1 Introduction

Channel state information (CSI) cannot be obtained perfectly because of the complexity of electromagnetic wave spreading and transmitting delay. Thus, CSI has estimation errors at the receiver. Janarthanan and Bhaskar (2013) and Khuong and Sofotasios (2013) analyzed the capacity and bit-error-rate in single-input multiple-output (SIMO) and multi-hop cooperation systems with CSI estimation errors, respectively.

Due to the broadcast nature of wireless links, it is difficult to prevent eavesdroppers from overhearing wireless communications (Liu Y et al., 2015; 2016). Thus, security issues play an important role in wireless networks. Physical layer security has been considered widely as an effective technology to prevent information from being intercepted (Shiu et al., 2011; Yang et al., 2015). In different fading scenarios, Sun et al. (2012), Zhang et al. (2014), Pan et al. (2015; 2016), Lei et al. (2015; 2016), and Liu H et al. (2016a) studied the secrecy performance over independent/correlated Rayleigh, log-normal, Rayleigh-log-normal, generalized-K, and generalized Gamma fading channels. In addition, maximal ratio combining (MRC) was used to improve the secrecy performance in He et al. (2011), Alves et al. (2012), Yang et al. (2013a; 2013b; 2013c), Wang et al. (2014a; 2014b), and Pan et al. (2015).

As an ideal diversity for linear systems, MRC is the best among the basic linear diversity combining schemes. However, due to the fact that the diversity weighting factors are proportional to the complex conjugate of the channel fading vector in time, when the pilot frequency range and channel bandwidth are similar, it is easy to cause a Gaussian channel estimation error, resulting in nonideal MRC diversity and degraded output-combined signal-to-noise ratio (SNR) (Gans, 1971). In addition, to isolate pilot from the signal in time, the signal and pilot are transmitted alternately. When this separation time is on the order of the reciprocal of the fading rate, the weighted factor obtained from the pilot will also have a Gaussian channel estimation error (Tomiuk et al., 1999).

For the situation where the receiver and the eavesdropper show Gaussian errors at the same time, Shrestha and Kwark (2014) and Hu and Tao (2015) extended the work of Gans (1971) and Tomiuk et al. (1999), and studied the secrecy outage probability in SIMO and multiple-input multiple-output (MIMO) wiretap channels. Zhao and Pan (2016) investigated the secrecy outage performance of decode-and-forward and randomize-and-forward cooperative systems, considering the MRC scheme with weighting errors.

However, Shrestha and Kwark (2014), Hu and Tao (2015), and Zhao and Pan (2016) did not consider the ergodic secrecy capacity (ESC), which is one of the most important parameters in physical layer security (Wang et al., 2014a; 2014b). Because of the rapid increase in the demand for wireless communication services, the capacity of fading channels is increasingly becoming a main concern in the design of wireless communication systems (Khatalin and Fonseka, 2006). In a Gaussian noise environment, the capacity is constant, because the carrier-to-noise ratio is constant. In fading channels, such as Rayleigh fading channels, the SNR of receivers varies with time. This offers an explanation why the capacity of fading channels has to be calculated in an average sense (Lee, 1990; Alouini and Goldsmith, 1999; Simon and Alouini, 2005). The derivation of a closed-form expression for the capacity over fading channels is of higher computational complexity compared to that of the outage probability, especially the ESC over wiretap channels, as logarithmic functions are presented in the integral equation (Rezki et al., 2014).

Though MRC has better combining performance than selection combining (SC), SC has a lower complexity. Thus, SC is also a common combining technology in practical applications. In recent years, it has been adopted widely to improve the secrecy performance in the physical layer (Ferdinand et al., 2013; Yang et al., 2013a; 2013b; 2013c; Elkashlan et al., 2015). However, due to the transmitting delay, the CSI obtained from the pilot at the receiver may be outdated, leading to an imperfect SC and the degradation of the combined SNR.

Motivated by the above observations, we analyze the secrecy performance of a SIMO wiretap system, where a source equipped with a single antenna transmits confidential messages to the destination equipped with M (M ≥ 1) antennas using the MRC/SC scheme to process the received multiple signals. Meanwhile, an eavesdropper, which is equipped with N (N ≥ 1) antennas, adopts the MRC/SC scheme to promote successful eavesdropping. We derive the exact and asymptotic closed-form expressions for the ESC over Rayleigh fading channels under two cases: (1) MRC with weighting errors and (2) SC with outdated CSI. Further, a high SNR slope and a high SNR power offset are derived, which govern ESC in the high SNR region.

According to the numerical results in Section 5, two key insights are obtained:

  1. 1.

    ESC rises with the increase of the number of antennas and the received SNR at the destination, and fades with the increase of those at the eavesdropper.

  2. 2.

    High SNR slope is constant, which means that high SNR slope is independent of the number of antennas and the received SNR at the destination and the eavesdropper. In contrast, high SNR power offset is correlated with the number of antennas at the destination and the eavesdropper.

2 System model

We consider a SIMO wiretap system, in which a source (S) equipped with a single antenna encodes the confidential messages into a transmitted codeword x = [x(1), x(2),…,x(n)] using the capacity achieving codebook for the wiretap channel, which is subject to an average power constraint \({1 \over n}\sum\limits_{i = 1}^n {E\left[ {\vert x(i){\vert ^2}} \right]} \, \leq {P_{\rm{S}}}\). S transmits x to the destination (D), which is equipped with M (M ≥ 1) antennas and adopts the MRC/SC scheme to improve its received SNR; an eavesdropper (E), which is equipped with N (N ≥ 1) antennas, also adopts the MRC/SC scheme to promote successful eavesdropping. Here, perfect secrecy can be achieved by using a proper coding scheme when the received SNR of D is higher than that of E. \({h_M} = {[{h_{{{\rm{D}}_1}}},{h_{{{\rm{D}}_2}}}, \ldots ,\,{h_{{{\rm{D}}_M}}}]^T}\) and \({h_N} = {[{h_{{{\rm{E}}_1}}},{h_{{{\rm{E}}_2}}}, \ldots ,\,{h_{{{\rm{E}}_N}}}]^T}\) are the channel gain vectors of S-D and S-E links, respectively. In this study, we consider a practical passive eavesdropping scenario, which means that the CSI of the S-E link is unavailable at S; thus, S has no choice but to encode the confidential data into codewords of a constant rate (Elkashlan et al., 2015). In contrast, both D and E are aware of their individual CSI accurately in the training period. Moreover, we assume that all considered channels in this study are subject to quasi-static Rayleigh fading, where the fading coefficients are constant over one fading block but vary independently from block to block, and the fading block lengths of S-D and S-E links are equal.

2.1 Maximal ratio combining scheme

If the receivers at D and E have full CSI of their own channels, the ideal combined signals of D and E are given by

$${y_{\rm{D}}} = \sum\limits_{u = 1}^M {{{h_{{{\rm{D}}_u}}^\ast} \over {{N_{{{\rm{D}}_u}}}}}} {y_{{{\rm{D}}_u}}},\quad u = 1,\,2, \ldots ,\,M,$$
((1))
$${y_{\rm{E}}} = \sum\limits_{v = 1}^N {{{h_{{{\rm{E}}_u}}^\ast} \over {{N_{{{\rm{E}}_u}}}}}} {y_{{{\rm{E}}_v}}},\quad v = 1,\,2, \ldots ,\,N,$$
((2))

respectively, where \(h_{{{\rm{D}}_u}}^\ast \) and \(h_{{{\rm{E}}_u}}^\ast \) represent the conjugate of \({h_{{{\rm{D}}_u}}}\) and \({h_{{{\rm{E}}_v}}}\), respectively. Likewise, \({y_{{{\rm{D}}_u}}}\) and \({N_{{{\rm{D}}_u}}}\), \({y_{{{\rm{E}}_v}}}\) and \({N_{{{\rm{E}}_v}}}\) represent the received signals and the mean square noise power of the uth and vth branches, respectively.

