 Research
 Open Access
 Published:
On performance analysis for optimum combining of DF relaying with fastfading multiple correlated CCIs, correlated sourcerelay, and thermal noise
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 96 (2019)
Abstract
This paper analyzes the outage probability (OP) and the average symbol error rate (SER) of decodeandforward (DF) relaying. The paper derives closedform expressions for the OP and the average SER with optimum combining (OC) considering fastfading multiple correlated CCIs, the correlated sourcerelay, and thermal noise. It is shown that the performance of the large distance between the source and the relay is better than that of the small distance, regardless of interference fading speed at the destination. We also show that given the sourcerelay distance, the performance of slowfading interference is basically better than that of fast fading, except in the low signaltonoiseratio (SNR) regime for the distance being small. In result, the sourcerelay distance is generally a more dominating factor for the performance than fading CCIs.
Introduction
Cooperative communications have been prominent because of diversity gain [1]. In cooperative networks, there are mainly two methods, such as the amplifyandforward (AF) relay network and the decodeandforward (DF) relay network [2]. Then with multiple copies, the destination can achieve cooperative diversity. In order to do so, we can use maximalratio combining (MRC) [3] (p., 316) or optimum combining (OC) [4]. MRC maximizes the signaltonoiseratio (SNR), while OC maximizes the signaltointerferenceplusnoise ratio (SINR). When cochannel interferers (CCIs) are present at the destination, OC reduces CCIs’ power and increases diversity [5]. Since the analytical expressions for the outage probability (OP) and the average symbol error rate (SER) are complex for derivation, some simplified models have been used [6, 7]. Usually, thermal noise is ignored for the tractability of analytical expressions, assuming CCIs being dense, and the system models are simplified. In this case, when the effect of thermal noise is greater than that of CCIs, the analysis of the simplified model might be incorrect [8]. Therefore thermal noise is considered, but fading is still assumed to be slow so that fading in phase 1 and 2 is unchanged and constant. In this case, the analysis is limited for slow fading with MRC [8] or with OC [9]. In addition, it has been assumed that the source and the relay are always far enough to be uncorrelated, which is not always true. Sometimes they become so close that the correlation between them occurs, with which the performance degrades to some extent.
Recently, there have been many research advances in the DF relay network: the opportunistic relaying (OR) in the presence of CCIs is investigated in [10]; a new transmission scheme for selective DF relaying networks is presented, considering the employment of different modulation levels at the transmitting nodes [11]; and a joint scheme (JS) has been proposed for a multiplerelay multipleinput multipleoutput (MIMO) network with a DF relaying strategy [12]. In [13], a novel distributed spacetime coding (DSTC) transmission scheme for a twopath successive DF relay network is proposed. The average SER is analyzed for a wirelesspowered threenode DF relaying system in Nakagamim fading environment [14].
