IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 4, APRIL

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1 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 9 Simultaneous Wireless Information and ower Transfer Under Different CSI Acquisition Schemes Chen-Feng iu, Marco Maso, Member, IEEE, Subhash akshminarayana, Member, IEEE, Chia-Han ee, Member, IEEE, and Tony Q. S. Quek, Senior Member, IEEE Abstract In this work, we consider a multiple-input singleoutput system in which an access point A performs a simultaneous wireless information and power transfer SWIT to serve a user terminal UT that is not equipped with external power supply. To assess the efficacy of the SWIT, we target a practically relevant scenario characterized by imperfect channel state information CSI at the transmitter, the presence of penalties associated to the CSI acquisition procedures, and non-zero power consumption for the operations performed by the UT, such as CSI estimation, uplink signaling and data decoding. We analyze three different cases for the CSI knowledge at the A: no CSI, and imperfect CSI in case of time-division duplexing and frequencydivision duplexing communications. Closed-form representations of the ergodic downlink rate and both the energy shortage and data outage probability are derived for the three cases. Additionally, analytic expressions for the ergodically optimal duration of power transfer and channel estimation/feedback phases are provided. Our numerical findings verify the correctness of our derivations, and also show the importance and benefits of CSI knowledge at the A in SWIT systems, albeit imperfect and acquired at the expense of the time available for the information transfer. Index Terms Simultaneous wireless information and power transfer SWIT, energy harvesting, wireless power transfer, TDD, FDD, analog feedback. I. INTRODUCTION IN conventional wireless systems, the limited battery capacity of mobile devices typically affects the overall network lifetime. Increasing the size of the battery might not be a feasible solution to address this problem, due to a consequent Manuscript received December 9, 03; revised May 9, 04, October 6, 04, and November 7, 04; accepted November 7, 04. Date of publication December, 04; date of current version April 7, 05. This work was supported in part by the SRG ISTD 0037, SUTD-MIT International Design Centre under Grant IDSF0006OH, and the SUTD-ZJU Research Collaboration Grant, and the Ministry of Science and Technology MOST, Taiwan under Grant MOST03--E The associate editor coordinating the review of this paper and approving it for publication was S. Bhashyam. C.-F. iu and C.-H. ee are with the Research Center for Information Technology Innovation, Academia Sinica, Taipei 5, Taiwan cfliu@ citi.sinica.edu.tw; chiahan@citi.sinica.edu.tw. M. Maso is with the Mathematical and Algorithmic Sciences ab, Huawei France Research Center, Boulogne-Billancourt 900, France marco. maso@huawei.com. S. akshminarayana is with the Singapore University of Technology and Design, Singapore subhash@sutd.edu.sg. T. Q. S. Quek is with the Singapore University of Technology and Design, Singapore 3868, and also with the Institute for Infocomm Research, A STAR, Singapore tonyquek@sutd.edu.sg. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 0.09/TWC reduction of the portability and increase in the cost of the equipment. For these reasons, the study of novel techniques to prolong the lifetime of the battery has triggered an increased interest in the wireless communications community. In this context, the study of the so-called wireless power transfer WT has recently gained prominence as means to implement a cable-less power transfer between devices [], either by resonant inductive coupling [] or by far-field power transfer [3]. The latter approach has seen increasing momentum in recent years, due to its promising potential for longer range transfers. A fundamental breakthrough in this context has been the design of rectifying antennas rectennas for microwave power transfer MT, key component to achieve an efficient radio frequency to direct current RF-to-DC conversion. This has brought to several technological advances, e.g., the design of flying vehicles powered solely by microwave [4], which confirmed that RF-to-DC conversion can not only be performed but also achieve remarkable efficiency. In this regards, commercial products exhibiting an efficiency larger than 50% are already available on the market [5]. Remarkably, performance