In the practical scenario, the combiner weights \(h_{{{\rm{D}}_u}}^\ast /{N_{{{\rm{D}}_u}}}\) and \(h_{{{\rm{E}}_u}}^\ast /{N_{{{\rm{E}}_u}}}\) cannot be obtained perfectly. A complex Gaussian error will result in the weighting factors, \(\hat h_{{{\rm{D}}_u}}^\ast \) and \(\hat h_{{{\rm{E}}_v}}^\ast \), which are the estimates of \(h_{{{\rm{D}}_u}}^\ast \) and \(h_{{{\rm{E}}_v}}^\ast \) derived from the pilot signal (Gans, 1971), respectively. The channel gain with weighting errors of D is given by Hu et al. (2015) as

$$\hat h_{{{\rm{D}}_u}}^\ast = \sqrt {{\rho_{\rm{D}}}} {h_{{{\rm{D}}_u}}} + \sqrt {1 + {\rho_{\rm{D}}}} {g_{{{\rm{D}}_u}}},$$
((3))

where ρD ∈ [0, 1] is a power coefficient defined by Eq. (58) in Gans (1971), and \({g_{{{\rm{D}}_u}}}\) is a random variable which experiences the same distribution as \({h_{{{\rm{D}}_u}}}\).

Let \({\gamma _{\rm{D}}} = \sum\limits_{u = 1}^M {(({P_{\rm{S}}}|{{\hat h}_{{{\rm{D}}_u}}}{|^2})/{N_{{{\rm{D}}_u}}})} \) and \({\gamma_{\rm{E}}}\; = \sum\limits_{v = 1}^N {(({P_{\rm{S}}}\vert {{\hat h}_{{{\rm{E}}_v}}}{\vert ^2})/{N_{{{\rm{E}}_v}}})}\) be the instantaneous SNRs at D and E, respectively. The probability density functions (PDFs) of γD and γ E are (Tomiuk et al., 1999)

$${f_{{\gamma_{\rm{D}}}}}(x) = \sum\limits_{i = 1}^M {A(i)} {{{x^{i - 1}}\exp \left( { - x/{{\bar \gamma}_{\rm{D}}}} \right)} \over {\Gamma (i)\bar \gamma_{\rm{D}}^i}},$$
((4))
$${f_{{\gamma_{\rm{E}}}}}(x) = \sum\limits_{j = 1}^N {B(j)} {{{x^{j - 1}}\exp \left( { - x/{{\bar \gamma}_{\rm{E}}}} \right)} \over {\Gamma (j)\bar \gamma_{\rm{E}}^j}},$$
((5))

respectively, where \({\bar \gamma_{\rm{D}}}\) and \({\bar \gamma_{\rm{E}}}\) are the average perantenna SNRs at D and E, respectively, Γ(·) is the Gamma function (Gradshteyn and Ryzhik, 2007), and A(·) and B(·) are

$$A(i) = \left( {\begin{array}{*{20}c}{M - 1} \\ {i - 1} \end{array}} \right){(1 - {\rho_{\rm{D}}})^{M - i}}\rho_{\rm{D}}^{i - 1},$$
((6))
$$B(j) = \left( {\begin{array}{*{20}c}{N - 1} \\ {j - 1} \end{array}} \right){(1 - {\rho_{\rm{E}}})^{N - j}}\rho_{\rm{E}}^{j - 1},$$
((7))

respectively, where ρE ∈ [0, 1] is the power correlation coefficient between the actual and estimated channel of S-E link..

Further, by applying the probability theory, we have

$$\begin{array}{*{20}c}{\int\nolimits_0^\infty {{f_{{\gamma_{\rm{D}}}}}(x){\rm{d}}x = \sum\limits_{i = 1}^M {A(i){1 \over {\gamma (i)\bar \gamma_{\rm{D}}^i}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}}} \\ { \cdot \int\nolimits_0^\infty {{x^{i - 1}}\exp \left( { - {x \over {{{\bar \gamma}_{\rm{D}}}}}} \right)} {\rm{d}}x = 1{.}} \end{array}$$
((8))

Using Eq. (3.326.2) in Gradshteyn and Ryzhik (2007), we can derive \(\sum\limits_{i = 1}^M {A(i) = 1}\). The cumulative PDFs of γD and γE can be given by (Shrestha and Kwark, 2014)

$${F_{{\gamma_{\rm{D}}}}}(x) = \sum\limits_{i = 1}^M {A(i)} \left( {1 - {{\Gamma (i,x/{{\bar \gamma}_{\rm{D}}})} \over {\Gamma (i)}}} \right)\;,$$
((9))
$${F_{{\gamma_{\rm{E}}}}}(x) = \sum\limits_{j = 1}^M {B(j)} \left( {1 - {{\Gamma (j,x/{{\bar \gamma}_{\rm{E}}})} \over {\Gamma (j)}}} \right)\;,$$
((10))

respectively, where T(·, ·) is the upper incomplete Gamma function (Gradshteyn and Ryzhik, 2007).

2.2 Selection combining scheme

In the SC scenario, before receiving confidential messages from S, D performs antenna selection using the CSI obtained from the pilot, and therefore the channel gain of the selected antenna can be written as \({h_{Ds}} = \mathop {\max }\limits_{u \in \left\{ {1,2,...,M} \right\}} \left| {{h_{Du}}} \right|\). Similarly, we have \({h_{Es}} = \mathop {\max }\limits_{v \in \left\{ {1,2,...,N} \right\}} \left| {{h_{Ev}}} \right|\).

After antenna selection, D receives the messages from S via the selected antenna. However, because of transmitting delay, the CSI of the received signal at D is normally different from the one during antenna selection. We assume that \({\tilde h_{{D_s}}}\) denotes the τd time-delayed channel coefficient version of \({h_{{D_s}}}\). The SNR at D can be written as \({\gamma _D} = \left( {{P_S}{{\left| {{{\tilde h}_{{B_s}}}} \right|}^2}} \right)/{N_0}\), where N0 denotes the AWGN’s power. Similarly, we have \({\gamma _E} = \left( {{P_S}{{\left| {{{\tilde h}_{{E_s}}}} \right|}^2}} \right)/{N_0}\), where \({\tilde h_{{E_s}}}\) denotes the τe time-delayed channel coefficient version of \({h_{{E_s}}}\).

3 Ergodic secrecy capacity of maximal ratio combining scheme

In this section, we consider that both D and E adopt MRC with weighting errors to improve their received SNR. The exact and asymptotic closed-form expressions for the ESC of MRC with weighting errors are derived.

The instantaneous secrecy capacity is given by (Shrestha and Kwark, 2014)

$${C_{\rm{S}}} = \left\{ {\begin{array}{*{20}c}{\ln (1 + {\gamma_{\rm{D}}}) - \ln (1 + {\gamma_{\rm{E}}}),\;\;{\gamma_{\rm{D}}} > {\gamma_{\rm{E}}},} \\ {0,\;\;{\gamma_{\rm{D}}} \leq {\gamma_{\rm{E}}}{.}\quad \quad \quad \quad \quad \quad \quad \quad \quad} \end{array}} \right.$$
((11))

Thus, we can write ESC as

$${\bar C_{\rm{S}}}({\gamma_{\rm{D}}},{\gamma_{\rm{E}}}) = \int\nolimits_0^\infty {\int\nolimits_0^\infty {{C_{\rm{S}}}{f_{{\gamma_{\rm{D}}}}}({\gamma_{\rm{D}}}){f_{{\gamma_{\rm{E}}}}}} ({\gamma_{\rm{E}}}){\rm{d}}{\gamma_{\rm{D}}}{\rm{d}}{\gamma_{\rm{E}}}_{.}}$$
((12))

