellations are normalized to ensure equal average symbol power. We denote by K the normalization factor. It is computed according to Equation (2.5), where P_xdenotes the power of the symbol x. Note that the constellations in Figure 2.4 are represented before normalization.

K = 1 M

X

x∈X

Px. (2.5)

The normalization factors obtained for the different constellations are gathered in Ta- ble 2.1.

2.1.5 OFDM

IEEE 802.11p specifies the use of Orthogonal Frequency-Division Multiplexing, a multi- carrier transmission technique that removes ISI over frequency selective channels.

Principle The total bandwidth is divided into Nscnarrow sub-bands such that the channel can be considered frequency flat on each subcarrier. In principle, OFDM is similar to Frequency Division Multiple Access (FDMA) in the sense that both techniques rely on a division of the bandwidth into smaller bands. However, OFDM is much more bandwidth efficient, as it uses orthogonal and overlapping carriers while FDMA requires the inser- tion of guard frequency intervals between the sub-bands to guarantee the independence

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2.1. TRANSMITTER 13

I Q

-3

-3 -1

-1

1 1

3 3

00 00 00 01 00 11 00 10

01 00 01 01 01 11 01 10

11 00 11 01 11 11 11 10

10 00 10 01 10 11 10 10

(a) 16-QAM constellation

I Q

-7

-7 -5

-5 -3

-3 -1

-1 1 1

3 3

5 5

7 7

000 000 000 001 000 011 000 010 000 110 000 111 000 101 000 100

001 000 001 001 001 011 001 010 001 110 001 111 001 101 001 100

011 000 011 001 011 011 011 010 011 110 011 111 011 101 011 100

010 000 010 001 010 011 010 010 010 110 010 111 010 101 010 100

110 000 110 001 110 011 110 010 110 110 110 111 110 101 110 100

111 000 111 001 111 011 111 010 111 110 111 111 111 101 111 100

101 000 101 001 101 011 101 010 101 110 101 111 101 101 101 100

100 000 100 001 100 011 100 010 100 110 100 111 100 101 100 100

(b) 64-QAM constellation

Figure 2.4: QAM signal constellations with Gray coding.

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Modulation K

BPSK 1

QPSK 1

√2

16-QAM 1

√10

64-QAM 1

√42

Table 2.1: Normalization factors for the different constellations

between the carriers.

In OFDM, the carriers are chosen among the eigenvectors of the channel, which ensures that they will remain orthogonal during the transfer through the channel. The complex exponential waveforms e^{j2πf t} match this criteria.

There is an infinite number of complex exponentials, and they are defined over an infinite time horizon. For practical reasons, only a finite number of time limited waveforms are considered. The frequency and time spacing must be chosen to maintain the eigenvector property and the orthogonality between the waveforms. Let us note that these requirements will not be met rigorously, but approximately. To fulfill the first requirement, we must choose the symbol period T large compared to the channel delay spread. To fulfill the second, the set of Nsc frequencies must be chosen such that (fn− fm)T = k, ∀(n, m) and where k is a non-zero integer.

The transmitted complex baseband signal is then given by:

u(t) = 1

√T

Nsc−1

X

n=0

S[n]e^j2πnt/TI_{[0,T ]}(t),

where S[n] is the n^th transmitted symbol. These symbols are obtained after a first modulation, such as PSK or QAM.

Discrete-time Description Now let us consider the discrete description of the transmitted signal, since most implementations are based on discrete-time baseband processing.

The bandwidth of the transmitted OFDM signal u(t) is approximately Nsc/T , and spread between−Nsc/2T and N_sc/2T when dealing with the baseband representation. According to the Nyquist-Shannon theorem, which states that a signal can be perfectly reconstructed when the sampling frequency is greater than twice the maximum frequency of the signal, we can sample u(t) at a rate F_s = N_sc/T without loss of information, where F_s is the sampling frequency. We also note T_s= 1/F_s, the sampling interval. The sampled version of the transmitted signal is given by:

u(kTs) = 1

√N

Nsc−1

X

n=0

S[n]e^j2πnk/N^sc.

