Performance Analysis of Piecewise-and-Forward Relay Network on Rayleigh Fading Channel

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Performance Analysis of

Piecewise-and-Forward Relay Network on Rayleigh Fading

Channel

Farnaz Gharari

This thesis is presented as part of Degree of Master of Science in Electrical Engineering

Blekinge Institute of Technology March 2015

Blekinge Institute of Technology Faculty of Engineering

Department of Applied Signal Processing Supervisor: Mrs. Thi My Chin Chu Examiner: Prof. Hans-Jürgen Zepernick

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Abstract

The rapid changes and developments in cellular and wireless networks aim to provide more reliable communications with high data rate transmission. So far, one solution which is being considered for improving the reliability and efficiency of mobile communications is cooperative communications. Implementing different relay scenarios leads to improvement of the secure, efficient, and fast transmission with a good quality of service.

The work of this thesis is on the piecewise-and-forward (PF) relay network. This relay protocol is implemented for decreasing the complexity of signal detection at the receiver.

Our focus is to investigate the performance of PF relay networks based on the statistical variations which are caused by the fading environment. We considered a relay network with one source and one destination while multiple parallel relays aid the transmission process. At the destination the maximum likelihood (ML) method is implemented for detecting the received signals. Therefore, the probability density function (PDF) of received signals should be used at the ML detector. We analytically investigate the PDF of the received signals, when the transmitted signals are faced with Rayleigh fading.

For verifying the theoretical calculations, we use Monte-Carlo simulations in MATLAB to evaluate the results. Afterwards, the bit error rate (BER) has been considered for investigating the performance of the PF relay network over a Rayleigh fading channel.

Furthermore, the BER performance of the PF relay network is compared with three well known relay networks, amplify-and-forward (AF), decode-and forward (DF), and estimate-and-forward (EF) relay networks.

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Acknowledgements

At the step of finalizing my master studies, I would like to express my special gratitude to my high-minded and mighty examiner Prof. Hans-Jürgen Zepernick and my supportive supervisor Mrs. Thi My Chinh Chu for their guidance, help, and support. Without their help and support it would have not been possible for me to do this thesis. I am really thankful for all that I learned from them. My warm thanks to my dear husband and my parents, for their love, support that helped me to study in my desired field.

Thanks to all my dear teachers, friends, and who were beside me during the journey of learning and studying. They helped me to promote my personal growth.

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Abbreviations

AF Ampify-and-Forward

BER Bit Error Rate

BPSK Binary Phase Shift Keying

CDF Cumulative Distribution Function CSI Channel State Information

DF Decode-and-Forward

EF Estimate-and-Forward

LLR Logarithm Likelihood Ratio

ML Maximum Likelihood

MMSE Minimum Mean Squared Error

MMSUE Minimum Mean Squared Uncorrelated Error

MRC Maximum Ratio Combining

OP Outage Probability

PF Piecewise-and-Forward

SEP Symbol Error Probability

SER Symbol Error Ratio

SNR Signal to Noise Ratio TWRC Two Way Relay Channel

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Introduction

Motivation

Nowadays, the demand for multimedia services in new generation of mobile cellular systems (3G and 4G/LTE), and wireless networks is rapidly growing. However, providing services with high data rate transmission faces many difficulties, which are inflicted by multipath fading channels. Therefore, a major challenge for designing a wireless communication system is how to obtain higher data rate, while still providing reliable transmission. To overcome this difficulty, cooperative communications is considered as a powerful technique to improve the reliability of the network. On other hand, it is an energy efficient transmission to expand the coverage area of wireless networks. These main factors of cooperative communications are the main motivation of this thesis. In this chapter, by bringing a summary of related works, the main contributions of the thesis are explained. The other sections present an introduction to the concept of cooperative communications and fundamentals of relay strategies.

1.1 Related Works

Many studies have been done in the area of cooperative communications after the ﬁrst introduction of it in [1]. The coding fundamentals and strategies of cooperative communications are comprehensively addressed in [2, 3]. After that, because of the great advantages of relay networks over one-way communication, the performance analysis of

1

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Chapter 1. Introduction 2 relay networks has been studied according to diﬀerent protocols and parameters [4–11].

The capacity of relay networks over a Rayleigh fading channel is investigated in [3–5].

By proposing different relaying protocols, due to the well defined structure of amplify- and-forward (AF) and decode-and-forward (DF) relay networks, they are implemented easily in cellular networks and wireless communication systems. Therefore, these relay networks have been studied extensively. The performance of these protocols in terms of capacity, diversity, outage probability, and bite error rate (BER) have been investigated over different fading channels in [12–24].

