- Research
- Open Access
Performance comparison of cooperative relay links with different relay processing strategies: Nakagami/Gamma approximation approaches
- Jia Shi^{1}Email author,
- Chen Dong^{1} and
- Lie-Liang Yang^{1}
https://doi.org/10.1186/1687-1499-2014-53
© Shi et al.; licensee Springer. 2014
- Received: 14 November 2013
- Accepted: 13 March 2014
- Published: 4 April 2014
Abstract
In this article, we investigate and compare the error performance of two-hop communication links (THCLs) with multiple relays, when distributed and cooperative relay processing schemes are respectively employed. Our main objectives include finding some general and relatively simple ways for estimating error performance and demonstrating the trade-off of using cooperative relay processing. One distributed relay processing and two cooperative relay processing schemes are compared. In the two cooperative relay processing schemes, one assumes the ideal relay cooperation, in which relays exchange information without consuming energy, while the other one assumes energy consumption for relay cooperation. In this paper, the error performance of the THCLs employing the considered relay processing schemes is investigated, when the channels from source to relays, the channels for information exchange and that from relays to destination experience various types of fading modeled by the Nakagami-m distributions. In order to derive the formulas for the bit error rate (BER) of the THCLs employing binary phase-shift keying (BPSK) modulation and various relay processing schemes, we introduce the Nakagami and Gamma approximation for finding the distribution functions of various variables encountered. Our studies show that the proposed approximation approaches are highly effective, which are capable of accurately predicting the BER of the THCLs supported by the different relay processing schemes.
Keywords
- Cooperative communications
- Relay communications
- Distributed processing
- Transmitter preprocessing
- Maximal-ratio combining
- Equal gain combining
- Nakagami fading
1 Introduction
It has been widely recognized that cooperative communications will play important roles in the future generations of wireless communication systems [1–4]. One type of cooperative communication systems is the relay-assisted, where distributed mobile nodes, often referred to as relays, are employed for attaining cooperative diversity, in order to enhance the reliability of wireless communications [5, 6]. The relay-assisted wireless communication systems have been investigated in the context of various relay protocols, which include amplify-and-forward (AF), decode-and-forward (DF), compress-and-forward (CF) protocols, etc. [5–7].
Along with the relay-assisted wireless communications, a lot of researches have addressed the bit error rate (BER) or symbol error rate (SER) analysis, when assuming communications over, such as, Rayleigh fading, Rician fading and Nakagami-m fading channels [8–15]. In the analyses, various cooperative relay scenarios have been considered, which include the classic three-node relay network [13, 14], serial or parallel multihop cooperative relay networks [9, 12, 15], etc. A lot of exact or approximate closed-form formulas have been derived for evaluating the BER/SER of considered scenarios. In [10], the exact average SER formulas have been obtained for the cooperative network, where a source sends messages to a destination with the aid of multiple AF relays, when assuming communications over flat Rayleigh fading channels. In [11], the SER analysis has been done in the context of the multihop cooperative relay networks over various types of fading channels, when both the number of relays and the number of hops may take arbitrary values.
In addition to BER/SER, the outage probability of cooperative wireless systems has been investigated, for example, in [16–19]. In more detail, the lower and asymptotic bounds of outage probability have been derived in [16] for dual-hop relay networks experiencing Rayleigh fading. By contrast, in [17, 19], the outage probability has been derived, when assuming communication over independent and non-identically distributed Nakagami-m fading channels. Furthermore, considering capacity/throughput, in [20–22], the authors have studied the capacity bound and rate region of two-hop relay networks. In [21], a traditional three-node relay system has been considered, and the upper and lower bounds of ergodic capacity have been derived, when various relay schemes are employed. In [22], the capacity bounds have been analyzed in the context of multinode ad hoc networks.
In the published references, such as [13, 18, 23, 24], on the relay communications employing cooperative relays, a typical assumption used is ideal cooperation among relays. Under this assumption, there is no energy consumption for the information exchange required by cooperation and, furthermore, other overheads required are also often ignored.
Against this background, in [25], we studied and compared various relay processing schemes in association with the two-hop communication links (THCLs), where multiple relays assisted one source to communicate with a destination. We demonstrated that the studies under ideal assumptions often result in misleading observations, when practical scenarios were considered. Continuing the work in [25], in this paper, we investigate further three types of relay processing schemes. The first one is the distributed relay processing (DRP), in which relays do not cooperate with each other, whereas process their signals independently in a distributed way to achieve the transmitter equal-gain combining (TEGC). The second scheme is the ideal centralized maximal ratio combining (MRC) and TEGC relay processing, which is termed as ICRP. In this scheme, signals received by the relays from the source are ideally forwarded to a so-called information exchange central unit (IECU) without consuming any energy. After the MRC-assisted detection at the IECU, the IECU broadcasts the detected data back to the relays also without consuming any energy and without error. Finally, the third scheme is the centralized MRC and TEGC relay processing, which is referred to as CRP and has the same structure as the ICRP, except the assumptions for information exchange. Specifically, after the relays receive the signals from the source, they convey the information to the IECU based on the principles of direct-sequence code-division multiple-access (DS-CDMA). After the IECU detects the signals, it broadcasts the information back to the relays also over non-ideal channels. The system needs to allocate energy for implementation of both the above processes.
