Appendix 1: upper bound for symbol error probability at the destination
Pair-wise error probability (PEP) at the destination can be expressed as
$$\begin{aligned} P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} } \right.} \right) & = P\left( {c_{i} \to c_{i} {\text{ at R}}\left| {c_{i} } \right.} \right) \times P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} } \right.} \right) \\ & \quad + P\left( {c_{i} \to c_{j} {\text{ at R}}\left| {c_{i} } \right.} \right) \times \left( {1 - P\left( {c_{i} \to c_{i} {\text{ at D}}\left| {c_{i} \to c_{j} {\text{ at R}},c_{i} } \right.} \right)} \right) \\ \end{aligned}$$
(35)
where \(P\left( {c_{i} \to c_{i} {\text{ at R}}\left| {c_{i} } \right.} \right)\) and \(P\left( {c_{i} \to c_{j} {\text{ at R}}\left| {c_{i} } \right.} \right)\) denote the probability of correct decoding probability and PEP at the relay respectively when CFNC symbol c
i
is sent. Also, \(P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} } \right.} \right)\) represents the PEP at the destination given that c
i
is sent and the decision of the relay is correct, whereas \(P\left( {c_{i} \to c_{i} {\text{ at D}}\left| {c_{i} \to c_{j} {\text{ at R}},c_{i} } \right.} \right)\) is the probability of making correct decision of the destination when c
i
is sent and the relay reaches an erroneous decision.
The PEP of the relay (by assuming that CSI is known at the relay and ML relaying is used) becomes
$$\begin{aligned} P\left( {c_{i} \to c_{j} {\text{ at R }}\left| {c_{i} } \right.,{\mathbf{h}}_{sr} } \right) & = P \left(\left| {\sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} ({\mathbf{x}}_{i} )_{k} } + z_{r} - \sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} ({\mathbf{x}}_{i} )_{k} } } \right|^{2}\right. \\ & \quad\left. \ge \left| {\sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} ({\mathbf{x}}_{i} )_{k} } + z_{r} - \sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} ({\mathbf{x}}_{j} )_{k} } } \right|^{2}\right) \\ & = P\left( {\left| {z_{r} } \right|^{2} \ge \left| {\sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} ({\mathbf{x}}_{i} )_{k} } + z_{r} - \sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} ({\mathbf{x}}_{j} )_{k} } } \right|^{2} } \right) \\ & = P\left( { \, - \left| {\sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right|^{2} \ge z_{r}^{*} \left( {\sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right) +\, z_{r} \left( {\sum\limits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right)^{*} } \right) \\ \end{aligned}$$
(36)
where “*” denotes the complex conjugation and the random variable \(z_{r}^{*} \left( {\sum\nolimits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right) + z_{r} \left( {\sum\nolimits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right)^{*}\) is Gaussian distributed with zero mean and variance of \(4\sigma^{2} \left| {\sum\nolimits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right|^{2}\). Therefore, the PEP in Eq. (36) can be found as
$$P\left( {c_{i} \to c_{j} {\text{ at R }}\left| {c_{i} } \right.,{\mathbf{h}}_{sr} } \right) = Q\left( {\frac{{\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right|}}{2\sigma }} \right)$$
(37)
where d
ijk
represents the kth component of the difference vector between the ith and jth decision vectors, \({\mathbf{d}}_{ij} = ({\mathbf{x}}_{i} - {\mathbf{x}}_{j} )\). This probability can be upper bounded using the Chernoff-bound as follows:
$$P\left( {c_{i} \to c_{j} {\text{ at R }}\left| {c_{i} } \right.,{\mathbf{h}}_{sr} } \right) \le \frac{1}{2}e^{{ - \frac{{\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right|^{2} }}{{8\sigma^{2} }}}}$$
(38)
Consequently, a bound on the average PEP can be obtained by averaging the upper-bound in Eq. (38) over fading gains of the users-to-relay links as:
$$P\left( {c_{i} \to c_{j} {\text{ at R }}\left| {c_{i} } \right.,{\mathbf{h}}_{sr} } \right) \le E_{{{\text{h}}_{sr} }} \left[ {e^{{ - \frac{{\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right|^{2} }}{{8\sigma^{2} }}}} } \right]$$