of stateof-the-art RF-to-DC converters can be even higher, i.e., more larger than 80%, resulting in an impressive potential DC-to-DC efficiency of 45% [6], [7]. A. Related Works Recent advances in signal processing and microwave technology have shown that far-field power transfer can offer interesting perspectives also in the context of traditional wireless communication systems. For instance, the same electromagnetic field could be used as a carrier for both energy and information, realizing the so-called simultaneous wireless information and power transfer SWIT. The potential of this approach has been first highlighted by the studies proposed in [8] and [9]. Therein the trade-off between the two transfers within SWIT was investigated, in the case of both flat fading and frequency-selective channels. After these pioneering works, several studies have been performed to assess the implementability of receivers that can make use of RF signals to both harvest energy and decode information [0], and analyze the performance of SWIT in many scenarios, e.g., multiantenna systems [], opportunistic networks [], wireless sensor networks [3]. In particular, the aforementioned tradeoff is investigated in multiple-input multiple-output MIMO systems for two information and power transfer architectures, i.e., time-switching and power-splitting, for both perfect [] and imperfect [4] channel state information CSI. A different IEEE. ersonal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 9 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 approach is considered in [5], where the trade-off between the information and the energy transfer is investigated in a multiuser network under two different constraints, i.e., a constraint expressed in terms of secrecy rate for the former and amount of harvested energy at the receiver for the latter. A line of work considering orthogonal frequency division multiplexing OFDM systems is presented in [6] [8], where the resource allocation policy at the transmitter is studied for both single and multi-user scenarios, in which the receiver adopts a power-splitting strategy and considers undesired interference as an additional energy resource. Finally, ad-hoc scenarios departing from the standard SWIT paradigm have been analyzed in works such as [9], [0], where it is assumed that the energy and information transfers are performed by two different devices, operating in frequency-division duplexing FDD mode. More precisely, in [9] the time allocation policy for the two transmitters is studied, under the assumption that the efficiency of the energy transfer is maximized by means of an energy beamformer that exploits a quantized version of the CSI received in the uplink by the energy transmitter. Along similar lines, [0] investigates the optimal time and power allocations strategies such that the amount of harvested energy is maximized, taking into account the impact of the CSI accuracy on the latter quantity. B. Summary of Our Contribution In general, most existing works for SWIT rely on ideal assumptions: a availability of perfect CSI at the transmitter, b no penalty for CSI acquisition, c no power consumption for signal decoding operations at the receiver. If these assumptions are relaxed and the resulting penalties and issues are considered, then the performance of wireless systems can decrease significantly. Studies taking into account these aspects have been proposed for conventional information transfers, and the effect of imperfect CSI acquisition, training, feedback as well as the resource allocation problem have been thoroughly analyzed []. Departing from these observations, in this work we aim at investigating the efficacy of SWIT for practically relevant scenarios, by relaxing the three aforementioned ideal assumptions. In particular, we target a multiple-input single-output MISO system consisting of a multi-antenna access point A which transfers both information symbols and energy to a single user terminal UT that does not have access to any external power source. Accordingly, in contrast to the previous contributions on this topic, in this work the power harvested by means of the WT is used by the UT to perform all the necessary signal processing operations for both information decoding and uplink communications. Additionally, we adopt a systematic approach and consider the three main possible scenarios for an A that engages in a downlink transmission in modern networks, to provide a more complete characterization of the considered