By developing the relay strategies and the geometric structure of cooperative networks, diﬀerent investigations are done for parallel, serial, and hybrid relay networks. Normally, in the presence of more than one relay in the system, the destination needs detectors for combing the received signals and detecting the correct transmitted signals without error.

At the destination of many relay networks, maximum ratio combining (MRC) can be used to combine the signals from different paths and different methods of detection can be used at the destination for detecting the transmitted signal. In [25], a performance analysis for AF relay network is presented with best relay selection over Rayleigh fading channels. Analytical expressions for the probability density function (PDF), cumulative distribution function (CDF), and the moment generating function (MGF) of the end-to- end signal to noise ratio (SNR) of the system are derived. Those expressions are used for maximum likelihood (ML) detection at the destination. Also, the average symbol error rate (SER), the outage probability (OP), and the average end-to-end SNR are obtained from these expressions. In [26], a weighted maximum ratio combing, which is called (C-MRC) is suggested for the DF relay strategy. This method of detection is similar to the ML method, but this work shows that the computational complexity of this method is less and the network can reach to a maximum diversity beyond this method. In [27], the ML method is used at the detector in different schemes for binary phase shift keying (BPSK) modulation, and the BER performance of the relay networks is examined. In [28,29], a method of detection based on SNR thresholds is investigating to decrease error propagation in a one-way relay network. The same method is used in [30] for a two-way relay channel (TWRC) to minimizing the end-to-end BER. In [31], threshold-based relaying aims for the DF network to minimize the BER. In this work, an approximate expression for the PDF of the relay is derived, which is used as a threshold based logarithm likelihood ratio (LLR) detector at the destination to investigate the

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end-to-end BER of the network.

The estimate-and-forward (EF) relay network implements estimation functions such as hard and soft detection, to transmit a more detectable information to the destination.

The basis of an EF relay network, for the ﬁrst time, was suggested by Cover and El Gamal [32]. According to the studies in [24,33–36], the advantage of EF in comparison with DF relay network is that it has lower BER at low SNRs, although the EF relay network can achieve the same gain as the DF relay network. In [37], the application of an EF relay network with diﬀerent mapping is investigated. It was shown that the EF relay network has good performance in lower SNRs. Abou-Faycal and Medard in [38]

studied an EF relay network for minimizing the error probability by the Lambert-W method. In this study, they did not consider the direct link from source to destination.

They used approximations and a look up table for their approximations, as they did not present an analytic investigation. In [39], the estimation of the EF relay network is based on scalar quantization of the received signal. The work in [40] uses the concept of minimum mean squared uncorrelated error (MMSUE) for the EF relay network. The results show that this scheme for the EF relay network has better performance than the AF and DF relay networks in both parallel and serial relay networks. Indeed, a numerical study of minimum mean squared error (MMSE), in a two-hop relay network was presented in [41]. The study in [42] is done for a TWRC wireless network with binary antipodal signaling. Their work shows that the Lambert-W function minimizes the probability of error. The authors of [43] investigated the operation of the EF relay network for a coded relay network by using the LLR as the estimation method.

Khormuji and Skoglund in [44] investigated an EF relay network by instantaneous relaying methodology. In this work, a Gaussian relay channel is studied with a direct link from source to destination and with perfect conditions for source to relay links.

The numerical investigations of this work showed that a relay with a piecewise linear mapping improves the relay performance over the AF relay network. In [37] and [45], for investigating EF in multiple relay networks, a piecewise scheme is suggested. The reason for this suggestion is the complex convolution which should be done by the detector to calculate the PDF of the received signals for the detection process. Therefore, this threshold based piecewise method is investigated by them to simplify the PDF calculation.

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Chapter 1. Introduction 4 As far as our studies have shown, most of the works on EF relay networks are done under perfect channel conditions. There is a lack of works for investigating the performance of this relay network over various protocols, environments, and channel conditions. In the case of channel properties, obviously, for a more practical investigation, the fading coefficient should be considered as a random variable. The argue of this thesis is that, as the PDF of the received signals defines the detection criteria at the ML detector, therefore, the expression of the PDF should be derived accurately by considering the random distribution of the channel fading coefficient.

1.2 Main Contributions of the Thesis

As mentioned, many studies have been extensively done on the performance of AF and DF relaying networks, for different relay typologies and under different channel conditions. On the other hand, as the EF protocol can use different estimation methods to improve the detection, it has attracted some researches to work on the best optimized methods and to analyze EF protocol performance. In this context, the piecewise- and-forward (PF) relay network introduces the linear piecewise method to simplify the complexity of the EF protocol for implementation. The work in [45] shows that the performance in terms of BER of a PF relaying network for low SNRs outperforms those of AF and DF relaying networks. However, the authors of [45] studied a very idealistic scenario of a communication system, where Rayleigh fading coefficients were kept constant.

