In this section, mathematical expression of the relay selection criteria and performance analyses of the considered schemes are provided.

### MRC-based model

In MRC-based relaying model, the end-to-end bit error rate (BER) in S → R_{
k
} → D link, \( {P}_{\mathrm{eq}}^{\mathrm{b}}\left({\gamma}_{{\mathrm{SR}}_k},\kern0.5em {\gamma}_{{\mathrm{R}}_k\mathrm{D}}\right) \) can be determined by [38]:

$$ {P}_{eq}^b\left({\gamma}_{S{R}_k},{\gamma}_{R_kD}\right)=\left[1-{P}_{S{R}_k}^b\left({\gamma}_{S{R}_k}\right)\right]{P}_{R_kD}^b\left({\gamma}_{R_kD}\right)+\left[1-{P}_{R_kD}^b\left({\gamma}_{R_kD}\right)\right]{P}_{S{R}_k}^b\left({\gamma}_{S{R}_k}\right) $$

(7)

In (7), \( {P}_{\mathrm{S}{\mathrm{R}}_k}^b \) and \( {P}_{{\mathrm{R}}_kD}^b \) are the BERs of S → R_{
k
} and R_{
k
} → D links which can be calculated by \( {P}_{S{R}_k}^b=Q\left(\sqrt{2{\gamma}_{S{R}_k}}\right) \) and \( {P}_{R_kD}^b=Q\left(\sqrt{2{\gamma}_{R_kD}}\right) \), respectively, for BPSK modulation where \( Q(x)=\left(1/\sqrt{2\pi}\right){\displaystyle {\int}_x^{\infty } \exp \left(-{t}^2/2\right)dt} \). The equivalent SNR of S → R_{
k
} → D link *γ*
_{eq, k
} can be expressed by [38]:

$$ {\gamma}_{eq,k}=\frac{1}{\eta}\left\{{Q}^{-1}\left[{P}_{eq}^b\left({\gamma}_{S{R}_k},{\gamma}_{R_kD}\right)\right]\right\} $$

(8)

where *η* = 2 for BPSK modulation. Therefore, the relay selection criterion for MRC-based system model can be expressed as:

$$ k=\underset{k^{\mathit{\prime}}\in \left\{1,2,\dots, M\right\}}{\mathrm{argmax}}{\gamma}_{eq,{k}^{\mathit{\prime}}} $$

(9)

where *k* denotes the index of the selected relay. At the destination, MRC technique is used to combine the signals *y*
_{SD} and \( {y}_{{\mathrm{R}}_k\mathrm{D}} \). Thus, the combined signal at the destination can be given as:

$$ {y}_D={\omega}_{SD}{y}_{SD}+{\omega}_{R_kD}{y}_{R_kD} $$

(10)

Here, *ω*
_{SD
} and \( {\omega}_{R_kD} \) are the weighting coefficients which are the functions of α_{SD} and \( {\beta}_{{\mathrm{R}}_k\mathrm{D}} \). Substituting \( {\omega}_{SD}={\alpha}_{{}_{SD}}^{*} \) and \( {\omega}_{R_kD}={\beta}_{{}_{R_kD}}^{*} \) into (10), the combined signal at the destination can be given as:

$$ {y}_D={\alpha}_{SD}^{*}{y}_{SD}+{\beta}_{R_kD}^{*}{y}_{R_kD} $$

(11)

where (.)* denotes the complex conjugate operator. The received SNR at the destination can be expressed as *γ*
_{
tot
} = *γ*
_{
b
} + *γ*
_{SD} where *γ*
_{
b
} = *γ*
_{
eq,k
} and the PDF of *γ*
_{
tot
} can be calculated by:

$$ p\left({\gamma}_{tot}\right)={\displaystyle {\int}_0^{\gamma_{tot}}{p}_{\gamma_{SD}}(x){p}_{\gamma_b}\left({\gamma}_{tot}-x\right)dx} $$

(12)

The end-to-end probability of error can be written as [39]:

$$ {P}_e={P}_{prop}{P}_{S{R}_k}+\left(1-{P}_{S{R}_k}\right){P}_{MRC}, $$

(13)

where *P*
_{
prop
} denotes the probability of error propagation and *P*
_{
MRC
} is the error probability of combined signals. *P*
_{prop} can be approximated for BPSK modulation by [39]:

$$ {P}_{prop}\approx {\overline{\gamma}}_b/\left({\overline{\gamma}}_b+{\overline{\gamma}}_{SD}\right) $$

