Multi-user mixed multi-hop relay network

Abstract

The present disclosure relates to a triple-hop multiuser relay network. Further, the relay network is comprised of mixed communication mediums (radiofrequency/free-space optical/radiofrequency), and utilizes a generalized order user scheduling scheme for determining the next source or destination to be selected for transmission. Closed-form expressions were achieved to describe outage probability, average symbol error probability, and channel capacity assuming Rayleigh and Gamma-Gamma fading models for the radiofrequency and free-space optical links, respectively. The effects of pointing errors on the free-space optical link were also considered. Additionally, a power allocation algorithm was proposed to optimize power allocation at each hop.

Claims

1. A system for wireless network communication, comprising: a first plurality of wireless devices at a source configured to transmit or receive communication via radiofrequency; a second plurality of wireless devices at a destination configured to transmit or receive communication via radiofrequency; a first relay configured to communicate with a selected one of the first plurality of wireless devices at the source via radiofrequency and communicate with a second relay via free space optical communication, the second relay being configured to communicate with the first relay via free space optical communication and with a selected one of the second plurality of wireless devices at the destination via radiofrequency; and processing circuitry configured to select a wireless device at the source with the largest partially modeled source signal-to-noise ratio as the selected one of the first plurality of wireless devices, select a wireless device at the destination with the largest partially modeled destination signal-to-noise ratio as the selected one of the second plurality of wireless devices, wherein the partially modeled source signal-to-noise ratio is based on transmitted power, additive white Gaussian noise, and Rayleigh fading model-based channel coefficients, and the partially modeled destination signal-to-noise ratio is based on transmitted power, additive white Gaussian noise, and Rayleigh fading model-based channel coefficients.

2. The system for wireless network communication of claim 1, wherein the radiofrequency communication of the first plurality of wireless devices at the source is mmWave radiofrequency communication.

3. The system for wireless network communication of claim 1, wherein the radiofrequency communication of the first relay is mmWave radiofrequency communication.

4. The system for wireless network communication of claim 1, wherein the first relay employs a decode and forward scheme.

5. The system for wireless network communication of claim 1, wherein the second relay employs a decode and forward scheme.

6. The system for wireless network communication of claim 1, wherein the radiofrequency communication of the second relay is mmWave radiofrequency communication.

7. The system for wireless network communication of claim 1, wherein the radiofrequency communication of the second plurality of wireless devices at the destination is mmWave radiofrequency communication.

8. The system according to claim 1, wherein the Rayleigh fading model-based channel coefficients are exponentially distributed random variables.

9. The system according to claim 1, wherein the processing circuitry is further configured to calculate a partially modeled signal-to-noise ratio for the free space optical communication, the calculated partially modeled signal-to-noise ratio being determined in accordance with a Gamma-Gamma fading model.

10. The system according to claim 9, wherein the Gamma-Gamma fading model is a unified Gamma-Gamma fading model including pointing errors.

11. The system according to claim 1, wherein a transmission power of the selected one of the first plurality of wireless devices is calculated as $P_{s,r_1}^{*}=\frac{d_{s,r_1}^{}{}_{\mathrm{out}}{P}_{\mathrm{tot}}}{{}_{\mathrm{out}}\left(d_{s,r_1}^{}+d_{r_2,d}^{}\right)+{\left[{\mathrm{ABd}}_{r_1,r_2}^{}{}_{\mathrm{out}}\right]}^{\frac{1}{2}}},$ where d.sub.s,r.sub.1.sup. is a distance between the source and the first relay, .sub.out is a predetermined outage threshold signal-to-noise ratio, P.sub.tot is a sum power constraint, d.sub.r.sub.2.sub.,d.sup. is a distance between the second relay and the destination, and d.sub.r.sub.1.sub.,r.sub.2.sup. is a distance between the first relay and the second relay.

12. The system according to claim 11, wherein an outage probability of the communicated signal from the selected one of the first plurality of wireless devices to the first relay is calculated as $F_{U_{\mathrm{Sel}},R_1}\left(\right)=K_1\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)\underset{k=0}{\overset{K_1-N_1}{.Math.}}\frac{\left(\begin{array}{c}{K}_1-N_1\\ k\end{array}\right){\left(-1\right)}^k}{\left(k+N_1\right)}[1-\exp \left(-\left(k+N_1\right){}_{u,r_1}\right],$ where N.sub.1 is an order of the selected one of the first plurality of wireless devices, K.sub.1 is at total number of the first plurality of wireless devices at the source, k is a number of a user, and is a partially modeled signal-to-noise ratio.

13. The system according to claim 12, wherein the transmission power of the selected one of the first plurality of wireless devices is determined such that the outage probability, among the first plurality of wireless devices, of the communicated signal is minimized.

14. A method of wireless network communication, comprising: selecting, by processing circuitry, one of a first plurality of wireless devices at a source, the selected one of the first plurality of wireless devices being a wireless device at the source with the largest partially modeled source signal-to-noise ratio; transmitting, by the processing circuitry, a communication of the selected one of the first plurality of wireless devices to a first relay via radiofrequency; receiving and decoding, by the processing circuitry, the transmitted radiofrequency communication from the selected one of the first plurality of wireless devices at the first relay; forwarding, by the processing circuitry, the decoded communication from the first relay to a second relay via free space optical communication; receiving and decoding, by the processing circuitry, the forwarded free space optical communication from the first relay at the second relay; selecting, by the processing circuitry, one of a second plurality of wireless devices at a destination, the selected one of the second plurality of wireless devices being a wireless device at the destination with the largest partially modeled destination signal-to-noise ratio; and transmitting, by the processing circuitry, the decoded communication from the second relay to the selected one of the second plurality of wireless devices at the destination via radiofrequency, wherein the partially modeled source signal-to-noise ratio is based on transmitted power, additive white Gaussian noise, and Rayleigh fading model-based channel coefficients, and the partially modeled destination signal-to-noise ratio is based on transmitted power, additive white Gaussian noise, and Rayleigh fading model-based channel coefficients.

15. The method of wireless network communication of claim 14, wherein the radiofrequency communication of the selected one of the first plurality of wireless devices at the source is mmWave radiofrequency communication.

16. The method of wireless network communication of claim 14, wherein the radiofrequency communication received by the first relay is mmWave radiofrequency.

