Wireless Powered DualHop MultiAntenna Relaying Systems: Impact of CSI and Antenna Correlation
Abstract
This paper investigates the impact of the channel state information (CSI) and antenna correlation at the multiantenna relay on the performance of wireless powered dualhop amplifyandforward relaying systems. Depending on the available CSI at the relay, two different scenarios are considered, namely, instantaneous CSI and statistical CSI where the relay has access only to the antenna correlation matrix. Adopting the powersplitting architecture, we present a detailed performance study for both cases. Closedform analytical expressions are derived for the outage probability and ergodic capacity. In addition, simple high signaltonoise ratio (SNR) outage approximations are obtained. Our results show that, antenna correlation itself does not affect the achievable diversity order, the availability of CSI at the relay determines the achievable diversity order. Full diversity order can be achieved with instantaneous CSI, while only a diversity order of one can be achieved with statistical CSI. In addition, the transmit antenna correlation and receive antenna correlation exhibit different impact on the ergodic capacity. Moreover, the impact of antenna correlation on the ergodic capacity also depends heavily on the available CSI and operating SNR.
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I Introduction
How to prolong the operation lifetime of energy constrained wireless devices has become a critical issue, especially with the explosive growth of sensor nodes due to the fast development of internet of things. Responding to this challenge, empowering wireless devices with energy harvesting capabilities has been proposed as a promising solution [1, 2, 3], where the wireless devices can be designed to scavenge energy from natural resources such as solar and wind. However, the energy harvested from such sources is random and intermittent, and highly depends on uncontrollable factors such as weather, making it undesirable for communication systems with stringent qualityofservice (QoS) requirements [4].
As a practically viable solution to address this challenge, a new paradigm has emerged where the wireless devices harvest energy from the radiofrequency (RF) signals [5]. Since the RF signals can be fully controlled, it is more reliable and efficient. The combination of wireless power and information transfer has resulted in a new topic, generally referred to as simultaneous wireless information and power transfer (SWIPT). The idea of SWIPT was initially proposed by Varshney in [6], where the author studied the fundamental tradeoff between the information capacity and the harvested energy under the ideal assumption that the circuit can decode the information and harvest energy at the same time. Later in [7], two practical receiver architectures were proposed, namely timeswitching and powersplitting. A more sophisticated dynamic power splitting scheme was proposed in [8]. The performance of wirelessly powered multiple antenna systems with energy beamforming was studied in [9, 10], the issue of imperfect channel state information (CSI) was addressed in [11], and a training based SWIPT system was studied in [12]. In addition, the performance of SWIPT in cellular system was considered in [13, 14] and the hybrid conventional battery and wireless energy transfer systems were investigated in [15, 16].
SWIPT also finds important applications in cooperative relaying system [17, 18, 19] which is a fundamental building block for many important deployment scenarios in 5G systems such as Internet of things [20]. The authors in [17] investigated the throughput performance of amplifyandforward (AF) halfduplex (HF) relaying network for both timeswitching and powersplitting receiver architectures in Rayleigh fading channels, and [21] extended the analysis to the more general Nakagamim fading channel. The authors in [22] considered the decodeandforward relaying system and studied the power allocation strategies for multiple sourcedestination pairs. Later in [25], the authors proposed a lowcomplexity antenna switching protocol for the implementation of SWIPT, while in [23], the authors studied the performance of relay selection in SWIPT systems. The performance of energy harvesting cooperative networks with randomly distributed users/relays was studied in [24, 25]. In [26], fullduplex (FD) relaying was introduced into SWIPT systems, while [27, 28] considered the extension of twoway relaying. More recently, the impact of multiple antennas on the performance of dualhop AF SWIPT relaying system was studied in [29], where it was shown that increasing the number of relay antennas can substantially improve the system performance. However, the conclusion of [29] is based on the assumption that the multiple antennas are uncorrelated and the instantaneous CSI is available at the relay. In practice, due to insufficient antenna spacing and lack of local scatters, the channels tend to exhibit spatial correlation. And in the cases that channels are rapidly timevarying, in order to reduce the complexity of channel estimation, the relay may obtain only statistical CSI rather than instantaneous CSI. While the effect of statistical CSI has been extensively investigated in conventional multiple antenna systems[30, 31], it has not been investigated in the dualhop multiantenna SWIPT systems.^{1}^{1}1In the conference paper [32], we have investigated the impact of CSI and antenna correlation on the outage probability of the dualhop multiantenna SWIPT systems. However, the detailed proof of the key results is not included due to space limitation, which will be presented here. Therefore, it is of great interest to investigate the achievable performance of SWIPT relaying systems in realistic scenarios taking into consideration the impact of antenna correlation and statistical CSI.