(14)

where \( {\overline{\gamma}}_b \) and \( {\overline{\gamma}}_{SD} \) are the expected values of *γ*
_{
b
} and *γ*
_{
SD
}, respectively. Similarly, \( {P}_{S{R}_k} \) is the probability of error in the S → R_{
k
} link which can be given by:

$$ {P}_{S{R}_k}={\displaystyle {\int}_0^{\infty}\frac{1}{2} erfc\left(\sqrt{x}\right){p}_{\gamma_{S{R}_k}}(x)dx} $$

(15)

Here, \( {p}_{\gamma_{S{R}_k}}\left(\cdot \right) \) is the PDF of \( {\gamma}_{S{R}_k} \) and *erfc*(.) is complementary error function. \( {p}_{\gamma_{S{R}_k}}(x) \) can be given as [28]:

$$ {p}_{\gamma_{\mathrm{S}{\mathrm{R}}_{\mathrm{k}}}}(x)=\frac{1}{x{\displaystyle {\prod}_{i=1}^N\varGamma \left({m}_i\right)}}{G}_{0,N}^{N,0}\left(\frac{x}{{\overline{\gamma}}_{\mathrm{S}{\mathrm{R}}_k}}{\displaystyle \prod_{i=1}^N{m}_i}\left|{}_{m_1,\dots, {m}_N}^{-}\right.\right), $$

(16)

where \( {\overline{\gamma}}_{S{R}_k} \) is \( {\overline{\gamma}}_{S{R}_k}=\left({E}_s/{N}_0\right){\displaystyle \prod_{i=1}^N{m}_i} \). By using:

$$ \mathrm{erfc}(x)=\frac{1}{\sqrt{\pi }}{G}_{1,2}^{2,0}\left(x\left|{}_{0,0.5}^1\right.\right) $$

(17)

with the help of [40] (Eq. 07.34.21.0011.01), the expression (15) can be calculated as:

$$ {P}_{\mathrm{S}{\mathrm{R}}_k}=\frac{1}{2\sqrt{\pi }{\displaystyle {\prod}_{i=1}^N\varGamma \left({m}_i\right)}}{G}_{2,N+1}^{N,2}\left(\frac{1}{{\overline{\gamma}}_{\mathrm{S}{\mathrm{R}}_k}}{\displaystyle \prod_{i=1}^N{m}_i}\left|{}_{m_1,\dots {m}_N,0}^{1,0.5}\right.\right) $$

(18)

The error probability of combined source-destination and relay-destination signals *P*
_{MRC} can be expressed as [39]:

$$ {P}_{MRC}={\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{1}{2} erfc\left(\sqrt{\gamma_{tot}}\right)p\left({\gamma}_{tot}\right)d{\gamma}_{tot}} $$

(19)

The error probability of the considered system can be obtained by substituting (14), (18), and (19) into (13).

### C-MRC-based model

The decision rule at the destination for C-MRC-based model can be written as [38]:

$$ \widehat{x}= \arg { \min}_{x\in {A}_x}{\left|{\omega}_{SD}{y}_{SD}+{\omega}_{R_kD}{y}_{R_kD}-\left({\omega}_{SD}{\alpha}_{SD}+{\omega}_{R_kD}{\beta}_{R_kD}\right)x\right|}^2 $$

(20)

where *A*
_{
x
} = {−1, + 1} for BPSK modulation. In the previous section, it is mentioned that, in the MRC technique, weighting coefficients were chosen as \( {\omega}_{SD}={\alpha}_{SD}^{*} \) and \( {\omega}_{R_kD}={\beta}_{R_kD}^{*} \) to maximize the SNR at the output of combiner. This choice may decrease the average SNR obtained at the destination especially when detection errors occur at the relay terminal. In order to solve this problem C-MRC technique is proposed in which *ω*
_{
SD
} is fixed to \( {\alpha}_{{}_{SD}}^{*} \), while \( {\omega}_{{\mathrm{R}}_k\mathrm{D}} \) is determined by [38]:

$$ {\omega}_{R_kD}\left({\alpha}_{S{R}_k},{\beta}_{R_kD}\right)=\left({\gamma}_{eq,k}/{\gamma}_{R_kD}\right){\beta}_{R_kD}^{*} $$

(21)

In (21), *γ*
_{
eq,k
} is the equivalent SNR given in (8).