After some mathematical manipulation, we can obtain

$$\begin{array}{*{20}c}{{{\bar C}_{\rm{S}}}({\gamma_{\rm{D}}},{\gamma_{\rm{E}}}) = \int\nolimits_0^\infty {\int\nolimits_0^{{\gamma_{\rm{D}}}} {{C_{\rm{S}}}{f_{{\gamma_{\rm{D}}}}}({\gamma_{\rm{D}}}){f_{{\gamma_{\rm{E}}}}}} ({\gamma_{\rm{E}}}){\rm{d}}{\gamma_{\rm{E}}}{\rm{d}}{\gamma_{\rm{D}}}\quad \quad \quad \quad \quad}} \\ { + \int\nolimits_0^\infty {\int\nolimits_{{\gamma_{\rm{D}}}}^\infty {{C_{\rm{S}}}{f_{{\gamma_{\rm{D}}}}}({\gamma_{\rm{D}}}){f_{{\gamma_{\rm{E}}}}}} ({\gamma_{\rm{E}}}){\rm{d}}{\gamma_{\rm{E}}}{\rm{d}}{\gamma_{\rm{D}}}}} \\ { = \int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{D}}}){f_{{\gamma_{\rm{D}}}}}({\gamma_{\rm{D}}})\cdot\int\nolimits_0^{\gamma {\rm{D}}} {{f_{{\gamma_{\rm{E}}}}}} ({\gamma_{\rm{E}}}){\rm{d}}{\gamma_{\rm{E}}}{\rm{d}}{\gamma_{\rm{D}}}\;\;\;\quad \quad \quad \quad}} \\ { - \int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{E}}}){f_{{\gamma_{\rm{E}}}}}({\gamma_{\rm{E}}})\cdot\int\nolimits_{{\gamma_{\rm{E}}}}^\infty {{f_{{\gamma_{\rm{D}}}}}} ({\gamma_{\rm{D}}}){\rm{d}}{\gamma_{\rm{D}}}{\rm{d}}{\gamma_{\rm{E}}},\;\;\;\;\quad \quad \quad \quad}} \end{array}$$
((13))

which provides a general form to obtain the closed-form expression for the ESC over fading channels. In the following, the ESC analysis of SC with outdated CSI also adopts this integral equation.

Let

$${\bar C_{\rm{D}}} = \int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{D}}}){f_{{\gamma_{\rm{D}}}}}({\gamma_{\rm{D}}})\int\nolimits_0^{{\gamma_{\rm{D}}}} {{f_{{\gamma_{\rm{E}}}}}} ({\gamma_{\rm{E}}}){\rm{d}}{\gamma_{\rm{E}}}{\rm{d}}{\gamma_{\rm{D}}},}$$

and

$${\bar C_{\rm{E}}} = \int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{E}}}){f_{{\gamma_{\rm{E}}}}}({\gamma_{\rm{E}}})\int\nolimits_{{\gamma_{\rm{E}}}}^\infty {{f_{{\gamma_{\rm{D}}}}}} ({\gamma_{\rm{D}}}){\rm{d}}{\gamma_{\rm{D}}}{\rm{d}}{\gamma_{\rm{E}}}{.}}$$

We establish Theorem 1 as follows:

Theorem 1 The closed-form expression for the ESC of MRC with weighting errors is derived as

$${\bar C_{{\rm{S\_MRC}}}} = {\bar C_{{\rm{D\_MRC}}}} - {\bar C_{{\rm{E\_MRC}}}},$$
((14))

where \({\bar C_{{\rm{D\_MRC}}}}\) and \({\bar C_{{\rm{E\_MRC}}}}\) will be derived in Eqs. (23) and (27), respectively.

Proof See Sections 3.1 and 3.2.

3.1 Derivation of \({\bar C_{{\rm{D\_MRC}}}}\)

We can rewrite \({\bar C_{{\rm{D\_MRC}}}}\) as

$${\bar C_{{\rm{D\_MRC}}}} = \int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{D}}}){f_{{\gamma_{\rm{D}}}}}({\gamma_{\rm{D}}}){F_{{\gamma_{\rm{E}}}}}} ({\gamma_{\rm{D}}}){\rm{d}}{\gamma_{\rm{D}}}{.}$$
((15))

Substituting the PDF of γD and the cumulative density function (CDF) of γE into Eq. (15), we have Eq. (16). I1 can be rewritten as Eq. (17). Eqs. (16) and (17) are shown on the next page.

We consider the integral equation given in Appendix B in Alouini and Goldsmith (1999) as

$$\begin{array}{*{20}c}{\int\nolimits_0^\infty {\ln (1 + x){x^{n - 1}}\exp ( - ux){\rm{d}}x} \;\;\;\quad \quad} \\ { = (n - 1)!\exp (u)\sum\limits_{l = 1}^n {{{\Gamma ( - n + l,u)} \over {{u^l}}},}} \end{array}$$
((18))
$${\bar C_{{\rm{D\_MRC}}}} = \sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {A(i)B(j)}} {1 \over {\Gamma (i)\bar \gamma_{\rm{D}}^i}}\underbrace {\int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{D}}})\gamma_{\rm{D}}^{i - 1}} \exp \left( { - {{{\gamma_{\rm{D}}}} \over {{{\bar \gamma}_{\rm{D}}}}}} \right)\;\;\left( {1 - {{\Gamma (j,{\gamma_{\rm{D}}}/{{\bar \gamma}_{\rm{E}}})} \over {\Gamma (j)}}} \right){\rm{d}}{\gamma_{\rm{D}}}}_{{I_1}}{.}$$
((16))
$${I_1}({\bar \gamma_{\rm{D}}},\,{\bar \gamma_{\rm{E}}},\,i,j) = \underbrace {\int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{D}}})\gamma_{\rm{D}}^{i - 1}} \exp \left( { - {{{\gamma_{\rm{D}}}} \over {{{\bar \gamma}_{\rm{D}}}}}} \right){\rm{d}}{\gamma_{\rm{D}}}}_{{Q_1}} - \underbrace {{1 \over {\Gamma (j)}}\int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{D}}})\gamma_{\rm{D}}^{i - 1}} \exp \left( { - {{{\gamma_{\rm{D}}}} \over {{{\bar \gamma}_{\rm{D}}}}}} \right)\Gamma (j,{\gamma_{\rm{D}}}/{{\bar \gamma}_{\rm{E}}}){\rm{d}}{\gamma_{\rm{D}}}}_{{Q_2}}{.}$$
((17))

where Γ(·, ·) is the complementary incomplete Gamma function, which is given by

$$\begin{array}{*{20}c}{\Gamma (a,z) = \exp ( - z) \cdot U(1 - a,\;1 - a,\;z)\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ { = \exp ( - z) \cdot {z^{a - 1}}{\cdot_2}{F_0}(1 - a,\;1;; - \;{z^{ - 1}}),\quad} \end{array}$$
((19))

where U(·, ·, ·) is the second kind of confluent hypergeometric function and 2F0(·, ¿;;·) is the hypergeometric function (Gradshteyn et al., 2007). Note that as the complementary incomplete Gamma function cannot be directly calculated in Matlab, we transform Γ(·, ·) in Eq. (18) into the form of hypergeometric function.

Using Eq. (18), we can derive Q1 as

$$\begin{array}{*{20}c}{{Q_1}\;({{\bar \gamma}_{\rm{D}}},i)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ { = (i - 1)!\exp \left( {{1 \over {{{\bar \gamma}_{\rm{D}}}}}} \right)\sum\limits_{p = 1}^i {{{\Gamma ( - i + p,\;1/{{\bar \gamma}_{\rm{D}}})} \over {{{(1/{{\bar \gamma}_{\rm{D}}})}^p}}}{.}} \quad} \end{array}$$
((20))

Expanding T(·, ·) into the form of the series in Q2, we have Eq. (21), as shown on the next page.

Substituting Eqs. (20) and (21) into Eq. (17), it follows

$${I_1}({\bar \gamma_{\rm{D}}},\;{\bar \gamma_{\rm{E}}},\,i,\;j) = {Q_1}({\bar \gamma_{\rm{D}}},i) - {Q_2}({\bar \gamma_{\rm{D}}},\;{\bar \gamma_{\rm{E}}},\,i,\;j){.}$$
((22))

Finally, substituting Eq. (22) into Eq. (16), we can obtain

$${\bar C_{{\rm{D\_MRC}}}} = \sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {A(i)} B(i){1 \over {\Gamma (i)\bar \gamma_{\rm{D}}^i}}{I_1}({{\bar \gamma}_{\rm{D}}},\;{{\bar \gamma}_{\rm{E}}},\,i,\;j)\;{.}}$$
((23))

3.2 Derivation of \({\bar C_{{\rm{E\_MRC}}}}\)

We can rewrite \({\bar C_{{\rm{E\_MRC}}}}\) as Eq. (24), as shown on the next page.