In this paper, assuming binary phase-shift keying (BPSK) baseband modulation, we analyze the BER of the THCLs employing the above-stated various relay processing schemes, when assuming that the source-to-relay (S-R) channels, the multiple-access (MA) channels from relays to IECU, the broadcast (BC) channels from IECU to relays, and the relay-to-destination (R-D) channels experience independent flat Nakagami-m fading. As the exact BER analyses for the considered scenarios are extremely hard - if they are not impossible, we propose some general and accurate approximation approaches for reaching our objectives. Specifically, the proposed approximation approaches include the Nakagami theoretical approximation (Nakagami-TAp), the Nakagami statistical approximation (Nakagami-SAp), and the Gamma approximation (Gamma-Ap). The principles of these approaches as well as their applications will become explicit in our forthcoming discourses. Finally, the error performance of the various relay processing schemes are demonstrated and compared based on the results obtained by both simulations and evaluation of the formulas analytically obtained. The results show that the error performance predicted from the formulas derived based on the approximation approaches agree well with that obtained by simulations for the various scenarios addressed.
The rest of the paper is organized as follows: Section 2 states the system model. Section 3 details the relay processing schemes. In Section 4, we analyze the average BER of the THCLs employing various relay processing schemes. Section 5 demonstrates the BER performance of the THCL systems. Finally, in section 6, we summarize our main findings.
2 System model
We assume that each of the communication terminals, including the source, destination, relays, and IECU, is equipped with one antenna for signal receiving and transmission. The source and destination are separated by a long distance and unable to communicate directly. Hence, information is transmitted from source to destination in two hops under the support of relays. We assume that the L relays from a cluster and are close to each other. When the relays are operated in cooperation mode, we assume that the IECU seats in the middle of the L relays and has small and similar distances from all the L relays. We also assume that the relays do not communicate with each other, instead, they receive signals from the source, share their information with the aid of the IECU and independently process and transmit their signals to the destination. By contrast, the IECU is assumed to communicate only with the relays, it does not receive signals from the source or transmit signals to the destination. Note that the IECU may be viewed as a signal processing unit, which implements multi-way relay [26, 27] to aid information exchange among relays. As shown in Figure 1, the relays forward their signals to the IECU based on the principles of DS-CDMA and, then, the IECU broadcasts the processed signal back to the relays. In this paper, we assume that all communication terminals are operated in half-duplex mode. We assume that a relay employs the channel state information (CSI) of both the S-R and R-D channels related to this relay. The IECU has the CSI required for carrying out MRC. Furthermore, when the CRP is employed, a relay is also assumed to have the CSI of the IECU’s BC channel to this relay.
where ${\mathit{y}}_{r}={[{y}_{{r}_{1}},{y}_{{r}_{2}},\dots ,{y}_{{r}_{L}}]}^{T}$ and ${y}_{{r}_{i}}$ represents the observation of the i th relay, ${\mathit{h}}_{\text{sr}}={[{h}_{{\text{sr}}_{1}},{h}_{{\text{sr}}_{2}},\dots ,{h}_{{\text{sr}}_{L}}]}^{T}$ contains the channel gains of the L S-R channels, α_{1} depends on the relative power allocated to the first hop, and ${\mathit{n}}_{r}={[{n}_{{r}_{1}},{n}_{{r}_{2}},\dots ,{n}_{{r}_{L}}]}^{T}$ is a length-L additive white Gaussian noise (AWGN) vector, each element of which obeys the complex Gaussian distribution with zero mean and a variance of 2σ^{2}, where ${\sigma}^{2}=1/\left(2{\stackrel{\u0304}{\gamma}}_{s}\right)$ with ${\stackrel{\u0304}{\gamma}}_{s}$ denoting the average sound-to-noise ratio (SNR) per symbol. From (1) we are implied that the average SNR of the first hop is ${\gamma}_{\text{sr}}={\alpha}_{1}{\stackrel{\u0304}{\gamma}}_{s}$ per relay.
Based on (1), the L relays carry out one of the three relay processing schemes, including the DRP, ICRP and the CRP. Note that, when the ICRP and CRP are considered, the relays use AF to send signals to the IECU. For communications between relays and destination, the DRP scheme achieving TEGC is always used to forward information to the destination, no matter which of the three relay processing schemes is employed.
where ${h}_{{r}_{i}d}$ represents the gain of the i th R-D channel, n_{ d } is the Gaussian noise added at the destination, which is distributed with zero mean and a variance of 2σ^{2}, while α_{2} is determined by the relative power allocated to the second hop. Based on (2), we can know that the average SNR of the i th, where i=1,2,…,L, R-D channel is ${\gamma}_{{r}_{i}d}={\alpha}_{2}E\left[\right|{\stackrel{~}{y}}_{{r}_{i}}{|}^{2}]{\stackrel{\u0304}{\gamma}}_{s}$, where ${\stackrel{\u0304}{\gamma}}_{s}$ again denotes the average SNR per symbol.