17. The method of wireless network communication of claim 14, wherein the radiofrequency communication transmitted by the second relay is mmWave radiofrequency.

18. The method of wireless network communication of claim 14, wherein the radiofrequency communication received by the selected one of the second plurality of wireless devices at the destination is mmWave radiofrequency.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) A more complete appreciation of the disclosure and many of the attendant advantages thereof will be readily obtained as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings, wherein:

(2) FIG. 1 is a schematic of a multiuser multi-hop mixed transmission modality network with generalized order user scheduling according to one or more aspects of the disclosed subject matter;

(3) FIG. 2 is a schematic of a multiuser triple-hop mixed RF/FSO/RF relay network with generalized order user scheduling according to one or more aspects of the disclosed subject matter;

(4) FIG. 3 is a graphical representation of system outage probability versus SNR of a multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of N.sub.1=N.sub.2 according to one or more aspects of the disclosed subject matter;

(5) FIG. 4 is a graphical representation of system outage probability versus SNR of a multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of K.sub.1=K.sub.2 according to one or more aspects of the disclosed subject matter;

(6) FIG. 5 is a graphical representation of system outage probability versus SNR of a multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of .sub.out with fixed average SNRs and varying average SNRs according to one or more aspects of the disclosed subject matter;

(7) FIG. 6 is a graphical representation of system outage probability versus order of selected user of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of average SNR according to one or more aspects of the disclosed subject matter;

(8) FIG. 7 is a graphical representation of system outage probability versus SNR of multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of .sub.out with and without power optimization according to one or more aspects of the disclosed subject matter;

(9) FIG. 8 is a graphical representation of system average symbol error probability versus SNR of a multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of according to one or more aspects of the disclosed subject matter;

(10) FIG. 9 is a graphical representation of system average symbol error probability versus SNR of a multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of , , and r according to one or more aspects of the disclosed subject matter;

(11) FIG. 10 is a graphical representation of system average symbol error probability versus SNR of a multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of (K.sub.1, N.sub.1) and (K.sub.2, N.sub.2) according to one or more aspects of the disclosed subject matter;

(12) FIG. 11 is a graphical representation of ergodic capacity versus SNR of a multiuser mixed RF/FSO/RF relay network with generalized order user scheduling for different values of N.sub.1=N.sub.2 according to one or more aspects of the disclosed subject matter; and

(13) FIG. 12 is a hardware block diagram of a server according to one or more exemplary aspects of the disclosed subject matter.

DETAILED DESCRIPTION

(14) The terms a or an, as used herein, are defined as one or more than one. The term plurality, as used herein, is defined as two or more than two. The term another, as used herein, is defined as at least a second or more. The terms including and/or having, as used herein, are defined as comprising (i.e., open language). Reference throughout this document to one embodiment, certain embodiments, an embodiment, an implementation, an example or similar terms means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the present disclosure. Thus, the appearances of such phrases or in various places throughout this specification are not necessarily all referring to the same embodiment. Furthermore, the particular features, structures, or characteristics may be combined in any suitable manner in one or more embodiments without limitation.

(15) Recently, incorporation of free-space optical communication in relay networks has been proposed to provide improved reliability in the last-mile of wireless communications. These networks, often known as dual-hop relay networks, transmit a source message from a source to a relay node over a RF link (licensed frequencies) and then forward the message to the destination over an FSO link (license-free). In such networks, relays provide greater diversity among nodes, expand the coverage area, and reduce the need for high-power transmitters. Further, this approach can be complemented via multiuser cooperation and opportunistic scheduling.

(16) Extensive work has been dedicated to the above, filling the connectivity gap in last-mile connectivity while conserving economic resources and saving bandwidth by exploiting optical communications. To this point, however, FSO communications have been incorporated into dual-hop networks with the FSO link connecting directly to a base station at a terminus. Therefore, a triple-hop mixed relay network with generalized user scheduling and power allocation algorithm, yet to be developed, is described in the present disclosure.

(17) FIG. 1 shows a triple-hop mixed RF/FSO/RF relay network consisting of K.sub.1 sources on a first hop 117 U.sub.k (k=1, 2, . . . , K.sub.1) (102, 103, 104), two un-coded type DF relays R.sub.i (i=1, 2) (105, 115) and K.sub.2 destinations on a third hop 119 D.sub.j (j=1, 2, . . . , K.sub.2) (112, 113, 114). The sources 110 are connected with a first relay 105 through RF links 117. The first relay 105 is connected with a second relay 115 via FSO link. The second relay 115 is connected with destinations 120 through RF links 119. The abovementioned communication modalities are non-limiting and are merely representative of a variety of communication modalities. The direct links between the sources and destinations are assumed to be in deep fade. Further, channel coefficients are held constant over an entire block of communication in accordance with a block fading model.

(18) Communication is operated in a half-duplex mode and to be conducted over three phases outlined above and repeated here: selected user U.sub.set.fwdarw.R.sub.1, R.sub.1.fwdarw.R.sub.2, and R.sub.2.fwdarw.D.sub.set. The received signal at R.sub.1 105 from the kth user can be expressed as
y.sub.k,r.sub.1=/{square root over (P.sub.k)}h.sub.k,r.sub.1x.sub.k,r.sub.1+n.sub.r.sub.1,(1)
where P.sub.k is the transmit power of the kth user, h.sub.k,r.sub.1 is the channel coefficient of the U.sub.k.fwdarw.R.sub.1 link, x.sub.k,r.sub.1 is the transmitted symbol of U.sub.k with {|x.sub.k,r.sub.1|.sup.2}=1, and n.sub.r.sub.1N(0, N.sub.01) is an additive white Gaussian noise (AWGN) term, where {} is the mathematical expectation. Using (1), the SNR at R.sub.1 105 due to U.sub.k 104 can be written as

(19) $\begin{array}{cc}{U}_k,R_1=\frac{P_k}{N_{01}}{.Math.h_{k,r_1}.Math.}^2.& \left(2\right)\end{array}$