Whereas, in a real environment, the fading channels always ﬂuctuate randomly. The reason is that after sending a signal, it will face diﬀerent changes in attenuation, delay, and phase. Obviously, in order to have a realistic investigation, these variations on the received signal should be considered in the case of theoretical calculations. The reason of this claim is that the quality of channel links in relay networks plays an important role in this communication process.

To the best of our knowledge, there are few works which have studied the PF protocol and its performance. Hence, the scope of this thesis is to extend the study of the PF protocol proposed in [45]. The focus of this work is to study the performance of relay networks by investigating the eﬀect of channel variations according a statistical approach. In this investigation, we derive the PDF of the received signals at the destination of a PF relay network subject to Rayleigh fading.

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The research of this thesis is organized as follows:

After bringing the motivation and reviewing the most related works, in Section 1.3, brief background of cooperative communications and relaying networks is presented. Chapter 2 is on the system model, description of relay function in EF relay network, the detection method at the destination, and presentation of the piecewise linear approximation for calculating the PDF of signals at the ML detector. Then, in Chapter 3, the PDF of the received signal over Rayleigh channel by considering the slow fading condition is derived.

The main emphasis of the investigation is on the effect of the random distribution of channel fading coefficients on this relay network. After deriving analytically the PDF of received signals at the destination, in Chapter 4, the PDF of the PF relay network is simulated over the same conditions for verifying the theoretical results. Specifically, by Monte-Carlo simulations, the BER performance versus different parameters is performed for different relay networks over Rayleigh fading channel. Finally, in Chapter 5, the thesis concludes the investigations.

1.3 Cooperative Communications

In general, cooperative communications is a wireless network with cellular or ad hoc topology in which wireless agents, which can be called users, can cooperate with each other to transmit their messages to the destination [24]. In such a system, each user can transmit its own message to the destination as well as act as a cooperative agent for another user (in this case, the user is called relay). Cooperative communications can significantly help to increase the effective quality of service of the communication network. This can be measured in terms of BER or outage probability (OP), which measure the robustness of the communication process to fading based on the SNR [46]. In Figure 1.1, a simple scenario for cooperative communications is shown. When the source S broadcasts the signal, it travels through two links. One direct link to the destination D and one link with the relay R for transmitting the signal to the destination. When the direct link is weak, the cooperation of users between sources and destinations can improve the communication. The destination can combine the received signals from the direct link and the relaying links to attain spatial diversity. However, the improvement in spatial diversity gain happens with the cost of loss of transmission spectrum as users in different roles need to occupy more spectrum and bandwidth [46].

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Chapter 1. Introduction 6

Figure 1.1: A simple three node cooperative relay network.

Relaying Strategies

In the simplest case, a cooperative relaying network consists of three nodes: one source S, one destination D and one relay R, which conducts the relaying communication. In the case of a multiple relay network, depending on the requirement and characteristics of the communication network, diﬀerent typologies can be used. For example, in the case of long distance between source and destination, serial relay transmission is used to provide power gain as shown in Figure 1.2. In this topology, signals from one relay propagate to

Figure 1.2: A serial multiple relay network.

the next relay and the channels are orthogonal to avoid signal interference. In the case of an outdoor environment, parallel relay transmission is being used, which increases robustness against multi-path fading, see Figure 1.3. In this topology, signals propagate through multiple relays in multiple parallel paths. Consequently, at the destination, diﬀerent schemes can be used for combining the received signals which provide power gain and diversity gain.

In cooperative relaying, based on the operation process of relays and based on the network requirement, complexity, and physical factors such as physical environment and

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Figure 1.3: A parallel multiple relay network.

distance, diﬀerent strategies can be implemented. The main relaying strategies which are deﬁned for implementation are as follows:

Amplify-and-Forward

In an AF relaying network, the relay simply amplifies the signal received from the source with a gain factor and forwards the resulting signal to the destination. However, the noise is amplified in the relay as well. This protocol can be implemented with two methods. The first is called orthogonal amplify-and-forward (OAF). In this proposal, in the first phase, the source transmits the signal and in the second phase, the relay amplifies the signal and sends it to the destination. This method is performed in two orthogonal channels to have a half duplex structure. Therefore, this method faces a bandwidth loss. The other method is introduced to overcome this drawback, which is called non-orthogonal amplify-and-forward (NOAF). In this method, the source is also active when the relay transmits the signal to the destination. This method uses the maximum system degree of freedom and is more efficient and offers higher data rate transmission [22].