Substituting the PDF of γE into I2 and using Eq. (18), we have

$$\begin{array}{*{20}c}{{I_2}(N,{{\bar \gamma}_{\rm{E}}},{\rho_{\rm{E}}})\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ { = \sum\limits_{j = 1}^N {B(j){1 \over {\Gamma (j)\bar \gamma_E^j}}\int\nolimits_0^\infty {\ln (1 + {\gamma_E})\gamma}_E^{j - 1}\quad}} \\ { \cdot \,\exp \left( { - {{{\gamma_E}} \over {{{\bar \gamma}_E}}}} \right)d{\gamma_E}\quad \quad \quad \quad \quad \quad \quad} \\ { = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {A(i)B(i){1 \over {\Gamma (j)\bar \gamma_E^j}}{I_1}({{\bar \gamma}_E},\;{{\bar \gamma}_D},\;j,\;i){.}}}} \end{array}$$
((25))

Considering that the PDFs of γD and γE are similar and the equations of \({\bar C_{{\rm{D\_MRC}}}}\) and I3 have the same structure, we can rewrite I3 as

$$\begin{array}{*{20}c}{{I_3}\;(M,\,N,\,{{\bar \gamma}_{\rm{D}}},\,\,{{\bar \gamma}_{\rm{E}}},\,{\rho_{\rm{D}}},{\rho_{\rm{E}}})\;\;\;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ { = \sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {A(i)B(j){1 \over {\Gamma (j)\bar \gamma_{\rm{E}}^j}}{I_1}({{\bar \gamma}_{\rm{D}}},\,\,{{\bar \gamma}_{\rm{E}}},\,j,\;i)\;{.}}}} \end{array}$$
((26))

Here, it is important to note the parameters’ order in I1(·, ·, ·, ·) and I3(·, ·, ·, ·).

Substituting Eqs. (25) and (26) into Eq. (24), we can derive the closed-form expression for \({\bar C_{{\rm{E\_MRC}}}}\) as

$$\begin{array}{*{20}c}{{{\bar C}_{{\rm{E\_MRC}}}} = {I_2}(N,\;{{\bar \gamma}_{\rm{E}}},\;{\rho_{\rm{E}}})\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ { - {I_3}(M,\;N,\;{{\bar \gamma}_{\rm{D}}},\;{{\bar \gamma}_{\rm{E}}},\;{\rho_{\rm{D}}},\;{\rho_{\rm{E}}})\;{.}} \end{array}$$
((27))

Finally, ESC can be obtained by substituting \({\bar C_{{\rm{D\_MRC}}}}\) and \({\bar C_{{\rm{E\_MRC}}}}\) into Eq. (13).

3.3 Asymptotic ergodic secrecy capacity of maximal ratio combining

In this section, we analyze the asymptotic ESC when \({\bar \gamma_{\rm{D}}} \rightarrow \infty\), while \({\bar \gamma_{\rm{E}}}\) is finite.

$$\begin{array}{*{20}c}{{Q_2}({{\bar \gamma}_{\rm{D}}},\;{{\bar \gamma}_E},\;i,\;j) = \sum\limits_{n = 0}^{j - 1} {{1 \over {\bar \gamma_E^nn!}}\int\nolimits_0^\infty {\ln \;(1 + {\gamma_D})\;\gamma_D^{i + n - 1} \cdot \exp \left[ { - \left( {{1 \over {{{\bar \gamma}_E}}} + {1 \over {{{\bar \gamma}_D}}}} \right){\gamma_D}} \right]}} d{\gamma_D}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ { = \sum\limits_{n = 0}^{j - 1} {{1 \over {\bar \gamma_E^nn!}}} (i + n - 1)!\exp \left( {{1 \over {{{\bar \gamma}_E}}} + {1 \over {{{\bar \gamma}_D}}}} \right)\; \cdot \;\sum\limits_{q = 1}^{i - n} {{{\Gamma ( - i - n + q,\;1/{{\bar \gamma}_E} + 1/{{\bar \gamma}_D})} \over {{{(1/{{\bar \gamma}_E} + 1/{{\bar \gamma}_D})}^q}}}{.}}} \end{array}$$
((21))
$$\begin{array}{*{20}c}{{{\bar C}_{{\rm{E\_MRC}}}} = \int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{E}}})\;{f_{{\gamma_{\rm{E}}}}}({\gamma_{\rm{E}}})\; \cdot \;[1 - F{\gamma_{\rm{D}}}]\;{\rm{d}}{\gamma_{\rm{E}}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}} \\ { = \underbrace {\int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{E}}})\;{f_{{\gamma_{\rm{E}}}}}({\gamma_{\rm{E}}})\;{\rm{d}}{\gamma_{\rm{E}}}}}_{{I_2}} - \underbrace {\int\nolimits_0^\infty {\ln (1 + {\gamma_{\rm{E}}})\;{f_{{\gamma_{\rm{E}}}}}({\gamma_{\rm{E}}}){F_{{\gamma_{\rm{D}}}}}({\gamma_{\rm{E}}})\;{\rm{d}}{\gamma_{\rm{E}}}}}_{{I_3}}{.}} \end{array}$$
((24))

Using \(\Gamma (n + 1,x)\; = \;n!\exp ( - x)\sum\limits_{r = 0}^n {{{{x^r}} \over {r!}}\;\;\;(n\; = \;0,\;1,\; \ldots \;)}\), and considering Eq. (8.352.4) in Gradshteyn and Ryzhik (2007), we can rewrite FγD(x) as

$${F_{{\gamma_{\rm{D}}}}}(x) = \sum\limits_{i = 1}^M {A(i)\;\left[ {1 - \exp \left( { - {x \over {{{\bar \gamma}_{\rm{D}}}}}} \right)\sum\limits_{r = 0}^{i - 1} {{{{x^r}} \over {r!\bar \gamma_{\rm{D}}^r}}}} \right]} \;{.}$$
((28))

For \(\sum\limits_{i = 1}^M {A(i)\; = 1}\), FγD(x) can be rewritten as

$${F_{{\gamma_{\rm{D}}}}}(x) = 1 - \sum\limits_{i = 1}^M {A(i)} \exp \left( { - {x \over {{{\bar \gamma}_{\rm{D}}}}}} \right)\sum\limits_{r = 0}^{i - 1} {{{{x^r}} \over {r!\bar \gamma_{\rm{D}}^r}}}{.}$$
((29))

Similarly, we can rewrite FγE(x) as

$$\begin{array}{*{20}c}{{F_{{\gamma_{\rm{E}}}}}(x) = 1 - \sum\limits_{j = 1}^N {B(j)} \exp \left( { - {x \over {{{\bar \gamma}_{\rm{E}}}}}} \right)\sum\limits_{m = 0}^{i - 1} {{{{x^m}} \over {m!\bar \gamma_{\rm{E}}^m}}}} \\ { = 1 - {\chi_{{\gamma_E}}}(x),\quad \quad \quad \quad \quad} \end{array}$$
((30))

where \({\chi_{{\gamma_{\rm{E}}}}}(x)\; = \;\sum\limits_{j = 1}^N {B(j)\exp \left( { - {x \over {{{\bar \gamma}_{\rm{E}}}}}} \right)} \sum\limits_{m = 0}^{j - 1} {{{{x^m}} \over {m!\bar \gamma_{\rm{E}}^m}}}\).

As the CDF of MRC with Gaussian errors can be regarded as the weighted sum of multiple CDFs of MRC with perfect combinations (Shrestha and Kwark, 2014), to simplify the derivation of the closed form of asymptotic ESC when \({\bar \gamma_{\rm{D}}} \rightarrow \infty\), we can write the asymptotic ESC by using Eq. (22) in Wang et al. (2014a), as

$$\bar C_{\rm{S}}^\infty = \underbrace {\int\nolimits_0^\infty {\ln {\gamma_{\rm{D}}}{f_{{\gamma_{\rm{D}}}}}({\gamma_{\rm{D}}})} {\rm{d}}{\gamma_{\rm{D}}}}_{{I_4}}{\gamma_{\rm{D}}} - \underbrace {\int\nolimits_0^\infty {{{{\chi_{{\gamma_{\rm{E}}}}}({\gamma_{\rm{E}}})} \over {1 + {\gamma_{\rm{E}}}}}{\rm{d}}{\gamma_{\rm{E}}}}}_{{I_5}}{.}$$
((31))

Considering the closed-form expressions for I4 and I5 in Eqs. (A1)–(A4) in the Appendix, the asymptotic ESC can be analyzed. To gain more insight, we evaluate the high SNR slope and the high SNR power offset, which determine the ESC in the high SNR regime.

