On the Performance of RF-FSO Links with and without Hybrid ARQ
This paper studies the performance of hybrid radio-frequency (RF) and free-space optical (FSO) links assuming perfect channel state information (CSI) at the receiver. Considering the cases with and without hybrid automatic repeat request (HARQ), we derive closed-form expressions for the message decoding probabilities as well as the throughput and the outage probability of the RF-FSO setups. We also evaluate the effect of adaptive power allocation and different channel conditions on the throughput and the outage probability. The results show the efficiency of the RF-FSO links in different conditions.
The next generation of communication networks must provide high-rate reliable data streams. To address these demands, a combination of different techniques are considered among which free-space optical (FSO) communication is very promising [1, 2, 3]. FSO systems provide fiber-like data rates through the atmosphere using lasers or light emitting diodes (LEDs). Thus, the FSO can be used for a wide range of applications such as last-mile access, fiber back-up, back-haul for wireless cellular networks, and disaster recovery. However, such links are highly susceptible to atmospheric effects and, consequently, are unreliable. An efficient method to improve the reliability in FSO systems is to rely on an additional radio-frequency (RF) link to create a hybrid RF-FSO communication system.
Typically, to achieve data rates comparable to those in the FSO link, a millimeter wavelength carrier is selected for the RF link. As a result, the RF link is also subject to atmospheric effects such as rain and scintillation. However, the good point is that these links are complementary because the RF (resp. the FSO) signal is severely attenuated by rain (resp. fog/clouds) while the FSO (resp. the RF) signal is not. Therefore, the link reliability and the service availability are considerably improved via joint RF-FSO based data transmission.
The performance of RF-FSO systems is studied in different papers, e.g., [3, 4, 5, 6, 7, 8, 9, 10], where the RF and the FSO links are considered as separate links and the RF link acts as a backup when the FSO link is down. In the meantime, there are works such as [11, 12, 13, 14, 15, 16] in which the RF and the FSO links are combined to improve the system performance. Also, [17, 18] study RF-FSO based relaying schemes with an RF source-relay link and an FSO [17, 18] or RF-FSO  relay-destination link. Moreover, the implementation of hybrid automatic repeat request (HARQ) in RF-based (resp. FSO-based) systems is investigated in, e.g., [19, 20, 21, 22, 23, 24, 25, 26] (resp. [27, 28, 29, 30, 31, 32]), while the HARQ-based RF-FSO systems have been rarely studied, e.g., [16, 33].
In this paper, we study the data transmission efficiency of RF-FSO systems from an information theoretic point of view. We derive closed-form expressions for the message decoding probabilities as well as the system throughput and outage probability (Lemmas 1-5, Eq. (III-B), (III-B), (III-B)). The results are obtained in the cases with and without HARQ. Also, we analyze the effect of adaptive power allocation between the RF and the FSO links on the throughput/outage probability (Lemmas 6-7 and Fig. 11). Finally, we investigate the effect of different channel conditions on the performance of RF-FSO setups and compare the results with the cases utilizing either the RF or the FSO link separately. Note that, while the results are presented for the RF-FSO setups, with proper refinements of the channel model, the same analysis as in the paper is useful for other coordinated data transmission schemes as well (see Section III.C for more discussions).
As opposed to [3, 4, 5, 6, 7, 8, 9, 10], we consider joint data transmission/reception in the RF and FSO links. Also, the paper is different from [11, 12, 13, 17, 18, 14, 15, 16, 20, 21, 19, 22, 23, 24, 25, 26, 31, 27, 28, 29, 30, 32, 33] because we study the performance of HARQ in joint RF-FSO links and derive new analytical/numerical results on the message decoding probabilities, optimal power allocation, and outage probability/throughput which have not been presented before. The derived results provide a framework for the analysis of RF-FSO links from different perspectives.
The numerical and the analytical results show that 1) depending on the relative coherence times of the RF and FSO links there are different methods for the analytical performance evaluation of the RF-FSO systems (Lemmas 1-5, Eq. (III-B), (III-B), (III-B)). 2) While adaptive power allocation improves the system performance, at high signal-to-noise ratios (SNRs), the optimal, in terms of throughput/outage probability, power allocation between the RF and the FSO links converges to uniform power allocation, independently of the links channel conditions (Lemma 6). Finally, 3) the joint implementation of the RF and FSO links leads to substantial performance improvement, compared to the cases with only the RF or the FSO link. For instance, consider the exponential distribution and the common relative coherence times of the RF and FSO links. Then, with the initial code rate nats-per-channel-use (npcu), a maximum of retransmission rounds of the HARQ and the outage probability , the joint RF-FSO based data transmission reduces the required power by and dB, compared to the cases with only the RF or the FSO link, respectively (see Fig. 10 for more details).