Decode-and-Forward

In a DF relaying network, the relay decodes the received signal and in the case of correct decoding it forwards the signal to the destination. This process does not add

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Chapter 1. Introduction 8 any additional information to the signal. This approach can be done in two different methods. The first method is called basic DF. In the first phase, the source forwards the message to the relay and the relay decodes the message. In the case of successful decoding, the relay forwards the message to the destination. The second method is selection DF. In this method, in the second phase, the source transmits its message to the destination directly. The destination compares the two decoded messages. Generally, a DF relay network needs to know source-destination and relay-destination channel state information (CSI), but AF relay network just needs to know channel state information of the source to relay link. This difference causes a better performance for the AF relay network. On the other hand, in some cases, for example, in the case of ideal channel conditions with high SNR, DF performs better than AF as it eliminates the noise [26].

Estimate-and-Forward

In an EF relaying network, which is also known as compress-and-forward (CF), the relay provides an estimation of the source message by diﬀerent transformation methods.

Actually, this estimation helps to detect the message at the destination. Mostly, EF is useful when the relay can not decode the source message easily. According to [26], the quality of the source to relay link has a great inﬂuence on the performance of the EF protocol. Moreover, based on [24], when the relay is near to the destination, the EF relay network has better BER performance compared to the DF relay network.

Piecewise-and-Forward

In the EF protocol, with multiple parallel relays, the destination combines the signals received from source and relays. To decode the signals, the destination applies diﬀerent detection methods such as MMSE or ML. However, in order to utilize these detection methods, it is required to obtain the statistical distributions, including the CDF or the PDF of the received signals. Due to the complexity of the transformation in the EF relay networks, a linear, piecewise method, was proposed in [37] and [45] to overcome the computational complexity of calculation of statistical distributions at the destination.

According to these works, a piecewise strategy is being implemented in relays to simplify the detection process at the destination. This relaying network is known as piecewise- and-forward (PF).

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The PF relay network is the relay strategy, which we use in our system model to investigate the eﬀect of Rayleigh fading channel on the system performance.

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Chapter 2 Preliminaries

In this chapter, we deﬁne the system model and detection method, which are used in the relay network of this thesis. Then, we study a general EF relay network and PF relay network for this system. This system model will be used in the next chapter for the further investigation of the PF relay network over Rayleigh fading channel.

2.1 System Model

The system model, which we investigate in this thesis, is a two-hop relay network with multiple parallel relays. The topology of this system is shown in Figure 2.1. We consider a communication network consists of one source S, one destination D and K relays,

Figure 2.1: A multiple relay network with K relays.

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where Ri, presents the i^threlay. For this relay network, binary phase shift keying (BPSK) is used as modulation scheme. Each relay participates in the process of communication to assist the transmission. Figure 2.2 shows this system with one relay in more detail.

Figure 2.2: Description of the implemented system with three nodes (source, relay, destination), and the noisy links with channel fading coeﬃcients.

For this model, according to Figure 2.3, time division multiple access (TDMA) is assumed where a signal is transmitting in two time slots. In this case, the relays work in half-duplex mode, and they can not receive and transmit simultaneously. In the ﬁrst time slot, the source broadcasts the BPSK symbol to the destination and the relays.

According to the deﬁned channel for each link and the added noise, the relays and destination receive the transmitted signals from the source. In the second time slot, only the relays transmit the signals to destination according to their assigned protocol.

Figure 2.3: Half-duplex transmission mode. In the ﬁrst time slot, the source transmits the signals, and in the second time slot, the relay transmits the signals.

After completing the transmission process in the two time slots, detection should be done at the destination. The following equations are deﬁned for the described process

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Chapter 2. Preliminaries 12 according to Figure 2.2:

ysri=

Psxshsri+ nsri (2.1)

ysd=

Psxshsd+ nsd (2.2)

yrid= yrihrid+ nrid (2.3)

where xs is the transmitted symbol by the source, and according to BPSK modulation xs=±1 and √

Psis the source transmitter power. The term yri denotes the output signal, which is transmitted from the relay to the destination, according the relay function f (ysri). hsri, hsd and hrid, are the channel fading coefficients from S to Ri, from S to D, and from Ri to D, respectively. According to the distribution of the channel fading, the variance of these channel coefficients are presented as Ω_sri = E{|hsri|²}, Ωsd= E{|hsd|²} and Ω_rid= E{|hrid|²}. In this chapter, we do not emphasis on a special fading channel, but we consider a general case for the fading channel coefficients. In Chapter 3, we continue our investigation based on Rayleigh fading. Note that nsri, nsd and nrid, are the corresponding adaptive white Gaussian noise (AWGN) samples with zero mean and variances σ²_sri= E{|nsri|²}, σ_sd² = E{|nsd|²}, and σ_rid² = E{|nrid|²}, respectively.

