1 Introduction

Spatial diversity, which combines multiple replicas of the received signal, has long been recognized as an effective technique to mitigate fading and its impact on the performance of communication systems [1, 2]. Receiver diversity combining methods such as maximal ratio combining (MRC), equal gain combing (EGC) and selection combing (SC) are often used to combine available diversity branches [1,2,3]. It is well known that, in the ideal case of independent diversity branches, MRC is the optimal technique in terms of maximizing the SNR at the combiner output [1,2,3]. However, for MRC, the multiple radio frequency (RF) chains associated with each branch increase the cost and the signal processing complexity of the receiver. On the other hand, SC is one of the simplest suboptimal combining scheme, in which the receiver selects the diversity branch having the highest instantaneous SNR [1,2,3]. The performance of MRC and SC techniques has been extensively investigated under various fading channels, including Rayleigh, Nakagami-m, and Rician (see references in [4,5,6]). Previous works have shown that the diversity gain depends on the statistical correlation among the fading envelopes of the signals received at the different branches [7,8,9]. Hence, the maximum diversity gain is obtained when the fading processes at the branches are independent. The impact of spatial correlation on the performance of MRC and SC techniques is well studied (see e.g., [8,9,10,11,12]). Furthermore, prior works are based on the assumption of a perfect scattering environment. Under this hypothesis, the covariance matrix of the channel vector has full-rank.

Recently, field measurements have shown that the lack of scatterers around the transmitter and the receiver may reduce the channel matrix rank [13, 14]. To characterize this phenomenon, a new channel model for multiple-input multiple-output (MIMO) systems has been proposed in [13]. This double-scattering model takes into account both rank-deficiency and spatial correlation by modeling the MIMO channel as a product of two statistically independent complex Gaussian matrices and three deterministic matrices. The latter correspond to the transmitter, receiver, and scatterer correlation matrices. The double Rayleigh (or Rayleigh-product) model is a special case of the double-scattering model, where all three correlation matrices are equal to the identity matrix [15]. In contrast to rich-scattering, analytical performance assessment under double-scattering channels are relatively rare. In [16], authors have presented a performance analysis of transmit beamforming (BF) systems in Rayleigh product channels. To characterize the statistical proprieties of the received signal-to-noise ratio (SNR), Jin et al. have derived new closed-form expressions for the cumulative distribution function (CDF), probability density function (PDF), and moments of the maximum eigenvalue of a product of independent complex Gaussian matrices. The performance of an interference-limited double scattering MIMO channel with optimum combining is presented in [17]. For mathematical simplicity, authors have assumed that the number of co-channel interferers is greater than or equal to the number of receive antennas and the impact of noise was neglected. This assumption has been relaxed in [18] where authors have investigated the performance of a MIMO-BF system in Rayleigh-product channels with arbitrary-power co-channel interference. In [15], Rayleigh-product MIMO channels with linear receivers are considered and exact closed-form expressions for the ergodic sum-rate are derived. The performance of double-scattering channels with diversity combining techniques has been investigated in [19, 20]. In [19], double Nakagami-m channels with MRC diversity are considered, while authors in [20] have focused on double Rayleigh fading channels with EGC. The double-scattering propagation was modeled as a double-generalized Gamma (dGG) distribution in [21], where an analytical framework has been derived to analyze the performance of a transmit antenna selection system operating in vehicle-to-vehicle communication channels.

In this paper, we consider the performance of SC and MRC combining techniques in uncorrelated double-Rayleigh channels. First, we derive new closed-form expressions and asymptotic expansions for the PDF and the CDF of the SNR at each combiner output. Based on these expressions, we then investigate the performance of SC and MRC in terms of outage probability, average symbol error rate (SER), and ergodic capacity. In addition, in order to obtain the diversity order and array gain of the considered techniques, asymptotic SER expressions are derived. Furthermore, simple expressions for the ergodic capacity in the asymptotic low and high SNR regimes are provided. The main contribution of this paper is the derivation of the closed-form expressions for the CDF and PDF of the SNR at both SC and MRC output. It is noted that the PDF of the SNR at the MRC combiner output in double Nakagami-m channels has been derived in [19]. In [19], the fading coefficient in each diversity branch is expressed as the product of two independent Nakagami-m fading coefficients. In contrast with [19] and as we will see, the fading in this paper is the sum of many products of two independent Rayleigh fading coefficients. Moreover, the MRC system considered in this paper differs from that studied in [16] where a MIMO-BF system is investigated. In fact, the SNR of the MIMO-BF approach depends explicitly on the maximum eigenvalue of the MIMO channel.

The notation adopted in this paper conforms to the following conventions. Scalars are denoted by lower case letters, vectors by bold lower case letters and matrices by bold upper case letters. The symbol \(\mathbf{I}_n\) denotes the identity matrix of size n. Superscript \(( \cdot )^{H}\) denotes the Hermitian operation, the Frobenius norm of a matrix \(\mathbf{A}\) is denoted by \(\Vert \mathbf{A}\Vert _{\text {F}}\) and \([\mathbf{A}]_{ij}\) refers to the (i, j)th element of a matrix \(\mathbf{A}\).

The remainder of this manuscript is organized as follows. In Sect. 2, we introduce the system and the double-Rayleigh channel models. Section 3 provides the exact expressions and the asymptotic expansions of the CDF and PDF of the SNR. The performance of SC and MRC combiners are investigated in Sect. 4. Finally, a conclusion to this work is provided in Sect. 5.

2 System Model

Consider \(N_r\)-branch diversity reception over Rayleigh-product fading channels. Following [15,16,17,18], we assume that the channel vector \(\mathbf{h}= [h_1, \ldots , h_{N_r}]^T\) has the statistical factorization given by

$$\begin{aligned} \mathbf{h} = \frac{1}{\sqrt{N_s}} \mathbf{H}_1 \mathbf{g}, \end{aligned}$$
(1)

where \(\mathbf{H}_1 =[h_{1,ij}] \in \mathbb {C}^{N_r \times N_s}\) and \(\mathbf{g} =[g_{i}] \in \mathbb {C}^{N_s \times 1}\) are mutually independent matrices with independent identically distributed (i.i.d.) entries that follow a zero-mean unit-variance complex-Gaussian distribution and \(N_s\) is the number of effective scatterers. Hereafter, we refer to this channel model by \((N_r, N_s)\). The generic model of (1) includes the extreme cases of the rich-scattering Rayleigh fading (\(N_s \rightarrow \infty\)) and the keyhole channel (\(N_s=1\)) [16]. Let \(E_s\) and \(N_0\) denote the average transmit energy per symbol and the one-sided power spectral density of the zero-mean additive white Gaussian, respectively.

In this paper, the MRC and SC combining schemes are considered. We assume that the fading channel is quasi-static, i.e. that fading coefficients remain invariant within one frame and change independently from one frame to another. It is also assumed that the channel state information (CSI) is perfectly available to the receiver.