(39)
\(E_{{{\mathbf{h}}_{sr} }} [.]\) represents the expectation operation with respect to the CSI vector \({\mathbf{h}}_{sr}\). Each fading coefficient \(h_{{s_{k} r}}\) is assumed to be a zero-mean complex Gaussian random variable with unit variance, which is denoted by CN (0, 1). Hence, the distribution of random variable \(\sum\nolimits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} }\) is CN
\(CN\left( {0, \sum\nolimits_{k = 1}^{N} {g_{k} } \left| {\theta_{k} } \right|^{ 2} d_{ijk}^{ 2} } \right)\), and the pdf of \(\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {g_{k} } h_{{s_{k} r}} \theta_{k} d_{ijk} } } \right|^{2}\) becomes exponential with the mean of \(\sum\nolimits_{k = 1}^{N} {g_{k} \left| {\theta_{k} } \right|^{ 2} } d_{ijk}^{ 2}\). Hence, the expectation term in Eq. (39) can be deduced as:
$$\begin{aligned} P\left( {c_{i} \to c_{j} {\text{ at R}}\left| {c_{i} } \right.} \right) & \le \int\limits_{0}^{\infty } {\frac{1}{2}e^{{ - \frac{t}{{8\sigma^{2} }}}} } \frac{1}{{\sum\nolimits_{k = 1}^{N} {g_{k} \left| {\theta_{k} } \right|^{2} d_{ijk}^{2} } }}e^{{\frac{ - t}{{\sum\nolimits_{k = 1}^{N} {g_{k} \left| {\theta_{k} } \right|^{2} d_{ijk}^{2} } }}}} dt \\ & \le \frac{0.5}{{\left( {1 + \frac{{\sum\nolimits_{k = 1}^{N} {g_{k} \left| {\theta_{k} } \right|^{2} d_{ijk}^{2} } }}{{8\sigma^{2} }}} \right)}} \\ \end{aligned}$$
(40)
Similar to Eq. (36), the PEP at the destination with complete CSI can be calculated, when it employs ML detection and the decoding of the relay is correct, as:
$$\begin{aligned} P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} ,h_{sd} ,h_{rd} } \right.} \right) & = P\left(\left| {\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} (x_{i} )_{k} } + z_{d} - \sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} (x_{i} )_{k} } } \right|^{2} \right.\\ & \quad \left.+\,\left| {\sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} (x_{i} )_{k} } + z_{d} - h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} (x_{i} )_{k} } } \right|^{2} \right.\\ & \quad\left. \ge \left| {\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} (x_{i} )_{k} } + z_{d} - \sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} (x_{j} )_{k} } } \right|^{2} \right.\\ & \quad \left.+\,\left| {\sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} (x_{i} )_{k} } + z_{d} - \sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} (x_{j} )_{k} } } \right|^{2} \right) \\ & = P\left( {\left| {z_{d} } \right|^{2} + \left| {z_{d} } \right|^{2} \ge \left| {\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } + z_{d} } \right|^{2} +\,\left| {\sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} d_{ijk} } + z_{d} } \right|^{2} } \right) \\ & = P\left( - \left| {\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } } \right|^{2} - \left| {\sqrt {g_{r} } h_{rd}\sum\limits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} \right.\\ & \quad \left.\ge 2\text{Re} \left\{ z_{d}^{*} \sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } \right\} + 2\text{Re} \left\{ z_{d}^{*} \sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} d_{ijk} } \right\} \right) \\\end{aligned}$$
(41)
Since z
d
is assumed to be CN (0, 2σ
2), the random variable \(2\text{Re} \{ z_{d}^{*} \sum\nolimits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } \} + 2\text{Re} \{ z_{d}^{*} \sqrt {g_{r} } h_{rd} \sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } \}\) is Gaussian with mean of zero and variance of \(4\sigma^{2} \left( {\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } } \right|^{2} + \left| {\sqrt {g_{r} } h_{rd} \sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} } \right)\). Hence, the PEP in Eq. (41) can be determined as
$$P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} ,{\mathbf{h}}_{sd} ,h_{rd} } \right.} \right) = Q\left( {\frac{{\sqrt {\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } } \right|^{2} + \left| {\sqrt {g_{r} } h_{rd} \sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} } }}{2\sigma }} \right)$$