system. Consequently, in this work, the performance and feasibility of the SWIT in a MISO system is studied for the three following cases: No CSI available at the A. Time-division duplexing TDD communications and CSI acquisition at the A by means of training symbols. FDD communications and CSI acquisition at the A by means of analog symbols feedback. We compare these three scenarios for three performance metrics of interest, namely, ergodic downlink rate, energy shortage probability, and data outage probability. Our contributions in this work are as follows: We derive closed-form representations for the three performance metrics of interest in all three scenarios and match them to the numerical results. We derive the approximations of the ergodically optimal duration of the WT phase in all the three scenarios as a portion of the channel coherence time. Additionally, for the TDD and the FDD scheme, we derive closed-form approximations for the ergodically optimal duration of the channel training/feedback phases, to maximize the downlink rate. We show that the TDD scheme can outperform the FDD scheme in SWIT systems in terms of both downlink rate and data outage probability. Our numerical findings verify the correctness of our derivations. More specifically, concerning the downlink rate, we show that the performance gap between the numerical optimal solutions and the results obtained by means of our approximations is very small for low to mid signal-to-noise ratio SNR values and negligible for high SNR values. Moreover, we show that both TDD and FDD outperform the non-csi case at any SNR value in terms of downlink rate. This confirms that CSI knowledge at the A is always beneficial for the information transfer in SWIT systems, despite both its imperfectness and the resources devoted to the channel estimation/feedback procedures. The correctness of our derivations is further verified when numerically evaluating both the energy shortage and data outage probability of the considered MISO system adopting SWIT, for which a perfect match of analytic and numerical results is achieved. Finally, it is worth noting that throughout our study TDD consistently outperforms FDD in terms of both downlink rate and data outage probability, confirming the potential of this duplexing scheme for the future advancements in modern networks. The rest of the paper is organized as follows. In Section II, we specify the system model. In Section III-A, III-B, and III-C, we specify the system model and derive the downlink rate for the non-csi, the TDD, and the FDD scheme. In Section IV, we derive the energy shortage and data outage probability for each scheme. In Section V, we show and discuss the numerical results. Finally, we conclude in Section VI. Notations: In this paper, we denote matrices as boldface upper-case letters, vectors as boldface lower-case letters. Additionally, we let [ ] be the conjugate transpose of a vector. All vectors are columns, unless otherwise stated. Furthermore, for a scalar c C we note by c its complex conjugate. We use x y to express the orthogonality between vectors x and y. We denote a circular symmetric Gaussian random vector with mean μ and covariance matrix Σ as CNμ, Σ. The chi-squared distribution with K degrees of freedom is denoted by χ K and its probability density function DF is given by f X x = Γ K K/ x K e x/ where Γq = 0 uq e u du is the Gamma function. The

3 IU et al.: SWIT UNDER DIFFERENT CSI ACQUISITION SCHEMES 93 non-central chi-squared distribution with K degrees of freedom and non-central parameter ν is denoted by χ K ν and its DF is given by f X x = e x+ν/ x K/4 / ν I K νx, I n, modified Bessel function of the first kind. Γq, r = r uq e u du is the upper incomplete Gamma function. Q M q, r = ξ M r exp ξ +q q M I M qξdξ is the generalized Marcum Q-function []. II. SYSTEM MODE We consider a point-to-point communication system consisting of an A with antennas and a UT with single antenna. We denote the downlink channel from the A to the UT as h =[h,,h ]. The channel is assumed to be block fading, with independent fading from block to block. The entries of the channel vector are complex Gaussian Rayleigh fading, hence h CN0, I.etT C be the coherence time length. For simplicity in the notation, we assume that the total number of symbols that can be transmitted within the coherence time is T C. The A transmits the symbol x C with a transmit power, i.e., E[ x ]=. The received signal at the UT is given by y = h x + n, where n CN0, is the thermal noise, modeled