Motivated by this, we consider a sourcerelaydestination dualhop system where the source and destination are equipped with a single antenna while the relay powered via RF energy harvesting is equipped with multiple antennas. Unlike [29], we assume that the antennas are spatially correlated which is practical in the situation that the space among antennas is insufficient or the local scatters are inadequate. In addition, we consider two scenarios depending on the available CSI at the relay node, i.e., instantaneous CSI or statistical CSI where the relay only have access to the antenna correlation information.
The main contributions are summarized as follows:

For the instantaneous CSI scenario, an exact analytical expression in integral form is derived for the outage probability, and a closedform lower bound for the outage probability is presented. Also, simple asymptotic approximations for the outage probability are presented in the high signaltonoise ratio (SNR) regime, which shows that full diversity can be achieved.

For the statistical CSI scenario, a suboptimal relay processing matrix is proposed. An exact analytical expression in integral form is derived for the outage probability, and simple high SNR approximations for the outage probability are presented, which reveals that only unit diversity order can be attained.

For both the instantaneous and statistical CSI scenarios, closedform upper bounds for the ergodic capacity are derived. In addition, the impact of transmit and receive antenna correlation on the ergodic capacity is characterized.

The outcomes of the paper suggest that the amount of available CSI at the relay has a significant impact on the system performance. Moreover, the impact of antenna correlation on the system performance is also coupled with the availability of CSI at the relay. Specifically, the transmit antenna correlation is detrimental in the case of instantaneous CSI while it is beneficial with statistical CSI, while the impact of receive antenna correlation is undetermined which depends heavily on the availability of CSI and the operating SNR.
The rest of the paper is organized as follows: Section II introduces the system model, Section III focuses on the instantaneous CSI scenario, while Section IV deals with the statistical CSI scenario. Numerical results are presented in Section V. Finally, Section VI concludes the paper and summarizes the main findings.
Notation: The upper bold case letters, lower bold case letters and lower case letters denote matrices, vectors and scalars respectively. For an arbitrarysize matrix , , and denote the conjugate transpose, conjugate and squareroot of respectively. denotes the Frobenius norm of a complex vector . denotes the th element of vector . denotes the cumulative distribution function (CDF) of a random variable . denotes . denotes the statistical expectation. is the gamma function[33, Eq. (8.310.1)]. is the exponential integral function [33, Eq. (8.211.1)]. is the psi function[33, Eq. (8.360)]. is the nth order modified Bessel function of the second kind [33, Eq. (8.407)]. is the Meijer’s Gfunction[33, Eq. (9.301)].
Ii System Model
We consider a dualhop energy harvesting relay system where a singleantenna source, S, communicates with a singleantenna destination, D, with the aid of a multiantenna relay, R, as illustrated in Fig. 1(a). It is assumed that the relay has no external power supply, and entirely relies on the energy harvested from the source signal. ^{2}^{2}2In this paper, we have assumed that the processing power required by the transmit/receive circuitry at the relay is negligible as compared to the relay transmission power. This assumption is justifiable when the transmission distances are relatively large such that the transmission energy becomes the dominant source of energy consumption [34]. As a matter of fact, similar assumption has been widely adopted in SWIPT relaying literature with both singleantenna relay [17] and multiantenna relay [35]. In addition, as in [17, 36], we assume that the relay is entirely powered via RF energy harvesting. By implementing multiple antennas at the relay, the energy harvesting efficiency can be significantly improved [37, 4], hence more energy can be harvested to support its power requirements. Therefore, we believe that adopting multiantenna relay is practically viable. We assume that the direct link between the source and destination does not exist due to obstacles or severe shadowing as in [17, 26, 29]. It is assumed that CSI is available at the destination node, while no CSI is available at the source node as in [29, 38]. Also, two different types of CSI assumptions are considered at the relay node, namely, instantaneous CSI and statistical CSI. In practice, CSI can be obtained by pilotassisted channel training or with the help of a suitable feedback mechanism.