For the MRC scheme, all the \(N_r\) receive antennas are combined to generate the output. The instantaneous received SNR after combining is then

$$\begin{aligned} \gamma _{\text {MRC}} = \bar{\gamma } \Vert \mathbf{h}\Vert _{\text {F}}^2, \end{aligned}$$
(2)

where \(\bar{\gamma } =E_s/N_0\). For the SC scheme, a single receive antenna, the one with the largest instantaneous SNR, is selected. Let \(\alpha\) be the index of the selected antenna. The instantaneous received SNR at the SC output is

$$\begin{aligned} \gamma _{\text {SC}} = \bar{\gamma } | h_{\alpha }|^2. \end{aligned}$$
(3)

In the remainder of this paper, we use the following notation

$$\begin{aligned} \gamma _{\text {i}} = \bar{\gamma } \beta _{\text {i}}, \quad \text {for}, \; \text {i} \in \{\text {MRC}, \text {SC}\}, \end{aligned}$$
(4)

where \(\beta _{\text {MRC}} = \Vert \mathbf{h}\Vert _{\text {F}}^2\) and \(\beta _{\text {SC}} = | h_{\alpha }|^2\).

3 Statistical Proprieties of the SNR

In this section, we first derive new expressions for the exact and asymptotic CDF and PDF of the SNR at the output of the MRC and SC combiners. These statistical functions will be used in the next section to evaluate system performance.

3.1 Exact Expressions for the CDF and the PDF

Proposition 1

The CDF and PDF of \(\gamma _{\text {MRC}}\) are, respectively, given by

$$\begin{aligned} F_{\gamma _{\text {MRC}}} (x) =&1- \sum _{k=0}^{N_r-1} \frac{2K_{N_s-k} \left( {2\sqrt{xN_{s}/\bar{\gamma } } } \right) }{\Gamma (N_s) k!} \left( \frac{x N_s}{\bar{\gamma }} \right) ^{(N_s+k)/2}, \end{aligned}$$
(5)

and

$$\begin{aligned} f_{\gamma _{\text {MRC}}}(x) =&\frac{2 x^{\frac{N_s+N_r}{2}-1} K_{N_s-N_r} \left( {2\sqrt{xN_{s}/\bar{\gamma } } } \right) }{\Gamma (N_r) \Gamma (N_s) } \left( \frac{ N_s}{\bar{\gamma }} \right) ^{(N_s+N_r)/2}, \end{aligned}$$
(6)

where \(K_v(\cdot )\) is the vth order modified Bessel function of the second kind.

Proof

See Appendix 6. \(\square\)

Proposition 2

The CDF and PDF of \(\gamma _{\text {SC}}\) are, respectively, given by

$$\begin{aligned} F_{\gamma _{\text {SC}}} (x) = 1+\sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{2 (-1)^k}{\Gamma (N_s)} \left( \frac{k x N_s}{\bar{\gamma }} \right) ^{N_s/2} K_{N_s} \left( {2\sqrt{k xN_{s}/\bar{\gamma } } } \right) , \end{aligned}$$
(7)

and

$$\begin{aligned} f_{\gamma _{\text {SC}}} (x ) =&\sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{2 k N_s(-1)^{k+1} }{\Gamma (N_s)\bar{\gamma }} \left( \frac{k x N_s}{\bar{\gamma }} \right) ^{(N_s-1)/2} K_{N_s-1} \left( {2\sqrt{k xN_{s}/\bar{\gamma } } } \right) . \end{aligned}$$
(8)

Proof

See Appendix 7. \(\square\)

Fig. 1
figure 1

CDF \(F_{\gamma _{\text {MRC}}} (\gamma )\) for different scenarios \((N_r,N_s)\) and \(\bar{\gamma }=5\) dB

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Fig. 2
figure 2

CDF \(F_{\gamma _{\text {SC}}} (\gamma )\) for different scenarios \((N_r,N_s)\) and \(\bar{\gamma }=5\) dB

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Figures 1 and 2 show the CDF of the SNR at the MRC and SC combiners for different channel models \((N_r,N_s)\) and \(\bar{\gamma }=5\) dB. The analytical results are obtained by (5) and (7) and the simulated curves are generated by Monte-Carlo simulations carried out over \(10^5\) independent channel realizations. We can see an excellent agreement between the analytical and simulated results.

3.2 Asymptotic Expressions for the CDF and the PDF

Here, we consider the high SNR regime and derive the first-order expansions for the CDF and the PDF given in (5)–(8). These asymptotic expressions will constitute a basis from which to evaluate the diversity order and the array gain of the considered systems.

To obtain such asymptotic expressions, we use the series representation of \(K_v(\cdot )\), given by [22, Eq. (8.446)], and express \(K_v(2\sqrt{z})\) as

$$\begin{aligned}&K_v(2\sqrt{z}) = \underbrace{\frac{1}{2} \sum _{l=0}^{v-1} (-1)^l \frac{(v-l-1)!}{l!} z^{l-v/2}}_{\text {Term I}} \nonumber \\&\quad + \underbrace{\frac{(-1)^{v+1}}{2} \sum _{l=0}^{\infty } \frac{1}{l! (v+l)!} z^{(l+v/2)} \left[ \ln z - \Psi (v,l)\right] }_{\text {Term II}}, \end{aligned}$$
(9)

where \(\Psi (v,l)=\psi (l+1) + \psi (v+l+1)\) and \(\psi (\cdot )\) is Euler’s digamma function.

3.2.1 Case I: MRC with \(N_r\ne N_s\)

Given the first order of the expansion in (9), we express the asymptotic PDF of \(\gamma _{\text {MRC}}\) as follows

$$\begin{aligned} f_{\gamma _{\text {MRC}}}^{\infty }(x) = \frac{\Gamma (|N_s-N_r|) }{\Gamma (N_r) \Gamma (N_s) } \left( \frac{N_s}{\bar{\gamma }}\right) ^{s} x^{s-1} + o \left( x^{s-1}\right) , \end{aligned}$$
(10)

where \(s=\min (N_s,N_r)\). Using Proposition I in [23], we approximate the CDF of \(\gamma _{\text {MRC}}\) in the high SNR regime as

$$\begin{aligned} F_{\gamma _{\text {MRC}}}^{\infty }(x) = \frac{\Gamma (|N_s-N_r|) }{s \Gamma (N_r) \Gamma (N_s) } \left( \frac{N_s}{\bar{\gamma }}\right) ^{s} x^{s} + o \left( x^{s}\right) . \end{aligned}$$
(11)

3.2.2 Case II: MRC with \(N_r = N_s\)

Using (9), we express the PDF as

$$\begin{aligned} f_{\gamma _{\text {MRC}}}^{\infty }(x) =\frac{ 1 }{ [\Gamma (N_s)]^2 } \left( \frac{ N_s}{\bar{\gamma }} \right) ^{N_s} x^{N_s-1} \ln \left( {\frac{\bar{\gamma }}{xN_{s}} } \right) . \end{aligned}$$
(12)