as a complex additive white Gaussian noise AWGN. We assume the UT does not have any external power source such as the battery and all the power required for the operations to be performed at the UT is provided by the A through the WT component of the SWIT. Accordingly, the UT is equipped with a circuit that can perform two different functions: a harvest energy from the received RF signal, b information decoding. As considered in previous literature [], we assume that the UT cannot harvest energy and decode information from the same signal, at the same time. Hence, a time switching strategy is adopted under which the A transmits the signals in two phases: the signal sent during the first phase has WT purposes and is used by the UT to harvest energy, whereas the signal sent in the second phase has information transfer purposes. Note that, throughout this work, we assume that the energy harvested in the first phase power transfer phase is the sole source of power for all the subsequent operations performed by the UT the exact details of these operations will be specified the following sections. Details of the ower Transfer hase: During each coherence time interval, the A first transmits the power wirelessly to the UT for ɛ<t C time slots. First, the A divides its power equally between its transmit antennas to perform the WT. Hence the -sized transmit symbol during this phase, denoted by x EH, is given by x EH = s, where s is a random vector with zero mean and covariance matrix E[ss ]=I. Thus, the power harvested at the UT is given by H = β h, Note that, the words energy and power are used interchangeably in this paper, for the sake of simplicity, in spite of their conceptual difference. The exact value of ɛ will be specified later depending upon the mode of operation. Fig.. Operations of the A and the UT during the coherence time in the non-csi scheme. where β [0, ] is a coefficient that measures the efficiency of the RF to direct current RF-to-DC power conversion [], [3]. Details of the Information Transmission hase: For the second phase, namely information transmission phase, we adopt a systematic approach and consider the three scenarios. In the first one, the A transmits the information symbols without the knowledge of CSI we will refer to this approach as non-csi scheme. In the second scheme, we consider a TDD communication, in which the downlink and uplink communications are performed over the same bandwidth. Accordingly, first the A acquires the CSI by evaluating a pilot sequence transmitted by the UT in the uplink, and then engages in the downlink transmission. In the last case, we consider an FDD communication, in which the downlink and uplink communications are performed over two separate bandwidths. Consequently, in this case, UT sends an analog feedback signal in the uplink, carrying the downlink channel estimation, to allow the A to acquire the CSI and subsequent transmit information symbols. Under the aforementioned settings, we analyze the performance of the system for two different metrics of interest, namely the downlink rate and outage probability. We provide a detailed analysis of these two metrics in the rest of the paper. III. ANAYSIS OF THE DOWNINK RATE In this section, we analyze the downlink rate for the three considered schemes. A. Non-CSI Scheme We first consider the case where the A transmits the information symbols without the knowledge of CSI. The schematic diagram of this scenario is shown in Fig.. Under this scheme, the system utilizes ɛ = α N T C symbols to transfer power and the remaining symbols to transmit information symbols, where 0 <α N <. The received signal during the information trans- mission phase is given by y = h x + n, where the x = s and s CN0, I. Note that in the absence of CSI, the A performs equal power allocation over all its antennas to transmit the information symbol. For the information decoding at the UT, we consider that the power consumption of the circuit components devoted to the decoding is proportional to the number of received symbols as typically considered in previous works on the subject [3]. Accordingly, we denote the power consumption per decoded symbol at the UT as D.