Note that, for the sake of comparison of the various relay processing schemes, the total transmission power of a symbol is constraint to P=1, regardless of using distributed or cooperative relay processing, and of the number of relays. If distributed relay processing is employed, the power allocated to the first and second hops is α_{1} and α_{2}, respectively, which satisfy α_{1}+α_{2}=1. By contrast, if the THCL system employs cooperative relay processing, a portion of power, which is expressed as α_{ r }, has to be allocated for information exchange among the relays. Furthermore, according to our previous discussion associated with Figure 1, information exchange requires both the MA transmission and BC transmission. Their corresponding power is expressed as α_{ma} and α_{bc}, respectively. Consequently, the relationships of α_{ r }=α_{ma}+α_{bc} and α_{1}+α_{2}+α_{ r }=1 are satisfied.
where the subscripts i,j associated with h_{ i j } are dependent on the specific channel considered. In (3), Ω=E[|h_{ i j }|^{2}] denotes the average power of the channel and m (m≥0.5) is the Nakagami-m fading parameter characterizing the severity of fading, the fading becomes less severe when the value of m increases.
We also consider the special scenarios, where the MA/BC channels only suffer from AWGN, in order to demonstrate that, even in this over-optimistic communication environments, the energy spent for relay cooperation may significantly degrade the achievable performance.
3 Relay processing
In this article, three types of relay processing schemes are investigated, which include (1) DRP: TEGC-assisted distributed relay processing; (2) ICRP: ideal centralized MRC- and TEGC-aided relay processing, which implements ideal relay cooperation; and (3) CRP: centralized MRC- and TEGC-aided relay processing, which requires energy for carrying out relay cooperation. Regardless of which of the above three schemes is employed, we assume that the DRP achieving TEGC is employed by the relays for forwarding information to the destination. For this sake, below, we first discuss the principles of the DRP.
3.1 DRP
During the R-D transmission, we assume that every relay has the CSI of the channel between it and the destination. Since there is no cooperation among relays, the relays can only carry out distributed transmitter preprocessing; every relay can only try to maximize the SNR of the link between it and the destination, which is optimum when considering the individual links. At the receiver, equal-gain combining (EGC) is achieved and, therefore, we have the TEGC-assisted DRP, which is simply referred to as DRP. Explicitly, the TEGC-assisted DRP is optimum in the sense of maximizing the SNR at the destination.
Explicitly, the diversity order achieved by the DRP is L.
where, for convenience of BER analysis, we defined ${h}_{\sum l}=\sum _{i{\in}_{l}\{1,\dots ,L\}}\left|{h}_{{r}_{i}d}\right|$, ${h}_{\sum q}=\sum _{j{\widehat{\in}}_{q}\{1,\dots ,L\}}\left|{h}_{{r}_{j}d}\right|$ and ${h}_{l,q}={h}_{\sum l}-{h}_{\sum q}$.
3.2 ICRP
When the ICRP is employed, we assume that information exchange among the L relays is ideal (error-free) and does not consume energy. This is a typical assumption used in many references, such as in [13, 18, 23, 24], considering cooperative relays. In this case, the total power P=1 is only consumed by the first (S-R) and second (R-D) hops. Therefore, we have α_{1}+α_{2}=1. At the IECU, the symbol transmitted by the source is detected with the aid of MRC. Then, the IECU returns the detected symbol to the relays without consuming energy. Finally, after the TEGC-assisted preprocessing, the relays forward the symbol received from the IECU to the destination.
implying that the IECU is capable of obtaining L-order of diversity for detection of the symbol transmitted by the source.
3.3 CRP
In the ICRP, it is assumed that information exchange among the relays does not consume energy, which is obviously impractical. In this subsection, we consider the CRP scheme, which takes into account of the energy spent for information exchange among the relays and, hence, is a practical relay processing scheme. By comparing the achievable performance of the CRP with that of the ICRP considered in Section 3.2, we will realize that using ideal assumptions for cooperation may greatly overestimate the achievable performance of cooperative wireless systems.
When the CRP scheme is employed, the L relays first transmit the signals received from the source to the IECU in the principles of AF relay. At the IECU, signals received from the L relays are detected in the MRC principles. Then, the IECU broadcasts the detected symbol back to the L relays. Finally, the relays use the DRP scheme to transmit their detected symbols to the destination.
where, by definition, y_{cu} is a length-N vector, when a spreading factor of N is used by the DS-CDMA, ${\mathit{s}}_{r}={\left[{s}_{{r}_{1}},{s}_{{r}_{2}},\dots ,{s}_{{r}_{L}}\right]}^{T}$, H_{ma}=C A_{ma}, where C=[c_{1},c_{2},…,c_{ L }] with ||c_{ i }||^{2}=1 is a (N×L) matrix containing the L spreading sequences assigned to the L relays and A_{ma}=diag{a_{1},a_{2},…,a_{ L }} with a_{ i } representing the fading gain of the i th channel. We assume that |a_{ i }| obeys the Nakagami-m distribution with the PDF expressed as ${f}_{\left|{a}_{i}\right|}\left(r\right)$ in the form of (3) and the parameters ${m}_{{\text{ma}}_{i}}$ and ${\Omega}_{{\text{ma}}_{i}}$. In (10), G_{ r }=diag{g_{1},g_{2},…,g_{ L }}, where ${g}_{i}=1/\sqrt{|{h}_{{\text{sr}}_{i}}{|}^{2}+2{\sigma}^{2}}$ (i=1,2,…,L). Finally, in (10), ${\mathit{n}}_{\text{cu}}={[{n}_{{\text{cu}}_{1}},{n}_{{\text{cu}}_{2}},\dots ,{n}_{{\text{cu}}_{N}}]}^{T}$, the elements of which obey the complex Gaussian distribution with zero mean and a variance of 2σ cu2 with ${\sigma}_{\text{cu}}^{2}=1/\left(2{\stackrel{\u0304}{\gamma}}_{s}{\beta}_{1}\right)$, where β_{1} is relate to the noise variance of the MA channels. Note that, it can be shown that the average SNR of each of the DS-CDMA channels is ${\gamma}_{\text{ma}}={\alpha}_{\text{ma}}{\stackrel{\u0304}{\gamma}}_{s}{\beta}_{1}/L$.
