On performance analysis for optimum combining of DF relaying with fast-fading multiple correlated CCIs, correlated source-relay, and thermal noise

Abstract

This paper analyzes the outage probability (OP) and the average symbol error rate (SER) of decode-and-forward (DF) relaying. The paper derives closed-form expressions for the OP and the average SER with optimum combining (OC) considering fast-fading multiple correlated CCIs, the correlated source-relay, 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 source-relay distance, the performance of slow-fading interference is basically better than that of fast fading, except in the low signal-to-noise-ratio (SNR) regime for the distance being small. In result, the source-relay distance is generally a more dominating factor for the performance than fading CCIs.

1 Introduction

Cooperative communications have been prominent because of diversity gain [1]. In cooperative networks, there are mainly two methods, such as the amplify-and-forward (AF) relay network and the decode-and-forward (DF) relay network [2]. Then with multiple copies, the destination can achieve cooperative diversity. In order to do so, we can use maximal-ratio combining (MRC) [3] (p., 316) or optimum combining (OC) [4]. MRC maximizes the signal-to-noise-ratio (SNR), while OC maximizes the signal-to-interference-plus-noise ratio (SINR). When co-channel 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 multiple-relay multiple-input multiple-output (MIMO) network with a DF relaying strategy [12]. In [13], a novel distributed space-time coding (DSTC) transmission scheme for a two-path successive DF relay network is proposed. The average SER is analyzed for a wireless-powered three-node DF relaying system in Nakagami-m fading environment [14].

In this paper, a DF protocol is considered. It is assumed that at the relay symbol-by-symbol 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 closed-form expressions for the OP and the average SER with OC considering fast-fading multiple correlated CCIs, the correlated source-relay, and thermal noise. Second, we investigate the effects of the source-relay distance and fast/slow-fading 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.

2 System and channel model

We define the full cooperative case as relaying with no symbol errors and the non-cooperative 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 non-cooperative 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, t₁ and t₂. One time interval t₁ is for phase 1 and the other t₂ is for phase 2. Therefore, a single transmission duration (STD) t_STD becomes t₁ + t₂. 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 source-relay channel correlation coefficient r_SR and the destination interferer channel correlation coefficient ${r}_{I_D}$ are defined in the following sections.)

Fig. 1

System and channel model

Under the above assumptions, for the first time slot t₁, i.e., in phase 1, the source transmits its data symbols. The received signal at the destination is expressed by:

$$ {y}^{\left(S,D,{t}_1\right)}=\sqrt{E_S^{t_1}}{g}_0^{\left(S,D,{t}_1\right)}{b}_0+\sum \limits_{i=1}^{N_{I_D}}\sqrt{E_{I_D}^{t_1}}{g}_i^{\left({I}_D,D,{t}_1\right)}{b}_i+{n}^{\left(S,D,{t}_1\right)} $$

(1)

where ${E}_S^{t_1}$, ${E}_{I_D}^{t_1}$, b₀, and b_i are the power transmitted by the source over the time slot t₁, the power transmitted by each interferer over the time slot t₁, 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:

$$ {y}^{\left(S,R,{t}_1\right)}=\sqrt{E_S^{t_1}}{g}_0^{\left(S,R,{t}_1\right)}{b}_0+\sum \limits_{j=1}^{N_{I_R}}\sqrt{E_{I_R}^{t_1}}{g}_i^{\left({I}_R,R,{t}_1\right)}{r}_i+{n}^{\left(S,R,{t}_1\right)} $$

(2)

where ${E}_{I_R}^{t_1}$ is each interferer power over the time slot t₁ 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 t₂, 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:

$$ {y}^{\left(R,D,{t}_2\right)}=\sqrt{E_R^{t_2}}{g}_0^{\left(R,D,{t}_2\right)}{b}_0+\sum \limits_{i=1}^{N_{I_D}}\sqrt{E_{I_D}^{t_2}}{g}_i^{\left({I}_D,D,{t}_2\right)}{b}_i+{n}^{\left(R,D,{t}_2\right)} $$

(3)