4 94 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 Fig.. Operations of the A and the UT during the coherence time in the TDD scheme. Since the power harvested in the first phase must be sufficient to decode all the information symbols, we have that α N T C H = α N T C D. Now, if we plug into this equation then, after some manipulations, we have that the minimum fraction of time that should be devoted to the power transfer, i.e., α N, given by D α N = β h. + D We now analyze the downlink rate for this scheme. We recall that the A can transmit α N symbols for the information transfer. Accordingly, using, the downlink rate obtained for the non-csi scheme is given by R NC = R NC α N = α N log + h = β h β h + D log + h. 3 B. TDD Scheme We switch our focus to the TDD scheme, whose schematic diagram is shown in Fig.. We recall that, in this case, the A should provide the UT with sufficient energy for the latter to be able not only to decode the received data but also to perform all the operations related the uplink signaling inherent to the TDD scheme. Accordingly, the system utilizes ɛ = α T T C symbols for power transfer, where 0 <α T <. This is followed by the CSI acquisition phase. Since we assume that all the operations are performed within the coherence time, the A can exploit the reciprocity of the downlink and uplink channels, inherent feature of the TDD scheme. This way, the channel estimated in the uplink can be used to design the beamformer for the downlink transmission. Accordingly, the UT transmits uplink pilots with power E for the next η T T C Z + symbol periods, with 0 <η T < and 0 <α T + η T. The signal received by the A during the ith symbol period in the uplink pilot transmission phase is given by y p T [i] = E h + w[i], where w[i] CN0, I is the Gaussian noise at the A. The A estimates the channel by a minimum variance unbiased MVU based estimator [4]. Thus, the channel estimate at the A is given by η T T C ĥ = E h + w [i] = h + w, 4 E η T T C i= N where w C, 0 η T T C E I denotes the estimation error. This is followed by the information transmission phase. The focus of this work is on the performance of the SWIT under non-ideal system assumptions and practical transmit schemes. Therefore, for simplicity we will assume that the A beamforms the signal carrying the information symbols with a matched filter precoder MF [5], optimal linear filter for maximizing the SNR. Accordingly, the A exploits the CSI estimate to design the desired beamforming vector, obtained as m T = ĥ/ ĥ. Then, the received signal at the UT is given by yt s = h ĥ + n, 5 ĥ s where s is the information symbol, with E[ s ]=.Asinthe previous case, the UT consumes a power D to decode every received information symbol. Thus, since the harvested power by the UT must be sufficient to send the pilot symbols and decode information at the UT, the condition α T T C H = η T T C E + α T η T T C D must be satisfied for the TDD scheme. Now, if we plug into this condition then, after some manipulations, we have that the minimum fraction of time that should be devoted to the power transfer, i.e., α T, is given by α T = η T E η T D + D β h. 6 + D Accordingly, we can use 6 to compute the downlink rate for the TDD scheme as R T = R T α T,η T = α T η T log + h ĥ N 0 ĥ = η T β h η T E β h + D log + h ĥ ĥ. 7 Note that the channel training time η T T c impacts both the accuracy of the estimated channel vector and the remaining available time for information transfer. As a consequence, let ηt be the duration of the portion of coherence time devoted to the channel training/estimation that maximizes the ergodic downlink rate, defined as [ ηt η T β h η T E =argmax ηt E h, w β h + D ] log + h ĥ. 8 ĥ The derivation of the exact value of ηt is very complicated. However, two approximations of this value, valid for high and low SNR, respectively, can be derived as stated in the following result. emma : At high SNR, ηt can be approximated as ηt where B = E h [+ E it can be approximated as ηt + T C E log e B T C E, 9 β h log h ]. AtlowSNR, + βt C E β + E. 0 roof: See Appendix A. emma provides a result whose interpretation is not trivial. In fact, several parameters are present in 9 and 0, thus