In this paper, a DF protocol is considered. It is assumed that at the relay symbolbysymbol decoding is executed, and at the destination, full decoding is carried out [2]. We also assume that multiple correlated CCIs are fast faded, the source and relay are correlated, and thermal noise is present. To the best of our knowledge, the performance analysis for this system has not been reported. First, we derive closedform expressions for the OP and the average SER with OC considering fastfading multiple correlated CCIs, the correlated sourcerelay, and thermal noise. Second, we investigate the effects of the sourcerelay distance and fast/slowfading CCIs on the performance.
The paper is organized as follows: Section 2 defines the system and channel model. In Section 3, the exact analytical expressions are derived for the OP and the average SER. Section 4 presents the analytical and simulation results, which we discuss in detail. The paper is concluded in Section 5.
System and channel model
We define the full cooperative case as relaying with no symbol errors and the noncooperative case as relaying with the symbol error probability being one. Let the probability of symbol errors at the relay be \( {P}_{\mathrm{e}}^{(R)} \). For the full cooperative case, \( {P}_{\mathrm{e}}^{(R)}=0 \), and for the noncooperative case, \( {P}_{\mathrm{e}}^{(R)}=1 \). For \( 0<{P}_{\mathrm{e}}^{(R)}<1 \), we say simply the cooperative case. We assume that the destination knows whether or not the relay sends the symbol with the probability one. We suppose a time division duplex (TDD) mode [1, 8]. The DF protocol is composed of two time slots, t_{1} and t_{2}. One time interval t_{1} is for phase 1 and the other t_{2} is for phase 2. Therefore, a single transmission duration (STD) t_{STD} becomes t_{1} + t_{2}. Note that under fast fading assumption, channel states change and are not constant over t_{STD}. The relay system consists of a source (S), a relay (R), a destination (D), interferers (\( {I}_R^{(j)} \), \( j=1,2,\cdots, {N}_{I_R} \)) at the relay, and interferers (\( {I}_D^{(i)} \), \( i=1,2,\cdots, {N}_{I_D} \)) at the destination. We model thermal noise as circularly symmetric additive white Gaussian noise (AWGN). Each channel is affected by AWGN. The system and channel model is depicted in Fig. 1. (The sourcerelay channel correlation coefficient r_{SR} and the destination interferer channel correlation coefficient \( {r}_{I_D} \) are defined in the following sections.)
Under the above assumptions, for the first time slot t_{1}, i.e., in phase 1, the source transmits its data symbols. The received signal at the destination is expressed by:
where \( {E}_S^{t_1} \), \( {E}_{I_D}^{t_1} \), b_{0}, and b_{i} are the power transmitted by the source over the time slot t_{1}, the power transmitted by each interferer over the time slot t_{1}, the source data symbol with unit average power, and each interferer data symbol with unit average power for \( i=1,\cdots, {N}_{I_D} \), respectively, and \( {N}_{I_D} \) is the number of interferers. Furthermore, the channel propagation parameters \( {g}_0^{\left(S,D,{t}_1\right)} \) and \( {g}_i^{\left({I}_D,D,{t}_1\right)} \), \( i=1,2,\cdots, {N}_{I_D} \), \( \sim \mathbf{\mathcal{CN}}\left(0,{1}^2\right) \) are Rayleigh faded, and thermal noise \( {n}^{\left(S,D\right)}\sim \mathbf{\mathcal{CN}}\left(0,{N}_0\right) \) is complex AWGN, where the notation \( \mathbf{\mathcal{CN}}\left(\mu, \Sigma \right) \) denotes the complex circularly symmetric normal distribution with mean μ and variance Σ. The received signal at the relay is expressed by:
where \( {E}_{I_R}^{t_1} \) is each interferer power over the time slot t_{1} and r_{i}, and \( j=1,\cdots, {N}_{I_R} \) are the interferer data symbols each with unit average power. The channel parameters \( {g}_i^{\left({I}_R,R,{t}_1\right)} \), \( i=1,2,\cdots, {N}_{I_R} \), \( \sim \mathbf{\mathcal{CN}}\left(0,{1}^2\right) \) are Rayleigh faded, and \( {n}^{\left(S,R,{t}_1\right)}\sim \mathbf{\mathcal{CN}}\left(0,{N}_0\right) \) is complex AWGN.
For the second time slot t_{2}, i.e., in phase 2, if the relay correctly decodes the symbol, then it forwards the symbol to the destination. In this case, the signal at the destination is expressed by:
where \( {E}_R^{t_2} \) is the transmitter power and \( {E}_{I_D}^{t_2} \) is each interferer power. The channel parameters \( {g}_0^{\left(R,D,{t}_2\right)} \) and \( {g}_i^{\left(I,D,{t}_2\right)} \), \( i=1,2\cdots, {N}_{I_D,} \), \( \sim \mathbf{\mathcal{CN}}\left(0,{1}^2\right) \) are Rayleigh faded, and \( {n}^{\left(R,D,{t}_2\right)}\sim \mathbf{\mathcal{CN}}\left(0,{N}_0\right) \) is complex AWGN.
Thus, assuming \( {E}_S={E}_S^{t_1}={E}_R^{t_2} \) and \( {E}_{I_D}={E}_{I_D}^{t_1}={E}_{I_D}^{t_2} \), for \( {P}_{\mathrm{e}}^{(R)}=0 \), the received signal at the destination is expressed by:
where \( {\mathbf{y}}_{\mathrm{full}\hbox{} \mathrm{co}}\triangleq \left[\begin{array}{c}{y}^{\left(S,D,{t}_1\right)}\\ {}{y}^{\left(R,D,{t}_2\right)}\end{array}\right] \), \( {\boldsymbol{g}}_0\triangleq \left[\begin{array}{c}{g_0}^{\left(S,D,{t}_1\right)}\\ {}{g_0}^{\left(R,D,{t}_2\right)}\end{array}\right] \), \( {\boldsymbol{g}}_i\triangleq \left[\begin{array}{c}{g_i}^{\left({I}_D,D,{t}_1\right)}\\ {}{g_i}^{\left({I}_D,D,{t}_2\right)}\end{array}\right] \), \( i=1,2,\cdots, {N}_{I_D} \) , and \( \mathrm{n}\triangleq \left[\frac{n^{\left(S,D,{t}_1\right)}}{n^{\left(R,D,{t}_2\right)}}\right] \) . Here, we assume g_{0}, g_{i}, \( i=1,2\cdots, {N}_{I_D} \), and n are (2 × 1) zeromean complex symmetric Gaussian random vectors. For \( {P}_{\mathrm{e}}^{(R)}=1 \), the received signal at the destination is expressed by:
OP and SER derivation
We first derive the OP, \( {P}_{\mathrm{out}}^{\left(\mathrm{full}\hbox{} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{full}\hbox{} \mathrm{co}\right)}\right) \), and the average SER, SER_{full − ∞}, for \( {P}_{\mathrm{e}}^{(R)}=0 \) and later derive the OP, \( {P}_{\mathrm{out}}^{\left(\mathrm{non}\hbox{} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{non}\hbox{} \mathrm{co}\right)}\right) \), and the average SER, \( {\mathrm{SER}}_{\mathrm{non}\hbox{} \infty } \), for \( {P}_{\mathrm{e}}^{(R)}=1 \). In order to obtain the decision \( {x}_{\mathrm{full}\hbox{} \mathrm{co}}={\mathrm{w}}_{\mathrm{full}\hbox{} \mathrm{co}}^{\dagger}\;{\mathrm{y}}_{\mathrm{full}\hbox{} \mathrm{co}} \), the weight vector w_{full ‐ co} is expressed by w_{full ‐ co} = R^{−1}g_{0} with the interferenceplusnoise correlation matrix R = N_{0}I_{2} + E_{I}GG^{†} and \( \mathrm{G}=\left[{\boldsymbol{g}}_1{\boldsymbol{g}}_2\cdots {\boldsymbol{g}}_{N_{I_D}}\right] \) [4]. The notation I_{2} is the (2 × 2) identity matrix, and the notation (•)^{†} is the conjugation and transposition. The instantaneous maximum output SINR at the destination is expressed as
The momentgenerating function (MGF) of γ_{full ‐ co} is given by:
where on the fifth line in the above equation, we use the general central quadratic form [15] of the MGF, the notation \( \left\mathbf{\mathcal{A}}\right \) is the determinant of a matrix \( \mathbf{\mathcal{A}} \), the sourcerelay channel parameter (2 × 2) correlation matrix \( \mathbb{E}\left[{\boldsymbol{g}}_0{\boldsymbol{g}}_0^{\dagger}\right] \) is denoted as Σ_{SR}, the power ratio Γ_{0} ≜ E_{S}/N_{0} is the SNR over each time slot, i.e., t_{1} or t_{2}, and the power ratio \( {\Gamma}_1\triangleq {E}_{I_D}/{N}_0 \) is the interferencetonoise ratio (INR) over each time slot, i.e., t_{1} or t_{2}. We express the (2 × 2) Hermitian matrix GG^{†} as the eigenvalue decomposition [16].
where β_{1} and β_{2} with β_{1} ≥ β_{2} are the nonzero ordered real eigenvalues of GG^{†} and U is the (2 × 2) unitary matrix. The MGF \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \) is given by:
where on the fourth line in the above equation, we use the fact that the random variable (RV) θ is uniformly distributed in the interval [−π, π) [17]. The MGF \( {M}_{\gamma_{\mathrm{co}}}(s) \) is simplified by the integration over the RV θ as:
where the notation a is the absolute value of a scalar a. The expectation over Β is obtained using the probability density function (PDF) f_{Β}(Β) [18] as:
where \( {\left[{a}_{i,j}\right]}_{i,j=1}^2 \) is a (2 × 2) matrix with elements a_{i, j}, i, j = 1, 2, and the constant K is given by:
The function Γ(·) is the gamma function. The values α_{1} and α_{2} with α_{1} ≥ α_{2} are the eigenvalues of the destination interferer channel parameter (2 × 2) correlation matrix \( {\boldsymbol{\Sigma}}_{I_D}\triangleq \mathbb{E}\left[{\boldsymbol{g}}_i{{\boldsymbol{g}}_i}^{\dagger}\right] \), \( i=1,2,\cdots, {N}_{I_D} \). Using the (2 × 2) determinant formula, the MGF \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \) is given by:
With some algebraic manipulations, the MGF \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \) is expressed as:
The MGF \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \) is further simplified as:
where
and we use the fact that \( {\boldsymbol{\Sigma}}_{I_D}=\mathbb{E}\left[{\boldsymbol{g}}_i{{\boldsymbol{g}}_i}^{\dagger}\right] \), \( i=1,2,\cdots, {N}_{I_D} \), has unit diagonal elements because the channel parameters \( {g}_i^{\left({I}_D,D,{t}_1\right)} \) and \( {g}_i^{\left({I}_D,D,{t}_2\right)} \), \( i=1,2,\cdots, {N}_{I_D} \), are distributed according to \( \kern0.5em \mathbf{\mathcal{CN}}\left(0,{1}^2\right) \), so that the trace of \( {\boldsymbol{\Sigma}}_{I_D} \) is \( \mathrm{tr}\left({\boldsymbol{\Sigma}}_{I_D}\right)=2={\alpha}_1+{\alpha}_2 \). The result in Eq. (15) is valid for the multiple correlated CCIs with \( {N}_{I_D}\ge 2 \). For the single interferer case with \( {N}_{I_D}=1 \), we obtain the simpler MGF \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \). Let the conditional MGF \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}\mid {\lambda}_1}(s) \) be the MGF \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \) conditioned on λ_{1} [19] and the RV λ_{1} be the random eigenvalue of R (the other eigenvalue of R is the constant N_{0}). (Note that λ_{1} and N_{0} are the eigenvalues of R = N_{0}I_{2} + E_{I}GG^{†}, and β_{1} and β_{2} are the eigenvalues of GG^{†}.) Then the MGF \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \) of γ_{full ‐ co} is derived as:
In the derivation of Eq. (17), we use \( {\lambda}_1\sim {\chi}_1^2 \), where the notation \( {\chi}_1^2 \) denotes a complex chisquared distribution with one complex degree of freedom; \( {f}_{\lambda_1}\left({\lambda}_1\right)=\left({\lambda}_1{N}_0\right){e}^{\left({\lambda}_1{N}_0\right)/{E}_{I_D}}/{E}_{I_D}^2 \), λ_{1} ≥ N_{0}. Now we have derived \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \) for all \( {N}_{I_D}\ge 1 \). From \( {M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(s) \), we obtain the characteristic function (CF) \( {\phi}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(t)={M}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}\left(\sqrt{1}\;t\right) \). The PDF \( {f}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}\left({\gamma}_{\mathrm{full}\hbox{} \mathrm{co}}\right) \) of γ_{full ‐ co} is obtained from \( {\phi}_{\gamma_{\mathrm{full}\hbox{} \mathrm{co}}}(t) \) by the Fourier transform, which is easily calculated using the fast Fourier transform (FFT). Then, \( {\mathtt{SER}}_{\mathrm{full}\hbox{} \mathrm{co}} \) with the coherent binary phase shift keying (BPSK) is calculated as [19]:
where \( Q(x)\triangleq 1/\sqrt{2\pi }{\int}_x^{\infty }{e}^{{y}^2/2} dy \). For \( {N}_{I_D}=1 \), SER_{full ‐ co} can be alternatively calculated by Eq. (10.20) in [19]. For a given threshold \( {\gamma}_{Th}^{\left(\mathrm{full}\hbox{} \mathrm{co}\right)} \), the OP \( {P}_{\mathrm{out}}^{\left(\mathrm{full}\hbox{} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{full}\hbox{} \mathrm{co}\right)}\right) \) is defined and is calculated as:
Next, we derive \( {P}_{\mathrm{out}}^{\left(\mathrm{non}\hbox{} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{non}\hbox{} \mathrm{co}\right)}\right) \) and SER_{non ‐ co} for \( {P}_{\mathrm{e}}^{(R)}=1 \). To obtain the decision \( {x}_{\mathrm{non}\hbox{} \mathrm{co}}={\mathrm{w}}_{\mathrm{non}\hbox{} \mathrm{co}}^{\dagger }{y}^{\left(S,D,{t}_1\right)} \), the weight w_{non ‐ co} is expressed by \( {\mathrm{w}}_{\mathrm{non}\hbox{} \mathrm{co}}={\mathrm{R}}_{\mathrm{non}\hbox{} \mathrm{co}}^{1}{g}_0^{\left(S,D,{t}_1\right)} \) with \( {\mathrm{R}}_{\mathrm{non}\hbox{} \mathrm{co}}={N}_0\kern0.36em +{E}_{\mathrm{I}}{\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}{\left({\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}\right)}^{\dagger } \). The notation \( {\left\langle {a}_i\right\rangle}_{i=1}^N \) is a (1 × N) matrix with elements a_{i}, \( i=1,2,\cdots, {N}_{I_D} \). The maximum instantaneous output SINR at the destination can be expressed as:
The RV \( X\triangleq {\left{g}_0^{\left(S,D,{t}_1\right)}\right}^2 \) is exponentially distributed with the PDF f_{X}(x) ≜ e^{−x}, x ≥ 0. The chisquareddistributed RV \( W\triangleq {\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}{\left({\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}\right)}^{\dagger}\sim {\chi}_{N_{I_D}}^2 \) has the PDF \( {f}_W(w)\triangleq 1/\left({N}_{I_D}1\right)!\kern0.5em \cdot {w}^{N_{I_D}1}{e}^{w} \), w ≥ 0, with \( {N}_{I_D} \) complex degree of freedom. The RV Y ≜ 1/Γ_{0} + Γ_{1}/Γ_{0}W has the PDF
with y ≥ 1/Γ_{0}. Then, the RV γ_{non ‐ co} = X/Y is ratio distributed, and the \( {f}_{\gamma_{\mathrm{non}\hbox{} \mathrm{co}}}\left({\gamma}_{\mathrm{non}\hbox{} \mathrm{co}}\right) \) is derived as:
Similarly as in the \( {P}_{\mathrm{e}}^{(R)}=0 \) case, with \( {f}_{\gamma_{\mathrm{non}\hbox{} \mathrm{co}}}\left({\gamma}_{\mathrm{non}\hbox{} \mathrm{co}}\right) \), we calculate \( {P}_{\mathrm{out}}^{\left(\mathrm{non}\hbox{} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{non}\hbox{} \mathrm{co}\right)}\right) \) and SER_{non ‐ co} for \( {P}_{\mathrm{e}}^{(R)}=1 \).
Now based on the total probability theorem, finally, we obtain a closedform expression for the OP P_{out}(γ_{Th}) at the destination as:
and the average SER at the destination is derived as:
With these exact analytical expressions, we can investigate the effects of the distance between the source and the relay, i.e., Σ_{SR} and fast/slowfading interference at the destination, i.e., \( {\boldsymbol{\Sigma}}_{I_D} \).
Results and discussion
We assume that the signals have the exponential correlation [20]. Thus, with the sourcerelay channel correlation coefficient r_{SR} ∈ [0, 1):
and with the destination interferer channel correlation coefficient \( {r}_{I_D}\in \left[0,1\right) \):
We define the total SNR as \( {\Gamma}_0^{\mathrm{total}}\triangleq {E}_S^{\mathrm{total}}/{N}_0 \) and the total INR \( {\Gamma}_1^{\mathrm{total}}\triangleq {E}_{I_D}^{\mathrm{total}}/{N}_0 \), where \( {E}_S^{\mathrm{total}}={E}_S^{t_1}+{E}_R^{t_2}=2{E}_S \) and \( {E}_{I_D}^{total}=\left(2{N}_{I_D}\right){E}_{I_D} \), where the factor 2 represents two time slots. The correlation coefficients are explained as follows: the smaller the r_{SR} is, the larger the distance between the source node and the relay node is. On the other hand, the smaller the \( {r}_{I_D} \) is, the more independent, i.e., the less correlated, the two channel coefficients \( {g}_i^{\left({I}_D,D,{t}_1\right)} \) and \( {g}_i^{\left({I}_D,D,{t}_2\right)} \) are, for a given i among \( i=1,2,\cdots, {N}_{I_D} \). This means that the maximum Doppler spread is larger so that the coherence time is smaller, i.e., fast fading [21]. Thinking in the opposite direction, i.e., slow fading, is also true.
First, we investigate the effect of the probability of symbol errors \( {P}_{\mathrm{e}}^{(R)} \) at the relay on the OP P_{out}(γ_{Th}) at the destination. We assume that with \( {N}_{I_D}=2 \), \( {\Gamma}_1^{total}=\left(2{N}_{I_D}\right){\Gamma}_1=3\kern0.5em \mathrm{dB}+3\kern0.5em \mathrm{dB}+4\kern0.5em \mathrm{dB}=10\kern0.5em \mathrm{dB} \) is fixed. We also assume that there are almost uncorrelated users (r_{SR} = 0.01) and slowfading multiple correlated CCIs (\( {r}_{I_D}=0.99 \)), which are the assumptions of the previous researches, i.e., independent users and flatfading interference over t_{STD}. In Fig. 2, the OP performance is shown for various \( {P}_{\mathrm{e}}^{(R)} \) values. We observe in Fig. 2 that the OP performance with \( {P}_{\mathrm{e}}^{(R)}\le 0.001 \) reaches that with the full cooperative case \( {P}_{\mathrm{e}}^{(R)}=0 \). Since this paper focuses on the performance analysis for the sourcerelay distance and fast/slow fading CCIs, from now on, we set \( {P}_{\mathrm{e}}^{(R)}=0.001 \). In Fig. 2, we also show the analytical and simulation results, which are in good agreement, so that the following analyses are based on the analytical expressions.
Next, we analyze the OP P_{out}(γ_{Th}) at the destination for various r_{SR} and \( {r}_{I_D} \) values. In Fig. 3, for the fixed \( {r}_{I_D}=0.99 \), i.e., slow fading CCIs (which is the previous research assumption), the OP P_{out}(γ_{Th}) at the destination is shown for various r_{SR}. We observe in Fig. 3 that as the correlation between the source and the relay becomes larger, the OP performance degrades severely and cooperative diversity decreases. It is shown in Fig. 4 that for the fixed \( {r}_{I_D}=0.01 \), i.e., fast fading CCIs (which is considered in this paper), the OP P_{out}(γ_{Th}) at the destination is shown for various r_{SR}. The results in Fig. 4 are similar with those in Fig. 3, but the patterns of the OP performance degradation are different. In order to investigate the difference, we plot the combination of Fig. 3 and Fig. 4 in Fig. 5. It is investigated in Fig. 5 that the performance