(42)
which is upper bounded by
$$P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} ,{\mathbf{h}}_{sd} ,h_{rd} } \right.} \right) \le \frac{1}{2}e^{{ - \frac{{\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } } \right|^{2} + \left| {\sqrt {g_{r} } h_{rd} \sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} }}{{8\sigma^{2} }}}}$$
(43)
Again, a bound on the average PEP at the destination can be obtained by averaging the upper-bound in Eq. (43) over the fading coefficients \(h_{{s_{k} d}}\), h
rd
. Since fading coefficients are zero-mean complex Gaussian random variables with unit variance, \(\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } } \right|^{2}\) and \(\left| {\sqrt {g_{r} } h_{rd} \sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2}\) are exponential random variables with a mean of \(\lambda_{1} = \sum\nolimits_{k = 1}^{N} {\gamma_{k} \left| {\theta_{k} } \right|^{2} d_{ijk}^{2} }\) and \(\lambda_{2} = g_{r} \sum\nolimits_{k = 1}^{N} {\left| {\theta_{k} } \right|^{2} d_{ijk}^{2} }\) , respectively. So, the bound on the average PEP at the destination can be obtained as:
$$\begin{aligned} P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} } \right.} \right) & \le \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {0.5\exp \left( { - \frac{{t_{1} + t_{2} }}{{8\sigma^{2} }}} \right)} } \frac{{\exp \left( { - \frac{{t_{1} }}{{\lambda_{1} }}} \right)}}{{\lambda_{1} }}\frac{{\exp \left( { - \frac{{t_{2} }}{{\lambda_{2} }}} \right)}}{{\lambda_{2} }}dt_{1} dt_{2} \\ & \le \frac{0.5}{{\left( {1 + \frac{{\sum\nolimits_{k = 1}^{N} {\gamma_{k} \left| {\theta_{k} } \right|^{2} d_{ijk}^{2} } }}{{8\sigma^{2} }}} \right)\left( {1 + \frac{{\left| {g_{r} \sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} }}{{8\sigma^{2} }}} \right)}} \\ \end{aligned}$$
(44)
A similar analysis can be conducted to determine\(P\left( {c_{i} \to c_{i} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} } \right.} \right)\). Toward that goal, we need to determine first, \(P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} } \right.} \right)\) as:
$$\begin{aligned} P\left( {c_{i} \to c_{i} {\text{ at D}}\left|{c_{i} \to c_{i} {\text{ at R}},c_{i},{\mathbf{h}}_{{{\mathbf{sd}}}} ,h_{rd} } \right.} \right) & =P\left( \, \left| {\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} }h_{{s_{k} d}} \theta_{k} ({\mathbf{x}}_{i} )_{k} } + z_{d} -\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}}\theta_{k} ({\mathbf{x}}_{i} )_{k} } } \right|^{2} \right.\\ &\quad \left.+ \left| {\sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N}{\theta_{k} ({\mathbf{x}}_{i} )_{k} } + z_{d} - h_{rd}\sum\limits_{k = 1}^{N} {\theta_{k} ({\mathbf{x}}_{i} )_{k} } }\right|^{2} \right.\\ & \quad \left.\le \left| {\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} ({\mathbf{x}}_{i} )_{k} } + z_{d} - \sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} ({\mathbf{x}}_{j} )_{k} } } \right|^{2} \right.\\ & \quad\left. + \left| {\sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} ({\mathbf{x}}_{i} )_{k} } + z_{d} - \sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} ({\mathbf{x}}_{j} )_{k} }} \right|^{2} \right) \\ & = P\left( \, \left| {z_{d} } \right|^{2} + \left| {z_{d} } \right|^{2} \ge \left| {\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } + z_{d} } \right|^{2} + \left| {\sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} d_{ijk} } + z_{d} } \right|^{2} \right) \\ & = P\left( \, - \left|{\sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } } \right|^{2} + \left| {\sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2}\right. \\ & \quad \left.\ge 2\text{Re} \left\{ z_{d}^{*} \sum\limits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } \right\} + 2\text{Re} \left\{ z_{d}^{*} \sqrt {g_{r} } h_{rd} \sum\limits_{k = 1}^{N} {\theta_{k} d_{ijk} }\right\} \, \right) \\ & = Q\left( {\frac{{ - \left| {\sum\nolimits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } } \right|^{2} + \left| {\sqrt {g_{r} } h_{rd} \sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} }}{{2\sigma \sqrt {\left| {\sum\nolimits_{k = 1}^{N} {\sqrt {\gamma_{k} } h_{{s_{k} d}} \theta_{k} d_{ijk} } } \right|^{2} + \left| {\sqrt {g_{r} } h_{rd} \sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} } }} \, } \right) \\ \end{aligned}$$