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:

$$ {\displaystyle \begin{array}{l}{\mathbf{y}}_{\mathrm{full}\hbox{-} \mathrm{co}}=\left[\begin{array}{c}{y}^{\left(S,D,{t}_1\right)}\\ {}{y}^{\left(R,D,{t}_2\right)}\end{array}\right]=\sqrt{E_S}\left[\begin{array}{c}{g}_0^{\left(S,D,{t}_1\right)}\\ {}{g}_0^{\left(R,D,{t}_2\right)}\end{array}\right]{b}_0+\sum \limits_{i=1}^{N_{\mathrm{I}}}\sqrt{E_{I_D}}\left[\begin{array}{c}{g}_i^{\left({I}_D,D,{t}_1\right)}\\ {}{g}_i^{\left({I}_D,D,{t}_2\right)}\end{array}\right]{b}_i+\left[\begin{array}{c}{n}^{\left(S,D,{t}_1\right)}\\ {}{n}^{\left(R,D,{t}_2\right)}\end{array}\right]\\ {}=\sqrt{E_S}{\boldsymbol{g}}_0{b}_0+\sum \limits_{i=1}^{N_{\mathrm{I}}}\sqrt{E_{I_D}}{\boldsymbol{g}}_i{b}_i+\mathbf{n}\end{array}} $$

(4)

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 g₀, g_i, $i=1,2\cdots, {N}_{I_D}$, and n are (2 × 1) zero-mean complex symmetric Gaussian random vectors. For ${P}_{\mathrm{e}}^{(R)}=1$, the received signal at the destination is expressed by:

$$ {\mathbf{y}}_{\mathrm{non}\hbox{-} \mathrm{co}}=\left[\begin{array}{c}{y}^{\left(S,D,{t}_1\right)}\\ {}0\end{array}\right]=\sqrt{E_S}\left[\begin{array}{c}{g}_0^{\left(S,D,{t}_1\right)}\\ {}0\end{array}\right]{b}_0+\sum \limits_{i=1}^{N_{\mathrm{I}}}\sqrt{E_{I_D}}\left[\begin{array}{c}{g}_i^{\left({I}_D,D,{t}_1\right)}\\ {}0\end{array}\right]{b}_i+\left[\begin{array}{c}{n}^{\left(S,D,{t}_1\right)}\\ {}0\end{array}\right] $$

(5)

3 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⁻¹g₀ with the interference-plus-noise correlation matrix R = N₀I₂ + E_IGG^† and $\mathrm{G}=\left[{\boldsymbol{g}}_1{\boldsymbol{g}}_2\cdots {\boldsymbol{g}}_{N_{I_D}}\right]$ [4]. The notation I₂ 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

$$ {\gamma}_{\mathrm{full}\hbox{-} \mathrm{co}}={E}_S{\boldsymbol{g}}_0^{\dagger }{\mathbf{R}}^{-1}{\boldsymbol{g}}_0 $$

(6)

The moment-generating function (MGF) of γ_{full ‐ co} is given by:

$$ {\displaystyle \begin{array}{l}{M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)={\mathbb{E}}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}\left[{e}^{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}s}\right]\\ {}={\mathbb{E}}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}\left[{e}^{E_S{\boldsymbol{g}}_0^{\dagger }{\mathbf{R}}^{-1}{\boldsymbol{g}}_0s}\right]\\ {}={\mathbb{E}}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}\left[{e}^{E_S{\boldsymbol{g}}_0^{\dagger }{\left({N}_0{\mathbf{I}}_2+{E}_{\mathrm{I}}{\mathbf{GG}}^{\dagger}\right)}^{-1}{\boldsymbol{g}}_0s}\right]\\ {}={\mathbb{E}}_{{\mathbf{GG}}^{\dagger }}\left[{\mathbb{E}}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}\mid {\mathbf{GG}}^{\dagger }}\left[{e}^{E_S{\boldsymbol{g}}_0^{\dagger }{\left({N}_0{\mathbf{I}}_2+{E}_{\mathrm{I}}{\mathbf{GG}}^{\dagger}\right)}^{-1}{\boldsymbol{g}}_0s}\right]\right]\\ {}={\mathbb{E}}_{{\mathbf{GG}}^{\dagger }}\left[\frac{1}{\left|{\mathbf{I}}_2-{E}_S\mathbb{E}\left[{\boldsymbol{g}}_0{\boldsymbol{g}}_0^{\dagger}\right]{\left({N}_0{\mathbf{I}}_2+{E}_{\mathrm{I}}{\mathbf{GG}}^{\dagger}\right)}^{-1}s\right|}\right]\\ {}={\mathbb{E}}_{{\mathbf{GG}}^{\dagger }}\left[\frac{1}{\left|{\mathbf{I}}_2-{\Gamma}_0{\boldsymbol{\Sigma}}_{SR}{\left({\mathbf{I}}_2+{\Gamma}_1{\mathbf{GG}}^{\dagger}\right)}^{-1}s\right|}\right]\end{array}} $$