where, for simplicity, we expressed h_{ T }=H_{ma}G_{ r }h_{sr}, which is a length-N vector and can be viewed as the equivalent source to IECU channel matrix. Similarly, in (11), the length-N noise vector ${\mathit{n}}_{T}=\sqrt{\frac{{\alpha}_{\text{ma}}}{L}}{\mathit{H}}_{\text{ma}}{\mathit{G}}_{r}{\mathit{n}}_{r}+{\mathit{n}}_{\text{cu}}$ contains the noise conflicted at both the relays and the IECU.
Based on z_{cu}, the IECU detects the symbol transmitted by the source, which is expressed as $\widehat{x}$.
where ${\widehat{n}}_{{r}_{i}}$ is the Gaussian noise added on the i th BC channel, which has zero mean and a variance of ${\sigma}_{\widehat{r}}^{2}=1/\left(2{\stackrel{\u0304}{\gamma}}_{s}{\beta}_{2}\right)$ per dimension, here β_{2} is related to the noise variance of the BC channels. In (13), the channel is assumed to experience Nakagami-m fading with $\left|{h}_{{\text{bc}}_{i}}\right|$ obeying the Nakagami-m distribution of (3) associated with the parameters ${m}_{{\text{bc}}_{i}}$ and ${\Omega}_{{\text{bc}}_{i}}$. From (13), we can know that the average SNR of a BC channel is ${\gamma}_{\text{bc}}={\alpha}_{\text{bc}}{\stackrel{\u0304}{\gamma}}_{s}{\beta}_{2}$.
Based on $\left\{{\u0177}_{{r}_{i}}\right\}$, the relays can make their decisions about the symbol transmitted by the IECU. Let the symbols detected by the L relays be expressed as $\left\{{x}_{{r}_{i}}\right\}$. They are forwarded respectively by the L relays to the destination, after the TEGC-assisted preprocessing, as detailed in section 3.1. Note again that, since the TEGC-assisted preprocessing invokes L relays for transmitting signals to the destination, a diversity order of L can be achieved by the CRP scheme.
Note that, if we let ${m}_{{\text{ma}}_{i}}\to \infty $ and ${m}_{{\text{bc}}_{i}}\to \infty $ in the PDFs for the MA and BC channels, then, the MA and BC channels are reduced to the AWGN channels [28].
4 Analysis of bit error rate
In this section, we analyze the BER of the THCL system employing the relay processing schemes considered in section 3, when BPSK baseband modulation is assumed. Our analysis is based on the assumptions that the S-R channels, the MA channels and BC channels for information exchange, and the R-D channels experience independent fading. Specifically, the S-R channels experience the independent and identically distributed (iid) Nakagami-m fading, the same occurs with the MA/BC channels and the R-D channels. However, the fading parameters characterizing the S-R channels, MA/BC channels and the R-D channels may be different. Additionally, we consider the special cases, where the S-R and R-D channels experience Nakagami-m fading, while the MA/BC channels are AWGN channels, in order to demonstrate that the energy spent for relay cooperation cannot be ignored even in this over-optimistic communication environments.
where γ_{ c } represents the average SNR and _{2}F_{1}(a,b;c;z) is the hypergeometric function defined as [31]${}_{2}{F}_{1}(a,b;c;z)=\sum _{k=0}^{\infty}\frac{{\left(a\right)}_{k}{\left(b\right)}_{k}{z}^{k}}{{\left(c\right)}_{k}k!}$ with (a)_{ k }=a(a+1)…(a+k−1) and (a)_{0}=1.
Note that, there are a range of special forms for (14), which can be found in [28–30].
4.1 Bit error rate of DRP
where ${P}_{b}^{\left(\text{S-R}\right)}$ is the average BER of S-R channels and ${P}_{b}^{\left(\text{TEGC}\right)}\left(l\right)$ is the average BER of the destination’s detection on the condition that l out of L relays send the destination correct bits, while the other q=L−l relays send the destination erroneous bits.
where P_{ b }(m,γ_{ c }) is given by (14).
Considering the transmission from the L relays to the destination, if there are l relays sending the destination correct bits and q=L−l relays sending the destination erroneous bits, the decision variable formed by the destination is given by (6), i.e. ${y}_{d}^{(l,q)}=\sqrt{\frac{{\alpha}_{2}}{L}}{h}_{l,q}x+{n}_{d}$ where ${h}_{l,q}={h}_{\sum l}-{h}_{\sum q}$ with ${h}_{\sum l}=\sum _{i{\in}_{l}\{1,\dots ,L\}}\left|{h}_{{r}_{i}d}\right|$ and ${h}_{\sum q}=\sum _{j{\widehat{\in}}_{q}\{1,\dots ,L\}}\left|{h}_{{r}_{j}d}\right|$. Hence, in order to derive the BER expression, we need to derive the PDF of h_{l,q}. Below, we derive this PDF by first introducing the Nakagami approximation for the PDFs of ${h}_{\sum l}$ and ${h}_{\sum q}$. Two types of Nakagami approximation approaches are proposed, which are the modified Nakagami theoretical approximation (Nakagami-TAp) and the Nakagami statistical approximation (Nakagami-SAp).