To derive the CDF, we use Proposition I in [23]. First, we note that \(\ln (x)\) is a slowlyFootnote 1 varying function at \(+\infty\), since \(\hbox {lim}_{x\rightarrow \infty } \ln (tx)/\ln (x)=1\). Then, by virtue of [23, Proposition I], we obtain

$$\begin{aligned} F_{\gamma _{\text {MRC}}}^{\infty }(x) =\frac{ 1 }{ N_s [\Gamma (N_s)]^2 } \left( \frac{ N_s}{\bar{\gamma }} \right) ^{N_s} x^{N_s} \ln \left( {\frac{\bar{\gamma }}{xN_{s}} } \right) . \end{aligned}$$
(13)

3.2.3 Case III: SC Scheme with \(N_s\le N_r\)

For the SC scheme, we can express the CDF as

$$\begin{aligned} F_{\gamma _{\text {SC}}}^{\infty }(x)= & {} \quad \underbrace{1+\sum _{k=1}^{N_r} \sum _{l=0}^{N_s} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{(-1)^{k+l}(N_s-l-1)!}{\Gamma (N_s) l!} \left( \frac{k x N_s}{\bar{\gamma }} \right) ^{l}}_{\text {Term I}} \nonumber \\&\quad + \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{(-1)^{N_s+k+1}}{\Gamma (N_s)} \sum _{l=0}^{\infty } \Biggl \{ \frac{1}{l! (N_s+l)!} \nonumber \\&\quad \times \left( \frac{k x N_s}{\bar{\gamma }} \right) ^{(N_s+l)} \left[ \ln \left( \frac{k x N_s}{\bar{\gamma }} \right) - \Psi (N_s,l) \right] \Biggr \}. \end{aligned}$$
(14)

Using [22, Eq. (0.154)], we have

$$\begin{aligned}&\sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^k = -1, \nonumber \\&\sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^l k =0, \; \text {for } 1 \le l \le N_r-1. \end{aligned}$$
(15)

Now, it is convenient to consider two cases \(N_s\le N_r\) and \(N_s > N_r\).

For \(N_s\le N_r\), using (15), we can easily see that Term I in (14) equates to zero. The CDF in the high SNR regime can be obtained from the first-order term as

$$\begin{aligned} F_{\gamma _{\text {SC}}}^{\infty }(x) =&\sum _{k=1}^{N_r} \Biggl \{ \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{(-1)^{k+N_s+1}}{\Gamma (N_s)\Gamma (N_s+1)} \nonumber \\&\times \left( \frac{k x N_s}{\bar{\gamma }} \right) ^{N_s} \ln \left( \frac{k x N_s}{\bar{\gamma }} \right) \Biggr \}. \end{aligned}$$
(16)

The asymptotic PDF is given by

$$\begin{aligned} f_{\gamma _{\text {SC}}}^{\infty }(x) =&\sum _{k=1}^{N_r} \Biggl \{ \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{(-1)^{k+1+N_s} }{[\Gamma (N_s)]^2 } \nonumber \\&\times \left( \frac{k N_s}{\bar{\gamma }} \right) ^{N_s} x^{N_s-1} \ln \left( \frac{k x N_s}{\bar{\gamma }} \right) \Biggr \}. \end{aligned}$$
(17)

3.2.4 Case IV: SC Scheme with \(N_s > N_r\)

For \(N_s > N_r\), Term I in (14) is not equal to zero. We can see that the leading unity term in Term I is canceled by the summation from \(l=0\) to \(l=N_r-1\). Hence, the CDF expansion is dominated by Term I for \(l=N_r\). The asymptotic CDF is then given by

$$\begin{aligned} F_{\gamma _{\text {SC}}}^{\infty }(x) = \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{(-1)^{N_r+k} (N_s-N_r-1)!}{\Gamma (N_s) N_r!} \left( \frac{k x N_s}{\bar{\gamma }} \right) ^{N_r}. \end{aligned}$$
(18)

The asymptotic PDF can be expressed as

$$\begin{aligned} f_{\gamma _{\text {SC}}}^{\infty }(x) =&\sum _{k=1}^{N_r} \Biggl \{ \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+N_r} \nonumber \\&\times \frac{ k N_s (N_s-N_r-1)! }{\Gamma (N_s)\bar{\gamma } (N_r-1)!} \left( \frac{k x N_s}{\bar{\gamma }} \right) ^{N_r-1} \Biggr \}. \end{aligned}$$
(19)

It is noted that for the four cases, the asymptotic CDF scales inversely to \(s=\min (N_s,N_r)\).

3.3 Generalized SNR Moments

The tth moment of the SNR at the MRC output is

$$\begin{aligned} \mathbb {E} \left\{ \gamma _{\text {MRC}}^t \right\} = \frac{\Gamma (N_s+t)\Gamma (N_r+t)}{\Gamma (N_r) \Gamma (N_s) } \left( \frac{\bar{\gamma }}{ N_s} \right) ^{t}. \end{aligned}$$
(20)

where \(\mathbb {E}\{\cdot \}\) refers to the expectation operator.

Proof

The tth moment of \(\beta _{\text {MRC}}\) is

$$\begin{aligned} \mathbb {E} \left\{ \beta _{\text {MRC}}^t \right\} =&\int _0^{\infty } x^t f_{\beta _{\text {MRC}}}(x) dx \nonumber \\ =&\frac{4 }{\Gamma (N_r) \Gamma (N_s) } \left( \frac{ 1}{N_s} \right) ^{t} \int _0^{\infty } z^{{N_s+N_r}-1+2t} K_{N_s-N_r} \left( 2z \right) d z. \end{aligned}$$
(21)

Using [22, Eq. (6.561.16)], we obtain

$$\begin{aligned} \mathbb {E} \left\{ \beta _{\text {MRC}}^t \right\} = \frac{\Gamma (N_s+t)\Gamma (N_r+t)}{\Gamma (N_r) \Gamma (N_s) } \left( \frac{1}{ N_s} \right) ^{t}. \end{aligned}$$
(22)

Finally, we use \(\mathbb {E} \left\{ \gamma _{\text {MRC}}^t \right\} = \bar{\gamma }^t \mathbb {E} \left\{ \beta _{\text {MRC}}^t \right\}\). \(\square\)

The tth moment of the SNR at the SC output is given by

$$\begin{aligned} \mathbb {E} \left\{ \gamma _{\text {SC}}^t \right\} = \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} \frac{\Gamma (N_s+t)\Gamma (t+1) }{\Gamma (N_s)} \left( \frac{\bar{\gamma }}{k N_s} \right) ^t. \end{aligned}$$
(23)

Proof

The tth moment of \(\beta _{\text {SC}}\) is given by

$$\begin{aligned} \mathbb {E} \left\{ \beta _{\text {SC}}^t \right\} =&\int _0^{\infty } x^t f_{\beta _{\text {SC}}}(x) dx \nonumber \\ =&\sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} \frac{4 }{\Gamma (N_s)} \left( \frac{1 }{k N_s} \right) ^t \int _0^{\infty } z^{N_s+2t} K_{N_s-1}(2 z) dz. \end{aligned}$$
(24)