This is in fact the N-point inverse Discrete Fourier Transform (DFT) of the symbol

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2.1. TRANSMITTER 15

sequence S[n]. Let s[k] denote the inverse DFT of S[n]:

s[k] = u(kTs) = 1

√Nsc N −1

X

n=0

S[n]e^j2πnk/N^sc.

In the case where Nsc = 2ⁱ, with i∈ N∼, this operation can be efficiently performed using the IFFT.

At the receiver, the inverse operation can be performed, using this time the Fast Fourier Transform (FFT):

R[n] = 1

√Nsc Nsc−1

X

k=0

r[k]e^−j2πnk/N^sc.

The mapped symbols S[n] are considered to be in the frequency domain and the transmitted samples s[k] to be in the time domain. S[n] is broken into N_sc parallel low rate streams before generating s[k]. Similarly, at the receiver, r[n] is broken into N_sc parallel low rate streams.

OFDM and Multipath Channels The signal passes through a multipath fading channel, modeled by a Finite Impulse Response (FIR) filter of length L_ch. The sampled noiseless received signal is given by:

r[m] = (h∗ b)[m] =

Lch−1

X

l=0

h[l]s[m− l]. (2.6)

The N-point DFT of the channel impulse response, denoted by H is given by:

H[n] = 1

√Nsc Lch−1

X

l=0

h[l]e^−j2πnl/N^sc.

Replacing the linear convolution of Equation (2.6) by a circular convolution, defined as











r[m] = (h ~ s)[m] ,˜

Nsc−1

X

l=0

h[i]s[j]

i≡ l (mod N^sc) and j ≡ m − l (mod N^sc) would ensure that

R[n] = H[n]S[n],˜

where it is clear that each subcarrier is affected by a single channel coefficient.

A linear convolution is equivalent to a circular convolution if the signals have an infinite time horizon or if one of the signals is periodic; neither condition is verified here. However, we can emulate a circular convolution by appending a Cyclic Prefix (CP). The samples

s[k] = s[N_sc− k], k = −(Lch− 1), . . . , −1,

are sent before s[0], . . . , s[Nsc− 1]. At the receiver, the CP is discarded. In order to be capable of generating the CP, L_ch must be smaller than Nsc. The number of subcarrier must be chosen accordingly. Moreover, adding the CP introduces an overhead of (L_ch− 1)/Nsc; choosing a high value for Nsc will reduce that overhead.

The implementation of the OFDM transmitter and receiver with CP is described in Figure 2.5. More detailed descriptions of OFDM systems can be found in [7] and [8].

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S[n] To DAC

S/P IFFT P/S

(a) OFDM transmitter

From ADC R[n]˜

S/P FFT P/S

(b) OFDM receiver

Figure 2.5: Implementation of OFDM modulation.

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2.2. CHANNEL MODEL 17

Advantages and Drawbacks of OFDM Systems OFDM is an advantageous method.

First, it has a high spectral efficiency. Secondly, it is relevant for dispersive channels. With the addition of the Cyclic Prefix, it suppresses ISI. Moreover, there are efficient solutions to implement it when using time-discrete processing, employing the FFT and IFFT. This implementation also has the benefit of distributing the complexity evenly between the transmitter and the receiver. In classical receivers, the gain in complexity is even higher, considering the fact that the complex equalizers commonly used to reduce the effect of ISI can now be replaced by simple one-tap equalizers.

OFDM has also a few disadvantages. OFDM systems have indeed a high Peak-to- Average Power Ratio (PAPR) and are subject to InterCarrier Interference (ICI). ICI can be caused by improper carrier synchronization. To alleviate this issue, a certain number of carriers are dedicated to carrier synchronization. ICI may also be caused by time-varying channels. However, we would need to consider communication nodes having a relative speed higher than 1440 km h−1 to start suffering from it, as shown in [9], which is unlikely in the field of vehicular communications.

2.2 Channel Model

In this section, we discuss the channel model. The channel is the medium between the transmitting antenna and the receiving antenna. The wireless signals’ characteristics are modified while they propagate through this medium. The distortions may be caused by the environment (presence of various objects such as buildings or trees), the weather, the movement of the transmitter or the receiver, etc. Typically, a channel depends on the settings of the location where the communication system operates. Wireless signals exchanged in a building will not be modified the same way as signals transmitted in a rural area for example. Each transmission system must be designed accordingly, so that it can operate in the presence of specific distortions. Therefore it is important to find a model that accounts for these alterations. At this stage we derive the channel’s impulse response H(f, t).