We rewrite the asymptotic ESC in a general form as (given by Eq. (24) in Wang et al. (2014a))

$$\bar C_{\rm{S}}^\infty \; = \;{S_\infty}(\ln {\bar \gamma_{\rm{D}}} - {\Omega_\infty})\;,$$
((32))

where S∞ is the high SNR slope in nat/(s·Hz) (3 dB) and Ω∞ is the high SNR power offset in 3 dB unit.

We can rewrite the high SNR slope as \({S_\infty } = \mathop {\lim }\limits_{{{\overline \gamma }_D} \to \infty } \overline C _S^\infty /\ln {\overline \gamma _D}\). Obviously, S∞ = 1. We can conclude that the number of antennas at D and E has no impact on the high SNR slope.

We can express the high SNR power offset Ω∞ as

$${\Omega_\infty} = \lim\limits_{{{\bar \gamma}_{\rm{D}}} \rightarrow \infty} \left( {\ln {{\bar \gamma}_{\rm{D}}} - {{\bar C_{\rm{S}}^\infty} \over {{S_\infty}}}} \right) = \Omega_\infty ^M + \Omega_\infty ^N,$$
((33))

where \(\Omega_\infty ^M = - \sum\limits_{i = 1}^M {A(i)\psi (i)}\) and \(\Omega_\infty ^N = {I_5}\). We find that the high SNR power offset is independent of \({\bar \gamma_{\rm{D}}}\). We highlight that \(\Omega_\infty ^M\) assesses the benefits of M on ESC, and \(\Omega_\infty ^N\) quantifies the loss of ESC due to eavesdropping.

4 Ergodic secrecy capacity of selection combining scheme

In this section, we investigate the ESC of SC when both D and E adopt SC with outdated CSI to improve their received SNR. The exact and asymptotic closed-form expressions for SC are derived.

4.1 Probability and cumulative density functions of selection combining with outdated imperfect channel state information

Considering Eq. (3), we have the PDF of γD using SC with outdated CSI as (given by Eq. (10) in Ferdinand et al. (2013))

$${f_{{\gamma_{\rm{D}}}}}(x) = \int\nolimits_0^\infty {{f_{{\gamma_{\rm{D}}}\vert {\gamma_{\rm{d}}}}}(x\vert z)} {f_{{\gamma_{\rm{D}}}}}(z){\rm{d}}z{.}$$
((34))

Note that the correlation expression of the actual and the outdated channels gain under the SC scheme has the same form as that under the MRC scheme with weighting errors (Ferdinand et al., 2013). In addition, ρD and ρE are power coefficients defined by Eq. (58) in Gans (1971). In Eq. (34), \({f_{{\gamma_{\rm{d}}}}}( \cdot )\) is the PDF of γD in the scenario with perfect CSI, given by

$$\begin{array}{*{20}c}{{f_{{\gamma_{\rm{D}}}}}(z) = M \cdot F_{{\gamma_{{\rm{d}},\kappa}}}^{M - 1}(z)\; \cdot \;{f_{{\gamma_{{\rm{d,}}\kappa}}}}(z)\quad \quad \quad \quad} \\ { = {M \over {{{\bar \gamma}_{\rm{D}}}}}\sum\limits_{{m_1} = 0}^{M - 1} {\left( {\begin{array}{*{20}c}{M - 1} \\ {{m_1}} \end{array}} \right)\;{{( - 1)}^{{m_1}}}}} \\ {{.}\exp \left( { - {{{m_1} + 1} \over {{{\bar \gamma}_{\rm{D}}}}}z} \right)\;,\quad \quad} \end{array}$$
((35))

where fγd,k(·) and Fγd,k(·) are the PDF and CDF of the receiving SNR at the kth antenna which experiences an exponential fading, respectively.

Using Eq. (13) in Ferdinand et al. (2013), we can derive the PDF of γD as

$$\begin{array}{*{20}c}{{f_{{\gamma_{\rm{D}}}}}(x) = \underbrace {M\sum\limits_{{m_1} = 0}^{M - 1} {{{( - 1)}^{{m_1}}}} \left( {\begin{array}{*{20}c}{M - 1} \\ {{m_1}} \end{array}} \right){1 \over {1 - {\rho_{\rm{D}}}}}{1 \over {{\alpha_{\rm{D}}}{{\bar \gamma}_{\rm{D}}}}}}_{{\Sigma_{{\Omega_{\rm{D}}}}}}\quad \quad \quad \quad \quad \quad} \\ {{.}\exp \left( { - {{{\varsigma_{\rm{D}}}x} \over {{{\bar \gamma}_{\rm{D}}}}}} \right) = \sum\limits_{{\Omega_{\rm{D}}}} {\exp \left( { - {{{\varsigma_{\rm{D}}}x} \over {{{\bar \gamma}_{\rm{D}}}}}} \right)\;,\quad \quad}} \end{array}$$
((36))

where \({\alpha_D} = {{{\rho_D}} \over {1 - {\rho_D}}} + {m_1} + 1\) and \({\varsigma_D} = {1 \over {1 - {\rho_D}}} - {{{\rho_D}} \over {{\alpha_D}{{(1 - {\rho_D})}^2}}}\).

The CDF of γD can be given by

$$\begin{array}{*{20}c}{{F_{{\gamma_{\rm{D}}}}}(x) = \int\limits_0^x {\sum\limits_{{\Omega_{\rm{D}}}} {\exp \left( {{{{\varsigma_D}} \over {{{\bar \gamma}_{\rm{D}}}}}u} \right)} {\rm{d}}u\quad \quad \quad \quad \quad}} \\ { = \sum\limits_{{\Omega_{\rm{D}}}} {{{{{\bar \gamma}_{\rm{D}}}} \over {{\varsigma_{\rm{D}}}}}\left[ {1 - \exp \left( {{{{\varsigma_{\rm{D}}}x} \over {{{\bar \gamma}_{\rm{D}}}}}} \right)} \right]} \;{.}} \end{array}$$
((37))

Similarly, we can derive the PDF and CDF of γE as

$$\begin{array}{*{20}c}{{f_{{\gamma_{\rm{E}}}}}(x) = \underbrace {N\sum\limits_{{n_1} = 0}^{N - 1} {{{( - 1)}^{{n_1}}}} \left( {\begin{array}{*{20}c}{N - 1} \\ {{n_1}} \end{array}} \right){1 \over {1 - {\rho_{\rm{E}}}}}{1 \over {{\alpha_{\rm{E}}}{{\bar \gamma}_{\rm{E}}}}}}_{{\Sigma_{{\Omega_{\rm{E}}}}}}\quad \quad \quad \quad} \\ {{.}\exp \left( { - {{{\varsigma_{\rm{E}}}x} \over {{{\bar \gamma}_{\rm{E}}}}}} \right)\; = \;\sum\limits_{{\Omega_{\rm{E}}}} {\exp \left( { - {{{\varsigma_{\rm{E}}}x} \over {{{\bar \gamma}_{\rm{E}}}}}} \right)} \;,} \end{array}$$
((38))
$${F_{{\gamma_{\rm{E}}}}}(x) = \sum\limits_{{\Omega_E}} {{{{{\bar \gamma}_{\rm{E}}}} \over {{\varsigma_{\rm{E}}}}}\left[ {1 - \exp \left( { - {{{\varsigma_{\rm{E}}}x} \over {{{\bar \gamma}_{\rm{E}}}}}} \right)} \right]} \;,$$
((39))

where \({\alpha_{\rm{E}}} = {{{\rho_{\rm{E}}}} \over {1 - {\rho_{\rm{E}}}}} + {n_1} + 1\) and \({\varsigma_{\rm{E}}} = {1 \over {1 - {\rho_{\rm{E}}}}} - {{{\rho_{\rm{E}}}} \over {{\alpha_{\rm{E}}}{{(1 - {\rho_{\rm{E}}})}^2}}}\).

4.2 Ergodic secrecy capacity of selection combining

Theorem 2 The closed-form expression for the ESC of SC with outdated CSI is derived as

$${\bar C_{{\rm{S\_SC}}}} = {\bar C_{{\rm{D\_SC}}}} - {\bar C_{{\rm{E\_SC}},}}$$
((40))

where \({\bar C_{{\rm{D\_SC}}}}\) and \({\bar C_{{\rm{E\_SC}}}}\) will be derived as Eqs. (44) and (48), respectively.