(7)

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 source-relay 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 Γ₀ ≜ E_S/N₀ is the SNR over each time slot, i.e., t₁ or t₂, and the power ratio ${\Gamma}_1\triangleq {E}_{I_D}/{N}_0$ is the interference-to-noise ratio (INR) over each time slot, i.e., t₁ or t₂. We express the (2 × 2) Hermitian matrix GG^† as the eigenvalue decomposition [16].

$$ {\mathbf{GG}}^{\dagger }=\mathbf{U}\mathbf{B} {\mathbf{U}}^{\dagger },\kern2.1em \mathbf{B} =\left[\begin{array}{cc}{\beta}_1& 0\\ {}0& {\beta}_2\end{array}\right] $$

(8)

where β₁ and β₂ with β₁ ≥ β₂ are the non-zero 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:

$$ {\displaystyle \begin{array}{l}{M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)\;\\ {}={\mathbb{E}}_{{\mathbf{GG}}^{\dagger }}\left[\frac{1}{\left|{\mathbf{I}}_2-{\Gamma}_0{\boldsymbol{\Sigma}}_{SR}{\left({\mathbf{I}}_2\kern0.36em +{\Gamma}_1{\mathbf{GG}}^{\dagger}\right)}^{-1}\kern0.5em s\right|}\right]\\ {}={\mathbb{E}}_{\mathbf{B}, \mathbf{U}}\left[\frac{1}{\left|{\mathbf{I}}_2-{\Gamma}_0{\boldsymbol{\Sigma}}_{SR}{\left({\mathbf{I}}_2\kern0.36em +{\Gamma}_1{\mathbf{GG}}^{\dagger}\right)}^{-1}\kern0.5em s\right|}\right]\\ {}={\mathbb{E}}_{\mathbf{B}}\left[{\mathbb{E}}_{\mathbf{U}\mid \mathbf{B}}\left[\frac{1}{\left|{\mathbf{I}}_2-{\Gamma}_0{\boldsymbol{\Sigma}}_{SR}{\left({\mathbf{I}}_2\kern0.36em +{\Gamma}_1{\mathbf{GG}}^{\dagger}\right)}^{-1}s\right|}\right]\right]\\ {}={\mathbb{E}}_{\mathbf{B}}\left[{\mathbb{E}}_{\mathbf{U}\mid \mathbf{B}}\left[\frac{\left(1+{\Gamma}_1{\beta}_1\right)\left(1+{\Gamma}_1{\beta}_2\right)}{\left(1+{\Gamma}_1{\beta}_1-{\Gamma}_0{\sigma}_1s\right)\left(1+{\Gamma}_1{\beta}_2-{\Gamma}_0{\sigma}_2s\right)+\left({\Gamma}_0{\sigma}_1s-{\Gamma}_0{\sigma}_2s\right)\left({\Gamma}_1{\beta}_2-{\Gamma}_1{\beta}_1\right){\sin}^2\theta}\right]\right]\end{array}} $$

(9)