However, as the results in [32] show, the above approximation may be very inaccurate.
Parameters κ for PDF of ${h}_{\sum l}$ obtained by Nakagami-TAp, where component distributions have parameter Ω _{ 0 } =1
l | m _{ 0 } | |||
---|---|---|---|---|
1.0 | 1.5 | 3.0 | 4.0 | |
2 | 0.842 | 0.879 | 0.914 | 0.924 |
3 | 0.825 | 0.867 | 0.914 | 0.929 |
4 | 0.812 | 0.862 | 0.915 | 0.930 |
5 | 0.807 | 0.859 | 0.918 | 0.931 |
6 | 0.803 | 0.857 | 0.919 | 0.933 |
When the Nakagami-SAp is employed, we approximate ${h}_{\sum l}$ as a Nakagami-m distributed random variable with its PDF ${f}_{{h}_{\sum l}}\left(y\right|{m}_{l},{\Omega}_{l})$ expressed in the form of (3), whose parameters m_{ l } and Ω_{ l } are obtained by simulations. Note that, the Nakagami-m PDF is not very sensitive to the values of m and Ω, especially, when these values are relatively large. For example, the PDFs of ${f}_{\left|{h}_{\mathit{\text{ij}}}\right|}\left(y\right|m,\Omega )$ do not have any noticeable differences, when Ω±0.01Ω and m±0.01m are applied. Hence, it is usually sufficient for us to derive m and Ω based on about 10^{3} to 10^{4} realizations of h_{ i j }. Hence, the time spent for using the Nakagami-SAp to obtain BER results can be significantly less than that required by using direct simulations. When using direct simulations, we know that at least 10^{7} (independent) realizations are required for a BER of about 10^{−5}, in order to generate sufficient accuracy.
Parameters (m _{ L } , Ω _{ L } ) for PDF of ${h}_{\sum L}=\sum _{l=1}^{L}\left|{h}_{l}\right|$ obtained by Nakagami-SAp
L | Ω _{ 0 } | m _{ 0 } | |||||
---|---|---|---|---|---|---|---|
0.5 | 1 | 1.5 | 2 | 2.5 | 3 | ||
2 | 0.125 | 0.929, 0.424 | 1.908, 0.443 | 2.929, 0.46 | 3,948, 0.469 | 4.918, 0.474 | 5.869, 0.477 |
0.25 | 0.929, 0.808 | 1.908, 0.886 | 2.929, 0.919 | 3,948, 0.938 | 4.918, 0.948 | 5.869, 0.958 | |
0.5 | 0.929, 1.617 | 1.908, 1.771 | 2.929, 1.838 | 3,948, 1.875 | 4.918, 1.895 | 5.869, 1.904 | |
1 | 0.929, 3.234 | 1.908, 3.542 | 2.929, 3.676 | 3,948, 3.751 | 4.918, 3.791 | 5.869, 3.819 | |
3 | 0.125 | 1.36, 0.843 | 2.851, 0.957 | 4.314, 1.007 | 5.872, 1.038 | 7.303, 1.052 | 8.809, 1.061 |
0.25 | 1.36, 1.687 | 2.851, 1.913 | 4.314, 2.015 | 5.872, 2.077 | 7.303, 2.105 | 8.809, 2.122 | |
0.5 | 1.36, 3.373 | 2.851, 3.826 | 4.314, 4.029 | 5.872, 4.154 | 7.303, 4.21 | 8.809, 4.244 | |
1 | 1.36, 6.746 | 2.851, 7.652 | 4.314, 8.059 | 5.872, 8.307 | 7.303, 8.419 | 8.809, 8.487 | |
4 | 0.125 | 1.776, 1.443 | 3.758, 1.67 | 5.731, 1.77 | 7.737, 1.828 | 9.693, 1.858 | 11.842, 1.871 |
0.25 | 1.776, 2.885 | 3.758, 3.34 | 5.731, 3.541 | 7.737, 3.656 | 9.693, 3.715 | 11.842, 3.742 | |
0.5 | 1.776, 5.77 | 3.758, 6.68 | 5.731, 7.082 | 7.737, 7.312 | 9.693, 7.431 | 11.842, 7.485 | |
1 | 1.776, 11.54 | 3.758, 13.36 | 5.731, 14.163 | 7.737, 14.624 | 9.693, 14.861 | 11.842, 14.97 | |
5 | 0.125 | 2.226, 2.207 | 4.656, 2.58 | 7.082, 2.746 | 9.696, 2.835 | 12.12, 2.891 | 14.66, 2.916 |
0.25 | 2.226, 4.414 | 4.656, 5.161 | 7.082, 5.491 | 9.696, 5.67 | 12.12, 5.782 | 14.66, 5.831 | |
0.5 | 2.226, 8.829 | 4.656, 10.322 | 7.082, 10.982 | 9.696, 11.34 | 12.12, 11.564 | 14.66, 11.662 | |
1 | 2.226, 17.658 | 4.656, 20.645 | 7.082, 21.964 | 9.696, 22.68 | 12.12, 23.128 | 14.66, 23.324 | |
6 | 0.125 | 2.647, 3.13 | 5.636, 3.688 | 8.538, 3.932 | 11.6, 4.067 | 14.511, 4.149 | 17.685, 4.189 |
0.25 | 2.647, 6.26 | 5.636, 7.376 | 8.538, 7.864 | 11.6, 8.314 | 14.511, 8.299 | 17.685, 8.378 | |
0.5 | 2.647, 12.519 | 5.636, 14.752 | 8.538, 15.728 | 11.6, 16.268 | 14.511, 16.597 | 17.685, 16.756 | |
1 | 2.647, 25.038 | 5.636, 29.505 | 8.538, 31.456 | 11.6, 32.536 | 14.511, 33.195 | 17.685, 33.512 |
where Γ(a,x) represents the incomplete gamma function given by (40).