For this system model, it is assumed that the receivers at the relays and the destination have perfect channel state information (CSI). As mentioned, after receiving the signals at the destination, all these signals are employed in the ML detector at the destination to detect the correct transmitted signal.

2.2 Detection Model at the Destination

As mentioned, in relay networks, diﬀerent methods for combining the received signals at the destination and detecting the transmitted signal can be implemented. The ML method combines the noisy signals at the destination in an optimal way. Furthermore, a ML detector considers the eﬀect of propagation errors, which are mainly caused by the fading channel. Therefore, the ML detector uses a coherent detection scheme in which the status of the channels and noise variances should be known to the destination.

According to the ML detector deﬁnition, ˆxd, denotes the detected symbol at destination, and is obtained as

xˆd= argmax

xs

{P (y_sd, yrd|x_s)} (2.4)

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Based on the Bayes’ rule, the decision criteria is written as

xˆd= argmax

xs

P (xs)P (ysd|xs)P (yrd|xs) P (ysd, yrd)

(2.5)

The decision at the detector is not dependent on the value of P (ysd, yrd), so it can be ignored. Indeed, for BPSK modulation, P (xs) for each symbols is equal and it is not aﬀected by the last decision of the detector. Therefore, (2.5) can be rewritten as

xˆd= argmax

xs

{P (ysd|xs)P (yrd|xs)} (2.6)

For our network with multiple relays, according to TDMA orthogonality and a uniform relay function for all the relays, the decision criteria becomes

xˆd= argmax

xs

{P (ysd|xs)

K i=1

P (yrid|xs)} (2.7)

These conditional probabilities are represented as the conditional PDF of the received signals at the destination and we can therefore write

xˆd= argmax

xs

{f(y_sd|x_s)

K i=1

f (yrid|x_s)} (2.8)

It is assumed that each transmitted signal from the relays is an independent and identi- cally distributed sample. Therefore, the joint distribution of them are combined at the destination. On the other hand, as BPSK is used, it can be written as

fsd(ysd)

K i=1

f_rid|x_s₌₊₁(yrid)^x^s⁼⁺¹≷

xs=−1fsd(ysd)

K i=1

f_rid|x_s₌₋₁(yrid) (2.9)

where fsd(.) denotes the PDF of received signals of the S → D link at the destination, and f_rid|x_s₌₊₁(.) and f_rid|x_s₌₋₁(.), respectively, present the PDF of the received signals of the Ri → D link at the destination when xs = +1 or xs = −1 is transmitted from the source. Sometimes, it is easier to write (2.9) in logarithmic form as

ln

fsd(ysd)

k i=1

frid|xs=+1(yrid)

xs=+1 xs≷=−1ln

fsd(ysd)

k i=1

frid|xs=−1(yrid)

(2.10)

As can be seen from (2.10), the PDF of yrid plays an important role in the decision process at the destination. This PDF requires the PDF of ysri and the relay function.

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Chapter 2. Preliminaries 14 Thus, the complete transmission process, such as channel link, noise, and relay function should be known at the destination for interpreting the conditional probabilities. In the next section, we study the EF relay network for this system model which is used for the PF relay network.

2.3 Estimate-and-Forward Relay Function

As has been already explained in the ﬁrst chapter, for the estimation relay function of an EF relay network, diﬀerent methods can be implemented such as Lambert-W, MMSE, and LLR. According to [37] and [45], when BPSK modulation is used with MMSE estimation, a hyperbolic tangent can be used as the estimation function at the relay. In this section, we investigate this EF function for our system.

According to the general deﬁnition, the MMSE estimation is the conditional expectation of xs, i.e.

xˆMMSE = fEF(ysri) = E[xs|ysri] (2.11) The MMSE has a Bayesian approach with quadratic cost function. This means that there is some prior knowledge about the desired parameters. Here, we consider hsri as a general term for the channel fading coefficients. By considering the statistical behavior channel fading coefficients as a random variable (RV) and according to the definition of conditional joint PDF, for the PDF of the S → Ri link, we have

P_X_s_|Y_sri_,H_sri(xs|ysri, hsri) = P_X_s_|Y_sri(xs|ysri)PHsri(hsri) (2.12)

By using the Bayes’ rule, we have

fEF(ysri) =

xs

xsP_X_s_|Y_sri_,H_sri(xs|ysri, hsri) =

xsxsPysri|xs(ysri|xs)PHsri(hsri)P (xs) Pysri|xs(ysri|xs)PHsri(hsri)P (xs)

(2.13) where PHsri(hsri) can be simpliﬁed from the numerator and the denominator as

fEF(ysri) =

xsxsPysri|xs(ysri|x_s)P (xs)

Pysri|xs(ysri|xs)P (xs) (2.14)

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The PDF of ysri in Gaussian noise is ysri∼ N (hsrixs, σ²_sri), i.e.