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:

$$ {\displaystyle \begin{array}{l}{M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)\\ {}={\mathbb{E}}_{\mathbf{B}}\left[{\mathbb{E}}_{\mathbf{U}\mid \mathbf{B}}\left[\frac{\left(1+{\Gamma}_1{\beta}_1\right)\left(1+{\Gamma}_1{\beta}_2\right)}{\left(1+{\Gamma}_1{\beta}_1-{\Gamma}_0{\sigma}_1s\right)\left(1+{\Gamma}_1{\beta}_2-{\Gamma}_0{\sigma}_2s\right)+\left({\Gamma}_0{\sigma}_1s-{\Gamma}_0{\sigma}_2s\right)\left({\Gamma}_1{\beta}_2-{\Gamma}_1{\beta}_1\right){\sin}^2\theta}\right]\right]\\ {}={\mathbb{E}}_{\mathbf{B}}\left[\frac{1}{2\pi }{\int}_{-\pi}^{\pi}\frac{\left(1+{\Gamma}_1{\beta}_1\right)\left(1+{\Gamma}_1{\beta}_2\right)}{\left(1+{\Gamma}_1{\beta}_1-{\Gamma}_0{\sigma}_1s\right)\left(1+{\Gamma}_1{\beta}_2-{\Gamma}_0{\sigma}_2s\right)+\left({\Gamma}_0{\sigma}_1s-{\Gamma}_0{\sigma}_2s\right)\left({\Gamma}_1{\beta}_2-{\Gamma}_1{\beta}_1\right){\sin}^2\theta}\kern0.5em d\theta \right]\\ {}={\mathbb{E}}_{\mathbf{B}}\left[\frac{\left|\left(1+{\Gamma}_1{\beta}_1\right)\left(1+{\Gamma}_1{\beta}_2\right)\right|}{\sqrt{\left(1+{\Gamma}_1{\beta}_1-{\Gamma}_0{\sigma}_1s\right)\left(1+{\Gamma}_1{\beta}_1-{\Gamma}_0{\sigma}_2s\right)\left(1+{\Gamma}_1{\beta}_2-{\Gamma}_0{\sigma}_2s\right)\left(1+{\Gamma}_1{\beta}_2-{\Gamma}_0{\sigma}_1s\right)}}\kern0.5em \right].\end{array}} $$

(10)

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:

$$ {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)=K\left|{\left[{\int}_0^{\infty}\frac{x^{N_{I_D}+j-3}{e}^{-x/{\alpha}_i}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx\right]}_{i,j=1}^2\right| $$

(11)

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:

$$ K=\frac{1}{\Gamma \left({N}_{I_D}\right)\Gamma \left({N}_{I_D}-1\right){\left|{\boldsymbol{\Sigma}}_D\right|}^{N_{I_D}}\left|{\left[{\left(-{\alpha}_j^{-1}\right)}^{i-1}\right]}_{i,j=1}^2\right|}. $$

(12)

The function Γ(·) is the gamma function. The values α₁ and α₂ with α₁ ≥ α₂ 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:

$$ {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)=K\left(\begin{array}{c}{\int}_0^{\infty}\frac{x^{N_{I_D}-2}{e}^{-x/{\alpha}_1}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx{\int}_0^{\infty}\frac{x^{N_{I_D}-1}{e}^{-x/{\alpha}_2}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx\\ {}-{\int}_0^{\infty}\frac{x^{N_{I_D}-1}{e}^{-x/{\alpha}_1}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx{\int}_0^{\infty}\frac{x^{N_{I_D}-2}{e}^{-x/{\alpha}_2}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx\end{array}\right) $$

(13)

With some algebraic manipulations, the MGF ${M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)$ is expressed as:

$$ {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)=K\left(\begin{array}{c}{\int}_0^{\infty}\frac{x^{N_{I_D}-2}{e}^{-x/{\alpha}_1}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx{\int}_0^{\infty }x\frac{x^{N_{I_D}-2}{e}^{-x/{\alpha}_2}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx\\ {}-{\int}_0^{\infty}\frac{x^{N_{I_D}-2}{e}^{-x/{\alpha}_2}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx{\int}_0^{\infty }x\frac{x^{N_{I_D}-2}{e}^{-x/{\alpha}_1}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} dx\end{array}\right) $$