where P_{ b }(m,γ_{ c }) is given by (14).
Finally, the average BER of the THCL system employing the DRP can be evaluated from (15) with the aid of (23) and (24).
4.2 Bit error rate of ICRP
In (25), the first (second) term at the right-hand side denotes the probability that the detection at the IECU is incorrect (correct) while the detection at the destination is correct (incorrect).
First, in (25), ${P}_{b}^{\left(\text{TEGC}\right)}$ is the average BER of detection at the destination, which has been analyzed in section 4.1. In the ICRP scenario, all the L bits to be transmitted by the relays are the same. Hence, we have ${P}_{b}^{\left(\text{TEGC}\right)}={P}_{b}^{\left(\text{TEGC}\right)}\left(L\right)$, which is given by (24).
where $\text{\xb5}=\sqrt{{\alpha}_{1}{\Omega}_{\text{sr}}{\stackrel{\u0304}{\gamma}}_{s}/({\alpha}_{1}{\Omega}_{\text{sr}}{\stackrel{\u0304}{\gamma}}_{s}+{m}_{\text{sr}})}$ by definition.
4.3 Bit error rate of CRP
when the BC channels are assumed to experience independent Nakagami-m fading with the parameters m_{bc} and Ω_{bc}. When the BC channels are AWGN channels, we simply have ${P}_{b}^{\left(\text{BC}\right)}=Q\left(\sqrt{2{\alpha}_{\text{bc}}{\beta}_{2}{\stackrel{\u0304}{\gamma}}_{s}/L}\right)$.
where we defined ${\mathit{h}}_{D}^{T}=[{h}_{{D}_{1}},{h}_{{D}_{2}},\dots ,{h}_{{D}_{L}}]={\mathit{h}}_{T}^{H}{\mathit{H}}_{\text{ma}}{\mathit{G}}_{r}$. However, it is extremely hard to derive the exact PDF of γ_{cu} from (31), due to the forwarded noise by the relays, as seen in (11). Consequently, we are incapable of deriving the exact average BER of ${P}_{b}^{\left(\text{IECU}\right)}$. In this paper, we propose the Gamma approximation (Gamma-Ap) for obtaining the PDF of γ_{cu}.
Note that, in performance analysis, the Gaussian approximation (Gaussian-Ap) is typically employed. However, for some scenarios, such as for the PDF of (31), where the concerned variables are always positive, the Gamma-Ap has the advantages over the Gaussian-Ap. First, the Gamma distribution [29], which can be obtained by the squares of Nakagami-m distributed variables, is defined in [0,∞), while the Gaussian distribution is defined in (−∞,∞). Second, for applying the Gaussian-Ap, usually a high number of component variables is required, so that their sum yields a symmetric distribution. By contrast, the Gamma-Ap does not impose this constraint, and can be applied for the sum of any number of component variables. Furthermore, as the number of components increases, the resultant Gamma distribution appears the Gaussian-like shape, but, in the range [0,∞). Hence, the Gamma-Ap (also including the Nakagami approximation, as they belong to the same family) represents a versatile approximation approach, which may find applications for a lot of problems in practice, including a lot of performance analysis problems in wireless communications.
where Ω_{cu}=E[ζ_{cu}], and ${m}_{\text{cu}}={\Omega}_{\text{cu}}^{2}/E\left[{({\zeta}_{\text{cu}}-{\Omega}_{\text{cu}})}^{2}\right]$.
Parameters for the Gamma PDF ${f}_{{\zeta}_{\text{cu}}}\left(\zeta \right)$ obtained based on the Gamma-Ap
L | ${\stackrel{\u0304}{\gamma}}_{s}$(dB) | m _{ sr } | m _{ ma } | m _{ cu } | Ω _{ cu } |
---|---|---|---|---|---|
4 | 1 | 1.0 | 1.0 | 2.573 | 18.121 |
4 | 2.0 | 3.0 | 7.347 | 27.007 | |
9 | 1.0 | 4.0 | 5.332 | 29.775 | |
14 | 3.0 | 1.5 | 7.730 | 33.445 | |
20 | 2.5 | 1.5 | 6.401 | 33.150 | |
6 | 3 | 1.5 | 1.0 | 4.872 | 37.363 |
7 | 1.0 | 1.0 | 4.506 | 43.158 | |
13 | 3.0 | 3.0 | 13.320 | 64.612 | |
18 | 1.0 | 4.0 | 5.189 | 49.028 | |
22 | 1.5 | 3.0 | 6.199 | 55.084 |
Furthermore, the average of the THCL system employing the CRP scheme can be evaluated from (29) associated with (22) (or (23) and (24)), (30) and (35).