In the case of vehicular communications, we are considering three phenomena. First the signals will be altered by noise caused by the electronics in the device. Then, frequency selectivity caused by reflections on various objects must be accounted for. Finally, it is important to consider the relative movement between the transmitter and the receiver, which leads to a time-varying channel.

OFDM systems split the total bandwidth into parallel streams that are easier to deal with. Modeling these subcarriers will allow us to work directly on them. In order to implement iterative receivers, a Markov model for the channel affecting the subcarriers is also derived.

2.2.1 Additive White Gaussian Noise

We first describe the additive noise. It accounts for the electronic noise such as thermal noise or shot noise. It is modeled by an additive white noise, that is, its spectral density is constant, and which amplitude has a Gaussian distribution, hence the denomination of Additive White Gaussian Noise (AWGN). As it is the case for many natural phenomenon, the Gaussian distribution of the noise is due to the central limit theorem. This theorem states that the mean of a sufficiently large number of independent random variables will be

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approximately normally distributed. Since the noise term corresponds to the accumulation of disturbances from many sources, the theorem applies. AWGN is assumed to be an uncorrelated process.

In the case of complex signals, the noise samples along the phase and quadrature components are considered independent. Moreover, they have the same variance. Note that in our case, the output of the transmitter, which is obtained by computing an IFFT is, in general, complex. As a consequence, the complex noise samples z[n] follow a circular sym- metric complex normal distribution, with variance σ², which we denote by z ∼ CN (0, σ²).

The real part zR and the imaginary part zI are statistically independent and Gaussian distributed, with variance σ²/2. The distribution of z is:

p_Z(z) = 1 πσ² exp

−|z|² σ²

We are not directly interested in the noise variance σ², but rather in the Signal-to- Noise Ratio (SNR). It is common to use the normalized SNR, expressed as Eb/N0 where E_b denotes the energy per bit and N₀ the noise spectral density. The noise power can be deduced from the normalized SNR:

σ² = E_s mR E_b

N0

linear

The normalized SNR is typically indicated in decibels (dB). The relation between the value in dB and the linear value is:

E_b N0

dB

= 10 log₁₀ Eb

N0

linear

If the signal s(t) is sent, we receive r(t) as:

r(t) = s(t) + z(t).

2.2.2 Multipath Channel

The signal sent by the transmitting antenna will reflect on many objects in the environment. A portion of these reflected signals will reach the receiver. As a consequence, it receive multiple copies of the original signal. Each copy arrives with a certain delay and a different phase and amplitude. Figure 2.6 illustrates this principle. This type of figures are designated as delay profiles. The copies of the signal interfere at the receiver, leading to either constructive or destructive interference. The channel is then called frequency- selective.

Such a channel is called time dispersive. An important metric of time dispersive channels is the Delay Spread (DS). It is defined as the maximum delay after which the received signal becomes negligible [10]. The coherence bandwidth is the range of frequencies over which the channel is considered constant. It can be obtained from the autocorrelation of the Fourier Transform of the delay profile. Channels with a small coherence bandwidth are called frequency selective. If a signal with a bandwidth larger than the coherence

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t s(t)

(a) Sent signal (pulse)

t r(t)

(b) received signal

Figure 2.6: Multipath delay profile. A pulse is sent at the transmitter; the received signal contains copies of the original signal with different characteristics in terms of phase and amplitude.

bandwidth is sent through the channel, it will suffer from ISI. As seen in Section 2.1.5, OFDM systems are designed to fight this phenomenon, by splitting the total bandwidth into N narrow sub-bands, whose widths are more likely to be smaller than the coherence bandwidth. However, if the delay spread is longer than the cyclic prefix, then the orthogonality assumption does not hold anymore and it is not possible to prevent ISI.

Multipath channels can be represented with a tap delay line. The impulse response h(τ ) of the channel is expressed as:

h(τ ) =

Lch−1

X

i=0

c_iδ(τ − τi),

where the coefficients ci account for the attenuation and phase shift affecting the paths.