(14)

The MGF ${M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)$ is further simplified as:

$$ {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)=K\left({\int}_0^{\infty }g\left(x;{\alpha}_1\right) dx{\int}_0^{\infty } xg\Big(x;2-{\alpha}_1\left) dx-{\int}_0^{\infty } xg\right(x;{\alpha}_1\left) dx{\int}_0^{\infty }g\right(x;2-{\alpha}_1\Big) dx\right) $$

(15)

where

$$ g\left(x;\alpha \right)=\frac{x^{N_{I_D}-2}{e}^{-x/\alpha}\left|1+{\Gamma}_1x\right|}{\sqrt{\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_1\right)\left(1+{\Gamma}_1x-s{\Gamma}_0{\sigma}_2\right)}} $$

(16)

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 λ₁ [19] and the RV λ₁ be the random eigenvalue of R (the other eigenvalue of R is the constant N₀). (Note that λ₁ and N₀ are the eigenvalues of R = N₀I₂ + E_IGG^†, and β₁ and β₂ are the eigenvalues of GG^†.) Then the MGF ${M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)$ of γ_{full ‐ co} is derived as:

$$ {\displaystyle \begin{array}{l}{M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s)={\int}_{N_0}^{\infty }{M}_{\gamma_{\mathrm{co}}\mid {\lambda}_1}(s){f}_{\lambda_1}\left({\lambda}_1\right)\;d{\lambda}_1\\ {}={\int}_{N_0}^{\infty}\frac{1}{\left(1-\frac{E_S}{N_0}s\right)\left(1-\frac{E_S}{\lambda_1}s\right)}{f}_{\lambda_1}\left({\lambda}_1\right)\;d{\lambda}_1\\ {}={\int}_{N_0}^{\infty}\frac{1}{\left(1-\frac{E_S}{N_0}s\right)\left(1-\frac{E_S}{\lambda_1}s\right)}\frac{1}{E_{I_D}^2}\left({\lambda}_1-{N}_0\right){e}^{-\left({\lambda}_1-{N}_0\right)/{E}_{I_D}}\;d{\lambda}_1\\ {}={\int}_{N_0}^{\infty}\frac{\left({\lambda}_1-{N}_0\right){e}^{-\left({\lambda}_1-{N}_0\right)/{E}_{I_D}}}{E_{I_D}^2\left(1-{\Gamma}_0s\right)\left(1-\frac{E_S}{\lambda_1}s\right)}d{\lambda}_1\end{array}} $$

(17)

In the derivation of Eq. (17), we use ${\lambda}_1\sim {\chi}_1^2$, where the notation ${\chi}_1^2$ denotes a complex chi-squared distribution with one complex degree of freedom; ${f}_{\lambda_1}\left({\lambda}_1\right)=\left({\lambda}_1-{N}_0\right){e}^{-\left({\lambda}_1-{N}_0\right)/{E}_{I_D}}/{E}_{I_D}^2$, λ₁ ≥ N₀. 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]:

$$ {\mathtt{SER}}_{\mathrm{full}\hbox{-} \mathrm{co}}={\int}_0^{\infty}\mathrm{Q}\left(\sqrt{2{\gamma}_{\mathrm{full}\hbox{-} \mathrm{co}}}\right){f}_{\mathrm{full}\hbox{-} \mathrm{co}}\left({\gamma}_{\mathrm{full}\hbox{-} \mathrm{co}}\right)d{\gamma}_{\mathrm{full}\hbox{-} \mathrm{co}} $$

(18)

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:

$$ {P}_{\mathrm{out}}^{\left(\mathrm{full}\hbox{-} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{full}\hbox{-} \mathrm{co}\right)}\right)\triangleq p\left({\gamma}_{\mathrm{full}\hbox{-} \mathrm{co}}\le {\gamma}_{Th}^{\left(\mathrm{full}\hbox{-} \mathrm{co}\right)}\right)={\int}_0^{\gamma_{Th}^{\left(\mathrm{full}\hbox{-} \mathrm{co}\right)}}{f}_{\mathrm{full}\hbox{-} \mathrm{co}}\left({\gamma}_{\mathrm{full}\hbox{-} \mathrm{co}}\right)d{\gamma}_{\mathrm{full}\hbox{-} \mathrm{co}} $$