Using [22, Eq. (6.561.16)], we obtain

$$\begin{aligned} \mathbb {E} \left\{ \beta _{\text {SC}}^t \right\} = \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} \frac{\Gamma (N_s+t)\Gamma (t+1) }{\Gamma (N_s)} \left( \frac{1}{k N_s} \right) ^t. \end{aligned}$$
(25)

The tth moment of \(\gamma _{\text {SC}}^t\) is obtained using \(\mathbb {E} \left\{ \gamma _{\text {SC}}^t \right\} = \bar{\gamma }^t \mathbb {E} \left\{ \beta _{\text {SC}}^t \right\}\). \(\square\)

4 Performance Analysis

4.1 Outage Probability

In wireless communications, the outage probability (\(P_\text {out}\)) is an important measurement for the quality-of-service (QoS). It is defined as the probability that the instantaneous received SNR falls below a threshold \(\gamma _{th}\), i.e.,

$$\begin{aligned} P_{\text {out}}= \text {Pr}\left( \gamma \le \gamma _{th} \right) = F_{\gamma }(\gamma _{th}). \end{aligned}$$
(26)

The exact \(P_\text {out}\) of the MRC and SC diversity combining schemes can be evaluated based on (5) and (7), respectively. Asymptotic expressions for the \(P_\text {out}\) for the different scenarios are derived using the CDF approximations in the high SNR regime given in Sect. 3.2.

Figures 3 and 4 show the \(P_\text {out}\) for MRC and SC as a function of the SNR when the threshold \(\gamma _{th}\) is fixed at 0 dB. In these figures, we compare the exact \(P_\text {out}\), the asymptotic approximations and MC simulated curves for different system configurations. For the rich-scattering i.i.d Rayleigh case (\(N_s=\infty\)), the \(P_\text {out}\) curves are obtained based on [4, Eq. (7.19) p. 199] for MRC and [4, Eq. (7.8) p. 194] for SC. We can see that the results obtained from analysis match closely those obtained by simulations. It is noted that the slopes of the \(P_\text {out}\) curves are determined by \(s=\min (N_s,N_r)\) and that \(P_\text {out}\) diminishes as s increases. Furthermore, we note that as \(N_s\) increases, the \(P_\text {out}\) approaches that of rich scattering i.i.d Rayleigh channels. Moreover, we observe that slopes for configurations (4, 2) and (2, 4), which have the same value of s, are identical. However, \(P_\text {out}\) for the (4, 2) system is better than that of (2, 4). The difference is more evident for MRC than for SC. This can be explained by the fact that the array gain increases as \(N_r\) increases.

Fig. 3
figure 3

The \(P_\text {out}\) of different (\(N_r\),\(N_s\)) systems with MRC for \(\gamma _{th}=0\) dB

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Fig. 4
figure 4

The \(P_\text {out}\) of different (\(N_r\),\(N_s\)) systems with SC for \(\gamma _{th}=0\) dB

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4.2 Average Symbol Error Rate

It is well known that the instantaneous SER of many digital modulations can be expressed as [2]

$$\begin{aligned} P_e(\gamma ) = \frac{a}{2} \text {erfc} \left( \sqrt{b \gamma }\right) , \end{aligned}$$
(27)

where a and b are specific modulation constants and \(\text {erfc}(\cdot )\) is the complementary error function which can be expressed in terms of the Meijer G-function as [25]

$$\begin{aligned} \text {erfc}(\sqrt{x}) = \frac{1}{\sqrt{\pi }} G_{1,2}^{2,0} \!\left( x \; \left| \begin{matrix} 1, &{} \\ 0, &{} \frac{1}{2} \end{matrix} \right. \right) . \end{aligned}$$
(28)

The average SER \(\bar{P}_e\) can be derived by averaging the instantaneous SER over the instantaneous SNRs, i.e.,

$$\begin{aligned} \bar{P}_e = \mathbb {E} \left\{ P_e(\gamma ) \right\} = \int _0^{\infty } P_e(\gamma ) f_{\gamma } (\gamma ) d \gamma . \end{aligned}$$
(29)

By substituting (6), (27), and (28) into (29) and calculating the resultant integral with the aid of [22, Eq. (7.821.3)], we obtain the closed-form SER expression for the MRC scheme

$$\begin{aligned} \bar{P}_{e,\text {MRC}} =&\mathbb {E} \left\{ P_e(\gamma ) \right\} = \int _0^{\infty } P_e(\gamma ) f_{\gamma } (\gamma ) d \gamma \nonumber \\ =&\int _0^{\infty } \frac{a K_{N_s-N_r} \left( {2\sqrt{z}} \right) }{\sqrt{\pi } \Gamma (N_r) \Gamma (N_s) } \left( z \right) ^{\frac{N_s+N_r}{2}-1} G_{1,2}^{2,0} \!\left( \frac{b \bar{\gamma }}{N_s} z \; \left| \begin{matrix} 1, &{}\\ 0, &{} \frac{1}{2} \end{matrix} \right. \right) d z \nonumber \\ =&\frac{a G_{3,2}^{2,2} \!\left( \frac{b \bar{\gamma }}{N_s} \; \left| \begin{matrix} 1-N_s, &{} 1-N_r, &{} 1 \\ 0, &{} \frac{1}{2} &{} \end{matrix} \right. \right) }{2 \sqrt{\pi } \Gamma (N_r) \Gamma (N_s)}. \end{aligned}$$
(30)

By substituting (8), (27), and (28) into (29) and proceeding as in the case of MRC, we can express the SER of the SC scheme as follows

$$\begin{aligned}\bar{P}_{e,\text {SC}} &= \mathbb {E} \left\{ P_e(\gamma ) \right\} = \int _0^{\infty } P_e(\gamma ) f_{\gamma } (\gamma ) d \gamma \nonumber \\\quad &= \sum _{k=1}^{N_r} \Biggl \{ \frac{a \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} }{\sqrt{\pi }\Gamma (N_s)} \int _0^{\infty } G_{1,2}^{2,0} \!\left( \frac{b \bar{\gamma } }{k N_s} z \; \left| \begin{matrix} 1, &{} \\ 0, &{} \frac{1}{2} \end{matrix} \right. \right) \left( z\right) ^{(N_s-1)/2} K_{N_s-1} \left( {2\sqrt{z } } \right) d z \Biggr \} \nonumber \\\quad &= \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} \frac{a G_{3,2}^{2,2} \!\left( \frac{b \bar{\gamma } }{k N_s} \; \left| \begin{matrix} 1-N_s, &{} 0, &{} 1 \\ 0, &{} \frac{1}{2} &{} \end{matrix} \right. \right) }{2 \sqrt{\pi }\Gamma (N_s)}. \end{aligned}$$
(31)

We now investigate the SER performance in the high SNR regime in order to derive the diversity order and the array gain of the two considered schemes. To this end, we consider the approximation of \(K_v(x)\) for small values of x.