Proof See Sections 4.2.1 and 4.2.2.

4.2.1 Derivation of \({\bar C_{{\rm{D\_SC}}}}\)

We have \({\bar C_{\rm{D}}}{\__{{\rm{SC}}}}\) as

$$\begin{array}{*{20}c} {{{\bar C}_{{\rm{D\_SC}}}}(M,N,{{\bar \gamma }_{\rm{D}}},{{\bar \gamma }_{\rm{E}}},{\rho _{\rm{D}}},{\rho _{\rm{E}}})\quad \quad \quad \quad \,} \\ { = \int_0^\infty {{\rm{ln(1 + }}{\gamma _{\rm{D}}}{\rm{)}}{f_{{\gamma _{\rm{D}}}}}({\gamma _{\rm{D}}}){F_{{\gamma _{\rm{E}}}}}({\gamma _{\rm{D}}}){\rm{d}}{\gamma _{\rm{D}}}.} } \\ \end{array} $$
((41))

Substituting the PDF of γD and the CDF of γE into the above equation, we can rewrite \({\bar C_{\rm{D}}}{\__{{\rm{SC}}}}\) as Eq. (42), shown on the next page.

To simplify the analysis, we consider the following integral equation:

$$\begin{array}{*{20}c} {{\Phi _1}(a) = \int_0^\infty {{\rm{ln}}} (1 + {\gamma _{\rm{D}}}){\rm{exp( - }}a{\gamma _{\rm{D}}}{\rm{)d}}{\gamma _{\rm{D}}}\,} \\ { = {\rm{exp(}}a{\rm{)}}{{\Gamma (0,a)} \over a},\quad \quad } \\ \end{array} $$
((43))

where a > 0.

$$\begin{array}{*{20}c} {{{\bar C}_{{\rm{D\_SC}}}}(M,N,{{\bar \gamma }_{\rm{D}}},{{\bar \gamma }_{\rm{E}}},{\rho _{\rm{D}}},{\rho _{\rm{E}}})\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,} \\ { = \sum\limits_{{\Omega _{\rm{D}}}} {\sum\limits_{{\Omega _{\rm{E}}}} {{{{{\bar \gamma }_{\rm{E}}}} \over {{\varsigma _{\rm{E}}}}}\int_0^\infty {{\rm{ln}}} (1 + {\gamma _{\rm{D}}}){\rm{exp}}\left( { - {{{\varsigma _{\rm{D}}}{\gamma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}} \right) \cdot \left[ {1 - {\rm{exp}}\left( { - {{{\varsigma _{\rm{E}}}{\gamma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right)} \right]} {\rm{d}}{\gamma _{\rm{D}}}\quad \quad \quad \quad \quad \quad \quad \quad \quad } } \\ { = \sum\limits_{{\Omega _{\rm{D}}}} {\sum\limits_{{\Omega _{\rm{E}}}} {{{{{\bar \gamma }_{\rm{E}}}} \over {{\varsigma _{\rm{E}}}}}\left\{ {\int_0^\infty {{\rm{ln}}} (1 + {\gamma _{\rm{D}}}){\rm{exp}}\left( { - {{{\varsigma _{\rm{D}}}{\gamma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}} \right){\rm{d}}{\gamma _{\rm{D}}} - \int_0^\infty {{\rm{ln(1 + }}{\gamma _{\rm{D}}}{\rm{)exp}}\left[ { - \left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}} + {{{\varsigma _{\rm{E}}}} \over {{\gamma _{\rm{E}}}}}} \right){\gamma _{\rm{D}}}} \right]} {\rm{d}}{\gamma _{\rm{D}}}} \right\}.} } } \\ \end{array} $$
((42))

Using the above integral equation, we can derive the closed-form expression for \({\bar C_{{\rm{D\_SC}}}}\) as

$$\begin{array}{*{20}c} {{{\bar C}_{{\rm{D\_SC}}}}(M,N,{{\bar \gamma }_{\rm{D}}},{{\bar \gamma }_{\rm{E}}},{\rho _{\rm{D}}},{\rho _{\rm{E}}})\quad \quad \quad \quad \quad \quad \quad } \\ { = \sum\limits_{{\Omega _{\rm{D}}}} {\sum\limits_{{\Omega _{\rm{E}}}} {{{{{\bar \gamma }_{\rm{E}}}} \over {{\varsigma _{\rm{E}}}}}\left[ {{\Phi _1}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}} \right) - {\Phi _1}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}} + {{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right)} \right].} } } \\ \end{array} $$
((44))

4.2.2 Derivation of \({\bar C_{{\rm{E\_SC}}}}\)

We can write \({\bar C_{{\rm{E\_SC}}}}\) as

$$\begin{array}{*{20}c} {{{\bar C}_{{\rm{E\_SC}}}} = \int_0^\infty {{\rm{ln}}} (1 + {\gamma _{\rm{E}}}){f_{{\gamma _{\rm{E}}}}}({\gamma _E})[1 - {F_{{\gamma _{\rm{D}}}}}({\gamma _{\rm{E}}})]{\rm{d}}{\gamma _{\rm{E}}}} \\ { = \underbrace {\int_0^\infty {{\rm{ln}}} (1 + {\gamma _{\rm{E}}}){f_{{\gamma _{\rm{E}}}}}({\gamma _{\rm{E}}})d{\gamma _{\rm{E}}}.}_{{\Xi _1}}\quad \quad } \\ { - \underbrace {\int_0^\infty {{\rm{ln}}} (1 + {\gamma _{\rm{E}}}){f_{{\gamma _{\rm{E}}}}}({\gamma _{\rm{E}}}){F_{{\gamma _{\rm{D}}}}}({\gamma _{\rm{E}}})d{\gamma _{\rm{E}}}.}_{{\Xi _2}}} \\ \end{array} $$
((45))

Using the integral Eq. (43), we can derive

$$\begin{array}{*{20}c} {{\Xi _1} = \sum\limits_{{\Omega _{\rm{E}}}} {\int_0^\infty {{\rm{ln}}} } (1 + {\gamma _{\rm{E}}}){\rm{exp}}\left( { - {{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}{\gamma _{\rm{E}}}} \right){\rm{d}}{\gamma _{\rm{E}}}} \\ { = \sum\limits_{{\Omega _{\rm{E}}}} {{\Phi _1}} \left( {{{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right).\quad \quad \quad \quad \quad \quad \quad } \\ \end{array} $$
((46))

Considering that the PDFs of γD and γE are similar, and that the expressions of \({\bar C_{{\rm{D\_SC}}}}\) and Ξ2 have the same structure, we can rewrite Ξ2 as

$${\Xi _2} = {\bar C_{{\rm{D\_SC}}}}(N,M,{\bar \gamma _{\rm{E}}},{\bar \gamma _{\rm{D}}},{\rho _{\rm{E}}},{\rho _{\rm{D}}}).$$
((47))

The closed-form expression for \({\bar C_{{\rm{E\_SC}}}}\) is derived as

$$\begin{array}{*{20}c} {{{\bar C}_{{\rm{E\_SC}}}} = \sum\limits_{{\Omega _{\rm{E}}}} {{\Phi _1}} \left( {{{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ { - {{\bar C}_{{\rm{D\_SC}}}}(N,M,{{\bar \gamma }_{\rm{E}}},{{\bar \gamma }_{\rm{D}}},{\rho _{\rm{E}}},{\rho _{\rm{D}}}).} \\ \end{array} $$
((48))

Finally, ESC can be obtained by substituting \({\bar C_{{\rm{E\_SC}}}}\) and \({\bar C_{{\rm{E\_SC}}}}\) into Eq. (13).