(45)
In order to obtain a bound on the \(P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} \to c_{i} {\text{ at R}},c_{i} } \right.} \right)\), we need to average Eq. (45) over CSI coefficients, which cannot be calculated analytically. For simplicity, we bound the fourth term in Eq. (35) as:
$$1 - P\left( {c_{i} \to c_{i} {\text{ at D}}\left| {c_{i} \to c_{j} {\text{ at R}},c_{i} } \right.} \right) \le 1$$
(46)
In parallel lines, the first term in Eq. (35) is also bounded as:
$$P\left( {c_{i} \to c_{i} {\text{ at R}}\left| {c_{i} } \right.} \right) \le 1$$
(47)
Therefore, the upper bound for PEP at the destination in Eq. (35) can be re-written by combining Eqs. (40), (44), (46) and (47) as:
$$P\left( {c_{i} \to c_{j} {\text{ at D}}\left| {c_{i} } \right.} \right) \le \frac{0.5}{{1 + \frac{{\sum\nolimits_{k = 1}^{N} {\gamma_{k} |\theta_{k} |^{2} d_{ijk}^{2} } }}{{8\sigma^{2} }}}}\frac{1}{{1 + \frac{{g_{r} \left| {\sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} }}{{8\sigma^{2} }}}} + \frac{0.5}{{1 + \frac{{\sum\nolimits_{k = 1}^{N} {g_{k} |\theta_{k} |^{2} d_{ijk}^{2} } }}{{8\sigma^{2} }}}}$$
(48)
As a result, the upper bound for SEP (\(\bar{P}_{e}^{D}\)) at the destination can be given as:
$$\bar{P}_{e}^{D} \le \frac{1}{8}\sum\limits_{i = 1}^{{2^{N} }} {\sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{{2^{N} }} {\frac{1}{{1 + \frac{{\sum\nolimits_{k = 1}^{N} {\gamma_{k} |\theta_{k} |^{2} d_{ijk}^{2} } }}{{8\sigma^{2} }}}}} } \frac{1}{{1 + \frac{{g_{r} \left| {\sum\nolimits_{k = 1}^{N} {\theta_{k} d_{ijk} } } \right|^{2} }}{{8\sigma^{2} }}}} + \frac{1}{{1 + \frac{{\sum\nolimits_{k = 1}^{N} {g_{k} |\theta_{k} |^{2} d_{ijk}^{2} } }}{{8\sigma^{2} }}}}$$
(49)
By expressing the signature θ
k
in the polar form as \(\theta_{k} = \sqrt {P_{k} } e^{{j\phi_{k} }}\), Eq. (49) can be re-written for N = 2 as:
$$\begin{aligned} {{\bar{P}}_{e}^{D}} &\le \frac{1}{8}\sum\limits_{i = 1}^{4} {\sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{4} {\frac{1}{{1 + \frac{{\sum\nolimits_{k = 1}^{2} {\gamma_{k} P_{k} d_{ijk}^{2} } }}{{8\sigma^{2} }}}}} } \frac{1}{{1 + \frac{{g_{r} \left| {\sum\nolimits_{k = 1}^{2} {\sqrt {P_{k} } e^{{j\phi_{k} }} d_{ijk} } } \right|^{2} }}{{8\sigma^{2} }}}} + \frac{1}{{1 + \frac{{\sum\nolimits_{k = 1}^{2} {g_{k} P_{k} d_{ijk}^{2} } }}{{8\sigma^{2} }}}} \\ & \quad = \frac{1}{8}\sum\limits_{i = 1}^{4} {\sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{4} {\frac{1}{{1 + \frac{{\sum\nolimits_{k = 1}^{2} {\gamma_{k} P_{k} d_{ijk}^{2} } }}{{8\sigma^{2} }}}}} } \frac{1}{{1 + \frac{{g_{r} \left( {P_{1} d_{ij1}^{2} + P_{2} d_{ij2}^{2} + 2\sqrt {P_{1} } d_{ij1} \sqrt {P_{2} } d_{ij2} \cos (\phi_{1} - \phi_{2} )} \right)}}{{8\sigma^{2} }}}} + \frac{1}{{1 + \frac{{\sum\nolimits_{k = 1}^{2} {g_{k} P_{k} d_{ijk}^{2} } }}{{8\sigma^{2} }}}} \end{aligned}$$
(50)
Appendix 2: KKT conditions for the ser bound minimization problem at the destination
It is important to note that the average SEP bound in Eq. (11) is convex in powers P
1 and P
2, the cosine of the phase difference δ ≡ cos (ϕ
1 − ϕ
2). In order to determine the signatures of the users, both P
k
and ϕ
k
need to be optimally decided by minimizing the bound in Eq. (11) under the total power constraint \(\sum\nolimits_{k = 1}^{2} {P_{k} = P_{T} }\) together with the fact that each user actively sends information (i.e., \(P_{k} > 0{\text{ for }}k = 1,2\)). Additionally, there is a box constraint on δ such that −1 ≤ δ ≤ 1. Therefore, we can state the determination of optimal signatures as a constrained optimization problem.