5 Performance results
In this section, we demonstrate a range of performance results for characterizing the achievable performance of THCL systems with the various relay processing schemes considered. Both numerical results evaluated from the formulas derived in the previous sections and simulation results are provided. Note that, for obtaining the results, we assume that all channels of the first and second hops experience independent fading. The MA/BC channels are either Gaussian channels for illustrating the best cases or iid fading channels. When the CRP is employed, we assume that the parameters β_{1} and β_{2} take a value of 10, which results in that the average SNR of the MA/BC channels is typically 10 dB higher than that of the S-R and R-D channels. Furthermore, for the DS-CDMA used for the MA transmission, the spreading codes are assumed random sequences with a spreading factor N=16.
Additionally, as shown in Figures 2 and 3, the BER performance improves as the fading becomes less severe. It also improves as the number of relays increases, owing to the increased spatial diversity.
Additionally, we may compare the best BER achieved in Figure 7, which assumed AWGN MA/BC channels, with Figure 4. In Figure 4, the curve corresponding to the parameters of m=1,L=6 shows that the BER at 11 dB is well below 10^{−5}. This BER is much lower than the best BER of 1.47×10^{−4} shown in Figure 7. From this comparison, we are implied that the BER predicted by applying ideal assumptions is far overoptimistic.
6 Conclusions
In this paper, we have studied the BER performance of THCL systems with various relay processing schemes, including the DRP, ICRP, and CRP. As the BER of the TEGC-assisted detection at the destination is hard to analyze, two approximation approaches have been proposed, which are the Nakagami-TAp and Nakagami-SAp. Our performance studies show that both the approximation approaches can be confidently used for predicting the BER of the schemes generating EGC-type decision variables, although the Nakagami-SAp may yield more accurate results than the Nakagami-TAp. For evaluation of the BER at the IECU, the Gamma-Ap is proposed for finding the distribution of the related SINR. From our studies, we can realize that these approximation approaches, especially, the Gamma-Ap, are highly general, which may find applications in performance analysis of a wide range of communication systems. Finally, the BER performance of the various relay processing schemes has been demonstrated and compared. From the performance results, we can conclude that, without considering the cost for relay cooperation, the ICRP always outperforms the DRP. However, in the practical scenarios where relay cooperation consumes energy, bandwidth, and implementation complexity, we find that the DRP often outperforms the CRP even in terms of the BER performance. Therefore, in relay communications, the performance predicted under the assumption of ideal relay cooperation may be too optimistic and, hence, unachievable in practice. Owing to its low complexity for implementation, in practice, the DRP constitutes a highly desirable relay processing scheme, especially, in the communication environments where a few, such as L≥4, of relays are available.
Appendix
Note that, when m_{ l } and m_{ q } are relatively big, we may approximate 2m_{ l }−1 and 2m_{ q }−1 (sometimes even m_{ l } and m_{ q }) to their nearest integers and still generate sufficient accuracy, as evidenced by the results shown in [12]. This is because the Nakagami-m PDFs are not very sensitive to the value of m, when it is relatively big.
Declarations
Acknowledgements
This work was presented in part at the IEEE VTC2012 Spring, 6-9 May 2012, Yokohama, Japan.
Authors’ Affiliations
References
- Berger S, Kuhn M, Wittneben A, Unger T, Klein A: Recent advances in amplify-and-forward two-hop relaying. IEEE Commun. Mag 2009, 47(7):50-56.View ArticleGoogle Scholar
- Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inform. Theory 2004, 50(12):3062-3080. 10.1109/TIT.2004.838089MathSciNetView ArticleGoogle Scholar
- Sendonaris A, Erkip E, Aazhang B: User cooperation diversity. part I. system description. IEEE Trans. Commun 2003, 51(11):1927-1938. 10.1109/TCOMM.2003.818096View ArticleGoogle Scholar
- Sendonaris A, Erkip E, Aazhang B: User cooperation diversity. part II. Implementation aspects and performance analysis. IEEE Trans. Commun 2003, 51(11):1939-1948. 10.1109/TCOMM.2003.819238View ArticleGoogle Scholar
- Bastami AH, Olfat A: Optimal SNR-based selection relaying scheme in multi-relay cooperative networks with distributed space-time coding. IET Commun 2010, 4(6):619-630. 10.1049/iet-com.2009.0441MathSciNetView ArticleGoogle Scholar
- Anghel P, Kaveh M: Exact symbol error probability of a cooperative network in a Rayleigh-fading environment. IEEE Trans. Wireless Commun 2004, 3(5):1416-1421. 10.1109/TWC.2004.833431View ArticleGoogle Scholar