Ii System Model and Problem Formulation
Ii-a System Model
Consider a joint RF-FSO system, as demonstrated in Fig. 1. The data sequence is encoded into parallel FSO and RF bit streams. The FSO link employs intensity modulation and direct detection while the RF link modulates the encoded bits and up-converts the baseband signal to a millimeter wavelength, in the range of GHz, RF carrier frequency111A millimeter wavelength carrier is often selected for the RF link to achieve data rates comparable to those in the FSO link. However, this is not a necessary assumption, and the theoretical results hold for different ranges of carrier frequencies.. Then, the FSO and the RF signals are simultaneously sent to the receiver. At the receiver, the received RF (FSO) signal is down-converted to baseband (resp. collected by an aperture and converted to an electrical signal via photo-detection) and the signals are sent to the decoder which decodes the received signals jointly (also, see [11, 12, 13, 14, 15, 16] for further discussions on the coding process in RF-FSO setups). In this way, as seen in the following, the diversity increases by the joint data transmission of the RF and FSO links, and one link can compensate the poor performance of the other link experiencing severe atmospheric effects.
We denote the instantaneous realizations of the fading coefficient of the RF link and the turbulence coefficient of the FSO link in time slot by and , respectively, and for simplicity we refer to both of them as channel coefficients. These channel coefficients are assumed to be known at the receiver which is an acceptable assumption in block-fading conditions [20, 21, 19, 22, 23, 24, 25, 26, 27, 28, 29]. Also, we define which are referred to as channel gain realizations in the following. We then assume no channel state information (CSI) feedback to the transmitter, except for the HARQ feedback bits. In each round of data transmission, if the data is correctly decoded by the receiver, an acknowledgement (ACK) is fed back to the transmitter, and the retransmissions stop. Otherwise, the receiver transmits a negative-acknowledgment (NACK). The feedback channel can be an RF, an FSO or an RF-FSO link, and is supposed to be error- and delay-free. However, we can follow the same procedure as in  and  to extend the results to the cases with delayed and erroneous feedback, respectively. The results are presented for asymptotically long codes where the block error rate converges to the outage probability. Finally, we assume perfect synchronization between the links which is an acceptable assumption in our setup with the RF and the FSO signals generated at the same transmit terminal, e.g., [11, 12, 13, 14, 15, 16, 33] (to study practical schemes for synchronization between the RF and FSO links, the reader is referred to ).
Let us define a packet as the transmission of a codeword along with all its possible retransmissions. As the most promising HARQ approach leading to highest throughput/lowest outage probability [20, 21, 23, 22, 24, 25, 26], we consider the incremental redundancy (INR) HARQ with a maximum of retransmissions, i.e., the message is retransmitted a maximum of times. Using INR HARQ, information nats are encoded into a parent codeword of length channel uses. The parent codeword is then divided into sub-codewords of length channel uses which are sent in the successive transmission rounds. Thus, the equivalent data rate, i.e., the code rate, at the end of round is where denotes the initial code rate. In each round, the receiver combines all received sub-codewords to decode the message. The retransmission continues until the message is correctly decoded or the maximum permitted transmission round is reached. Note that setting represents the cases without HARQ, i.e., open-loop communication.
Finally, the results are valid for different ranges of RF and FSO links operation frequencies. Thus, for generality and in harmony with, e.g., [20, 21, 22, 23, 24, 25, 26], we do not specify the frequency ranges of the RF and FSO links and present the code rates in the nats-per-channel-use (npcu) units. The results of the simulation figures can be easily mapped to the bit-per-channel-use scale if the code rates are scaled by . Moreover, the results are represented in nats-per-second/Hertz (npsH), if each channel use is associated with a time-frequency unit. Also, for given bandwidths of the RF and FSO links, we can follow the same approach as in [35, Chapter 9.3] to present the results in bits- or nats-per-second/Hertz.
Ii-B Problem Formulation
respectively, where is the accumulated mutual information (AMI) at the end of round . Also, denotes the probability that the data is not correctly decoded up to the end of the -th round. In this way, the throughput and the outage probability of HARQ protocols are monotonic functions of the probabilities . This is because the system performance depends on the retransmission round in which the codewords are correctly decoded. Moreover, the probability is directly linked to the AMI which is a random variable and function of the channel realizations experienced in rounds As such, to analyze the throughput and the outage probability, the key point is to determine the AMIs as functions of channel realization(s) and find their corresponding cumulative distribution functions (CDFs)222The CDF and the probability distribution function (PDF) of a random variable are denoted by and respectively. . Then, having the CDFs, the probabilities and, consequently, the considered performance metrics are obtained. Therefore, instead of concentrating on (1)-(2), we first find the CDFs for the INR-based RF-FSO system.