$$\begin{aligned} & \mathop {\text{minimize}}\limits_{{P_{1} ,P_{2} ,\delta }} f_{0} (P_{1} ,P_{2} ,\delta ) =\, \frac{1}{8}\sum\limits_{i = 1}^{4} \mathop{\sum}\limits_{\mathop{ j = 1}\limits_{j \ne i}}^{4} {\frac{1}{{1 +\,\frac{{\sum\limits_{k = 1}^{2} {\gamma_{k} P_{k} d_{ijk}^{2} } }}{{8\sigma^{2} }}}}}\frac{1}{{1 +\,\frac{{g_{r} \left( {P_{1} d_{ij1}^{2} + P_{2} d_{ij2}^{2} + 2\sqrt {P_{1} } d_{ij1} \sqrt {P_{2} } d_{ij2} \delta } \right)}}{{8\sigma^{2} }}}} + \frac{1}{{1 +\,\frac{{\sum\limits_{k = 1}^{2} {g_{k} P_{k} d_{ijk}^{2} } }}{{8\sigma^{2} }}}} \\ & {\text{such that}} \\ &f_{1} (P_{1} ,P_{2} ,\delta ) = P_{T} - \sum\limits_{k = 1}^{2} {P_{k} = 0} \\ &f_{2} (P_{1} ,P_{2} ,\delta ) = P_{1} > 0, \hfill \\ &f_{3} (P_{1} ,P_{2} ,\delta ) = P_{2} > 0, \, \\ &f_{4} (P_{1} ,P_{2} ,\delta ) = \delta + 1 \ge 0, \\ &f_{5} (P_{1} ,P_{2} ,\delta ) = 1 - \delta \ge 0, \\ \end{aligned}$$
(51)
Since the objective function is convex and the constraints are affine, the optimization in Eq. (51) is a convex program, and its unique optimal global solution exists, which satisfies Karush–Kuhn–Tucker (KKT) conditions:
$$\begin{aligned} \nabla f_{0} \left( {{\mathbf{p}}^{*} } \right) - \sum\limits_{i = 1}^{5} {\lambda_{i}^{*} \nabla f_{i} \left( {{\mathbf{p}}^{*} } \right)} & = {\mathbf{0}}, \\ f_{1} \left( {{\mathbf{p}}^{*} } \right) & = 0, \\ f_{i} \left( {{\mathbf{p}}^{*} } \right) & > 0,{\text{ for }}i = 2,3 \\ f_{i} \left( {{\mathbf{p}}^{*} } \right) & \ge 0,{\text{ for }}i = 4,5 \\ \lambda_{i}^{*} & \ge 0,{\text{for }}i = 1,2,3,4,5 \\ \lambda_{i}^{*} f_{i} \left( {{\mathbf{p}}^{*} } \right) & = 0,{\text{for }}i = 1,2,3,4,5 \\ \end{aligned}$$
(52)
where \({\mathbf{p}} = [P_{1} ,P_{2} ,\delta ]^{t}\) is the vector of user signature parameters, and \({\mathbf{p}}^{*}\) represents the corresponding optimal vector.