(19)

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:

$$ {\displaystyle \begin{array}{l}{\gamma}_{\mathrm{non}\hbox{-} \mathrm{co}}={E}_S{g_0^{\left(S,D,{t}_1\right)}}^{\dagger }{\mathrm{R}}_{\mathrm{non}\hbox{-} \mathrm{co}}^{-1}{g}_0^{\left(S,D,{t}_1\right)}\\ {}\kern4em ={E}_S{\left|{g}_0^{\left(S,D,{t}_1\right)}\right|}^2{\mathrm{R}}_{\mathrm{non}\hbox{-} \mathrm{co}}^{-1}\\ {}\kern4em =\frac{E_S{\left|{g}_0^{\left(S,D,{t}_1\right)}\right|}^2}{N_0\kern0.36em +{E}_{I_D}{\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}{\left({\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}\right)}^{\dagger }}\\ {}\kern4em =\frac{X}{1/{\Gamma}_0\kern0.36em +{\Gamma}_1/{\Gamma}_0\;W}.\end{array}} $$

(20)

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 chi-squared-distributed 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/Γ₀ + Γ₁/Γ₀W has the PDF

$$ {f}_Y(y)\triangleq \frac{\Gamma_0}{\Gamma_1}{\left(\left({\Gamma}_0y-1\right)/{\Gamma}_1\right)}^{N_{I_D}-1}{e}^{-\left(\left({\Gamma}_0y-1\right)/{\Gamma}_1\right)}/\left({N}_{I_D}-1\right)! $$

(21)

with y ≥ 1/Γ₀. 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:

$$ {\displaystyle \begin{array}{l}{f}_{\gamma_{\mathrm{non}\hbox{-} \mathrm{co}}}\left({\gamma}_{\mathrm{non}\hbox{-} \mathrm{co}}\right)={\int}_{1/{\Gamma}_0}^{\infty }{yf}_X\left({\gamma}_{\mathrm{non}\hbox{-} \mathrm{co}}y\right){f}_Y(y) dy\\ {}={\int}_{1/{\Gamma}_0}^{\infty }{ye}^{-{\gamma}_{\mathrm{non}\hbox{-} \mathrm{co}}y}\frac{\Gamma_0}{\Gamma_1}\frac{1}{\left({N}_{I_D}-1\right)!}\kern0.5em {\left(\left({\Gamma}_0y-1\right)/{\Gamma}_1\right)}^{N_{I_D}-1}{e}^{-\left(\left({\Gamma}_0y-1\right)/{\Gamma}_1\right)} dy.\end{array}} $$

(22)

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 closed-form expression for the OP P_out(γ_Th) at the destination as:

$$ {P}_{\mathrm{out}}\left({\gamma}_{Th}\right)={P}_{\mathrm{out}}^{\left(\mathrm{full}\hbox{-} \mathrm{co}\right)}\left({\gamma}_{Th}\right)\left(1-{P}_{\mathrm{e}}^{(R)}\right)+{P}_{\mathrm{out}}^{\left(\mathrm{non}\hbox{-} \mathrm{co}\right)}\left({\gamma}_{Th}\right){P}_{\mathrm{e}}^{(R)} $$

(23)

and the average SER at the destination is derived as:

$$ \mathtt{SER}={\int}_0^{\infty}\mathrm{Q}\left(\sqrt{2\gamma}\right){f}_{\gamma}\left(\gamma \right) d\gamma \kern0.5em ={\mathtt{SER}}_{\mathrm{full}\hbox{-} \mathrm{co}}\left(1-{P}_e^{(R)}\right)+{\mathtt{SER}}_{\mathrm{non}\hbox{-} \mathrm{co}}{P}_e^{(R)}. $$

(24)

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/slow-fading interference at the destination, i.e., ${\boldsymbol{\Sigma}}_{I_D}$.