To find the CDFs, we utilize the properties of the RF and the FSO links to derive the AMIs as in (III). Since there is no closed-form expression for the CDFs of the AMIs, we need to use different approximation techniques. In the first method, we use the central limit Theorem (CLT) to approximate the contribution of the FSO link on the AMI by an equivalent Gaussian random variable. Using the CLT, we find the mean and the variance of the equivalent random variable for the exponential, log-normal and Gamma-Gamma distributions of the FSO link as given in (III-A)-(III-A), (III-A)-(III-A) and (-A)-(-A), respectively. With the derived means and variances of the Gaussian variable, we find the CDF of the AMIs in Lemmas 1-5 (see Section III.A for details). An alternative approximation approach, presented in Section III.B, is to approximate the PDF of the AMI in the FSO link and then find the PDF of the joint RF-FSO setup, as given in (III-B), (III-B), (III-B). Finally, we use the properties of the AMIs to analyze the optimal power allocation between the RF and the FSO links as given in Lemmas 6-7. In Section IV, we validate the accuracy of the approximations and evaluate the throughput/outage probability of the RF-FSO system for different channel conditions.
Iii Analytical Results
In RF-FSO systems, it was demonstrated by, e.g., [12, 36, 37, 38], that the RF link experiences very slow variations and the coherence time of the RF link is in the order of times larger than the coherence time of the FSO link. Here, we start the discussions by considering the setup as illustrated in Fig. 2 where the RF link remains constant in the retransmissions (quasi-static channel [20, 22, 23, 25, 26]) while in each retransmission round of HARQ different channel realizations are experienced in the FSO link. However, note that this is not a necessary condition because 1) the same analysis holds for the cases with shorter coherence time of the RF link, compared to the coherence time of the FSO link and 2) as seen in Section III.B, we can derive the results in the cases with few, possibly 1, channel realizations of the FSO link during the packet transmission.
Here, and are, respectively, the transmission powers in the RF and FSO links. Also, and ’s denote the channel gain realizations of the RF and the FSO links in different retransmission rounds, respectively. Then, represents the relative symbol rates of the RF and FSO links which is a design parameter selected by the network designer. With no loss of generality, we set in the following, while the results can be easily extended to the cases with different values of . Also, (III) is based on the fact that the achievable rate of an FSO link is given by with being the instantaneous received SNR and denoting a constant term such that for heterodyne detection and for intensity modulation/direct detection (IM/DD) , [41, Eq. 26], [42, Eq. 7.43], 333In , is proved as a tight lower bound on the capacity in the cases with an average power constraint. Then, [40, 43] show that the formula of the kind is an asymptotically tight lower bound on the achievable rates for the cases with an average power constraint, a peak power constraint, as well as combined peak and average power constraints.. In the following, we set which corresponds to the cases with heterodyne detection. In the meantime, setting , it is straightforward to extend the results to the cases with IM/DD. Finally, as the noise variances at the receiver are set to 1, we define as the SNR.
Iii-a Performance Analysis in the Cases with Considerably Different Coherence Times for the RF and FSO Links
Here, we consider the cases where the coherence times of the RF and FSO links are considerably different. Motivated by, e.g., [12, 36, 37, 38], we concentrate on the cases with shorter coherence time of the FSO link, compared to the RF link. Meanwhile, the same analysis is valid, if the RF link experiences shorter coherence time than the FSO link.
Considering the conventional channel conditions of the RF and FSO links and different values of , there is no closed-form expression for the CDF of Therefore, we use the CLT to approximate by the Gaussian random variable where and are the mean and variance determined based on the FSO link channel condition.
Reviewing the literature and depending on the channel condition, the FSO link is commonly considered to follow exponential, log-normal or Gamma-Gamma distributions, e.g., [12, 27, 44]. For the exponential distribution of the FSO link, i.e., with being the long-term channel coefficient, we have
Here, denotes the expectation operator. Also, and are obtained by partial integration. Then, denoting the Euler constant by , is given by variable transform some manipulations and the definition of Gamma incomplete function and the generalized hypergeometric function
For the log-normal distribution of the FSO link, i.e., where and represent the long-term channel parameters, the mean is rephrased as
Here, and represent the error and the Gaussian functions, respectively. Moreover, holds for the log-normal distribution and comes from the linearization technique
which is found by the derivative of at point Also, following the same procedure as in (III-A), the variance is determined as
Finally, we consider the Gamma-Gamma distribution for the FSO link in which the channel gain follows
with denoting the modified Bessel function of the second kind of order and being the Gamma function. Moreover, and are the distribution shaping parameters which can be expressed as functions of Rytov variance [27, 44].