Note that in high SNR regime, we obtain *γ*
_{
eq
} ≈ *γ*
_{min} where \( {\gamma}_{\min } = \min \left\{{\gamma}_{{\mathrm{R}}_k\mathrm{D}},\kern0.5em {\gamma}_{{\mathrm{SR}}_k}\right\} \). And finally, the best relay can be chosen by the help of (9).

### VN-MRC model

The main idea of VN-based model is the introduction of virtual noise component at the destination. In this approach, detection errors at the relay are modeled as the addition of virtual noise [14]. In this case, we rewrite the received signal at the destination in the second phase as:

$$ \begin{array}{l}{y}_{R_kD}=\sqrt{E_s}\widehat{x}\;{\beta}_{R_kD}+{n}_{R_kD}\\ {}\kern1.75em =\sqrt{E_s}x\;{\beta}_{R_kD}+\sqrt{E_s}{e}_r\;{\beta}_{R_kD}+{n}_{R_kD}\\ {}\kern1.75em =\sqrt{E_s}x\;{\beta}_{R_kD}+{n}_v+{n}_{R_kD},\end{array} $$

(22)

where *n*
_{v} is the virtual noise which is independent from additive white Gaussian noise \( {n}_{R_kD} \) and modeled by a zero-mean complex random variable. The variance of term *e*
_{
r
} in (22) is \( {N}_0/{\left|{\alpha}_{S{R}_k}\right|}^2 \) [14]. The combined signal at the destination in case of VN-based detection can be given as [14]:

$$ {y}_D={\omega}_{SD}{y}_{SD}+{\omega}_{R_kD}{y}_{R_kD} $$

(23)

where the weighting coefficients are determined by:

$$ {\omega}_{SD}={\alpha}_{SD}^{*}/{N}_0 $$

(24)

$$ {\omega}_{R_kD}={\beta}_{R_kD}^{*}/{N}_0\left(1+{\left|{\beta}_{R_kD}\right|}^2/{\left|{\alpha}_{S{R}_k}\right|}^2\right). $$

(25)

Note that this combining approach eliminates the decoding complexity in the classical ML-based detectors that increases exponentially with the number of the components in the signal constellation. The conditional BER of the proposed system for a given relay in VN with MRC-based model can be expressed as:

$$ {P}_k=\left(1-Q\left(\sqrt{2{\gamma}_{SR}}\right)\right)Q\left(\sqrt{\frac{2{\left({\gamma}_{SD}+{\gamma}_{EQ}\right)}^2}{\gamma_{SD}+{\gamma}_{EQ}^{\mathit{\prime}}}}\right)+Q\left(\sqrt{2{\gamma}_{SR}}\right)Q\left(\sqrt{\frac{2{\left({\gamma}_{SD}-{\gamma}_{EQ}\right)}^2}{\gamma_{SD}+{\gamma}_{EQ}^{\mathit{\prime}}}}\right) $$

(26)

where:

$$ {\gamma}_{EQ}=\frac{{\left|{\alpha}_{S{R}_k}\right|}^2{\left|{\beta}_{R_kD}\right|}^2}{\left({\left|{\alpha}_{S{R}_k}\right|}^2+{\left|{\beta}_{R_kD}\right|}^2\right){N}_0},{\gamma}_{EQ}^{\mathit{\prime}}=\frac{{\left|{\alpha}_{S{R}_k}\right|}^4{\left|{\beta}_{R_kD}\right|}^2}{\left({\left|{\alpha}_{S{R}_k}\right|}^2+{\left|{\beta}_{R_kD}\right|}^2\right){N}_0}. $$

(27)

The conditional BER expression given in (26) can be used as the best relay selection criterion which can be given by:

$$ k=\underset{k^{\mathit{\prime}}\in \left\{1,2,\dots, M\right\}}{ \arg \max }{P}_{k^{\mathit{\prime}}}. $$

(28)

Note that this selection criterion in (28) will also be employed in VN-ML and VN-LLR models given in the following sections.