4.2.1 Case I: MRC with \(N_r\ne N_s\)

By invoking [26, Propo. 1], we express the average SER in the high SNR regime as

$$\begin{aligned} \bar{P}_{e,\text {MRC}}^{\infty } (\bar{\gamma }) = \frac{a N_s^{s} \Gamma (|N_s-N_r|) \Gamma (s+\frac{1}{2})}{2\sqrt{\pi }s \Gamma (N_r) \Gamma (N_s) } \left( b \bar{\gamma } \right) ^{-s} + o \left( \bar{\gamma }^{-s}\right) . \end{aligned}$$
(32)

Given (32), we conclude that the diversity order for the MRC scheme, when \(N_r\ne N_s\) is

$$\begin{aligned} G_d^{\text {MRC}} = \min (N_s,N_r) =s, \end{aligned}$$
(33)

and the array gain is

$$\begin{aligned} G_a^{\text {MRC}} = b \left( \frac{a 2^s N_s^{s} \Gamma (|N_s-N_r|) \Gamma (s+\frac{1}{2})}{\sqrt{\pi }s \Gamma (N_r) \Gamma (N_s) } \right) ^{-1/s}. \end{aligned}$$
(34)

4.2.2 Case II: MRC with \(N_r = N_s\)

When \(N_r=N_s\), we cannot derive the diversity order using [26, Propo. 1]. In this case, the average SER in the high SNR regime can be obtained by replacing the complementary function by its integral representation and exploiting [22, Eq. (4.352.1)], as suggested in [27], .i.e.

$$\begin{aligned}&\bar{P}_{e,\text {MRC}}^{\infty } (\bar{\gamma }) = \int _0^{\infty } P_e(x) f_{\gamma _{\text {MRC}}}^{\infty }(x) d x \nonumber \\&\quad = - \frac{ a \sqrt{b} (N_s/\bar{\gamma })^{N_s} }{ \sqrt{\pi } [\Gamma (N_s)]^2 } \int _{1}^{\infty } \int _0^{\infty } e^{-b t^2 \frac{\bar{\gamma }}{N_s} x} \left( \frac{\bar{\gamma }}{N_s} \right) ^{N_s-1/2} x^{N_s-1/2} \ln (x) d x d t \nonumber \\&\quad = \frac{ a \Gamma (N_s + \frac{1}{2}) }{2 N_s \sqrt{\pi } [\Gamma (N_s)]^2 } \ln \left( \frac{b \bar{\gamma } }{N_s} \right) \left( \frac{N_s}{b \bar{\gamma } } \right) ^{N_s}. \end{aligned}$$
(35)

The diversity order is given by [28]

$$\begin{aligned} G_d^{\text {MRC}} = - \lim _{\bar{\gamma } \rightarrow \infty } \frac{\log \bar{P}_{e,\text {MRC}}^{\infty }}{\log \bar{\gamma }}. \end{aligned}$$
(36)

Given (35) and using \(\lim _{x \rightarrow \infty } \log (c x)/ \log (x)=1\), we conclude that

$$\begin{aligned} G_d^{\text {MRC}} \approx N_s-1 = s-1. \end{aligned}$$
(37)

Here, we note that given (35), the diversity behavior cannot be described by an integer [29]. In this case, the diversity order falls between \(N_s-1\) and \(N_S\) [29, 30].

4.2.3 Case III: SC Scheme with \(N_s\le N_r\)

In this case, the asymptotic SER can be expressed as

$$\begin{aligned} \bar{P}_{e,\text {SC}}^{\infty } (\bar{\gamma })&= \int _0^{\infty } P_e(x) f_{\gamma _{\text {SC}}}^{\infty }(x) d x \nonumber \\ =&\sum _{k=1}^{N_r} \Biggl \{ \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{(-1)^{k+1+N_s} a \Gamma (N_s + \frac{1}{2}) }{2 N_s \sqrt{\pi } [\Gamma (N_s)]^2 } \ln \left( \frac{b \bar{\gamma } }{k N_s} \right) \left( \frac{k N_s}{b \bar{\gamma } } \right) ^{N_s} \Biggr \}. \end{aligned}$$
(38)

For this case, the diversity order is

$$\begin{aligned} G_d^{\text {SC}} \approx N_s-1 =s-1. \end{aligned}$$
(39)

4.2.4 Case IV: SC Scheme with \(N_s > N_r\)

When \(N_s > N_r\), the asymptotic SER of the SC scheme is given by

$$\begin{aligned} \bar{P}_{e,\text {SC}}^{\infty } (\bar{\gamma })&= \int _0^{\infty } P_e(x) f_{\gamma _{\text {SC}}}^{\infty }(x) d x \nonumber \\ &=\sum _{k=1}^{N_r} \Biggl \{ \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{a (-1)^{k+N_r} \left( b \bar{\gamma } \right) ^{-N_r} \Gamma (N_r+\frac{1}{2}) (N_s-N_r-1)! \left( k N_s\right) ^{N_r}}{2 N_r! \Gamma (N_s) \sqrt{\pi }} \Biggr \}. \end{aligned}$$
(40)

Given (40), we conclude that the diversity order is

$$\begin{aligned} G_d^{\text {SC}} = N_r =s. \end{aligned}$$
(41)
Fig. 5
figure 5

SER of QPSK versus SNR \(\bar{\gamma }\) in \((4,N_s)\) channels with MRC

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To validate the derived SER expressions, MC simulations have been carried out for different Rayleigh product \((N_r,N_s)\) channels. The results are obtained for quadrature phase-shift keying (QPSK) modulation (\(a=2, b=0.5\)). The average SER of MRC and SC schemes are, respectively, shown in Figs. 5 and 6, where exact SER, asymptotic approximations and MC results are compared. For further comparison, the average SER of rich-scattering i.i.d Rayleigh channels is also presented. For this case, the SER curves for MRC and SC are obtained form [31, Eq. (12)] and [31, Eq. (23)], respectively. For all scenarios, there is excellent agreement between the exact and asymptotic analytical SER expressions and MC results. It is noted that slopes of SER curves are proportional to the diversity order and as \(N_s\) increases, the average SER approaches that of rich scattering i.i.d Rayleigh channels. Moreover, for \(N_s \ge 4\), the slopes of SER curves are identical to that of the rich-scattering case. In fact, for \(N_s \ge 4\), we have \(s=N_r=4\) which corresponds to the diversity order of MRC and SC in i.i.d Rayleigh channels.