For the Gamma-Gamma distribution, we can use (10) and, e.g., the approximation schemes of (-A)-(42) to derive the mean and variance as expressed in equations (-A)-(-A) of the Appendix. Intuitively, (III-A)-(III-A), (-A)-(-A) indicate that, with considerably different coherence times of the RF and FSO links, there are mappings between the performance of RF-FSO systems with exponential, log-normal and Gamma-Gamma distributions of the FSO link, in the sense that with proper scaling of the channel parameters the same outage probability/throughput is achieved in these conditions. Finally, note that Intuitively, this means that for asymptotically large values of , i.e., significantly shorter coherence time of the FSO link compared to the one in the RF link, the AMI of the FSO link converges to its ergodic capacity . Thus, in this case the RF-FSO link is mapped to an equivalent RF link in which successful decoding of the rate equal to the ergodic capacity of the FSO link is always guaranteed.
Having and , we find the CDFs as follows. Consider Rayleigh-fading conditions for the RF link where the fading coefficients follow and, consequently, Using (III) and the mean and variance of , the CDFs of the AMIs are given by
where comes from the CDF of Gaussian distributions and CLT. Also, for the exponential, log-normal and the Gamma-Gamma distribution of the FSO link the mean and variance are given by (III-A)-(III-A), (III-A)-(III-A) and (-A)-(-A), respectively. Therefore, the final step to derive the throughput and the outage probability is to find (III-A) while it does not have closed-form expression. The following lemmas propose several approximation/bounding approaches for the CDF of the AMIs and, consequently, the throughput/outage probability.
Lemma 1: The throughput and the outage probability of the HARQ-based RF-FSO setup are approximately given by
respectively, with defined in (III-A).
To find the approximations, we implement with
which leads to
Here, is obtained by applying the same linearization technique as in (III-A) on the Gaussian function of (III-A) at point . Then, using (III-A) in (1)-(2), one can find the throughput and outage probability, as stated in the lemma. ∎
As the second-order approximation of Lemma 1, the outage expression (15) is rephrased as
for large values of . Thus, using (III-A) and for large values of in (16), it is found that at high SNRs the outage probability of the joint RF-FSO link decreases with the power of RF and FSO signals exponentially, if the FSO link follows an exponential distribution.
Along with the approximation scheme of Lemma 1, Lemmas 2-5 derive upper and lower bounds of the system performance assuming that the mean and variance of the equivalent Gaussian random variable are calculated accurately.
Lemma 2: The performance of the RF-FSO system is upper-estimated, i.e., the throughput is upper bounded and the outage probability is lower bounded, via the following inequality
with given in (III-A).
As mentioned before and in [21, 22, 23, 24], the throughput (resp. the outage probability) of the HARQ-based systems is a decreasing (resp. increasing) function of the probabilities (resp. ). Thus, the throughput (resp. the outage probability) is upper bounded (resp. lower bounded) by lower bounding On the other hand, because the function is a decreasing function and is concave in , the CDFs of the AMIs are lower bounded if is replaced by its first order Taylor expansion at any point. Considering the Taylor expansion of at , we can write
Here, comes from partial integration and Also, the last equality is obtained by some manipulations and the definition of the error function. ∎
Lemma 3: An under-estimate of the performance of the RF-FSO system is given by
where is defined in (III-A).
To derive an under-estimate of the system performance, i.e., a lower bound of the throughput and an upper bound of the outage probability, we use (III-A) to upper bound the probability terms by
In (III-A), is based on partial integration. Also, follows from and variable transform . Finally, the last equality is obtained by manipulations and the definition of the -th order exponential integral function . Note that the bound is reasonably tight for different values of . Then, the appropriate value of can be determined numerically such that the difference between the exact and the bounded probabilities is minimized. ∎
Lemma 4: The performance of the RF-FSO system is upper-estimated via the following inequality
with given in (III-A).
Using and the same arguments as in Lemmas 2-3, the upper-estimate is found as
where the last equality is obtained with the same procedure as in (III-A). Note that, due to the considered bounding approach, the under-estimate is tight at low SNRs of the RF link. ∎
Lemma 5: An under-estimate of the performance of the RF-FSO system is given by
where is defined in (III-A).
To derive an under-estimate of the system performance, we use (III-A) to upper bound the probability terms by