(20) According to the generalized order user scheduling, the source with the N.sub.1.sup.th best .sub.U.sub.k.sub.,R.sub.1 or equivalently, the N.sub.1.sup.th largest |h.sub.k,r.sub.1|.sup.2 among the other sources 110 is selected to transmit its message to R.sub.1 105 in the first communication phase 117. In other words, the source 110 is selected such that .sub.U.sub.Sel.sub.R.sub.1=N.sub.1.sup.th max{.sub.U.sub.k.sub.,R.sub.1}. Processing circuitry is configured to perform the selection process at R.sub.1 according the selection scheme described herein. The subcarrier intensity modulation (SIM) scheme is employed at the relay R.sub.1 105, where a standard RF coherent/noncoherent modulator and demodulator can be used for transmitting and recovering the source data. At R.sub.1 105, after filtering by a bandpass filter (BPF), a direct current (DC) bias is added to the filtered RF signal to ensure that the optical signal is non-negative. Then, the biased signal is sent to a continuous wave laser driver. The retransmitted optical signal at R.sub.1 105 is written as
y.sub.r.sub.1.sup.Opt={square root over (P.sub.Opt)}(1+y.sub.Sel,r.sub.1),(3)
where P.sub.Opt denotes the average transmitted optical power and it is related to the relay electrical power P.sub.r by the electrical-to-optical conversion efficiency .sub.1 as P.sub.Opt=.sub.1 P.sub.r.sub.1, where M denotes the modulation index and .sub.Sel,r.sub.1 is the RF received signal at R.sub.1 105 from the selected source (see article by Lee, E et al, Performance analysis of the asymmetric dual-hop relay transmission with mixed RF/FSO links published in IEEE Transactions on Information Theory, in 2004, and incorporated herein by reference). The optical signal at R.sub.2 115 received from R.sub.1 105 at the second phase of communication 118 can be expressed as
.sub.r.sub.1.sub.,r.sub.2=g.sub.r.sub.1.sub.,r.sub.2{{square root over (P.sub.Opt)}[1+({square root over (P.sub.Sel)}h.sub.Sel,r.sub.1x.sub.Sel,r.sub.1+n.sub.r.sub.1)]}+n.sub.r.sub.2,(4)
where n.sub.r.sub.2N(0, N.sub.02) is an AWGN term at R.sub.2 115. Moreover, the channel coefficients of the R.sub.1.fwdarw.R.sub.2 link which is given by g.sub.r.sub.1.sub.,r.sub.2 is modeled as g.sub.r.sub.1.sub.,r.sub.2=g.sub.ag.sub.f, where g.sub.a and g.sub.f are the average gain and the fading gain of the FSO link, respectively, and are given by

(21) $\begin{array}{cc}\{\begin{array}{c}{g}_a={\left[\mathrm{erf}\left(\frac{\sqrt{}q}{\sqrt{2}{d}^{\mathrm{FSO}}}\right)\right]}^2{10}^{-d^{\mathrm{FSO}}/10},\\ g_f\mathrm{GGamma}\left(,\right),\end{array}& \left(5\right)\end{array}$
where q is the aperture radius, is the divergence angle of the beam, d.sup.FSO is the distance between the FSO transmitter and receiver, is the weather-dependent attenuation coefficient, and GGamma(, ) represents a Gamma-Gamma random variable with parameters and (see article by Zhang, W et al, Soft-switching hybrid FSO/RF links using short-length raptor codes: design and implementation published in IEEE Journal on Selected Areas in Communications, in 2009, and incorporated herein by reference). Assuming spherical wave propagation, the parameters and in the Gamma-Gamma distribution, which represent the fading turbulence conditions, are related to the physical parameters as follows:

(22) $\begin{array}{cc}={\left[\exp \left\{\frac{0.49{}^2}{{\left[1+0.18{}^2+0.56{}^{12/5}\right]}^{7/6}}\right\}-1\right]}^{-1},& \left(6\right)\\ ={\left[\exp \left\{\frac{0.51{{}^2\left[1+0.69{}^{12/5}\right]}^{-5/6}}{{\left[1+0.9{}^2+0.62{}^2{}^{12/5}\right]}^{5/6}}\right\}-1\right]}^{-1},& \left(7\right)\end{array}$
where .sup.2=0.5 C.sub.n.sup.2.sup.7/6(d.sup.FSO).sup.11/6, .sup.2=q.sup.2/d.sup.FSO, =2/.sup.FSO is the wavelength, and C.sub.n.sup.2 is the weather-dependent index of refraction structure parameter (see article by He, B et al, Bit-interleaved coded modulation for hybrid RF/FSO systems published in IEEE Transactions on Communications, in 2009, and incorporated herein by reference).

(23) When the DC component is filtered out at R.sub.2 115 and an optical-to-electrical conversion is performed, assuming M=1, the received signal can be expressed as follows:
.sub.r.sub.1.sub.,r.sub.2=g.sub.r.sub.1.sub.,r.sub.2{square root over (P.sub.Ele)}({square root over (P.sub.Sel)}h.sub.Sel,r.sub.1x.sub.Sel,r.sub.2+n.sub.r.sub.1)+n.sub.r.sub.2,(8)
where P.sub.Ele=.sub.2P.sub.Opt=.sub.1.sub.2P.sub.r.sub.1 is the electrical power received at R.sub.2 115 and .sub.2 is the optical-to-electrical conversion efficiency.

(24) From (8), the SNR at R.sub.2 115 can be written as

(25) $\begin{array}{cc}{R}_2=\frac{U_{\mathrm{Sel}},R_1{R}_1,R_2}{\mathrm{Sel},R_1+R_1,R_2+1},\mathrm{where}{}_{U_{\mathrm{Sel}},R_1}=\frac{P_{\mathrm{Sel}}}{N_{01}}{.Math.h_{\mathrm{Sel},r_1}.Math.}^2,{}_{R_1,R_2}=\frac{{}_1{}_2{P}_{r_1}}{N_{02}}{.Math.g_{r_1,r_2}.Math.}^2,& \left(9\right)\end{array}$
and P.sub.r.sub.1 is the transmit power at R.sub.1 105. The SNR in (9) can be rewritten using the standard approximation .sub.R.sub.2 min(.sub.U.sub.Sel.sub.,R.sub.1,.sub.R.sub.1.sub.,R.sub.2) (see article by Ansari, I S et al, Impact of point errors on the performance of mixed RF/FSO dual-hop transmission systems published in IEEE Wireless Communications Letters, in 2013 and an article by Ansari, I S et al, On the performance of mixed RF/FSO variable gain dual-hop transmission systems with pointing errors, published at IEEE Vehicular Technology Conference, in 2013, and incorporated herein by reference) as