Fig. 6
figure 6

SER of QPSK versus SNR \(\bar{\gamma }\) in \((4,N_s)\) channels with SC

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4.3 Ergodic Capacity

Here, we derive closed-form expressions for the ergodic capacity defined as the expectation of the instantaneous mutual information, i.e.

$$\begin{aligned} C = \mathbb {E} \left\{ \log _2 (1+\gamma ) \right\} = \int _0^{\infty } \log _2 (1+\gamma ) f_{\gamma } (\gamma ) d \gamma . \end{aligned}$$
(42)

For the MRC scheme, the ergodic capacity is given by

$$\begin{aligned} C = \int _0^{\infty } \log _2 (1+\gamma ) \frac{2 \gamma ^{\frac{N_s+N_r}{2}-1} }{\Gamma (N_r) \Gamma (N_s) } \left( \frac{ N_s}{\bar{\gamma }} \right) ^{\frac{N_s+N_r}{2} } K_{N_s-N_r} \left( {2\sqrt{\gamma N_{s}/\bar{\gamma } } } \right) d \gamma . \end{aligned}$$
(43)

Using [32, Eq. (8.4.6.5)], [22, Eq. (9.31.2)] and, [22, Eq. (7.821.3)], we can express the ergodic capacity as

$$\begin{aligned} C = \frac{1 }{\Gamma (N_r) \Gamma (N_s) \ln 2 } G_{2,4}^{4,1} \left( \frac{N_s}{\bar{\gamma }} \; \left| \begin{matrix} 0, &{} 1 &{} &{} \\ N_s, &{} N_r, &{} 0, &{} 0 \end{matrix} \right. \right) . \end{aligned}$$
(44)

For the SC scheme, the ergodic capacity is given by

$$\begin{aligned} C&= \int _0^{\infty } \log _2 (1+\gamma ) f_{\gamma _{\text {SC}}}(\gamma ) d \gamma \nonumber \\ =&\sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} \frac{2 }{\Gamma (N_s) } \left( \frac{k N_s}{\bar{\gamma }} \right) ^{\frac{N_s+1}{2}} \nonumber \\&\times \int _0^{\infty } \log _2 (1+\gamma ) \gamma ^{\frac{N_s-1}{2}} K_{N_s-1} \left( {2\sqrt{k \gamma N_{s}/\bar{\gamma } } } \right) d \gamma . \end{aligned}$$
(45)

Using [32, Eq. (8.4.6.5)], [22, Eq. (9.31.2)] and [22, Eq. (7.821.3)], we can express the ergodic capacity as

$$\begin{aligned} C = \sum _{k=1}^{N_r} \Biggl \{ \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} \frac{1 }{\Gamma (N_s) \ln 2 } G_{2,4}^{4,1} \left( \frac{k N_s}{\bar{\gamma }} \; \left| \begin{matrix} 0, &{} 1 &{} &{} \\ N_s , &{} 1, &{} 0, &{} 0 \end{matrix} \right. \right) \Biggr \}. \end{aligned}$$
(46)

4.3.1 High SNR Capacity

For high SNR values, the capacity can be expressed as [16, 33]

$$\begin{aligned} C (\bar{\gamma }) = S_{\infty } \left( { {\frac{\left. {\bar{\gamma }} \right| _\mathrm{dB}}{3~\mathrm{dB}}} - \mathcal{L}_{\infty } }\right) + o\left( 1 \right) , \end{aligned}$$
(47)

where 3 dB\(=10 \log _{10}(2)\) and the \(S_{\infty }\) and \(\mathcal{L}_{\infty }\) are defined as

$$\begin{aligned}&S_{\infty } = \mathop {\lim }\limits _{\bar{\gamma }\rightarrow \infty }{\frac{C(\bar{\gamma })}{\log _{2} (\bar{\gamma }) }}, \end{aligned}$$
(48)
$$\begin{aligned}&\mathcal{L}_{\infty } = \mathop {\lim }\limits _{\bar{\gamma }\rightarrow \infty } \left( {\log _{2} (\bar{\gamma }) - {\frac{ C(\bar{\gamma }) }{S_{\infty } }}}\right) . \end{aligned}$$
(49)

Given (53), it was shown in [33] that \(S_{\infty }=1\) bit/s/Hz, regardless of \(N_r\) and \(N_s\). The high SNR power offset \(\mathcal{L}_{\infty }\) is derived by substituting (53) in (49) as [16]

$$\begin{aligned} \mathcal{L}_{\infty } = - \log _{2}(e) {\frac{ d}{ d t}} \left( \mathbb {E} \left[ \beta _\text {i}^{t} \right] \right) \biggr \vert _{t=0}. \end{aligned}$$
(50)

For the MRC scheme, by substituting (22) in (50) and calculating the resultant derivative with the aid of the identity \(d \Gamma (z)/dz= \psi (z) \Gamma (z)\), we express \(\mathcal{L}_{\infty }^{\text {MRC}}\) as

$$\begin{aligned} \mathcal{L}_{\infty }^{\text {MRC}} = - \log _{2}(e) \left[ \psi (N_r) + \psi (N_s) -\ln N_s \right] . \end{aligned}$$
(51)

For the SC technique, \(\mathcal{L}_{\infty }^{\text {SC}}\) is given by

$$\begin{aligned} \mathcal{L}_{\infty }^{\text {SC}} = \log _{2}(e) \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k} \left[ \psi (N_s) + \psi (1) -\ln (k N_s) \right] . \end{aligned}$$
(52)
Fig. 7
figure 7

Ergodic capacity of \((N_r,3)\) channels with MRC

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Fig. 8
figure 8

Ergodic capacity of \((N_r,3)\) channels with SC

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Exact ergodic capacity expressions and analytical high SNR capacity approximations are compared with MC simulated capacity results in Figs. 7 and 8. We observe perfect agreement between analytical curves and simulation results. As expected, for all cases, the high SNR slopes of capacity curves are unity, since \(S_{\infty }=1\) bit/s/Hz. Thus, increasing \(N_r\) serves only to improve the high SNR power offset \(\mathcal{L}_{\infty }\).

4.3.2 Low SNR Capacity

Authors in [34,35,36] have shown that taking the first-order expansion of the ergodic capacity as \(\bar{\gamma }\rightarrow 0\) to investigate the channel impact in the low SNR regime is not relevant [37]. In this SNR regime, to characterize the tradeoff between channel capacity, bandwidth and power, it is appropriate to consider the normalized transmit energy per information bit \(E_b/N_0\) rather the SNR [34,35,36,37]. As discussed in [36], for low values of \(E_b/N_0\), the capacity is approximated by [34,35,36,37]

$$\begin{aligned} \mathrm{C}\left( {\frac{E_{b} }{N_{0} }} \right) \approx S_{0}\log _{2}\left( {\frac{\frac{E_{b} }{N_{0} }}{ {\frac{E_{b} }{N_{0} }}_{\min } }} \right) , \end{aligned}$$
(53)

where \(({E_b }/{N_0}_{\min })\) is the minimum \(E_b/N_0\) required for reliable communication and \(S_0\) is the spectral efficiency slope in bits/s/Hz. From [36], the two parameters are given by

$$\begin{aligned} \frac{E_b}{N_0}_{\min } = \frac{\ln 2}{\mathbb {E} \left\{ \beta _\text {i} \right\} }, \end{aligned}$$
(54)

and

$$\begin{aligned} S_0 = \frac{2 \left( \mathbb {E} \left\{ \beta _\text {i} \right\} \right) ^2}{\mathbb {E} \left\{ \beta _\text {i}^2 \right\} }. \end{aligned}$$
(55)