The KKT conditions in Eq. (52) result in the following equations as:
$$\begin{aligned} & - \left( {1 + \frac{{\gamma_{1} P_{1} }}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{\gamma_{1} }}{{4\sigma^{2} }}} \right)\left( {1 + \frac{{g_{r} P_{1} }}{{2\sigma^{2} }}} \right)^{ - 1} - \left( {1 + \frac{{\gamma_{1} P_{1} }}{{2\sigma^{2} }}} \right)^{ - 1} \left( {1 + \frac{{g_{r} P_{1} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{g_{r} }}{{4\sigma^{2} }} - \left( {1 + \frac{{g_{1} P_{1} }}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{1} }}{{4\sigma^{2} }}} \right) - \left( {1 + \frac{{g_{1} P_{1} + g_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{1} }}{{4\sigma^{2} }}} \right) \\ & \quad - \left( {1 + \frac{{\gamma_{1} P_{1} + \gamma_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{\gamma_{1} }}{{4\sigma^{2} }}} \right)\left[ {\left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} + 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 1} + \left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} - 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 1} } \right] \\ & \quad - \left( {1 + \frac{{\gamma_{1} P_{1} + \gamma_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 1} \left[ {\left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} + 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{r} \left( {1 + \frac{{\sqrt {P_{2} } \delta }}{{\sqrt {P_{1} } }}} \right)}}{{8\sigma^{2} }}} \right) + \left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} - 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{r} \left( {1 - \frac{{\sqrt {P_{2} } \delta }}{{\sqrt {P_{1} } }}} \right)}}{{8\sigma^{2} }}} \right)} \right] \\ & \quad - \lambda_{1}^{*} \left( { - 1} \right) - \lambda_{2}^{*} \left( 1 \right) = 0 \\ \end{aligned}$$
(53)
$$\begin{aligned} & - \left( {1 + \frac{{\gamma_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{\gamma_{2} }}{{4\sigma^{2} }}} \right)\left( {1 + \frac{{g_{r} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 1} - \left( {1 + \frac{{\gamma_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 1} \left( {1 + \frac{{g_{r} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{g_{r} }}{{4\sigma^{2} }} - \left( {1 + \frac{{g_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{2} }}{{4\sigma^{2} }}} \right) - \left( {1 + \frac{{g_{1} P_{1} + g_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{2} }}{{4\sigma^{2} }}} \right) \\ & \quad - \left( {1 + \frac{{\gamma_{1} P_{1} + \gamma_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{\gamma_{2} }}{{4\sigma^{2} }}} \right)\left[ {\left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} + 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 1} + \left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} - 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 1} } \right] \\ & \quad - \left( {1 + \frac{{\gamma_{1} P_{1} + \gamma_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 1} \left[ {\left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} + 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{r} \left( {1 + \frac{{\sqrt {P_{1} } \delta }}{{\sqrt {P_{2} } }}} \right)}}{{8\sigma^{2} }}} \right) + \left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} - 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{r} \left( {1 - \frac{{\sqrt {P_{1} } \delta }}{{\sqrt {P_{2} } }}} \right)}}{{8\sigma^{2} }}} \right)} \right] \\ & \quad \lambda_{1}^{*} \left( { - 1} \right) - \lambda_{3}^{*} \left( 1 \right) = 0 \\ \end{aligned}$$
(54)
$$\begin{aligned} & - \left( {1 + \frac{{\gamma_{1} P_{1} + \gamma_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 1} \left[ {\left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} + 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{g_{r} 2\sqrt {P_{1} P_{2} } }}{{8\sigma^{2} }}} \right) + \left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} - 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} \left( {\frac{{ - g_{r} 2\sqrt {P_{1} P_{2} } }}{{8\sigma^{2} }}} \right)} \right] \\ & \quad - \lambda_{4}^{*} + \lambda_{5}^{*} = 0 \\ \end{aligned}$$
(55)