4 Results and discussion

We assume that the signals have the exponential correlation [20]. Thus, with the source-relay channel correlation coefficient r_SR ∈ [0, 1):

$$ {\boldsymbol{\Sigma}}_{SR}=\left[\begin{array}{cc}1& {r}_{SR}\\ {}{r}_{SR}& 1\end{array}\right] $$

(25)

and with the destination interferer channel correlation coefficient ${r}_{I_D}\in \left[0,1\right)$:

$$ {\boldsymbol{\Sigma}}_{I_D}=\left[\begin{array}{cc}1& {r}_{I_D}\\ {}{r}_{I_D}& 1\end{array}\right]. $$

(26)

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 slow-fading multiple correlated CCIs (${r}_{I_D}=0.99$), which are the assumptions of the previous researches, i.e., independent users and flat-fading 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 source-relay 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.

Fig. 2

The outage probability, P_out(γ_Th), for various ${P}_{\mathrm{e}}^{(R)}$ values in the presence of almost uncorrelated users (r_SR = 0.01), and slow-fading multiple correlated CCIs (${r}_{I_D}=0.99$), and thermal noise, with ${N}_{I_D}=2$, ${\Gamma}_1^{\mathrm{total}}=10\kern0.5em \mathrm{dB}$, and γ_Th = 3 dB

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 fast-fading CCIs on the performance as the SNR Γ₀ loss in decibel compared with slow-fading 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 fast-fading 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, DF-relaying OC more easily cancels out almost flat-fading CCIs (${r}_{I_D}=0.99$) than fast-fading CCIs (${r}_{I_D}=0.01$). Slow-fading CCIs represent highly correlated CCIs, and fast-fading CCIs represent weakly correlated CCIs. In result, the source-relay 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 DF-relaying OC system, Fig. 6 shows the SER of the BPSK modulation versus the SNR Γ₀. 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 Γ₀ at the SER of 10⁻⁴. We also observe that the results in Fig. 6 are consistent with those in Fig. 5.

Fig. 3

The outage probability, P_out(γ_Th), for various r_SR values in the presence of slow-fading multiple correlated CCIs (${r}_{I_D}=0.99$), and thermal noise, with ${N}_{I_D}=2$, ${\Gamma}_1^{\mathrm{total}}=10\kern0.5em \mathrm{dB}$, γ_Th = 3 dB, and ${P}_{\mathrm{e}}^{(R)}=0.001$

Fig. 4

The outage probability, P_out(γ_Th), for various r_SR values in the presence of fast-fading multiple correlated CCIs (${r}_{I_D}=0.01$), and thermal noise, with ${N}_{I_D}=2$, ${\Gamma}_1^{\mathrm{total}}=10\kern0.5em \mathrm{dB}$, γ_Th = 3 dB, and ${P}_{\mathrm{e}}^{(R)}=0.001$

Fig. 5

The outage probability, P_out(γ_Th), for various r_SR and ${r}_{I_D}$ values in the presence of fading multiple correlated CCIs, and thermal noise, with ${N}_{I_D}=2$, ${\Gamma}_1^{total}=10\kern0.5em \mathrm{dB}$, γ_Th = 3 dB, and ${P}_e^{(R)}=0.001$

Fig. 6

SER for various r_SR and ${r}_{I_D}$ values in the presence of fading multiple correlated CCIs, and thermal noise, with ${N}_{I_D}=2$, ${\Gamma}_1^{total}=10\kern0.5em \mathrm{dB}$, γ_Th = 3 dB, and ${P}_e^{(R)}=0.001$

Now, we discuss the difference between OC and non-OC. In order to achieve cooperative diversity, OC maximizes the SINR, reduces CCIs’ power, and increases diversity. On the other hand, non-OC, 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.

5 Conclusion

In this paper, we investigated the effects of the source-relay distance and fast/slow-fading CCIs on the performance of the DF-relaying 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 closed-form 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 slow-fading interference is basically better than that of fast fading, except in the low SNR regime for the distance being small. In result, the source-relay 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.