The transmitted signal s(t) is considered to go through a Linear Time Invariant (LTI) filter, whose impulse response is h(τ ). The received signal r(t) is thus obtained by a convolution, as:

r(t) = (h∗ s)(t) =

Lch−1

X

i=0

c_is(t− τi). (2.7)

Equation (2.7) indicates that L_chcopies of the signal reach the receiver, with delays τ_i and multiplied by coefficients ci. These coefficients being complex, the signal suffers both from attenuation and phase shift. The parameter L_ch is chosen according to the delay spread. A channel with a large delay spread will be modeled using more paths than a channel with a short delay spread. Since the actual number of paths is infinite, we have to define when the amplitude of the incoming paths becomes negligible. It corresponds to the discrete time representation of the Power Delay Profile (PDP). In the previous equations, the channel is considered to be time invariant. This is a valid model if the environment, as well as the transmitter and the receiver, are all motionless. However, this assumption is rarely fulfilled. If any of the wireless propagation systems component is in motion, a

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Tap Channel A Channel B

Delay (ns) Power (dB) Delay (ns) Power (dB)

1 0 0.0 0 -2.5

2 310 -1.0 300 0.0

3 710 -9.0 8900 -12.8

4 1090 -10.0 12900 -10.0

5 1730 -15.0 17100 -25.2

6 2510 -20.0 20000 -16.0

Table 2.2: ITU channel model for vehicular test environment

model that accounts for time variations is required. The time-varying impulse response of the channel is given by:

h(t, τ ) =

Lch−1

X

i=0

c_i(t)δ(τ− τi(t)).

The corresponding received signal is in Equation (2.8). In this case, both the complex coefficients ci(t) and the delays τi(t), for i = 0 . . . Lch− 1, are time dependent and the transmitted signal s(t) goes through a Linear Time Varying (LTV) filter.

r(t) = (h∗ s)(t) =

L_ch−1

X

i=0

c_i(t)s(t− τi(t)). (2.8)

In the remainder of this thesis, the discrete-time version of the signals and the impulse response of the channel will be used. The discrete-time impulse response h[m] is such that h[m] = 0 if m < 0 or m≥ Lch. The discrete-time received signal is given in Equations (2.9) and (2.10), respectively in the time invariant and the time-varying case:

r[n] = h[m]∗ s[n] ,

Lch−1

X

m=0

x[n− m] · h[m] (2.9)

r[n] = h[n, m]∗ s[n] ,

Lch−1

X

m=0

x[n− m] · h[n, m]. (2.10)

As mentioned above, the channel model will depend on the application. The Interna- tional Telecommunication Union (ITU) has defined various models for the wireless channels. These ITU channel models are designed for the following environments: indoor office, outdoor-to-indoor pedestrian, and vehicular test environment. These are empirical models based on field measurements, since modeling the exact nature of a channel is virtually impossible. We are interested in the vehicular test environment model; two DS are considered, a low and a medium one, in channel models A and B respectively. Both models are summarized in Table 2.2 [10].

2.2.3 Doppler Shift

When either the receiver, the transmitter, or any object on which the received signal reflects is in motion, we observe a shift of the received frequency, called the Doppler shift. This leads to time-varying channels. The rate of the variations depends on the

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relative speed between the components of the wireless transmission system. Above a certain velocity, the channel changes within a frame. As a spreading in the time domain led to frequency selectivity, a spreading in the frequency domain leads to time selectivity.

The maximum Doppler shift is given by:

f_d= v

λ = f_c·v c,

where c is the speed of light, and λ and fcare respectively the wavelength and the frequency of the transmitted signal. v designates the relative speed between the communication nodes.

In the presence of Doppler shift, the autocorrelation function of the channel is:

R(τ ) = J₀(2πf_dτ ), (2.11)

where J₀(·) denotes the zeroth order Bessel function of the first kind. From this autocorrelation function, it is possible to derive the Doppler power spectral density [8]:

S(f ) = 1

πfd

r 1−

f f_d

2, for|f|< fd.