(26) $\begin{array}{cc}{R}_2=\frac{U_{\mathrm{Sel}},R_1{R}_1,R_2}{U\mathrm{Sel},R_1+R_1,R_2+1}.& \left(10\right)\end{array}$

(27) The signal received at D.sub.j 114 from R.sub.2 115 in the third phase of communication 119 can be written as
.sub.r.sub.2.sub.,d.sub.j=/{square root over (P.sub.r.sub.2)}h.sub.r.sub.2.sub.,jx.sub.d.sub.j+n.sub.d.sub.j,(11)
where P.sub.r.sub.2 is the transmit power at R.sub.2 115, h.sub.r.sub.2.sub.,j is the channel coefficient of the R.sub.2.fwdarw.D.sub.j link, x.sub.d.sub.j is the transmitted symbol of d.sub.j with {|x.sub.d.sub.j|.sup.2}=1, and n.sub.d.sub.jN(0, N.sub.03) is an AWGN term. Using (11), the SNR at D.sub.j 114 can be written as

(28) $\begin{array}{cc}{}_{R_2,D_j}=\frac{P_{r_2}}{N_{03}}{.Math.h_{r_2,j}.Math.}^2.& \left(12\right)\end{array}$

(29) According to generalized order user scheduling, the destination 120 with the N.sub.2.sup.th best .sub.R.sub.2.sub.,D.sub.j or equivalently, the N.sub.2.sup.th largest |h.sub.r.sub.2.sub.,j|.sup.2 among the other destinations 120 is selected to receive its message from R.sub.2 115 in the third communication phase 119. In other words, the destination is selected such that .sub.R.sub.2.sub.,D.sub.Sel=N.sub.2.sup.th max{.sub.R.sub.2.sub.,D.sub.j}. Processing circuitry is configured to perform the selection process at R.sub.2 according the selection scheme described herein.

(30) The channel coefficients of the RF links 117, 119 h.sub.k,r.sub.1 (k=1= . . . =K.sub.1) and h.sub.r.sub.2.sub.,j (j=1= . . . =K.sub.2) follow the Rayleigh fading model and, therefore, channel gains |h.sub.k,r.sub.1|.sup.2 and |h.sub.r.sub.2.sub.,j|.sup.2 are exponentially distributed random variables with mean powers .sub.k,r.sub.1 and .sub.r.sub.2.sub.,j, respectively. Therefore, the probability density functions (PDFs) of .sub.U.sub.k.sub.,R.sub.1, and .sub.R.sub.2.sub.,D.sub.j are given by

(31) ${U_k,R_1}_{}\left(\right)={}_{k,r_1}\exp \left(-{}_{k,r_1}\right),\mathrm{where}{}_{k,r_1}=1/{\stackrel{\_}{}}_{k,r_1}\mathrm{and}$${\stackrel{\_}{}}_{k,r_1}=\frac{P_k}{N_0}\mathrm{��}\left\{{.Math.h_{k,r_1}.Math.}^2\right\}=\frac{P_k}{N_{01}}{}_{k,r_1},\mathrm{and}\mathrm{by}$${R_2,D_j}_{}\left(\right)={}_{r_2,j}\exp \left(-{}_{r_2,j}\right),\mathrm{where}$${}_{r_2,j}=1/{\stackrel{\_}{}}_{r_2,j}\mathrm{and}{\stackrel{\_}{}}_{r_2,j}=\frac{P_k}{N_{03}}\mathrm{��}\left\{{.Math.h_{r_2,j}.Math.}^2\right\}=\frac{P_{r_2}}{N_{03}}{}_{r_2,j},$
respectively. Regarding the second hop, it is assumed that the FSO link 118 experiences a unified Gamma-Gamma fading model including the pointing errors effect whose SNR PDF (see article by Ansari, I S et al, Impact of point errors on the performance of mixed RF/FSO dual-hop transmission systems published in IEEE Wireless Communications Letters, in 2013, and incorporated herein by reference), is given by

(32) $\begin{array}{cc}{f}_{{}_{R_1,R_2}}\left(\right)=\frac{{}^2}{r\left(\right)\left(\right)}{G}_{1,3}^{3,0}\left[{\left({}_{r_1,r_2}\right)}^{\frac{1}{r}}.Math.\begin{array}{c}{}^2+1\\ {}^2,,\end{array}\right],& \left(13\right)\end{array}$
is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver (i.e. when .fwdarw., non-pointing error). r is the parameter defining the type of detection technique (i.e. r=1 represents heterodyne detection and r=2 represents intensity modulation (IM)/direct detection (DD)). and are the fading parameters related to the atmospheric turbulence conditions with lower values indicating severe atmospheric turbulence conditions. (.) is the Gamma function,

(33) ${}_{r_1,r_2}=1/{\stackrel{\_}{}}_{r_1,r_2}\mathrm{where}$${\stackrel{\_}{}}_{r_1,r_2}=\frac{{}_1{}_2{P}_{r_1}}{N_{02}}\mathrm{��}\left\{{.Math.g_{r_1,r_2}.Math.}^2\right\}=\frac{{}_1{}_2{P}_{r_1}}{N_{02}}{}_{r_1,r_2}$
and G(.) is the Meijer G-function (see textbook by Gradshteyn, I S and Ryzhik, I M, Tables of Integrals, Series and Products, published by Academic Press, in 2000, and incorporated herein by reference).