Since \(P_{ 1}^{*} > 0\) and \(P_{ 2}^{*} > 0\), both \(\lambda_{ 2}^{*}\) and \(\lambda_{ 3}^{*}\) become zero due to complementarity conditions. Also, because of the complementarity conditions \(\lambda_{i}^{*} f_{i} \left( {{\mathbf{p}}^{*} } \right) = 0\) for i = 4 and 5, \(\lambda_{ 4}^{*}\) and \(\lambda_{ 5}^{*}\) can be shown to be zero, otherwise δ should be equal to either −1 or 1. We can prove this by contradiction. Assuming that \(\delta^{*}\) is −1, \(\lambda_{ 5}^{*}\) has to be zero due to complementarity slackness. Therefore, the following expression from Eq. (55) can be stated:
$$\left[ {\left( {1 + \frac{{\gamma_{1} P_{1} + \gamma_{2} P_{2} }}{{2\sigma^{2} }}} \right)^{ - 1} \left( {\frac{{g_{r} 2\sqrt {P_{1} P_{2} } }}{{2\sigma^{2} }}} \right)} \right] \times \left[ { - \left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} - 2\sqrt {P_{1} P_{2} } } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} + \left( {1 + \frac{{g_{r} (P_{1} + P_{2} + 2\sqrt {P_{1} P_{2} } )}}{{2\sigma^{2} }}} \right)^{ - 2} } \right] = \lambda_{4}^{*}$$
(56)
Since \(\left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} - 2\sqrt {P_{1} P_{2} } } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} > \left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} + 2\sqrt {P_{1} P_{2} } } \right)}}{{2\sigma^{2} }}} \right)^{ - 2}\), the LHS of Eq. (56) is less than zero, which is a contradiction because \(\lambda_{ 4}^{*} \ge 0\). Similarly, we can show that \(\lambda_{ 4}^{*}\) should be zero. Hence, by putting \(\lambda_{ 4}^{*} = \lambda_{ 5}^{*} = 0\) in Eq. (55), we obtain the following expression:
$$\left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} + 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 2} = \left( {1 + \frac{{g_{r} \left( {P_{1} + P_{2} - 2\sqrt {P_{1} P_{2} } \delta } \right)}}{{2\sigma^{2} }}} \right)^{ - 2}$$
(57)
The only solution of Eq. (57) is δ = 0, which implies
$$\phi_{2}^{*} - \phi_{1}^{*} = (2k + 1)\pi /2{\text{ for any integer }}k$$
(58)
Combining Eq. (57) with Eqs. (53) and (54), the optimal powers \(P_{ 1}^{*}\) and \(P_{ 2}^{*}\) should satisfy the following expressions:
$$\begin{aligned} & 0.5\left( {1 + \frac{{\gamma_{1} P_{1}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{\gamma_{1} }}{{2\sigma^{2} }}\left( {1 + \frac{{g_{r} P_{1}^{*} }}{{2\sigma^{2} }}} \right)^{ - 1} +\,0.5\left( {1 + \frac{{\gamma_{1} P_{1}^{*} }}{{2\sigma^{2} }}} \right)^{ - 1} \left( {1 + \frac{{g_{r} P_{1}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{g_{r} }}{{2\sigma^{2} }} \\ & \quad \quad +\,0.5\left( {1 + \frac{{\gamma_{1} P_{1}^{*} + \gamma_{2} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{\gamma_{1} }}{{2\sigma^{2} }}\left( {1 + \frac{{g_{r} \left( {P_{1}^{*} + P_{2}^{*} } \right)}}{{2\sigma^{2} }}} \right)^{ - 1} +\,0.5\left( {1 + \frac{{g_{1} P_{1}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{g_{1} }}{{2\sigma^{2} }} +\,0.5\left( {1 + \frac{{g_{1} P_{1}^{*} + g_{2} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{g_{1} }}{{2\sigma^{2} }} \\ & \quad = 0.5\left( {1 + \frac{{\gamma_{2} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{\gamma_{2} }}{{2\sigma^{2} }}\left( {1 + \frac{{g_{r} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 1} +\,0.5\left( {1 + \frac{{\gamma_{2} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 1} \left( {1 + \frac{{g_{r} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{g_{r} }}{{2\sigma^{2} }} \\ & \quad \quad +\,0.5\left( {1 + \frac{{\gamma_{1} P_{1}^{*} + \gamma_{2} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{\gamma_{2} }}{{2\sigma^{2} }}\left( {1 + \frac{{g_{r} \left( {P_{1}^{*} + P_{2}^{*} } \right)}}{{2\sigma^{2} }}} \right)^{ - 1} +\,0.5\left( {1 + \frac{{g_{2} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{g_{2} }}{{2\sigma^{2} }} +\,0.5\left( {1 + \frac{{g_{1} P_{1}^{*} + g_{2} P_{2}^{*} }}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{g_{2} }}{{2\sigma^{2} }} \\ \end{aligned}$$
(59)
Appendix 3
The first and second derivative of f(x) in Eq. (28) can be determined as