5 IU et al.: SWIT UNDER DIFFERENT CSI ACQUISITION SCHEMES 95 Fig. 3. Operations of the A and the UT during the coherence time in the FDD scheme. understanding their impact on the accuracy of the proposed approximations is rather complex. However, some interesting insights can drawn from emma, if we focus on the approximations that are introduced to derive the final results. First, we note that the impact of D on the accuracy of the results is likely negligible, due the fact that D by construction. Subsequently, let us focus on the quantity λ = η T T C E h, introduced in 39. If we fix at the denominator of λ then it is straightforward to see that the latter increases with an increase in E and. Now, consider the low SNR case. In this case, is very large, thus the approximation λ 0 is adopted. In practice, the accuracy of this approximation depends on the value of and E, i.e., the lower those values are, the more accurate the approximation is. Switching our focus to the high SNR analysis, we observe an opposite behavior. In fact, in this case is very small, hence the approximation λ 0 is introduced. Thus, the accuracy of the approximation is greater when E and are large. Concerning the latter parameter, i.e., number of antennas at the A, we note that an analysis of its impact on the accuracy of the results in emma is extremely interesting, given its relevance in a MISO systems. Accordingly, a detailed discussion on this aspect will be provided in Section V. C. FDD Scheme We now consider the FDD scheme, whose schematic diagram is illustrated in Fig. 3. Differently from the TDD case, the downlink and uplink channels are in general uncorrelated in the FDD scheme. Therefore, two separate channel estimation procedures have to be performed at the UT and the A, to be able to provide to the latter with the CSI w.r.t. the downlink channel. Accordingly, the operations in the FDD scheme are as follows. First, the A transfers power to the UT for a period of ɛ = α F T C, with 0 <α F <. As before, we recall that the A should provide the UT with sufficient energy for the latter to be able not only to decode the received data but also to perform all the operations related the uplink signaling inherent to the FDD scheme. Afterwards, a downlink channel training phase takes place, in which the A sends pilot sequences of η F T C Z + symbols with power to the UT for estimating the downlink channel, with 0 <η F <. Finally, the UT feeds back in the uplink the estimated CSI in analog form over the subsequent τ F T C Z + symbols, where 0 <τ F < and 0 <α F + η F + τ F. Note that, in this work we adopt a simplified model for the uplink communication, for the sake of simplicity of the analysis, and matters of space economy. Specifically, we assume that the feedback signal sent by the UT to the A experiences an AWGN channel. We note that, this follows the typical approach proposed in the literature for first studies on CSI acquisition schemes based on analog feedback signals [], [6]. Now, let us analyze the aforementioned steps in detail. Consider the lth antenna. We denote the pilot sequence sent over it as e l =[e l [],e l [η F T C ]] C η F T C,l [,]. Naturally, the sequences adopted in this phase are known at both ends of the communications. In particular, without loss of generality, we assume orthogonality between pilot sequences sent over different antennas, i.e., e i e j,fori j. Thus, to guarantee their orthogonality, and estimate independent channel coefficients, a lower bound on the minimum sequence size must be satisfied, i.e., η F T C. Moreover, the A equally divides the power among its antennas, yielding e l = and thus e = = e = η F T C. Then, the signal received by the UT during the downlink channel training phase is given by y p UT,F = e h + + e h + w UT, where w UT CN0, I ηf T C is the thermal noise at the UT. The UT in turn multiplies the received signal y p UT,F by e l / e l to estimate the lth channel coefficient, h l. Similar to the previous section, the downlink channel coefficients are estimated by an MVU based estimator. The estimated channel vector at the UT can be written as ĥut =[ĥut,,...,ĥut,] = h + ŵ UT, with ŵ UT CN 0, η F T C I estimation error vector at the UT. The power consumption at UT to decode a pilot sequence sent from one of the transmit antennas is modeled similarly to the previous case, i.e., proportional to D. Accordingly, the total power consumed in decoding the pilot symbols is given by η F T C D. At this stage, the UT encodes each coefficient by means of a sequence f l =[f l [],,f l [τ F T C ]] C τ F T C, l [,], such that the sequences form an orthogonal set, i.e., f i f j, for i j, and f = = f = τ F T C F. As before, the adopted sequences are known at both ends of the communications. In particular, to guarantee their orthogonality and encode independent channel coefficients, a lower bound on the minimum sequence size must be satisfied, i.e., τ F T C. After the encoding, the signal to be fed back by the UT to the A is obtained as the sum of all the obtained sequences at the previous step, i.e., x f F = f ĥut, + + f ĥ UT,. Consequently, its transmission requires a power given by F ĥut, + + ĥut, = F ĥut. Then, the received signal by the A is given by y f A,F = f ĥ UT, + + f ĥ UT, + w A, where w A CN0, I τf T C is the thermal noise at the A. Now, the latter multiplies the received sequence by f k / f to estimate h k. Thus, the estimated channel vector at the A is obtained as ĥa = N h + ŵ UT + ŵ A, where ŵ A C, 0 τ F T C F I. In