P_X_s_|Y_sri(xs|ysri) = 1

2πσ_sri² exp

−(ysri− hsrixs)² 2σ²_sri

(2.15)

By considering BPSK modulation with antipodal signals, xs∈ {−1, +1} and after doing simpliﬁcations, the ﬁnal result for the EF function at each relay is

fEF(ysri) = exp

ysrihsri

σ_sri²

− exp

−ysrihsri

σ_sri²

exp

ysrihsri

σ_sri²

+ exp

−ysrihsri

σ_sri²

= tanh

ysrihsri

σ_sri²

(2.16)

After completing the estimation process, when a signal is transmitted from each relay to the destination, it is normalized with the transmit power of the relay, Pri. We represent this power transmit coeﬃcient for the relay with β, which is formulated as

β =

Pri

E{|fEF(ysri)|²} (2.17)

In Figure 2.4, the EF relay function is compared with the AF and DF relay functions

Figure 2.4: Comparison of the MMSE estimated EF relay by AF and DF relay functions.

for an ideal case. For this plot, BPSK with SN R = 10dB is considered for the source to relay link. It can be seen from Figure 2.4 that the EF relay function falls between AF

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Chapter 2. Preliminaries 16 and DF. This property is used to decrease the complexity of the PDF of calculation at the detector. Actually, this complexity makes this approach less applicable. In [37] and [47], piecewise linear approximations are suggested to make the PDF computation more appealing at the detector. In [45], the tangent hyperbolic form of the EF relay function is used for implementing a threshold based piecewise model. In the next section, we investigate this proposed piecewise model over Rayleigh fading channel.

2.4 Piecewise-and-Forward Relay Network

Due to the complexity of the hyperbolic function, in [37], a linear piecewise model is proposed to simplify the calculation of the PDF at the detector. However in [45], a piecewise model based on adaptive SNR thresholds is proposed. According to this model, based on the observation of the EF relay function shown in Figure 2.4, the hyperbolic tangent function can be approximated by a linear function when the inputs are very small. So, it approaches to a constant when the input goes to positive and negative inﬁnity [47]. Therefore, the EF relay function is replaced with a piecewise model with the following segments:

yri =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−1 if ysri ≤ T

ysri

T if−T < ysri< T 1 if ysri ≥ T

(2.18)

where T is a threshold for approximating yri. In [45], T is determined as the median value of the ysri in the first quadrant. Referring to the definition, a median value, represented by m, for a distribution such as F (x) = P (X ≤ x) will be defined as

P (X ≤ m) ≤ 1

2 (2.19)

where m ∈ R. By using this deﬁnition, the following approximation for the value of T is given in [45]:

tanh

Υ_sriT

2

≤ 1

2 (2.20)

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For our system, the instantaneous SNR Υ_sri is calculated as

Υ_sri= E{y_sri² }

E{n²_sri} = E{h²_sri}Ps

E{nsri} = Ω_sriPs

σ_sri² (2.21)

and T has the following approximation:

T = σsriln 3

PsΥ_sri (2.22)

We will use this PF relay scheme in the next chapter to investigate its performance over the Rayleigh fading channel.

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Chapter 3 Piecewise-and-Forward Relay Network over Rayleigh Fading Channel

In wireless communication, the properties of the channel have an important role in the signal propagation performance. Channel fading in multi-path environments can cause many variations and affects on the signal propagation behavior that should be estimated and predicted for a correct and optimized transmission. In this chapter, our emphasis is on investigating the effect of random channel coefficients on the PF relay network.

However, as the detection depends on the PDF of the received signal at the detector, we analytically investigate the eﬀect of the variation of the fading coeﬃcients on the PF relay network for Rayleigh fading.

3.1 Rayleigh Fading Channel as a Realistic Model

Multiple paths along the communication channel cause variation in the signal magnitude.

Fading is known as one of the most important impairments, which describes the impact of ﬂuctuation of signal strength during propagation over a communication channel. In general, fading can be fast or slow. Fast fading happens when many rapid ﬂuctuations occur during a short period or signal transmission. The reason of this fading is coming from the propagation of the signal inside an environment which contains objects such

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as buildings and trees that cause signals to be scattered. The Rayleigh distribution is introduced as the best model, which fits to the behavior in an environment when there is no line of sight (NLOS). Slow fading, may be caused by shadowing effects of objects such as buildings and trees for slow moving mobiles or objects. In slow fading, the channel coefficients are constant during a frame period but they are changing independently from frame to frame [45].