### VN-ML model

After choosing the best relay with the help of criterion in (28), ML technique can be used at the destination for detection. In [11,12], LLR of the ML detection for uncoded DF cooperation with binary modulations is given as:

$$ \mathrm{L}\mathrm{L}\mathrm{R}={t}_0+\psi \left({t}_1\right) $$

(29)

where:

$$ \psi \left({t}_1\right)= \ln \frac{\left(1-{\varepsilon}_r\right){e}^{t_1}+{\varepsilon}_r}{\varepsilon_r{e}^{t_1}+\left(1-{\varepsilon}_r\right)}. $$

(30)

Here, *ε*
_{
r
} is the average BER at the *k*th relay, and it can be calculated for BPSK signaling as [41]:

$$ {\varepsilon}_r=Q\left(\sqrt{\frac{2{E}_s}{N_0}}\left|{\alpha}_{S{R}_k}\right|\right), $$

(31)

*t*
_{0} and *t*
_{1} are log-likelihood ratios of direct and relay links, respectively, for BPSK modulation, which can be given by [41]:

$$ {t}_0=\frac{4\sqrt{E_s}\Re \left\{{y}_{SD}{\alpha}_{SD}^{*}\right\}}{N_0} $$

(32)

$$ {t}_1=\frac{4\sqrt{E_s}\Re \left\{{y}_{R_kD}{\beta}_{R_kD}^{*}\right\}}{N_0}. $$

(33)

As mentioned before, in some scenarios, performance analysis of ML technique can be really hard because of the nonlinear behavior of the ML detection given in (29). To overcome this problem, an alternative detector is proposed in [12] which is called as detector with piecewise-linear (PL) combiner. This approximation can be expressed as:

$$ \psi \left({t}_1\right)\approx {\psi}_{PL}\left({t}_1\right)=\left\{\begin{array}{c}\hfill {T}_1,\kern5em {t}_1\ge {T}_1\hfill \\ {}\hfill {t}_1,\kern2em -{T}_1<{t}_1<{T}_1\hfill \\ {}\hfill -{T}_1,\kern5em {t}_1\le -{T}_1\hfill \end{array}\right\} $$

(34)

where *T*
_{1} = ln[(1 − *ε*
_{
r
})/*ε*
_{
r
}] and *ε*
_{
r
} < 1/2.

### VN-LLR model

In this approach, after the relay selection is realized by using Equation 28, the soft information is transmitted to the destination by amplifying (scaling) and forwarding the LLRs at the output of the decoder (decoder-LLR) [17] instead of making binary hard decisions.

After choosing the best relay with the help of (28), the signal at the relay is given as [18]:

$$ {x}_R(i)={l}_R{E}_{LLR}\kern0.5em ,\kern0.75em i = 1,\dots, Q $$

(35)

where *Q* is the length of source packet, *l*
_{
R
} is the decoder-LLR at the relay and *E*
_{LLR} is the energy of *l*
_{
R
}. *l*
_{
R
} and *E*
_{LLR} can be calculated as:

$$ {l}_R=\frac{4\sqrt{E_s}}{N_0}\Re \left\{{y}_{S{R}_k}{\alpha}_{S{R}_k}\right\}, $$

(36)

$$ {E}_{LLR}=\frac{E_sQ}{{\displaystyle \sum_{i=1}^Q{\left|{l}_R\right|}^2}}. $$

(37)

In the second phase, the relay terminal transmits LLR information *l*
_{R} to the destination. Thus, the received signal at the destination can be expressed as:

$$ {y}_{R_kD}=\sqrt{E_{LLR}}{l}_R\kern0.1em {\beta}_{R_kD}+{n}_{R_kD} $$

(38)

where \( {n}_{{\mathrm{R}}_k\mathrm{D}} \) is the noise term which is AWGN component modeled as complex random variable with zero mean and variance of *N*
_{0}/2 per dimension. Finally, the optimal signal combination at the destination is expressed as:

$$ {l}_{coop}={l}_{SD}+{l}_{R_kD} $$

(39)

where *l*
_{
SD
} and \( {l}_{R_kD} \) are the LLRs of source-destination and *k*th relay-destination channels, respectively, and can be calculated by using [17] as:

$$ {l}_{R_kD}=\frac{16{E}_s{\left|{\alpha}_{S{R}_k}\right|}^2{\left|{\beta}_{R_kD}\right|}^2{E}_{LLR}}{16{E}_s{\left|{\alpha}_{S{R}_k}\right|}^2{\left|{\beta}_{R_kD}\right|}^2{E}_{LLR}^2+{N}_0^2{\left|{\beta}_{R_kD}\right|}^2}\Re \left\{{y}_{R_kD}{\beta}_{R_kD}^{*}\right\} $$

(40)

$$ {l}_{SD}=\frac{4\sqrt{E_s}}{N_0}\Re \left\{{y}_{SD}{\alpha}_{SD}^{*}\right\} $$

(41)