Using (22), we obtain the tth moment of \(\beta _{\text {MRC}}\) which yields

$$\begin{aligned} \frac{E_b}{N_0}_{\min }^{\text {MRC}} = \frac{\ln 2}{N_r}, \end{aligned}$$
(56)

and

$$\begin{aligned} S_0^{\text {MRC}} = \frac{2 N_r N_s}{(N_r+1)(N_s+1)}. \end{aligned}$$
(57)

From (56), we can observe that \(\frac{E_b}{N_0}_{\min }^{\text {MRC}}\) is inversely proportional to the number of available branches \(N_r\) and is independent of \(N_s\). Furthermore, the minimum required received bit energy above noise level, which is a fundamental feature for channels with additive Gaussian noise [34,35,36,37], is equal to

$$\begin{aligned} \frac{E_b}{N_0}_{\min }^{\text {MRC},r} = N_r \frac{E_b}{N_0}_{\min }^{\text {MRC}} = -1.59 \; \text {dB} \end{aligned}$$
(58)

It is noted that \(N_r\) and \(N_s\) do not affect this metric.

For the SC scheme, the identity (25) leads to

$$\begin{aligned} \frac{E_b}{N_0}_{\min }^{\text {SC}} = \frac{\ln 2}{\sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} k^{-1}}, \end{aligned}$$
(59)

and

$$\begin{aligned} S_0^{\text {SC}} = \frac{N_s}{N_s+1} \frac{ \left( \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} k^{-1} \right) ^2 }{\sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} k^{-2} }. \end{aligned}$$
(60)

Here, we can see that \(\frac{E_b}{N_0}_{\min }^{\text {SC}}\) depends only on \(N_r\), and unlike the MRC case, \(\frac{E_b}{N_0}_{\min }^{\text {SC},r}\) also depends on \(N_r\).

Fig. 9
figure 9

Low-SNR ergodic capacity for \((4,N_s)\) channels with MRC

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Fig. 10
figure 10

Low-SNR ergodic capacity for \((N_r,3)\) channels with MRC

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Fig. 11
figure 11

Low-SNR ergodic capacity for \((2,N_s)\) and \((4,N_s)\) channels with SC

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In Figs. 9, 10 and 11, we compare the low SNR capacity approximations with MC simulated ergodic capacity. The ergodic capacity and its low SNR approximation are depicted versus \(\frac{E_b}{N_0}_{\min }^{r}\) for different channel configurations. As we can see, the low SNR approximations are quite tight for all considered cases. Figure 9 illustrates the impact of \(N_S\) on the low SNR capacity and confirms the intuitive result that the capacity improves as \(N_s\) increases to converge to that of i.i.d Rayleigh channels. In Fig. 10, the low SNR approximations are presented for different values on \(N_r\). As expected, the low SNR slope increases with \(N_R\), whereas \(\frac{E_b}{N_0}_{\min }^{\text {MRC},r}=-1.59\) dB regardless of \(N_R\) and \(N_s\). For \(N_s=20\) and \(N_s=\infty\), the low SNR slope is about 1.56 bits/s/Hz per 3 dB. However, for \(N_s=1\) (keyhole channel), the slope is reduced to 1.02 dB. Hence, the discrepancy between the two extreme cases is about 35%. In Fig. 11, we consider the SC combiner and investigate the impact of \(N_r\) and \(N_s\) on the the low SNR capacity. It is noted that the minimum received bit energy above noise level required for reliable communication depends on \(N_r\).

5 Conclusion

An analytical performance assessment of the MRC and SC diversity schemes over Rayleigh product channels was presented. Exact closed-form expressions and asymptotic expansions of the CDF and PDF of the SNR at each combiner output were derived. Exact expressions were the starting point to derive closed-form formulas for the OP, the average SER and the channel capacity. Based on the asymptotic approximations of the PDFs, we evaluated the asymptotic SER which accurately reveals the diversity order and the array gain. We have shown that the diversity gain depends on the minimum of the numbers of scatterers and available branches. Furthermore, when the number of scatterers becomes sufficiently large, the system performance approaches that of i.i.d Rayleigh channels. We have also characterized the channel capacity in high and low SNR regimes by deriving closed-form expressions for key parameters such as high SNR power offset \(\mathcal{L}_{\infty }\), high SNR slope in \(S_{\infty }\), the minimum required transmit energy per information bit \(\frac{E_b}{N_0}_{\min }\) and the spectral efficiency slope \(S_0\).

6 Proof of Proposition 1

Let \(\mathbf{g} \mathbf{g}^H = \mathbf{U}{} \mathbf{S}{} \mathbf{V^H}\) be the singular value decomposition of \(\mathbf{g} \mathbf{g}^H\), where \(\mathbf{S} \in \mathbb {C}^{N_s \times N_s}\) is a diagonal matrix, and \(\mathbf{U}\) and \(\mathbf{V}\) are two \(N_s \times N_s\) unitary matrices. We can express \(\Vert \mathbf{H}_1 \mathbf{g}\Vert _{\text {F}}^2\) as

$$\begin{aligned} \Vert \mathbf{H}_1 \mathbf{g}\Vert _{\text {F}}^2 = {{\text {tr}}\left( \mathbf{H}_1 \mathbf{U}{} \mathbf{S}{} \mathbf{V^H} \mathbf{H}_1^H \right) }. \end{aligned}$$
(61)

Given that \(\mathbf{U}\) and \(\mathbf{V}\) are unitary matrices, conditioned on \(\mathbf{g}\), we have

$$\begin{aligned} \mathbf{H}_1 \mathbf{U}{} \mathbf{S}{} \mathbf{V^H} \mathbf{H}_1^H |\mathbf{g} \sim \mathbf{H}_1 \mathbf{S} \mathbf{H}_1^H |\mathbf{g}, \end{aligned}$$
(62)

where “\(\sim\)“ denotes equivalent distribution. Since \(\mathbf{H}_1 \mathbf{g}\) is a vector, \(\mathbf{H}_1 \mathbf{g} \mathbf{g}^H \mathbf{H}_1^H\) is a rank one Hermitian matrix with only one non-zero eigenvalue, which corresponds to \(S_{11} = \Vert \mathbf{g}\Vert _{\text {F}}^2\). Hence, it can easily be observed that, conditioned on \(\mathbf{g}\), we have

$$\begin{aligned} \Vert \mathbf{h}\Vert _{\text {F}}^2 \sim {{\text {tr}}\left( \mathbf{H}_1 \mathbf{S} \mathbf{H}_1^H \right) } \sim \Vert \mathbf{g} \Vert _{\text {F}}^2 \Vert \mathbf{h}_{1,1}\Vert _{\text {F}}^2. \end{aligned}$$
(63)