6 96 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 particular, we note that ŵ UT and ŵ A are independent by definition. Finally, the A can exploit the knowledge of ĥa to derive the desired MF as before, given by ĥa / ĥa, and use it as beamforming vector while transmitting the information symbols for the remaining α F η F τ F T C symbols. The received information symbol at the UT is given by yut,f s = h ĥ A s + n, ĥa where s is the information symbol, with E[ s ]=. Concerning the energy required to perform all the operations at the UT, as a matter of fact, since the harvested energy must be sufficient to decode the received pilot sequences, feedback the estimated CSI, and decode the subsequent information, we have that the condition α F T C H = η F T C D + τ F T C F ĥut / + α F η F τ F T C D must be satisfied. Therefore, if we plug into this condition then, after some manipulations, we have that the minimum duration of the energy transfer/harvesting phase for this case, i.e., α F, should be α F = τ F F ĥut τ F D + D β h + D. 3 Now, we can use 3 to compute the downlink rate for the FDD scheme as R F = R F α F,η F,τ F = α F η F τ F log + h ĥ A ĥa = η F τ F β h τ F F ĥut η F D β h + D log + h ĥ A. 4 ĥa In this case, two parameters describe the duration of the channel estimation phase, i.e., η F and τ F, related to the channel estimation procedures at the UT and the A, respectively. In practice, these parameters impact both the accuracy of the estimated channel vectors and the remaining available time for information transfer at the A. As a consequence, let ηf,τ F be the optimal couple of parameters that maximizes the ergodic downlink rate, defined as [ ηf,τf = argmax ηf,τ F E h,ŵ log + h ĥ A ĥa η F τ F β h τ F F ĥut η F D β h, + D 5 where ŵ =ŵ A, ŵ UT. As before, the derivation of the exact value of ηf and τ F is very complicated. Nevertheless, two approximations of this value, valid for high and low SNR, respectively, can be derived as stated in the following result. ] emma : At high SNR, ηf and τ F can be approximated as ηf + F F β τ F, 6 τf log e, 7 B 5 T C F + F β [ where B 5 = E h log h ]. At low SNR, they can be approximated as τ F η F + + F T C β + F 4β β ++ F η F T C η F T C, 8 T C β + F + F η F T C + + T C β + F. 9 F roof: See Appendix B. Despite the complexity of 6, 7, 8 and 9, some interesting insights can drawn from the approximations adopted in the derivation in Appendix B following an approach similar to what has been done for emma. As before, the impact of D on the accuracy of the results is likely negligible, due the fact that D by construction. Now, consider T the quantities λ = C h and λ = η F T C h η F +, τ F F introduced in 49 and 5 respectively. We first focus on the low SNR case. Therein, the approximations λ, λ 0 are adopted. In this case, a smaller F improves the accuracy of these approximations, whereas no clear insight can be drawn for. Conversely, in the high SNR case, the approximation λ 0 is adopted. Differently from the previous case, the accuracy of this approximation increases with F. A further approximation is introduced in this part of the study, i.e., η F T C 0 in 50. Accordingly, an additional insight on the impact of the number of antennas on the accuracy of the result in emma can be drawn, i.e., the smaller the larger the accuracy. Interestingly, this is in contrast with the impact of the same parameter in the TDD case and highlights the expected larger penalty for CSI acquisition that FDD pays w.r.t. TDD as the number of antennas grows. A more detailed discussion on its impact on the accuracy of the results in emma, is deferred to Section V, where a comparative study of the downlink rate of the three considered schemes is provided. IV. ANAYSIS OF THE OUTAGE ROBABIITY In this section, we will study the outage probability for the considered system as a function of the parameters introduced so far, and the downlink rate. In the considered practical SWIT implementation two possible outage events can occur: The harvested energy is not sufficient for all the operations at the UT channel estimation, pilot transmission/csi feedback and information decoding, i.e., the UT experiences an energy shortage.