4.3 Asymptotic ergodic secrecy capacity of selection combining

In the scenario where γD ≫ 1, we have ln (1 + γD) ≈ lnγD. When \({\bar \gamma _{\rm{D}}} \to \infty \), we have \(\bar C_{\rm{D}}^\infty \) as

$$\begin{array}{*{20}c} {\bar C_{\rm{D}}^\infty (M,N,{{\bar \gamma }_{\rm{D}}},{{\bar \gamma }_{\rm{E}}},{\rho _{\rm{D}}},{\rho _{\rm{E}}})\quad \quad \quad \,} \\ { = \int_0^\infty {{\rm{ln}}} \,{\gamma _{\rm{D}}}{f_{{\gamma _{\rm{D}}}}}({\gamma _{\rm{D}}}){F_{{\gamma _{\rm{E}}}}}({\gamma _{\rm{D}}}){\rm{d}}{\gamma _{\rm{D}}}.} \\ \end{array} $$
((49))

We consider the integral equation given by Eq. (4.352.1) in Gradshteyn and Ryzhik (2007):

$$\begin{array}{*{20}c} {{\Phi _2}(a) = \int_0^\infty {{\rm{ln}}\,{\gamma _{\rm{D}}}\,{\rm{exp}}\,{\rm{( - }}a{\gamma _{\rm{D}}}{\rm{)d}}{\gamma _{\rm{D}}}\,\,\,} } \\ { = - {1 \over a}({\rm{ln}}\,a - \psi (1)),} \\ \end{array} $$
((50))

where a > 0 and ψ(·) is the Digamma function (Gradshteyn and Ryzhik, 2007).

Using Eq. (50), we can derive

$$\begin{array}{*{20}c} {\bar C_{\rm{D}}^\infty (M,N,{{\bar \gamma }_{\rm{D}}},{{\bar \gamma }_{\rm{E}}},{\rho _{\rm{D}}},{\rho _{\rm{E}}})\quad \quad \quad \quad \quad \quad } \\ {\sum\limits_{{\Omega _{\rm{D}}}} {\sum\limits_{{\Omega _{\rm{E}}}} {{{{{\bar \gamma }_{\rm{E}}}} \over {{\varsigma _{\rm{E}}}}}\left[ {{\Phi _2}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}} \right) - {\Phi _2}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}} + {{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right)} \right].} } } \\ \end{array} $$
((51))

Considering FγD(x) ≈ 0 when \({\bar \gamma _{\rm{D}}} \to \infty \) and 1−FγD (γE) ≈ 1 given by Wang et al. (2014a; 2014b), we can write \(\bar C_{\rm{E}}^\infty \) as

$$\begin{array}{*{20}c} {\bar C_{\rm{E}}^\infty = \int_0^\infty {{\rm{ln}}\,{\rm{(1 + }}{\gamma _{\rm{E}}}{\rm{)}}{f_{{\gamma _{\rm{E}}}}}({\gamma _{\rm{E}}})[1 - {F_{{\gamma _{\rm{D}}}}}({\gamma _{\rm{E}}})]} {\rm{d}}{\gamma _{\rm{E}}}\quad } \\ { = \int_0^\infty {{\rm{ln}}\,{\rm{(1 + }}{\gamma _{\rm{E}}}{\rm{)}}{f_{{\gamma _{\rm{E}}}}}({\gamma _{\rm{E}}})} {\rm{d}}{\gamma _{\rm{E}}} = \sum\limits_{{\Omega _{\rm{E}}}} {{\Phi _1}} \left( {{{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right).} \\ \end{array} $$
((52))

The closed-form expression for the asymptotic ESC of SC \((\bar C_{{\rm{SC}}}^\infty )\) can be derived by substituting \(\bar C_{\rm{D}}^\infty \) and \(\bar C_{\rm{E}}^\infty \) into Eq. (13):

$$\begin{array}{*{20}c} {\bar C_{{\rm{SC}}}^\infty = \sum\limits_{{\Omega _{\rm{D}}}} {\sum\limits_{{\Omega _{\rm{E}}}} {{{{{\bar \gamma }_{\rm{E}}}} \over {{\varsigma _{\rm{E}}}}}\left[ {{\Phi _2}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}} \right) - {\Phi _2}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}} + {{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right)} \right]} } } \\ { - \sum\limits_{{\Omega _{\rm{E}}}} {{\Phi _1}} \left( {{{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right),\quad \quad \quad \quad \quad \quad \quad } \\ \end{array} $$
((53))

where

$${\Phi _2}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}} \right) = {{{{\bar \gamma }_{\rm{D}}}} \over {{\varsigma _{\rm{D}}}}}\left[ {{\rm{ln}}\,\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}} \right) - \psi (1)} \right],$$
((54))
$$\begin{array}{*{20}c} {{\Phi _2}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}} + {{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right) = - {{\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}} + {{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right)}^{ - 1}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \,} \\ { \cdot \left[ {{\rm{ln}}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}} + {{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right) - \psi (1)} \right].} \\ \end{array} $$
((55))

The asymptotic ESC of the SC scheme has the same form as Eq. (32), similar to that of the MRC scheme. When \({\bar \gamma _{\rm{D}}} \to \infty \), we have

$$\matrix{ {\mathop {\lim }\limits_{{{\overline \gamma }_D} \to \infty } \left[ {\Phi \left( {{{{\varsigma _D}} \over {{{\overline \gamma }_D}}}} \right) - {\Phi _2}\left( {{{{\varsigma _D}} \over {{{\overline \gamma }_D}}} + {{{\varsigma _E}} \over {{{\overline \gamma }_E}}}} \right)} \right]} \cr { = \mathop {\lim }\limits_{{{\overline \gamma }_D} \to \infty } \left[ {{\Phi _2}\left( {{{{\varsigma _D}} \over {{{\overline \gamma }_D}}}} \right) - {\Phi _2}\left( {{{{\varsigma _E}} \over {{{\overline \gamma }_E}}}} \right)} \right]} \cr { = \mathop {\lim }\limits_{{{\overline \gamma }_D} \to \infty } - {{{{\overline \gamma }_D}} \over {{\varsigma _D}}}\left[ {\ln {{{\varsigma _D}} \over {{{\overline \gamma }_D}}} - \psi (1)} \right]} \cr { = \mathop {\lim }\limits_{{{\overline \gamma }_D} \to \infty } \;{{{{\overline \gamma }_D}} \over {{\varsigma _D}}}\left( {\ln {{\overline \gamma }_D} + \psi (1) - \ln {\varsigma _D}} \right).} \cr }$$
((56))

Considering Eqs. (56) and (53), we can derive Eqs. (57) and (58) (shown on the bottom of this page), where

$${\sum _{{\Omega _{D1}}}} = M\sum\limits_{{m_1} = 0}^{M - 1} {{{\left( { - 1} \right)}^{{m_1}}}} \left( {\matrix{ {M - 1} \cr {{m_1}} \cr } } \right){1 \over {1 - {\rho _D}}} \cdot {1 \over {{\alpha _D}}},$$

and

$${\sum _{{\Omega _{D1}}}} = N\sum\limits_{{n_1} = 0}^{N - 1} {{{\left( { - 1} \right)}^{{n_1}}}} \left( {\matrix{ {N - 1} \cr {{n_1}} \cr } } \right){1 \over {1 - {\rho _E}}} \cdot {1 \over {{\alpha _E}}}.$$