$$\begin{aligned} f^{'} \left( x \right) & = - \left( {1 + \frac{{g_{1} x}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2g_{1}^{2} }}{{4\sigma^{4} }} - \left( {1 + \frac{{\gamma_{1} x}}{{2\sigma^{2} }}} \right)^{ - 1} \left( {1 + \frac{{g_{r} x}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2g_{r}^{2} }}{{4\sigma^{4} }} - \left( {1 + \frac{{\gamma_{1} x}}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{2\gamma_{1} g_{r} }}{{4\sigma^{4} }}\left( {1 + \frac{{g_{r} x}}{{2\sigma^{2} }}} \right)^{ - 2} \\ & \quad - \left( {1 + \frac{{\gamma_{1} x}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2\gamma_{1}^{2} }}{{4\sigma^{4} }}\left( {1 + \frac{{g_{r} x}}{{2\sigma^{2} }}} \right)^{ - 1} - \left( {1 + \frac{{g_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2g_{2}^{2} }}{{4\sigma^{4} }} - \left( {1 + \frac{{\gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 1} \left( {1 + \frac{{g_{r} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2g_{r}^{2} }}{{4\sigma^{4} }} \\ & \quad - \left( {1 + \frac{{\gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{2\gamma_{2} g_{r} }}{{4\sigma^{4} }}\left( {1 + \frac{{g_{r} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 2} - \left( {1 + \frac{{\gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2\gamma_{2}^{2} }}{{4\sigma^{4} }}\left( {1 + \frac{{g_{r} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 1} \\ & \quad - \left( {1 + \frac{{g_{1} x + g_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2g_{1} (g_{1} - g_{2} )}}{{4\sigma^{4} }} + \left( {1 + \frac{{g_{1} x + g_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2g_{2} (g_{1} - g_{2} )}}{{4\sigma^{4} }} \\ & \quad - \left( {1 + \frac{{\gamma_{1} x + \gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2\gamma_{1} (\gamma_{1} - \gamma_{2} )}}{{4\sigma^{4} }}\left( {1 + \frac{{g_{r} (P_{T} )}}{{2\sigma^{2} }}} \right)^{ - 1} + \left( {1 + \frac{{\gamma_{1} x + \gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{2\gamma_{2} (\gamma_{1} - \gamma_{2} )}}{{4\sigma^{4} }}\left( {1 + \frac{{g_{r} (P_{T} )}}{{2\sigma^{2} }}} \right)^{ - 1} \\ \end{aligned}$$
(60)
$$\begin{aligned} f^{''} (x) & = \left( {1 + \frac{{g_{1} x}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6g_{1}^{3} }}{{8\sigma^{6} }} + \left( {1 + \frac{{\gamma_{1} x}}{{2\sigma^{2} }}} \right)^{ - 1} \left( {1 + \frac{{g_{r} x}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6g_{r}^{3} }}{{8\sigma^{6} }} + \left( {1 + \frac{{\gamma_{1} x}}{{2\sigma^{2} }}} \right)^{ - 2} \left( {1 + \frac{{g_{r} x}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{6\gamma_{1} g_{r}^{2} }}{{8\sigma^{6} }} \\ & \quad + \left( {1 + \frac{{\gamma_{1} x}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{6\gamma_{1}^{2} g_{r} }}{{8\sigma^{6} }}\left( {1 + \frac{{g_{r} x}}{{2\sigma^{2} }}} \right)^{ - 2} + \left( {1 + \frac{{\gamma_{1} x}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6\gamma_{1}^{3} }}{{8\sigma^{6} }}\left( {1 + \frac{{g_{r} x}}{{2\sigma^{2} }}} \right)^{ - 1} \\ & \quad - \left( {1 + \frac{{g_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6g_{2}^{3} }}{{8\sigma^{6} }} - \left( {1 + \frac{{\gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 1} \left( {1 + \frac{{g_{r} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6g_{r}^{3} }}{{8\sigma^{6} }} \\ & \quad - \left( {1 + \frac{{\gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 2} \frac{{6\gamma_{2} g_{r}^{2} }}{{8\sigma^{6} }}\left( {1 + \frac{{g_{r} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} - \left( {1 + \frac{{\gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 3} \frac{{6\gamma_{2}^{2} g_{r} }}{{8\sigma^{6} }}\left( {1 + \frac{{g_{r} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 2} \\ & \quad - \left( {1 + \frac{{\gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6\gamma_{2}^{3} }}{{8\sigma^{6} }}\left( {1 + \frac{{g_{r} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 1} + \left( {1 + \frac{{g_{1} x + g_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6g_{1} (g_{1} - g_{2} )^{2} }}{{8\sigma^{6} }} \\ & \quad - \left( {1 + \frac{{g_{1} x + g_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6g_{2} (g_{1} - g_{2} )^{2} }}{{8\sigma^{6} }} + \left( {1 + \frac{{\gamma_{1} x + \gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6\gamma_{1} (\gamma_{1} - \gamma_{2} )^{2} }}{{8\sigma^{6} }}\left( {1 + \frac{{g_{r} (P_{T} )}}{{2\sigma^{2} }}} \right)^{ - 1} \\ & \quad - \left( {1 + \frac{{\gamma_{1} x + \gamma_{2} (P_{T} - x)}}{{2\sigma^{2} }}} \right)^{ - 4} \frac{{6\gamma_{2} (\gamma_{1} - \gamma_{2} )^{2} }}{{8\sigma^{6} }}\left( {1 + \frac{{g_{r} (P_{T} )}}{{2\sigma^{2} }}} \right)^{ - 1} \\ \end{aligned}$$
(61)