We focus on the power splitting receiver architecture as in [17]. Hence, an entire transmission block is divided into two time slots with duration of each. During the first time slot, the relay listens to the source transmission, and splits the received signal into a power stream for energy harvesting and an information stream for information forwarding according to a power splitting ratio . During the second time slot, the relay forwards the processed information to the destination using the harvested power.
Let the vector and vector represent the channel coefficients of the sourcerelay and relaydestination links, respectively. We assume that channels are correlated due to insufficient antenna spacing, as such, and , where and denote the receive and transmit correlation matrices, respectively. and are mutually independent zeromean complex circular symmetric Gaussian random vectors with unit variance. Without loss of generality, the correlation matrices are assumed to be full rank.^{3}^{3}3To simplify the analytical derivation, we only provide the results for the fullrank case. However, the results for the arbitrary rank case can be obtained in a similar fashion, albeit with more involved mathematical manipulations.
As a result, the total harvested energy at the relay during the first time slot can be computed as
(1) 
where denotes the energy conversion efficiency, denotes the source transmit power, is the distance between the source and relay, denotes the pass loss exponent.
Also, the signal received by the information receiver of the relay can be written as
(2) 
where denotes the source symbol with unit power, and denotes the additive white Gaussian noise (AWGN) satisfying with I being the identity matrix.
During the second time slot, the relay employs the AF protocol to forward the transformed signal to the destination using the harvested energy. Depending on the availability of CSI at the relay, we consider two separate scenarios, namely, instantaneous CSI and statistical CSI.
Iia Instantaneous CSI
In this case, it is assumed that the relay knows both and . With the AF protocol, the relay simply scales the received signal by a transformation matrix . To meet the power constraint at the relay, it is required that , where . According to [39], the optimal transformation matrix maximizing the endtoend SNR can be expressed as
(3) 
where is the power constraint factor and is given by
(4) 
Therefore, the received signal at the destination can be expressed as
(5) 
where denotes the distance between the relay and the destination node, denotes the AWGN at the destination with .
Define , then the endtoend signaltonoise ratio (SNR) can be written as
(6) 
IiB Statistical CSI
In this case, it is assumed that the relay only knows the channel correlation matrices and . With only channel correlation information available, analytical characterization of the optimal relay processing matrix appears to be difficult. Hence, we adopt a heuristic Rank1 processing matrix as motivated by the instantaneous CSI scenario,^{4}^{4}4While the rank1 processing matrix is a heuristic choice, we believe that the achievable system performance with rank1 constraint should come close to that of the optimal solution. The main reason is that this paper assumes single antenna at the both the source and destination, hence, the single stream transmission in both the first hop and second hop is a reasonable choice, as demonstrated in prior works [40, 41]. i.e.,
(7) 
where , and is the power constraint.
With the Rank1 constraint, we have the following important observation:
Theorem 1
With the Rank1 assumption, the optimal and maximizing the ergodic capacity turn out to be the eigenvectors associated with the maximum eigenvalues of the correlation matrix and , respectively.
Proof: See Appendix A.
Now, consider the following eigenvalue decomposition of the correlation matrices and , then, we have
(8) 
where and denote the th and th element of and , respectively. And and are the th and th largest eigenvalues of and , respectively. Then, the endtoend SNR can be derived as
(9) 
Iii Instantaneous CSI Scenario
In this section, we focus on the scenario with instantaneous CSI, and present a detailed analysis on the system performance in terms of key performance measures, namely, the outage probability and the ergodic capacity.
Iiia Outage Probability
Mathematically, the outage probability is defined as the probability of the instantaneous SNR falls below a predefined threshold , i.e.,
(10) 
Proposition 1
The exact outage probability can be expressed as
(11) 
where , , , .
Proof: The desired result can be obtained by following the similar lines as in the proof [17].