Further, by applying probability theory, we have

$$\begin{array}{*{20}c} {\int_0^\infty {{f_{{\gamma _{\rm{D}}}}}(x){\rm{d}}x} = \sum\limits_{{\Omega _{\rm{D}}}} {\int_0^\infty {{\rm{exp}}} \,\left( { - {{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}x} \right){\rm{d}}x\quad \quad \quad \quad } } \\ { = \sum\limits_{{\Omega _{\rm{D}}}} {{{{{\bar \gamma }_{\rm{D}}}} \over {{\varsigma _{\rm{D}}}}} = \sum\limits_{{\Omega _{{\rm{D1}}}}} {{1 \over {{\varsigma _{\rm{D}}}}} = 1.} } } \\ \end{array} $$
((59))
$$\begin{array}{*{20}c} {{S_\infty } = \underset{{{\bar \gamma }_{\rm{D}}}{\lim } \to \infty } {{\sum\limits_{{\Omega _{\rm{D}}}} {\sum\limits_{{\Omega _{\rm{E}}}} {{{{{\bar \gamma }_{\rm{E}}}} \over {{\varsigma _{\rm{E}}}}}\left\{ {{\Phi _2}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}}} \right) - {\Phi _2}\left( {{{{\varsigma _{\rm{D}}}} \over {{{\bar \gamma }_{\rm{D}}}}} + {{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right)} \right\}} - \sum\limits_{{\Omega _{\rm{E}}}} {{\Phi _1}} \left( {{{{\varsigma _{\rm{E}}}} \over {{{\bar \gamma }_{\rm{E}}}}}} \right)} } \over {{\rm{ln}}\,{{\bar \gamma }_{\rm{D}}}}}} \\ { = \underset {{{\bar \gamma }_{\rm{D}}}{\lim } \to \infty } {{\sum\limits_{{\Omega _{{\rm{D1}}}}} {\sum\limits_{{\Omega _{\rm{E}}}} {{{{{\bar \gamma }_{\rm{E}}}} \over {{\varsigma _{\rm{E}}}{\varsigma _{\rm{D}}}}}({\rm{ln}}\,{{\bar \gamma }_{\rm{D}}} + \psi (1) - {\rm{ln}}\,{\varsigma _{\rm{D}}})} } } \over {{\rm{ln}}\,{{\bar \gamma }_{\rm{D}}}}} = \sum\limits_{{\Omega _{{\rm{D1}}}}} {\sum\limits_{{\Omega _{{\rm{E1}}}}} {{1 \over {{\varsigma _{\rm{E}}}{\varsigma _{\rm{D}}}}},} } } \\ \end{array} $$
((57))
$$\matrix{ {{\Omega _\infty } = \mathop {\lim }\limits_{{{\overline \gamma }_D} \to \infty } \left( {\ln {{\overline \gamma }_D} - \overline C _{SC}^\infty } \right) = \mathop {\lim }\limits_{{{\overline \gamma }_D} \to \infty } \left( {\ln {{\overline \gamma }_D} - \overline C _D^\infty + \overline C _E^\infty } \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \cr { = \mathop {\lim }\limits_{{{\overline \gamma }_D} \to \infty } \left\{ {\ln {{\overline \gamma }_D} - \sum\limits_{{\Omega _{D1}}} {\sum\limits_{{\Omega _E}} {{{{{\overline \gamma }_E}} \over {{\varsigma _E}{\varsigma _D}}}\left[ {\ln {{\overline \gamma }_D} + \psi (1) - {{\ln }_{{\varsigma _D}}}} \right] + \overline C _E^\infty } } } \right\}} \cr { = - \sum\limits_{{\Omega _{D1}}} {\sum\limits_{{\Omega _{E1}}} {{1 \over {{\varsigma _E}{\varsigma _D}}}\psi (1) + \sum\limits_{{\Omega _{D1}}} {\sum\limits_{{\Omega _{E1}}} {{1 \over {{\varsigma _D}{\varsigma _E}}}\ln {{\overline \varsigma }_D} + \overline C _E^\infty } .\;\;\;\;\;\;\;\;\;\;\;\;\;\;} } } } \cr } $$
((58))

Similarly, we can obtain \(\sum\limits_{{\Omega _{{\rm{E1}}}}} {{1 \over {{\varsigma _{\rm{E}}}}} = 1} \).

Finally, we can achieve S∞ = 1 and \({\Omega _\infty } = - \psi (1) + \sum\limits_{{\Omega _{{\rm{D1}}}}} {{1 \over {{\varsigma _{\rm{D}}}}}{\rm{ln}}{\mkern 1mu} {\varsigma _{\rm{D}}} + \bar C_{\rm{E}}^\infty } \).

5 Numerical results and discussions

5.1 Maximal ratio combining scheme

In this subsection, we run Monte-Carlo simulations to validate our exact and asymptotic expressions for ESC under the MRC scheme. In each simulation, S sends 105 bits to D. In MRC simulation, we can use Eq. (23) in Gans (1971) or Eq. (3) which reveals the correlation between the actual and estimated channels to build the SNR of the receivers.

Fig. 1 plots ESC vs. \({\bar \gamma _{\rm{D}}}\) for various \({\bar \gamma _{\rm{E}}}\)’s over Rayleigh fading channels with weighting errors. It is evident that for a fixed \({\bar \gamma _{\rm{E}}}\), ESC increases with \({\bar \gamma _{\rm{D}}}\), as the channel state of the S-D link improves. It is shown that ESC can be improved when \({\bar \gamma _{\rm{E}}}\) degrades, because the channel state of the S-E link gets worse. Fig. 2 plots ESC vs. \({\bar \gamma _{\rm{D}}}\) for various M’s and ρE’s. There is a pronounced decrease when ρE increases. This can be explained by the fact that the increasing ρE can improve the channel estimation at E. In addition, it is easy to observe that ESC can be improved when M increases, because of the increased MRC diversity gain at D.

Fig. 1
figure 1

ESC vs. \({\bar \gamma _{\rm{D}}}\) of MRC with weighting errors for M = N = 2, and ρD = ρE = 0.5

Full size image
Fig. 2
figure 2

ESC vs. \({\bar \gamma _{\rm{D}}}\) of MRC with weighting errors for N = 2, ρD = 0.5, and \({\bar \gamma _{\rm{E}}} = 5\) dB

Full size image

It can be seen from Figs. 1 and 2 that the asymptotic ESC matches very well the simulation and the exact analytical results in the high \({\bar \gamma _{\rm{D}}}\) regime.

Fig. 3 plots ESC vs. ρD for various N’s. It is clear that ESC increases with ρD. It is because increasing ρD can improve the channel estimation at D. One can also see that ESC decreases when N increases, because of the improved MRC diversity gain at E.

Fig. 3
figure 3

ESC vs. ρD of MRC with weighting errors for M = 3, ρE = 0.5, and \({\bar \gamma _{\rm{D}}} = {\bar \gamma _{\rm{E}}} = 5\) dB

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Further, it is obvious that simulation and analytical results match very well in Figs. 1–3.

5.2 Selection combining scheme

In this section, we run Monte-Carlo simulation to validate our exact and asymptotic expressions for ESC under the SC scheme. In each simulation, S sends 105 bits to D.

Fig. 4 indicates that ESC decreases when \({\bar \gamma _{\rm{E}}}\) increases, as the channel state of the S-E link, which degrades ESC, improves. Note that when \({\bar \gamma _{\rm{D}}}\) increases, the channel state of the S-D link improves. This explains why ESC increases with \({\bar \gamma _{\rm{D}}}\).

Fig. 4
figure 4

ESC vs. \({\bar \gamma _{\rm{D}}}\) of SC with outdated CSI for M = 3, N = 2, ρD = 0.5, ρE = 0.6, N0 = 1, and PS = 3 dB

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Fig. 5 illustrates the ESC of the SC scheme against \({\bar \gamma _{\rm{D}}}\) for various (M, N) combinations.

Fig. 5
figure 5

ESC vs. \({\bar \gamma _{\rm{D}}}\) of SC with outdated CSI for \({\bar \gamma _{\rm{E}}} = 5\) dB, ρD = 0.5, ρE = 0.6, N0 = 1, and PS = 3 dB

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Obviously, there is an upward trend in ESC when M increases and N decreases, because the diversity gain of D increases and that of E decreases.

In addition, Figs. 4 and 5 show that our asymptotic ESC matches the analysis and simulation very well in the high \({\bar \gamma _{\rm{D}}}\) regime, proving the accuracy of our derivations. Further, it is clear that −Ω∞ of SC increases when M increases and N and \({\bar \gamma _{\rm{E}}}\) decrease. Ω∞ and are independent of \({\bar \gamma _{\rm{D}}}\). The reason can be explained by Eqs. (57) and (58).

Fig. 6 compares the secrecy performance of the SC and MRC schemes for various ρE’s. In the scenario with a fixed ρE, we can observe that the ESC of the MRC scheme outperforms the one of the SC scheme in the high ρD region. There is a cutoff, at about ρD = 0.15, of the lines for the MRC and SC schemes when ρE = 0.4.

Fig. 6
figure 6

ESC vs. ρD of MRC with weighting errors/SC with outdated CSI for M = N = 3, \({\bar \gamma _{\rm{D}}} = {\bar \gamma _{\rm{E}}} = 5\) dB, N0 = 1, and PS = 1 dB

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6 Conclusions

In this paper, we have analyzed the ESC of the MRC/SC scheme in SIMO wiretap systems over Rayleigh fading channels with imperfect CSI. The exact and asymptotic closed-form expressions for ESC have been derived and verified through simulations. Further, the high SNR slope and high SNR power offset predict ESC accurately in the high SNR region.