(34) The end-to-end (e2e) SNR at the selected destination can be written using the standard approximation .sub.D min(.sub.R.sub.2, .sub.R.sub.2.sub.,D.sub.Sel) as

(35) 0$\begin{array}{cc}{}_D=\frac{{}_{R_2}{}_{R_2,D_{\mathrm{Sel}}}}{{}_{R_2}+{}_{R_2,D_{\mathrm{Sel}}}}.& \left(14\right)\end{array}$

(36) In an exemplary embodiment, as seen in FIG. 2, K.sub.1 sources 210 are users of wireless devices. A user U.sub.Sel 204 is selected from a group of source users 210, as the user with the N.sub.1.sup.th best SNR value, to transmit its signal to a first relay 201. The signal is transmitted via RF link on a first hop 217. Each source user (U.sub.1 202, U.sub.2 203, . . . , U.sub.K.sub.1 204) is equipped with a single antenna. The first relay 201 is equipped with a single antenna 205 and a single photo-aperture transmitter 206. A second relay 211 is equipped with a single antenna 215 and a photo-aperture transmitter 216. Deploying a DF scheme, the first relay 201, with support from processing circuitry configured to decode a transmitted signal, receives the signal from the user U.sub.Sel 204 and transmits the signal to a second relay 211 on a second hop 218 via FSO link. The second relay 211 receives the signal from the first relay 201 and, deploying a DF scheme, with support from processing circuitry configured to decode the transmitted signal, transmits the signal to a destination user D.sub.Sel 214 via RF link on a third hop 219. The destination user D.sub.Sel 214 is selected from a group of destination users 220, as the user with the N.sub.2.sup.th best SNR value, to receive its signal from the second relay 211. Each destination user (D.sub.1 212, D.sub.2 213, . . . , D.sub.K.sub.2 214) is equipped with a single antenna.

System Performance Metrics (Analytical Solutions)

(37) To evaluate system performance, the statistics of the e2e SNR provided in (14) must be determined.

(38) To this end, the outage probability is defined as the probability that the SNR at a selected destination drops below a predetermined outage threshold .sub.out, or P.sub.out=Pr[.sub.d.sub.out], where Pr[.] is the probability operation and .sub.out is a predetermined outage threshold. The outage probability can be obtained from the cumulative distribution function (CDF) of the e2e SNR as P.sub.out=F.sub.D(.sub.out). This CDF can be written in terms of CDFs of the three hops' SNRS as

(39) $\begin{array}{cc}{F}_{{}_D}\left(\right)=1-\left\{\left(1-F_{{}_{U_{\mathrm{Sel}},R_1}}\left(\right)\right)\left(1-F_{{}_{R_1,R_2}}\left(\right)\right)\left(1-F_{{}_{R_2,D_{\mathrm{Sel}}}}\left(\right)\right)\right\},\mathrm{where}{F}_{{}_{U_{\mathrm{Sel}},R_1}}\left(\right),F_{{}_{R_1,R_2}}\left(\right),F_{{}_{R_2,D_{\mathrm{Sel}}}}\left(\right)& \left(15\right)\end{array}$
are the CDFs of the first hop, second hop, and third hop SNRs, respectively.

(40) The CDF of the first hop begins from the PDF according to generalized order user selection, wherein the PDF represents the N.sub.1.sup.th best SNR or, the source of the N.sub.1.sup.th best SNR as selected by the first relay. The CDF is rewritten as

(41) $\begin{array}{cc}{F}_{{}_{U_{\mathrm{Sel}},R_1}}\left(\right)=K_1\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)\underset{k=0}{\overset{K_1-N_1}{.Math.}}\frac{\left(\begin{array}{c}{K}_1-N_1\\ k\end{array}\right){\left(-1\right)}^k}{\left(k+N_1\right)}\left[1-\exp \left(-\left(k+N_1\right){}_{u,r_1}\right)\right].& \left(19\right)\end{array}$
where the users on the third hop have been assumed to have independent identical distributed channels.

(42) The CDF of the second hop is determined from the PDF of the FSO link incorporating the Gamma-Gamma fading model and including point errors. The CDF is rewritten as

(43) $\begin{array}{cc}{F}_{{}_{R_1,R_2}}\left(\right)={\mathrm{AG}}_{r+1,3r+1}^{3r,1}\left[\frac{B}{{\stackrel{\_}{}}_{r_1,r_2}}.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right],\mathrm{where}A=\frac{r^{+-22}}{{\left(2\right)}^{r-1}\left(\right)\left(\right)},B=\frac{{\left(\right)}^r}{r^{2r}},{}_1=\frac{{}^2+1}{r},...,\frac{{}^2+r}{r},& \left(20\right)\end{array}$
comprises of r terms and

(44) ${}_2=\frac{{}^2}{r},...,\frac{{}^2+r-1}{r},\frac{}{r},...,\frac{+r-1}{r},\frac{}{r},...,\frac{+r-1}{r}$
comprises of 3r terms.

(45) Similar to the first hop, the CDF of the third hop begins from the PDF according to generalized order user selection, wherein the PDF represents the N.sub.2.sup.th best SNR or, the destination of the N.sub.2.sup.th best SNR as selected by the second relay. The CDF is rewritten as

(46) $\begin{array}{cc}{F}_{{}_{R_2,D_{\mathrm{Sel}}}}\left(\right)=K_2\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)\underset{j=0}{\overset{K_2-N_2}{.Math.}}\frac{\left(\begin{array}{c}{K}_2-N_2\\ j\end{array}\right){\left(-1\right)}^j}{\left(j+N_2\right)}\left[1-\exp \left(-\left(j+N_2\right){}_{r2,u}\right)\right],& \left(22\right)\end{array}$
where the users on the third hop have been assumed to have independent identical distributed channels.

(47) Following the substitution of the CDFs from each hop ((19), (20), (22)) into (15), the full e2e CDF can be written as