- Liu KJR, Sadek AK, Su WF, Kwasinski A: Cooperative Communications and Networking. New York: Cambridge University Press; 2009.Google Scholar
- Amin O, Ikki S, Uysal M: On the performance analysis of multirelay cooperative diversity systems with channel estimation errors. IEEE Trans. Vehicular Technol 2011, 60(5):2050-2059.View ArticleGoogle Scholar
- Dong C, Yang L-L, Hanzo L: Performance analysis of multihop-diversityaided multihop links. IEEE Trans Vehicular Technol 2012, 61(6):2504-2516.View ArticleGoogle Scholar
- Anghel P, Kaveh M: Exact symbol error probability of a cooperative network in a Rayleigh-fading environment. IEEE Trans Wireless Commun 2004, 3(5):1416-1421. 10.1109/TWC.2004.833431View ArticleGoogle Scholar
- Ribeiro A, Cai X, Giannakis G: Symbol error probabilities for general cooperative links. IEEE Trans. Wireless Commun 2005, 4(3):1264-1273.View ArticleGoogle Scholar
- Yang L-L, Chen H-H: Error probability of digital communications using relay diversity over Nakagami-m fading channels. IEEE Trans. Wireless Commun 2008, 7(5):1806-1811.View ArticleGoogle Scholar
- Barua B, Ngo H, Shin H: On the SEP of cooperative diversity with opportunistic relaying. IEEE Commun. Lett 2008, 12(10):727-729.View ArticleGoogle Scholar
- Himsoon T, Siriwongpairat W, Su W, Liu K: Differential modulations for multinode cooperative communications. IEEE Trans. Signal Process 2008, 56(7):2941-2956.MathSciNetView ArticleGoogle Scholar
- Boyer J, Falconer D, Yanikomeroglu H: Multihop diversity in wireless relaying channels. IEEE Trans. Commun 2004, 52(10):1820-1830. 10.1109/TCOMM.2004.836447View ArticleGoogle Scholar
- Zhao Y, Adve R, Lim TJ: Outage probability at arbitrary SNR with cooperative diversity. IEEE Commun. Lett 2005, 9(8):700-702. 10.1109/LCOMM.2005.1496587View ArticleGoogle Scholar
- Suraweera HA, Smith PJ, Armstrong J: Outage probability of cooperative relay networks in Nakagami-m fading channels. IEEE Commun. Lett 2006, 10(12):834-836.View ArticleGoogle Scholar
- Bletsas A, Shin H, Win M: Outage optimality of opportunistic amplify-and-forward relaying. IEEE Commun. Lett 2007, 11(3):261-263.View ArticleGoogle Scholar
- Sagias N, Lazarakis F, Tombras G, Datsikas C: Outage analysis of decode-and-forward relaying over Nakagami- m fading channels. IEEE Signal Process. Lett 2008, 15: 41-44.View ArticleGoogle Scholar
- Kramer G, Gastpar M, Gupta P: Cooperative strategies and capacity theorems for relay networks. IEEE Trans. Inform Theory 2005, 51(9):3037-3063. 10.1109/TIT.2005.853304MathSciNetView ArticleGoogle Scholar
- Host-Madsen A, Zhang J: Capacity bounds and power allocation for wireless relay channels. IEEE Trans. Inform. Theory 2005, 51(6):2020-2040. 10.1109/TIT.2005.847703MathSciNetView ArticleGoogle Scholar
- Host-Madsen A: Capacity bounds for cooperative diversity. IEEE Trans. Inform. Theory 2006, 52(4):1522-1544.MathSciNetView ArticleGoogle Scholar
- Chen Y, Kishore S, Li J: Wireless diversity through network coding. IEEE Wireless Commun. Network. Conf. (WCNC 2006) 2006, 3: 1681-1686.View ArticleGoogle Scholar
- Laneman J, Wornell GW: Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks. IEEE Trans. Inform. Theory 2003, 49(10):2415-2425. 10.1109/TIT.2003.817829MathSciNetView ArticleGoogle Scholar
- Shi J, Yang L-L: Performance of two-hop communication links employing various relay processing schemes. In IEEE VTC 2012-Spring 6–9 May. Piscataway: IEEE; 2012.Google Scholar
- Gunduz D, Yener A, Goldsmith A, Poor H: The multi-way relay channel. In IEEE International Symposium on Information Theory (ISIT 2009). Piscataway: IEEE; 2009:339-343.View ArticleGoogle Scholar
- Shi J, Yang L-L: Performance of multiway relay DS-CDMA systems over Nakagami-m fading channels. In IEEE 73rd Vehicular Technology Conference (VTC 2011-Spring). Piscataway: IEEE; 2011:1-5.Google Scholar
- Yang L-L, Hanzo L: Performance of generalized multicarrier DS-CDMA, over Nakagami-m fading channels. IEEE Trans. Commun 2002, 50(6):956-966. 10.1109/TCOMM.2002.1010615View ArticleGoogle Scholar
- Simon MK, Alouini M-S: Digital Communication Over Fading Channels: A Unified Approach to Performance Analysis. New York: Wiley; 2000.View ArticleGoogle Scholar
- Yang L-L: Multicarrier Communications. Chichester: Wiley; 2009.Google Scholar
- Gradshteyn IS, Ryzhik IM: Table of Integrals, Series, and Products. New York: Academic Press; 1980.Google Scholar
- Nakagami M: The m-distribution, a general formula of intensity of rapid fading. In Statistical Methods in Radio Wave Propagation: Proceeding of a Symposium, June. Oxford: Pergamon Press; 1958.Google Scholar
- Proakis JG: Digital Communications, 3rd ed. New York: McGraw-Hill; 1995.Google Scholar
- Simon MK, Alouini MS: A unified approach to the probability of error for noncoherent and differentially coherent modulations over generalized fading channels. IEEE Trans. Commun 1998, 46(12):1625-1638. 10.1109/26.737401View ArticleGoogle Scholar
- Rotar V: Probability and Stochastic Modeling. Boca Raton: Chapman and Hall/CRC; 2012.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.