Since the entries of \(\mathbf{H}_1\) are zero-mean unit-variance complex-Gaussian random variables, \(\Vert \mathbf{h}_{1,1}\Vert _{\text {F}}^2\) is a Chi-squared variate with \(2N_r\) degrees of freedom. Thus, using a change of variable, we can show that the conditioned CDF and PDF of the received SNR are given by

$$\begin{aligned}&F_{\beta _{\text {MRC}}} (x|\mathbf{g} ) = 1- e^{-\frac{x N_s}{ \Vert \mathbf{g} \Vert _{\text {F}}^2}} \sum _{l=0}^{N_r-1} \frac{1}{l!} \left( \frac{x N_s}{ \Vert \mathbf{g} \Vert _{\text {F}}^2} \right) ^l, \nonumber \\&f_{\gamma _{\text {MRC}}} (x|\mathbf{g} ) = \frac{N_s^{N_r} x^{N_r-1} e^{-x N_s/( \Vert \mathbf{g} \Vert _{\text {F}}^2)}}{\Gamma (N_r)\left( \Vert \mathbf{g} \Vert _{\text {F}}^2 \right) ^{N_r}}. \end{aligned}$$
(64)

The PDF of the Chi-squared distributed random variable \(Z=\Vert \mathbf{g} \Vert _{\text {F}}^2\) is

$$\begin{aligned} f_{Z} (z) = \frac{1}{\Gamma (N_s)} z^{N_s-1} e^{-z}. \end{aligned}$$
(65)

The unconditional CDF of \(\beta _{\text {MRC}}\) can be obtained as follows

$$\begin{aligned} F_{\beta _{\text {MRC}}} (x ) =&\int _0^{+\infty } F_{\beta _{\text {MRC}}} (x|\mathbf{g} ) f_{Z} (z) dz \nonumber \\ =&1- \sum _{l=0}^{N_r-1} \Biggl \{ \frac{(x N_s)^{l}}{\Gamma (N_s) ^l l!} \int _0^{+\infty } \exp \left( {-z-\frac{x N_s}{ z}}\right) z^{N_s-l-1} dz \Biggr \}. \end{aligned}$$
(66)

Using [22, Eq. (3.471.9)], we obtain the closed-form CDF expression

$$\begin{aligned} F_{\beta _{\text {MRC}}} (x ) = 1- \sum _{l=0}^{N_r-1} \frac{2 \left( {x N_s}{} \right) ^{\frac{N_s+l}{2}}}{\Gamma (N_s) l!} K_{N_s-l} \left( {2\sqrt{xN_{s} } } \right) . \end{aligned}$$
(67)

The unconditional PDF of \(\gamma _{\beta _{\text {MRC}}}\) can be obtained as

$$\begin{aligned}&f_{\beta _{\text {MRC}}} (x ) = \int _0^{+\infty } f_{\beta _{\text {MRC}}} (x|\mathbf{g} ) f_{Z} (z) dz \nonumber \\&\quad = \int _0^{+\infty } \frac{N_s^{N_r} x^{N_r-1}}{ \Gamma (N_r) \Gamma (N_s) } {z^{N_s-N_r-1} \exp \left( {-z}-\frac{x N_s}{ z} \right) } dz. \end{aligned}$$
(68)

Using [22, Eq. (3.471.9)], we obtain the closed-form PDF expression

$$\begin{aligned}&f_{\beta _{\text {MRC}}} (x ) = \frac{2 N_s^{\frac{N_s+N_r}{2} } x^{\frac{N_s+N_r-2}{2} }}{\Gamma (N_r) \Gamma (N_s) } K_{N_s-N_r} \left( {2\sqrt{xN_{s} } } \right) . \end{aligned}$$
(69)

Using \(F_{\gamma _{\text {MRC}}} (x)=F_{\beta _{\text {MRC}}}(x/\bar{\gamma })\) and \(f_{\gamma _{\text {MRC}}} (x)=F_{\beta _{\text {MRC}}}(x/\bar{\gamma })/\bar{\gamma }\), we obtain the identities of Proposition 1.

7 Proof of Proposition 2

Let us first consider the conditioned CDF and PDF of the received SNR at any antenna i given \(\mathbf{g}\). By substituting \(N_r=1\) into (64), we obtain the CDF and the PDF of \(\beta ^i=|h_i|^2\), i.e.

$$\begin{aligned}&F_{\beta ^i} (x|\mathbf{g} ) = 1- e^{-\frac{x N_s}{ \Vert \mathbf{g} \Vert _{\text {F}}^2}}, \nonumber \\&f_{\beta ^i} (x|\mathbf{g} ) = \frac{N_s e^{-x N_s/ \Vert \mathbf{g} \Vert _{\text {F}}^2}}{ \Vert \mathbf{g} \Vert _{\text {F}}^2 }. \end{aligned}$$
(70)

Given \(\mathbf{g}\), the conditioned \(\beta ^i\) are independent and identically distributed since the rows of \(\mathbf{H}_1\) are i.i.dFootnote 2. Hence, thanks to order statics, the conditioned CDF and PDF of \(\beta _{\text {SC}}\) can be expressed as [25]

$$\begin{aligned}&F_{\beta _{\text {SC}}} (x|\mathbf{g}) = \left[ F_{\beta ^i} (x|\mathbf{g}) \right] ^{N_r}, \nonumber \\&f_{\beta _{\text {SC}}} (x|\mathbf{g}) = N_r f_{\beta ^i} (x|\mathbf{g} ) \left[ F_{\beta ^i} (x|\mathbf{g}) \right] ^{N_r-1}. \end{aligned}$$
(71)

The unconditional CDF of \(\beta _{\text {SC}}\) is given by

$$\begin{aligned} F_{ \beta _{\text {SC}}} (x )&= \int _0^{+\infty } F_{ \beta _{\text {SC}}} (x|\mathbf{g} ) f_{Z} (z) dz \nonumber \\&= 1+ \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) \frac{2 (-1)^k \left( {k x N_s} \right) ^{\frac{N_s}{2}} }{\Gamma (N_s)} K_{N_s} \left( {2\sqrt{k xN_{s} } } \right) . \end{aligned}$$
(72)

To compute the PDF, we take the derivative of the CDF using [16, Eq. (99)]

$$\begin{aligned} \frac{d }{dx} ( (\alpha x)^{v/2} K_v(2\sqrt{\alpha x}) = -\alpha (\alpha x)^{(v-1)/2} K_{v-1}(2\sqrt{\alpha x}), \end{aligned}$$
(73)

The unconditional PDF of \(\beta _{\text {SC}}\) is given by

$$\begin{aligned} f_{\beta _{\text {SC}}} (x ) = \sum _{k=1}^{N_r} \left( {\begin{array}{c}N_r\\ k\end{array}}\right) (-1)^{k+1} \frac{2 k N_s }{\Gamma (N_s) } \left( {k x N_s} \right) ^{\frac{N_s-1}{2}} K_{N_s-1} \left( {2\sqrt{k xN_{s} } } \right) . \end{aligned}$$
(74)

The CDF and PDF of \(\gamma _{\text {SC}}\) are obtained in a straightforward manner from (72) and (74).