(48) $\begin{array}{cc}{F}_{{}_D}\left(\right)=\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\underset{k=0}{\overset{K_1-N_1}{.Math.}}\frac{\left(\begin{array}{c}{K}_1-N_1\\ k\end{array}\right){\left(-1\right)}^k}{\left(k+N_1\right)}\left\{1-\exp \left({}_1\right)-A\left(G_{r+1,3r+1}^{3r,1}\left[{}_0.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right]\left[1-\exp \left(-{}_1\right)\right]\right)\right\}+\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\underset{j=0}{\overset{K_2-N_2}{.Math.}}\frac{\left(\begin{array}{c}{K}_2-N_2\\ j\end{array}\right){\left(-1\right)}^j}{\left(j+N_2\right)}\left\{1-\exp \left(-{}_2\right)-A\left(G_{r+1,3r+1}^{3r,1}\left[{}_0.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right]\left[1-\exp \left(-{}_2\right)\right]\right)\right\}-\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\underset{k=0}{\overset{K_1-N_1}{.Math.}}\frac{\left(\begin{array}{c}{K}_1-N_1\\ k\end{array}\right){\left(-1\right)}^k}{\left(k+N_1\right)}\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\underset{j=0}{\overset{K_2-N_2}{.Math.}}\frac{\left(\begin{array}{c}{K}_2-N_2\\ j\end{array}\right){\left(-1\right)}^j}{\left(j+N_2\right)}\left\{1-\exp \left(-{}_1\right)-\exp \left(-{}_2\right)+\exp \left(-\left[{}_1+{}_2\right]\right)-A\left(G_{r+1,3r+1}^{3r,1}\left[{}_0.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right]\left[1-\exp \left(-{}_1\right)-\exp \left(-{}_2\right)+\exp \left(-\left[{}_1+{}_2\right]\right)\right]\right)\right\}+{\mathrm{AG}}_{r+1,3r+1}^{3r,1}\left[{}_0.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right],\mathrm{where}{}_1=\left(k+N_1\right){}_{u,r_1},{}_0=\frac{B}{{\stackrel{\_}{}}_{r_1,r_2}},\mathrm{and}{}_2=\left(j+N_2\right){}_{r_2,u}.& \left(23\right)\end{array}$
The CDF in (23) is used to determine several performance measures as closed-form expressions.

(49) To determine the exact average symbol error probability (ASEP), the ASEP is expressed in terms of the CDF of .sub.D as

(50) $\begin{array}{cc}\mathrm{ASEP}=\frac{a\sqrt{b}}{2\sqrt{}}{}_0^{}\frac{\exp \left(-b\right)}{\sqrt{}}{F}_{{}_D}\left(\right)d,& \left(24\right)\end{array}$
where a and b are modulation-specific parameters (see article by McKay, M R et al, Performance analysis of MIMO-MRC in double-correlated Rayleigh environments, published in IEEE Transactions on Communications, in 2007, incorporated herein by reference). A SIM scheme is adopted, allowing known digital modulation techniques such as phase shift keying to be used. Therefore, the error probability computing method, used for RF wireless communication systems, can be used to evaluate the error probability performance in FSO systems. Upon combination of equations, ASEP can be written as

(51) $\begin{array}{cc}\mathrm{ASEP}=\frac{a\sqrt{b}}{2\sqrt{}}\left\{\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\underset{k=0}{\overset{K_1-N_1}{.Math.}}\frac{\left(\begin{array}{c}{K}_1-N_1\\ k\end{array}\right){\left(-1\right)}^k}{\left(k+N_1\right)}\left(\frac{\left(1/2\right)}{b^{\frac{1}{2}}}-\frac{\left(1/2\right)}{{\left(b+{}_1\right)}^{\frac{1}{2}}}-A\left[b^{-\frac{1}{2}}{G}_{r+2,3r+1}^{3r,2}\left[\frac{{}_0}{b}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]-{\left(b+{}_1\right)}^{-\frac{1}{2}}{G}_{r+2,3r+1}^{3r,2}\left[\frac{{}_1}{\left(b+{}_1\right)}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]\right]\right)+\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\underset{j=0}{\overset{K_2-N_2}{.Math.}}\frac{\left(\begin{array}{c}{K}_2-N_2\\ j\end{array}\right){\left(-1\right)}^j}{\left(j+N_2\right)}\left(\frac{\left(1/2\right)}{b^{\frac{1}{2}}}-\frac{\left(1/2\right)}{{\left(b+{}_2\right)}^{\frac{1}{2}}}-A\left[b^{-\frac{1}{2}}{G}_{r+2,3r+1}^{3r,2}\left[\frac{{}_0}{b}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]-{\left(b+{}_2\right)}^{-\frac{1}{2}}{G}_{r+2,3r+1}^{3r,2}\left[\frac{{}_0}{\left(b+{}_2\right)}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]\right]\right)-\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\underset{k=0}{\overset{K_1-N_1}{.Math.}}\frac{\left(\begin{array}{c}{K}_1-N_1\\ k\end{array}\right)}{\left(k+N_1\right)}{\left(-1\right)}^k\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\underset{j=0}{\overset{K_2-N_2}{.Math.}}\frac{\left(\begin{array}{c}{K}_2-N_2\\ j\end{array}\right){\left(-1\right)}^j}{\left(j+N_2\right)}\left(\frac{\left(1/2\right)}{b^{\frac{1}{2}}}-\frac{\left(1/2\right)}{{\left(b+{}_1\right)}^{\frac{1}{2}}}-\frac{\left(1/2\right)}{{\left(b+{}_2\right)}^{\frac{1}{2}}}+\frac{\left(1/2\right)}{{\left(b+{}_1+{}_2\right)}^{\frac{1}{2}}}-A\left[b^{-\frac{1}{2}}{G}_{r+2,3r+1}^{3r,2}\left[\frac{{}_0}{b}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]-{\left(b+{}_1\right)}^{-\frac{1}{2}}{G}_{r+2,3r+1}^{3r,2}\left[\frac{{}_0}{\left(b+{}_1\right)}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]-G_{r+2,3r+1}^{3r,2}\left[\frac{{}_0}{\left(b+{}_2\right)}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]{\left(b{}_2\right)}^{-\frac{1}{2}}+G_{r+2,3r+1}^{3r,2}\left[\frac{{}_0}{\left(b+{}_1+{}_2\right)}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]{\left(b+{}_1+{}_2\right)}^{-\frac{1}{2}}\right]\right)+{\mathrm{Ab}}^{-\frac{1}{2}}{G}_{r+2,3r+1}^{3r,2}\left[\frac{{}_0}{b}.Math.\begin{array}{c}\frac{1}{2},1,{}_1\\ {}_2,0\end{array}\right]\right\}.& \left(25\right)\end{array}$

(52) Because the coherence time of the FSO fading channel is in the order of milliseconds, a single fade can obliterate millions of bits at gigabits/second data rates. Therefore, the exact average (i.e., ergodic) channel capacity represents the best achievable capacity of an optical wireless link. Using a PDF-based method, the ergodic capacity can be expressed in terms of the PDF of .sub.D as