Although the integral in (11) does admit a closedform solution, it can be efficiently evaluated through numerical integration. Alternatively, we can use the following tight lower bound of the outage probability.
While Theorem 2 gives an efficient method for evaluating a tight lower bound of the outage probability of the system, the expression in (12) provides little insights. Motivated by this, we now look into the high SNR regime, and derive a simple approximation for the outage probability, which enables the characterization of the achievable diversity order.
Theorem 3
In high SNR regime, the outage probability of the system can be approximated as
(13) 
Proof: See Appendix C.
Theorem 3 indicates that the system achieves the full diversity of , which implies that antenna correlation does not reduce the diversity order, instead, it degrades the system performance by affecting the coding gain. Moreover, it is observed that decays as rather than , which suggests that the outage probability decays much slower than the conventional case where the relay has a constant power source, as previously reported in [29]. The reason of this phenomenon is possibly that the transmit power at the relay is a random variable in SWIPT systems, which causes higher outage probability compared to the conventional cases.
IiiB Ergodic Capacity
We now study the ergodic capacity of the system. To start with, we first rewrite the endtoend SNR in (6) as
(14) 
where and . Hence, the ergodic capacity can be expressed as
(15) 
Since the probability density function (PDF) of is not available, exact characterization of the ergodic capacity is mathematically intractable. As such, we direct our efforts to seek tight capacity bounds. And we have the following key result:
Theorem 4
The ergodic capacity of the system with the AF protocol can be upper bounded as
(16) 
where
(17) 
Proof: See Appendix D.
IiiC Impact of Correlation on Ergodic Capacity
We now study the impact of antenna correlation on the ergodic capacity. Before delving into the details, we first introduce the following definition given in [42, Ch. 11]:
Definition 1
For two vectors , we use to denote vector majorize vector , if , and , where and are the ordered elements of and .
The following proposition characterizes the impact of correlation on the ergodic capacity.
Proposition 2
The ergodic capacity of the system with instantaneous CSI is a Schurconcave function with respect to the eigenvalues of the transmit correlation matrix, i.e., let and be two vectors comprised of the eigenvalues of the transmit correlation matrix satisfying , then we have
(18) 
In addition, the high SNR ergodic capacity is a Schurconcave function, while the low SNR ergodic capacity is a Schurconvex function with respect to the eigenvalues of the receiver correlation matrix, i.e., let and be two vectors comprised of the eigenvalues of the receive correlation matrix satisfying , then we have
(19) 
Proof: See Appendix E.
Proposition 2 indicates that transmit correlation has a detrimental effect on the ergodic capacity, the stronger the correlation, the lower the ergodic capacity. Such an observation is in consistent with those reported in the MIMO literature. Unfortunately, similar characterization on the impact of receive correlation is not available. In contrast, whether receive correlation is beneficial or detrimental depends on the operating SNR. At the high SNR regime, receive correlation leads to a reduction of the ergodic capacity, while at the low SNR regime, receive correlation contributes to an increase of the ergodic capacity.
Iv Statistical CSI Scenario
In this section, we present a detailed analysis on the system performance when the relay has only access to statistical CSI.
Iva Outage Probability
Due to the correlation between the random variables and , the analysis becomes much more involved.
Theorem 5
With both transmit and receive correlation, the outage probability of the system with only statistical CSI at the relay is given by
(20) 
Proof: See Appendix F.
To the best of the authors’ knowledge, (20) does not admit a closedform expression. To gain further insights, we now derive a simple high SNR approximation for the outage probability, which enables the characterization of the achievable diversity order.
Theorem 6
Theorem 6 suggests, with only the knowledge of correlation matrices at the relay, the system only achieves a diversity of one. Compared with the outage performance with instantaneous CSI in Section III where full diversity order is achieved, we can conclude the diversity order does not depend on the antenna correlation but is determined by the availability of CSI at the relay, and the correlation affects the system performance by affecting the coding gain. In addition, we observe that in (21) is a monotonic increasing function with respect to which implies strong transmit correlation is beneficial in terms of the outage performance with only statistical CSI.