For our relay network, we consider slow fading model with Rayleigh distribution. Accord- ing to the central limit theorem for an ideal case and when there are enough scatterers, the Gaussian model is used for modeling the channel fading distribution. If there are not suﬃcient scattering components, the channel model distributions is modeled with a zero mean and a phase between 0 and 2/π radians. The envelop of the Rayleigh fading distribution, which is denoted as the random variable R, has a PDF as follows:

fR(r) = 2r Ω exp

−r² Ω

, r ≥ 0 (3.1)

where Ω = E{R²}. Given this fading channel, we investigate the performance of a PF relay network.

3.2 Analytical Investigation of the PDF for ML Detector at the Destination

According to the system model, which we have already mentioned in Figure 2.2, we consider a two-hop relay network with multiple parallel relays. As mentioned in (2.10), the ML detector at the receiver needs the PDF of the received signals to detect the transmitted signal. Therefore, the PDF of the received signals should be calculated.

According to (2.10), the PDF of the received signals, ysri, from the source at the relay Ri, and the PDF of received signals, ysd, from the source at the destination are independent and they should be calculated separately. Due to the half-duplex transmission with multiple relays, the process of calculating the PDF of ysri is complicated. Hence, we start with the calculation of this PDF.

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Chapter 3. Piecewise-and-Forward Relay Network over Rayleigh Fading Channel 20 3.2.1 The PDF of the Received Signal at the i^th Relay

We deﬁne γsri = hsri

√Ps as a new variable to simplify the calculation. Then, (2.1) can be written as

ysri= γsrixs+ nsri (3.2)

where nsri is zero-mean complex Gaussian noise with variance σ_sri² . The PDF of nsri is given by

fnsri(y) = 1

2πσ²_sri exp

−y² 2σ_sri²

(3.3)

The PDF of the received signal at the i^th relay, Ri, conditioned on the source signal xs

and channel fading coeﬃcient hsri, is presented as

fYsri|Xs(ysri|γsri, xs) = 1

2πσ²_sri exp

−(ysri− γsrixs)² 2σ_sri²

(3.4)

Given BPSK modulation, the signals are mapped as xs ∈ {−1, +1}. Thus, from (3.4), we have

fYsri|Xs(ysri|xs={±1}) =

_+∞

−∞ fYsri|Xs(ysri|xs={±1})fγsri(γsri) dγsri (3.5) According to (3.1), the PDF of Rayleigh channel coeﬃcient hsri from the source to the i^th relay Ri is given by

fHsri(hsri) = 2hsri

Ω_sri exp

−h²_sri Ω_sri

, hsri ≥ 0 (3.6)

where Ω_sri is the channel power gain. To calculate (3.5), ﬁrstly, the PDF of γsri should be calculated as

fγsri(γsri) = 2γsri

PsΩ_sriexp

−γ_sri² PsΩ_sri

, γsri ≥ 0 (3.7)

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We ﬁrst consider the case that xs= +1 is being transmitted. From (3.4) and (3.7), we rewrite (3.5) for the case that the source sends xs= +1 as

fYsri|Xs(ysri|xs= +1) =

_+∞

0

⎧⎨

⎩ 2γsri

PsΩ_sri

2πσ_sri² exp

−(ysri− γsri)² 2σ²_sri

× exp

dγsri (3.8)

By using [48, E. 3.462.5], and then by performing simpliﬁcations, we have

f_Y_sri_|X_s(ysri|xs = +1) = 2 (Ω_sriPs+ 2σ_sri² )

2πσ_sri²

ysri

2

2σ_sri² Ω_sriPsπ Ω_sriPs+ 2σ_sri²

× exp

−y_sri² Ω_sriPs+ 2σ²_sri

1− erf

−ysri

2σ²

2σ²_sriΩ_sriPs

Ω_sriPs+ 2σ²_sri

+σ²_sriexp

−y_sri² 2σ²_sri

(3.9)

Similarly, when the source transmits xs =−1, we have

fYsri|Xs(ysri|γsri, xs=−1) = 1

2πσ_sri² exp

−(ysri+ γsri)² 2σ_sri²

(3.10)

Substituting (3.10) and (3.7) into (3.5), we have

f_Y_sri_|X_s(ysri|x_s=−1) =

_+∞

0

⎧⎨

⎩ 2γsri

PsΩ_sri

2πσ_sri² exp

−(y_sri+ γsri)² 2σ²_sri

× exp

dγsri (3.11)

Finally, the PDF of the received signal at the i^th relay when the source sends xs =−1 can be calculated as

f_Y_sri_|X_s(ysri|xs=−1) = 2 (Ω_sriPs+ 2σ_sri² )