(53) $\begin{array}{cc}C=\frac{1}{\ln \left(2\right)}{}_0^{}\ln \left(1+\right)f_{{}_D}\left(\right)d.& \left(26\right)\end{array}$

(54) Following multiple derivations and integrations described in detail in the cited references, the exact ergodic capacity can be written as

(55) 0$\begin{array}{cc}C=\frac{1}{\ln \left(2\right)}\{\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\underset{k=0}{\overset{K_1-N_1}{.Math.}}\frac{\left(\begin{array}{c}{K}_1-N_1\\ k\end{array}\right){\left(-1\right)}^k}{{}_1}(-\exp \left({}_1\right)E_i\left(-{}_1\right)-A\left\{\frac{1}{{}_0}\left[G_{r+2,3r+2}^{3r+2,1}\left[{}_0.Math.\begin{array}{c}0,1,{}_1\\ {}_2,0\end{array}\right]-G_{1,0:2,2:r,3r}^{0,1:1,2:3r,0}\left[\frac{1}{{}_1},{}_0{}_1.Math.\begin{array}{c}1\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}{}_1\\ {}_2\end{array}\right]\right]G_{1,0:2,2:r+1,3r+1}^{0,1:1,2:3r,1}\left[\frac{1}{{}_1},{}_0{}_1.Math.\begin{array}{c}2\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right]{\left({}_1\right)}^2\}\right)+\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\underset{j=0}{\overset{K_2-N_2}{.Math.}}\left(\begin{array}{c}{K}_2-N_2\\ j\end{array}\right){\left(-1\right)}^j{\left({}_2\right)}^{-1}(-\exp \left({}_2\right)E_i\left(-{}_2\right)-A\{\frac{1}{{}_0}\left[G_{r+2,3r+2}^{3r+2,1}\left[{}_0.Math.\begin{array}{c}0,1,{}_1\\ {}_2,0\end{array}\right]-G_{1,0:2,2:r,3r}^{0,1:1,2:3r,0}\left[\frac{1}{{}_2},{}_0{}_2.Math.\begin{array}{c}1\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}{}_1\\ {}_2\end{array}\right]]+{\left({}_2\right)}^2{G}_{1,0:2,2:r+1,3r+1}^{0,1:1,2:3r,1}\left[\frac{1}{{}_2},{}_0{}_2.Math.\begin{array}{c}2\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right]\}\right)-\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\underset{k=0}{\overset{K_1-N_1}{.Math.}}\frac{\left(\begin{array}{c}{K}_1-N_1\\ k\end{array}\right){\left(-1\right)}^k}{{}_1}\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\underset{j=0}{\overset{K_2-N_2}{.Math.}}{\left(-1\right)}^j\frac{\left(\begin{array}{c}{K}_2-N_2\\ j\end{array}\right)}{{}_2}(-\exp \left({}_1\right)E_i\left(-{}_1\right)-\exp \left({}_2\right)E_i\left(-{}_2\right)+\exp \left({}_1+{}_2\right)E_i\left(-\left({}_1+{}_2\right)\right)-A\left\{\frac{1}{{}_0}\left[G_{r+2,3r+2}^{3r+2,1}\left[{}_0.Math.\begin{array}{c}0,1,{}_1\\ {}_2,0\end{array}\right]-G_{1,0:2,2:r,3r}^{0,1:1,2:3r,0}\left[\frac{1}{{}_1},{}_0{}_1.Math.\begin{array}{c}1\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}{}_1\\ {}_2\end{array}\right]-G_{1,0:2,2:r,3r}^{0,1:1,2:3r,0}\left[\frac{1}{{}_2},{}_0{}_2.Math.\begin{array}{c}1\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}{}_1\\ {}_2\end{array}\right]+G_{1,0:2,2:r,3r}^{0,1:1,2:3r,0}\left[\frac{1}{\left[{}_1+{}_2\right]},{}_0\left[{}_1+{}_2\right].Math.\begin{array}{c}1\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}{}_1\\ {}_2\end{array}\right]\right]+{\left({}_1\right)}^2{G}_{1,0:2,2:r+1,3r+1}^{0,1:1,2:3r,1}\left[\frac{1}{{}_1},{}_0{}_1.Math.\begin{array}{c}2\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right]+{\left({}_2\right)}^2{G}_{1,0:2,2:r+1,3r+1}^{0,1:1,2:3r,1}\left[\frac{1}{{}_2},{}_0{}_2.Math.\begin{array}{c}2\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}1,{}_1\\ {}_2,0\end{array}\right]-{\left[{}_1+{}_2\right]}^2{G}_{1,0:2,2:r,3r}^{0,1:1,2:3r,0}\left[\frac{1}{\left[{}_1+{}_2\right]},{}_0\left[{}_1+{}_2\right].Math.\begin{array}{c}1\\ -\end{array}.Math.\begin{array}{c}1,1\\ 1,0\end{array}.Math.\begin{array}{c}{}_1\\ {}_2\end{array}\right]\}+\frac{1}{{}_0}{G}_{r+2,3r+2}^{3r+2,1}\left[{}_0.Math.\begin{array}{c}0,1,{}_1\\ {}_2,0\end{array}\right]\right\},& \left(36\right)\end{array}$
where E.sub.i(.) is an exponential integral function, and G[Z.sub.1, Z.sub.2|.|.|.] is the extended generalized bivariate Meijer G-function.

System Performance Metrics (Asymptotic Solutions)

(56) Due to the complexity of the above expressions, approximations of these expressions are required to appropriately evaluate the impact of changes in performance parameters and gain system insight. Detailed derivation of the approximate solutions can be found in the cited references.

(57) The asymptotic outage probability can be written at the high SNR regime as P.sub.out(G.sub.cSNR).sup.G.sup.d, where G.sub.c and G.sub.d are the coding gain and diversity order of the system, respectively (see textbook by Simon, M K and Alouini, M-S, Digital Communication over Fading Channels, published by John Wiley & Sons, Inc., in 2005, and incorporated herein by reference). G.sub.c represents a horizontal shift in the outage probability and G.sub.d refers to a change in the slope of the outage probability vs. SNR curve.