IvB Ergodic Capacity
In this section, we look into the ergodic capacity and have the following key result:
Theorem 7
With both transmit and receive correlation, the ergodic capacity of the system with only statistical CSI at the relay is upper bounded as
(22) 
where
(23) 
Proof: The result can be obtained by following similar lines as in proof of Theorem 4, along with some simple algebraic manipulations, hence is omitted.
IvC Impact of Correlation on Ergodic Capacity
We now study the impact of correlation on the ergodic capacity, and we have the following results:
Proposition 3
The ergodic capacity of the system with statistical CSI is a Schurconvex function with respect to the eigenvalues of the transmit correlation matrix, i.e., let and be two vectors comprised of the eigenvalues of the transmit correlation matrix satisfying , then we have
(24) 
For the special case , is a Schurconcave function with respect to the eigenvalues of the receive correlation matrix, i.e., let and be two vectors comprised of the eigenvalues of the receive correlation matrix satisfying , then we have
(25) 
Proof: See Appendix H.
Please note, if , then the impact of receive correlation is uncertain as shown in Appendix H. In contrast to the instantaneous CSI scenario, Proposition 3 indicates that transmit correlation is always beneficial with only statistical CSI. The reason is that, with instantaneous CSI, a higher correlation reduces the array gain, which leads to a performance degradation. While with only statistical CSI, a higher correlation provides more information about the channel and enables power focusing, which contributes to performance enhancement. To this end, it is worth emphasizing that the above claims are established based on the rank1 assumption. Hence, the impact of antenna correlation on the ergodic capacity with optimal relay processing matrix maybe different.
V Numerical Results And Discussion
In this section, we present numerical results to demonstrate the correctness of the analytical expressions presented in Section III and Section IV, and investigate the impact of key parameters on the system performance. Unless otherwise specified, we suppose and are exponentiallycorrelated[43] with different correlation factors and respectively, dB, , , and m.^{5}^{5}5Please note, the choice of distances is simply for illustration purpose, and the developed analytical results are applicable for arbitrary system configurations.
To measure the impact of CSI on the system performance, we also include the case where no CSI is available at the relay as a benchmark scheme, where the relay processing matrix is given by , and the corresponding endtoend SNR can be expressed as
(26) 
Due to the challenge of characterizing the statistical properties of , the results for the no CSI case are obtained through MonteCarlo simulations.
Fig. 2(a) and Fig. 2(b) show achievable outage probability and ergodic capacity performances of three different cases, i.e.,instantaneous CSI, statistical CSI and no CSI, when , and . It can be observed from 2(a) that the proposed lower bounds are very tight across the entire range of SNR, and the high SNR approximations are quite accurate, especially in the high SNR regime. In addition, we see that full diversity of is achieved with instantaneous CSI, yet the diversity order remains one with only statistical CSI or no CSI, which is consistent with our analytical results presented in Theorem 3 and Theorem 6. Similarly, Fig. 2(b) indicates that the proposed analytical upper bounds in Theorem 4 and Theorem 7 are sufficiently tight across the entire SNR region of interest. Finally, both figures demonstrate the intuitive results that the system performance improves with the available amount of CSI at the relay.
Fig. 3 investigates the impact of on the system performance when and . Intuitively, we observe that increasing the number of antennas significantly improves the outage probability and ergodic capacity performance. Moreover, the gain of increasing differs substantially for scenarios with different CSI assumptions. For instance, from Fig. 3(a), the outage probability reduction is most pronounced for the instantaneous CSI scenario when increases from two to four. The reason is that, with instantaneous CSI, affects the diversity order, while for the other two scenarios, only provides some coding gain, hence the overall benefit is relatively small. Similar observations can be made from Fig. 3(b) in terms of ergodic capacity gain.
In Fig. 2 and Fig. 3, a fixed power splitting ratio is used. Since the power splitting is a key design parameter, it is of interest to investigate its impact on the system performance. Fig. 4 illustrates the effect of optimal power splitting ratio on the system performance. The optimal power splitting ratio is obtained via numerical method based on the analytical expressions presented in Theorem 4 and 7. Interestingly, we observe that the performance gap between the ergodic capacity curves associated with the optimal and those with fixed is rather insignificant, especially for the case with statistical CSI.