of the large distance between the source and the relay is better than that of the small distance, regardless of interference fading speed at the destination. We define the impact of fastfading CCIs on the performance as the SNR Γ_{0} loss in decibel compared with slowfading CCIs. We observe in Fig. 5 that given the distance between the source and the relay (r_{SR} = 0.01 or 0.99), the performance of fastfading interference at the destination is basically worse than that of slow fading, except in the low SNR regime for the distance being small (r_{SR} = 0.99). The exception is explained as follows: since the small distance results in lost diversity, in the low SNR regime, the weak power correlated signals transmitted by the source and the relay are more vulnerable to highly correlated CCIs (\( {r}_{I_D}=0.99 \)) than almost uncorrelated CCIs (\( {r}_{I_D}=0.01 \)). (Note that if we ignored thermal noise, we could not observe the exception in the low SNR regimes.) In other words, besides the exception, DFrelaying OC more easily cancels out almost flatfading CCIs (\( {r}_{I_D}=0.99 \)) than fastfading CCIs (\( {r}_{I_D}=0.01 \)). Slowfading CCIs represent highly correlated CCIs, and fastfading CCIs represent weakly correlated CCIs. In result, the sourcerelay distance (r_{SR}) is generally a more dominating factor than the fading CCIs (\( {r}_{I_D} \)) at the destination, when the performance of OC for these systems is analyzed. It is also shown in Fig. 5 that the previous research assumption (\( {r}_{I_D}=0.01 \), \( {r}_{I_D}=0.99 \)) is the most optimistic. In order to further investigate the effects of various r_{SR} and \( {r}_{I_D} \) on the average SER for the DFrelaying OC system, Fig. 6 shows the SER of the BPSK modulation versus the SNR Γ_{0}. It is clearly shown in Fig. 6 that there is the gap between the most optimistic case (\( {r}_{I_D}=0.01 \), \( {r}_{I_D}=0.99 \)) and the most conservative case (r_{SR} = 0.99, \( {r}_{I_D}=0.01 \)). The gap is about 8 dB in the SNR Γ_{0} at the SER of 10^{−4}. We also observe that the results in Fig. 6 are consistent with those in Fig. 5.
Now, we discuss the difference between OC and nonOC. In order to achieve cooperative diversity, OC maximizes the SINR, reduces CCIs’ power, and increases diversity. On the other hand, nonOC, such as MRC, maximizes only the SNR so that a smaller output SINR is produced and the performance is degraded severely in the presence of CCIs.
Conclusion
In this paper, we investigated the effects of the sourcerelay distance and fast/slowfading CCIs on the performance of the DFrelaying OC system. Conditioned on the probability of symbol errors at the relay, we first developed the MGF of the instantaneous maximum output SINR. Using the total probability theorem, we then derived closedform expressions for the OP and the average SER at the destination. With these analytical expressions, it was shown that the performance of the large distance between the source and the relay is better than that of the small distance, regardless of interference fading speed at the destination. Furthermore, we also showed that given the distance, the performance of slowfading interference is basically better than that of fast fading, except in the low SNR regime for the distance being small. In result, the sourcerelay distance is generally a more dominating factor than the fading CCIs. Finally, we presented the average SER performance, which showed the gap between the most optimistic case and the most conservative case.
Abbreviations
 AF:

Amplifyandforward
 AWGN:

Additive white Gaussian noise
 BPSK:

Binary phase shift keying
 CCIs:

Cochannel interferers
 DF:

Decodeandforward
 DSTC:

Distributed spacetime coding
 FFT:

Fast Fourier transform
 MGF:

Momentgenerating function
 MIMO:

Multipleinput multipleoutput
 MRC:

Maximalratio combining
 OC:

Optimum combining
 OP:

Outage probability
 OR:

Opportunistic relaying
 PDF:

Probability density function
 SER:

Symbol error rate
 SINR:

Signaltointerferenceplusnoise ratio
 SNR:

Signaltonoiseratio
 STD:

Single transmission duration
 TDD:

Time division duplex
References
 1.
A. Nosratinia, T.E. Hunter, A. Hedayat, Cooperative communication in wireless networks. IEEE Commun. Mag. 42(10), 74–80 (2004)
 2.
J.N. Laneman, D.N.C. Tse, G.W. Wornell, Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theory 50(12), 3062–3080 (2004)
 3.
W.C. Jakes Jr. et al., Microwave Mobile Communications (Wiley, New York, 1974)
 4.
J. Winters, Optimum combining in digital mobile radio with cochannel interference. IEEE Trans. Veh. Technol. 33, 144–155 (1984)
 5.
N. Suraweera, N.C. Beaulieu, The impact of imperfect channel estimations on the performance of optimum combining in decodeandforward relaying in the presence of cochannel interference. IEEE Wireless Commun. Lett. 3(1), 18–21 (2014)
 6.
Q.T. Zhang, X.W. Cui, Outage probability for optimum combining of arbitrarily faded signals in the presence of correlated Rayleigh interferers. IEEE Trans. Veh. Technol. 53(4), 1043–1051 (2004)
 7.
P.D. Rahimzadeh, N.C. Beaulieu, Limits to performance of optimum combining with dense multiple correlated antennas. IEEE Trans. Commun. 58(7), 2014–2022 (2010)
 8.
H. Yu, I.H. Lee, G.L. Stüber, Outage probability of decodeandforward cooperative relaying systems with cochannel interference. IEEE Trans. Wirel. Commun. 11(1), 266–274 (2012)
 9.
N. Suraweera, N.C. Beaulieu, Outage probability of decodeandforward relaying with optimum combining in the presence of cochannel interference and Nakagami fading. IEEE Wireless Commun. Lett. 2(5), 495–498 (2013)
 10.
N. Wu, H. Li, Performance analysis of SNRbased decodeandforward opportunistic relaying in the presence of cochannel interference. IEEE Trans. Veh. Technol. 65(9), 7244–7257 (2016)
 11.
H.U. Sokun, H. Yanikomeroglu, On the spectral efficiency of selective decodeandforward relaying. IEEE Trans. Veh. Technol. 66(5), 4500–4506 (2017)
 12.
Y. Zhang, J. Ge, Joint antennaandrelay selection in MIMO decodeandforward relaying networks over Nakagamim fading channels. IEEE Signal Process. Lett. 24(4), 456–460 (2017)
 13.
M.S. Gilan, A. Olfat, New beamforming and spacetime coding for twopath successive decode and forward relaying. IET Commun. 12(13), 1573–1588 (2018)
 14.
P. Kumar, K. Dhaka, Performance analysis of wireless powered DF relay system under Nakagamim fading. IEEE Trans. Veh. Technol. 67(8), 7073–7085 (2018)
 15.
M. Schwartz, W.R. Bennett, S. Stein, Communication Systems and Techniques (McGrawHill, New York, 1996)
 16.
R.A. Horn, C.R. Johnson, Matrix Analysis, 1st ed (Cambridge Univ. Press, Cambridge, 1990)
 17.
K. Chung, An analytical expression for performance of optimum combining with multiple correlated CCIs and two antennas. IEEE Commun. Lett. 16(4), 458–461 (2012)
 18.
M. Chiani, M.Z. Win, A. Zanella, On the capacity of spatially correlated MIMO Rayleighfading channels. IEEE Trans. Inf. Theory 49(10), 2363–2371 (2003)
 19.
M.K. Simon, M.S. Alouini, Digital Communication over Fading Channels, 1st Ed (Wiley, Hoboken, 2000)
 20.
V. Aalo, Performance of maximalratio diversity systems in a correlated Nakagami fading environment. IEEE Trans. Commun. 43, 2360–2369 (1995)
 21.
A. Goldsmith, Wireless Communications (Cambridge University Press, New York, 2005)
Acknowledgements
Not applicable.
Funding
Not applicable.
Availability of data and materials
Not applicable.
Author information
Affiliations
Contributions
KC analyzed the outage probability (OP) and the average symbol error rate (SER) of decodeandforward (DF) relaying. KC derived closedform expressions for the OP and the average SER with optimum combining (OC) considering fast fading multiple correlated CCIs, the correlated sourcerelay, and thermal noise. KC read and approved the final manuscript.
Corresponding author
Correspondence to Kyuhyuk Chung.
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
Keywords
 Decodeandforward relaying
 Fastfading correlated multiple CCIs
 Correlated sourcerelay
 Optimum combining
 Rayleigh fading
 Thermal noise
 Outage probability
 Symbol error rate