(58) Consider the case of identical sources' channels (.sub.1,r.sub.1=.sub.2,r.sub.1= . . . =.sub.K.sub.1.sub.,r.sub.1=.sub.u,r.sub.1) and identical destinations' channels (.sub.r.sub.1.sub.,1=.sub.r.sub.1.sub.,2= . . . =.sub.r.sub.2.sub.,K.sub.2=.sub.r.sub.2.sub.,u). The approximate CDF of each hop separately is determined to calculate the approximate CDF of the e2e SNR.

(59) Regarding the first hop link, using a Taylor series representation of the exponential term in the CDF to simplify and integrate, the CDF is written as

(60) $\begin{array}{cc}{F}_{{}_{U_{\mathrm{Sel}},R_1}}\left(\right)\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right){K_1\left({}_{u,r_1}\right)}^{K_1-N_1+1}\frac{{}^{K_1-N_1+1}}{\left(K_1-N_1+1\right)}.& \left(38\right)\end{array}$

(61) Regarding the second hop link, the CDF is written as

(62) $\begin{array}{cc}{F}_{{}_{R_1,R_2}}\left(\right){\left(\frac{}{{\stackrel{\_}{}}_{r_1,r_2}}\right)}^{\frac{v}{r}},& \left(40\right)\end{array}$
where is constant and is written as

(63) $\begin{array}{cc}=A\underset{k=1}{\overset{6}{.Math.}}\frac{\underset{j=1,jk}{\overset{6}{.Math.}}\left(b_j-b_k\right)\left(b_k\right)}{\underset{j=2}{\overset{3}{.Math.}}\left(a_j-b_k\right)\left(1+b_k\right)}{B}^{{}^{b}k/2},& \left(43\right)\end{array}$

(64) The third hop link, similar to the first hop link, is simplified as

(65) $\begin{array}{cc}{F}_{{}_{R_2,D_{\mathrm{Sel}}}}\left(\right)\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right){K_2\left({}_{r_2,u}\right)}^{K_2-N_2+1}\frac{{}^{K_2-N_2+1}}{\left(K_2-N_2+1\right)}.& \left(45\right)\end{array}$

(66) To obtain the diversity order and coding gain of the system, the CDF of (16) can be simplified, at high SNR values, to be
F.sub.D()F.sub.U.sub.Sel.sub.,R.sub.1()+F.sub.R.sub.1.sub.,R.sub.2()+F.sub.R.sub.2.sub.,D.sub.Sel(),(46)

(67) Substituting values into (46), the approximate outage probability, at high SNR values, can be written as

(68) $\begin{array}{cc}{P}_{\mathrm{out}}^{}=\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right){K_1\left({\stackrel{\_}{}}_{u,r_1}\right)}^{-\left(K_1-N_1+1\right)}\frac{{\left({}_{\mathrm{out}}\right)}^{K_1-N_1+1}}{\left(K_1-N_1+1\right)}+{\left(\frac{{}^{-\frac{r}{v}}}{y_{\mathrm{out}}}{\stackrel{\_}{}}_{r_{1,}{r}_2}\right)}^{-\frac{v}{r}}+\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right){K_2\left({\stackrel{\_}{}}_{r_2,u}\right)}^{-\left(K_2-N_2+1\right)}\frac{{\left({}_{\mathrm{out}}\right)}^{K_2-N_2+1}}{\left(K_2-N_2+1\right)}.& \left(47\right)\end{array}$

(69) From (47), it is observed that the performance of the considered relay network will be dominated by the worst link among the available three links (first RF link, FSO link, second RF link). This domination depends on the parameters of these links. Therefore, the diversity order of the triple-hop mixed RF/FSO/RF relay network with generalized order user scheduling is equal to min(K.sub.1N.sub.1+1, /r, K.sub.2N.sub.2+1). Based on the value of the diversity order, one of the following three cases represents the overall system performance.

(70) Case 1 One hop is dominant, and the coding gain is written as

(71) $\begin{array}{cc}{G}_c=\{\begin{array}{cc}{\left[\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\frac{{\left({}_{\mathrm{out}}\right)}^{K_1-N_1+1}}{\left(K_1-N_1+1\right)}\right]}^{-\frac{1}{K_1-N_1+1}},& G_d=K_1-N_1+1,\\ \frac{{}^{-\frac{r}{v}}}{y_{\mathrm{out}}},& G_d=\frac{v}{r},\\ {\left[\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\frac{{\left({}_{\mathrm{out}}\right)}^{K_2-N_2+1}}{\left(K_2-N_2+1\right)}\right]}^{-\frac{1}{K_2-N_2+1}},& G_d=K_2-N_2+1\end{array}.& \left(48\right)\end{array}$

(72) Case 2 Two hops are dominant, and the coding gain is written as

(73) $\begin{array}{cc}{G}_c=\{\begin{array}{cc}\frac{1}{2}\{{\left[\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\frac{{\left({}_{\mathrm{out}}\right)}^{K_1-N_1+1}}{\left(K_1-N_1+1\right)}\right]}^{-\frac{1}{K_1-N_1+1}}+& G_d=K_1-N_1+1=K_2-N_2+1,\\ {\left[\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\frac{{\left({}_{\mathrm{out}}\right)}^{K_2-N_2+1}}{\left(K_2-N_2+1\right)}\right]}^{-\frac{1}{K_2-N_2+1}}\},& \\ \frac{1}{2}\left\{{\left[\left(\begin{array}{c}{K}_1-1\\ N_1-1\end{array}\right)K_1\frac{{\left({}_{\mathrm{out}}\right)}^{K_1-N_1+1}}{\left(K_1-N_1+1\right)}\right]}^{-\frac{1}{K_1-N_1+1}}+\frac{{}^{-\frac{r}{v}}}{{}_{\mathrm{out}}}\right\},& G_d=K_1-N_1+1=\frac{v}{r},\\ \frac{1}{2}\left\{{\left[\left(\begin{array}{c}{K}_2-1\\ N_2-1\end{array}\right)K_2\frac{{\left({}_{\mathrm{out}}\right)}^{K_2-N_2+1}}{\left(K_2-N_2+1\right)}\right]}^{-\frac{1}{K_2-N_2+1}}+\frac{{}^{-\frac{r}{v}}}{{}_{\mathrm{out}}}\right\},& G_d=K_2-N_1+1=\frac{v}{r},\end{array}& \left(49\right)\end{array}$