Fig. 5 illustrates the impact of antenna correlation on the ergodic capacity with instantaneous CSI at different operating SNRs. For a fixed , the ergodic capacity is decreasing function with respect to as shown in both figures, which indicates the transmit antenna correlation is always detrimental to the ergodic capacity, as analytically predicted in Proposition 2. In contrast, for a fixed , we observe that the impact of receive antenna correlation on the ergodic capacity behaves quite different, for instance, when dB, stronger correlation increases the ergodic capacity, while when dB, stronger correlation decreases the ergodic capacity.
Fig. 6 investigates the impact of antenna correlation on the ergodic capacity with statistical CSI. We observe that the transmit antenna correlation is always beneficial to the ergodic capacity, which is in consistent with the conclusion drawn in Proposition 3. As for the impact of receive antenna correlation, we observe it boosts the ergodic capacity at low SNRs which is similar to the scenario with instantaneous CSI. However, at high SNRs, we see a rather surprising behavior that the ergodic capacity first improves and then degrades when the correlation increases, indicating that there exists an optimal correlation point where the ergodic capacity is maximized.
Vi Conclusion
We have investigated the impact of antenna correlation and relay CSI on the performance of dualhop wireless powered multiantenna relay systems. Specifically, for both the instantaneous CSI and statistical CSI scenarios, we derived closedform analytical expressions for the outage probability as well as simple high SNR approximations. In addition, we presented closedform upper bound for the ergodic capacity, and characterized the impact of antenna correlation on the ergodic capacity. Our findings suggest that, the availability of CSI at the relay has a critical impact on the system performance. With instantaneous CSI, full diversity order can be achieved, while unit diversity order can be achieved with only statistical CSI. In addition, it is shown that the antenna correlation does not affect the achievable diversity order. Moreover, the impact of antenna correlation on the ergodic capacity depends on the available CSI and operating SNR, for instance, transmit antenna correlation is detrimental with instantaneous CSI while it is beneficial with only statistical CSI; receive antenna correlation improves ergodic capacity at lower SNRs, while it may degrade the ergodic capacity at moderate or high SNRs.
Appendix A Proof of Theorem 1
To maximize the ergodic capacity, the optimization problem can be formulated as
(27) 
Since and are decoupled, the optimal and can be separately handled. As such, we first fix , hence, the original problem becomes
(28) 
where and .
Since and 28) can be rewritten as , (
(29) 
where . Noticing that is unitary invariant, by taking , we reformulate the problem as
(30) 
Let , where , is the diagonal matrix with the th diagonal element being , and all other diagonal elements being . Now, noticing that and is a concave function with respect to
(31) 
Now, since is unitary invariant, we have and
(32) 
Hence, we have
(33) 
where the equality holds if and only if , indicating that the optimal is a diagonal matrix. Therefore, the optimal can be expressed as , where is a diagonal matrix. Since is a Rank1 matrix, there is only one nonzero diagonal element in which equals to one. To this end, it is easy to show that the optimal is the eigenvector associated with the biggest eigenvalue of . Using the same method, the optimal can be obtained.
Appendix B Proof of Theorem 2
Noticing that the endtoend SNR (6) can be upper bounded as
(34) 
Then, (11) can be rewritten as
(35) 
To solve the integral, we look into the PDFs of and and we have
(36) 
Since follows the exponential distribution, the PDF of is the sum of weighted exponential random variables. With the help of[45], the PDF of can be written as
(37) 
Similarly, the PDF of is given by
(38) 
Then, the right tail probability of can be obtained as
(39)  
(40) 
Appendix C Proof of Theorem 3
To prove Theorem 3, we need the following lemma:
Lemma 1
For an arbitrary dimensional square matrix with eigenvalues , with decreasing order, i.e if . We denote , represents the determinant of with the th row and th column removed, then
(42) 
for any real number .
Proof: We observe that (42) can be written as a determinant. And if , the determinant becomes zero since the determinant contains two identical rows.
To proceed with the proof of Theorem 3, as , the endtoend SNR can be tightly upper bounded as
(43) 
After some tedious manipulations, the outage probability can be rewritten as
(44) 
Note that and are hypoexponential random variables, using the similar method as in [46], the outage probability can be written in the integral form as