2πσ_sri²

−ysri

2

2σ²_sriΩ_sriPsπ Ω_sriPs+ 2σ²_sri

× exp

−y_sri² Ω_sriPs+ 2σ_sri²

1− erf

ysri

2σ²

2σ_sri² Ω_sriPs

+σ_sri² exp

−y_sri² 2σ²_sri

(3.12)

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Chapter 3. Piecewise-and-Forward Relay Network over Rayleigh Fading Channel 22 3.2.2 The PDF of Transmitted Signal at the i^th Relay

Based on the PF relay protocol, the output signal of relay Ri is segmented into three regions as follows:

yri =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−1 if ysri ≤ T

ysri

T if−T < y_sri< T 1 if ysri ≥ T

Therefore, fYri(yri) = 0 for yri < −1 and yri > +1. The PDF of Yri at the instant value Yri =−1 is equal to the probability that Ysri≤ −T :

f_Y_ri_|X_s(yri|xs= +1, yri =−1) =

_−T

−∞ f_Y_sri_|X_s

y_sri|x_s₌₊₁

dysri = I1 (3.13)

Substituting (3.9) into (3.13), we have

I1 =

_−T

−∞

2 (Ω_sriPs+ 2σ_sri² )

2πσ_sri²

×

ysri

2

2σ²_sriΩ_sriPsπ Ω_sriPs+ 2σ²_srie

−y2sri (ΩsriPs+2σ2sri)

1− erf

−ysri

2σ²

+σ²_sriexp

−y²_sri 2σ_sri²

dysri (3.14)

By the help of Mathematica, the integral in (3.14) becomes

I1 = 0.5

⎧⎨

⎩

⎡

⎣1 − erf

⎛

⎝T 2σ²_sri

⎞

⎠

⎤

⎦ −$

Ω_sriPs

Ω_sriPs+ 2σ² exp

−T² Ω_sriPs+ 2σ²

×

⎡

⎣1 − erf

⎛

⎝ T2σ²

2σ²Ω_sriPs

Ω_sriPs+ 2σ²

⎞

⎠

⎤

⎦

⎫⎬

⎭ (3.15)

For the second case of−1 < Yri < 1, as both Ysri and Yri increase monotonically to each other, i.e., Yri= Ysri/T , using for the method transformation variable in [49], the PDF of Yri in region [-1 1] is calculated as follows:

I2 = fYri|Xs(yri|xs= +1, −1 < yri< 1) = T fYsri|Xs(ysri = T yri|xs= +1) (3.16)

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Substituting (3.9) into (3.16), we get

I2 = 2T

(Ω_sriPs+ 2σ_sri² )

2πσ_sri²

×

T ysri

2

2σ_sri² Ω_sriPsπ Ω_sriPs+ 2σ_sri² exp

−T²y_sri² Ω_sriPs+ 2σ_sri²

×

1− erf

−T ysri

2σ²

2σ_sri² Ω_sriPs

Ω_sriPs+ 2σ_sri²

+ σ²_sriexp

−T²y²_sri 2σ_sri²

(

(3.17)

In the case Yri = +1, the PDF of Yri at the instant value of yri = +1 is equal to the probability that Ysri ≥ T :

f_Y_ri_|X_s(yri|x_s= +1, yri=−1) =

_+∞

T f_Y_sri_|X_s

y_sri|x_s₌₊₁

dysri= I3 (3.18)

Substituting (3.9) into (3.18), we have

I3 = f_Y_ri_|X_s(yri|x_s= +1, yri = +1) =

_∞

T

2 (Ω_sriPs+ 2σ²_sri)

2πσ_sri²

×

ysri

2

2σ²_sriΩ_sriPsπ Ω_sriPs+ 2σ²_sriexp

−y_sri² Ω_sriPs+ 2σ²_sri

×

1− erf

−ysri

2σ²

+ σ_sri² exp

−y_sri² 2σ_sri²

(

dysri (3.19)

where erf (.) is the error function for variable x given as erf (x) = ^√²_π )_x

0 e^−t²dt. After performing the simpliﬁcation on (3.19), I3 is obtained as

I3 = 0.5

⎧⎨

⎩

⎡

⎣1 − erf

⎛

⎝T 2σ_sri²

⎞

⎠

⎤

⎦ +

$ Ω_sriPs

Ω_sriPs+ 2σ²

× exp

−T² Ω_sriPs+ 2σ²

⎡⎣1 + erf

⎛

⎝ T2σ²

2σ²Ω_sriPs

Ω_sriPs+ 2σ²

⎞

⎠

⎤

⎦

⎫⎬

⎭ (3.20)

Performance Analysis of Piecewise-and-Forward Relay Network on Rayleigh Fading Channel