Here f represents the frequency shift, relative to the carrier frequency. The Doppler spectral density shows the extent of the spreading in frequency caused by the Doppler shifts.

Considering the relatively low velocities reached by vehicles, the Doppler shifts remain relatively small. Let us consider for instance a vehicle driving at v = 120 km h−1, trying to communicate with a fixed road side unit. This gives f_d = 656 Hz, since the symbols are transmitted at f_c = 5.9 GHz. The maximum Doppler shift is small compared to the carrier frequency. Generally, fd varies between 1 Hz and 1 kHz. As a consequence, ICI is not an issue [9]. However, time selectivity will introduce fading, which is an issue that cannot be overlooked. Similarly to the coherence bandwidth, we define the coherence time as the time during which the channel may be considered constant.

The time-varying impulse response of the channel h[n] can be modeled by Jakes spec- trum:

h[n] = r 2

L_ch

Lch

X

i=1

(cos βi+ j sin βi) cos (2πfdTsn cos αi+ θi) , where

• Lch is the number of paths,

• βi is a random variable introduced to ensure that the I and Q components are uncorrelated,

• fdis the maximum Doppler shift,

• Ts is the sampling time,

• αⁱ is the angle of arrival of the scatterers. Note that the Doppler shift on the i^th path is f_dcos(α_i),

• θi is the initial phase of the i^th path.

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0 1 2 3 4 5

0 0.1 0.2 0.3 0.4 0.5

a pA(a)

Measurements Theoretical

Figure 2.7: Distribution of the channel coefficients of a single subcarrier. The actual distribution (bar chart) is compared to the theoretical one (line plot).

Combined with the time invariant model for the multipath propagation, we obtain:

h[n, m] = h[m]· h[n],

where h[m] corresponds to the impulse response described in Section 2.2.2. Adding the noise coefficients, we obtain:

r[n] = h[n, m]∗ s[n] + z[n].

2.2.4 Modeling of the OFDM Subcarriers

It was shown in Section 2.1.5 that, in an OFDM system, each subcarrier goes through a flat fading, time-varying channel. Performing equalization on the subcarriers is thus less complex than performing it directly on the received signal. The model used for a single OFDM subcarrier (say, the k^th subcarrier) is the following:

rk[n] = hk[n]· s[n] + z[n]. (2.12) Note in Equation (2.12) that there is no more convolution, but instead the transmitted sample s[n] is multiplied by a single complex coefficient hk[n] = a[n]e^jφ[n], with a[n] > 0 and φ[n]∈ [0, 2π].

The coefficients h_k[n] form a stochastic process. Finding the properties of this stochastic process is essential, considering that the receiver will be designed based on that model.

Simulations showed that the amplitudes a of the channel coefficients on a single OFDM subcarrier are Rayleigh distributed. This can be observed on Figure 2.7, where the simu- lation results are plotted against the theoretical model. The probability density function

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0 0.2 0.4 0.6 0.8 1

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

τ

R(τ)

Figure 2.8: Autocorrelation function of a Rayleigh fading channel with relative motion between the transmission system’s components, for fd= 5 Hz.

(pdf) of a Rayleigh distributed random variable is given by:

p_A(a) = a ρ² exp

− a² 2ρ²

, a > 0.

The Rayleigh distribution is characterized by the parameter ρ, which is determined using measurements. It is related to the Root Mean Square (RMS) of the received signal:

Arms =p

E(A²) =√ 2ρ.

The autocorrelation function (acf) of the Rayleigh fading channel is linked to the maximum Doppler shift, as seen in Section 2.2.3:

R(τ ) = J₀(2πf_dτ ).

The acf of the channel is depicted on Figure 2.8.

In the remainder of this thesis, unless mentioned otherwise, we will consider a flat fading, time-varying channel. This model is suitable for the subcarriers, as observed in this section.

2.2.5 Markov Modeling of the Channel

Channel Modeling Using Markov Models Receivers based on the Forward-backward algorithm, as the one that is implemented here (see Chapter 3), require an iterative description of the correlated channel coefficients. As a consequence, classical, non-Markovian representations of Rayleigh fading channels are not suitable for our receiver. Hence, we

Receiver Design for Vehicular Communications