7 IU et al.: SWIT UNDER DIFFERENT CSI ACQUISITION SCHEMES 97 The harvested energy is sufficient to perform all the operations at the UT, but the achieved downlink rate is smaller than a target value, i.e., the UT experiences a data outage. We first focus on the case for which energy shortage occurs. Subsequently, we analyze the case for which the harvested energy is sufficient for all the operations at the UT, and compute the data outage probabilities for the three transmit schemes considered in this work. Before we proceed, we remark that, the analytic expressions derived in this section for the outage probabilities as a function of the system parameters are very complicated, and straightforward inference on their behavior is difficult to be drawn. Consequently, as before we defer the discussion on the outage as a function of the system parameters for all the cases considered in this work to Section V. A. Energy Shortage robability Non-CSI Scheme: Referring to, for any given value for α N, the energy shortage probability for the non-csi case can be expressed mathematically as { E,out αn β h } N α N =r < α N D γ, α N D α N β =, 0 Γ where γq, r = r 0 uq e u du is the lower incomplete Gamma function. The closed-form expression of this probability is derived by considering the cumulative distribution function CDF of χ if we note that h χ. TDD Scheme: Referring to 6, for any given value for α T and η T, the energy shortage probability for the TDD case, denoted by E,out T α T,η T, can be expressed mathematically as E,out T α T,η T { αt β h } =r < α T η T D + η T E γ, η T E + α T η T D α T β =. Γ Using the same approach as for 0, the closed-form expression of the probability in is computed. 3 FDD Scheme: Consider 3. For any given value for α F, η F and τ F, the energy shortage probability for the FDD case can be stated mathematically as { α F β h E,out F α F,η F,τ F =r < τ F F ĥut + α F τ F D }. The following result provides a closed-form expression of. However, for the sake of the simplicity of the representation of the result, let us denote σ = N0 η F T C, ρ = τf F τ α F β τ F F, ρ = F F τ α F β τ F F + F F α F β τ F F, and ρ 3 = αf τ F D α F β τ F F. emma 3: The energy shortage probability for the FDD scheme, as in, can be computed as E,out F α F,η F,τ F Q ρ σ θ 4, ρ σ θ 4 + ρ 3 = dθ θ 4 =0 e θ θ4 Γ roof: The outage probability can be evaluated as follows. First, applying the law of total probability, i.e., given a random variable A, r =E A [r A], wehave [ { = EŵUT r h ρ ŵ UT } ] <ρ ŵ UT + ρ 3 ŵ UT. 4 From 4, it can be easily deduced that h ρ ŵ UT ŵut χ ρ ŵ UT. Therefore, substituting the DF of h ρ ŵ UT ŵut into 4, we can rewrite [ ] 4 = EŵUT Q ρ ŵ UT, ρ ŵ UT + ρ 3. 5 Since ŵ UT CN0, σi, we have ŵ UT = σθ 4, where Θ 4 χ. Substituting the DF of ŵ UT into 5, we derive the RHS of 3, and this concludes the proof. At this stage, if we focus on 0,, and 3, we note that the energy shortage probability in the three considered cases clearly depends on the values of α N, α T,η T, and α F,η F,τ F respectively. However, drawing meaningful insights from these results is extremely difficult, due to their complexity. Accordingly, we will investigate this aspect in Section V, by means of suitable numerical analyses. B. Data Outage robability for the Non-CSI Scheme We now compute the data outage probability for the non-csi scheme. Given α N and a specific target downlink rate R NC,the data outage probability can be stated mathematically as { D,out αn β h N α N,R NC =r α N D, α N log + } h <R NC, that is the probability that the harvested energy is sufficient for the decoding operations at the UT, but the achieved downlink rate is smaller than R NC. Now, let us rewrite D,out N as { αn D r h < N } 0 R NC α N. 6 α N β The intersection between the two events in 6 is non-empty when R NC α N. 7 α N D α N β < If this condition is not satisfied, then 6 in this case is equal to 0. In is worth noting that, assuming R NC 0, the data outage probability would be 0 only in case of extremely low value

8 98 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 of, given that typically D, as previously discussed. This is in line with what could be expected in a wireless communication system, in which the data outage probability tends to 0 as the SNR at the receiver increases. If this is not the case, and 7 is satisfied, then 6 can be computed as D,out N α N,R NC, γ = R NC α N Γ γ, α N D α N β Γ where we made use of the CDF of the χ distribution., 8 C. Data Outage robability for the TDD Scheme We switch our focus back to the TDD scheme. For given values of α T, η T, and a target downlink rate R T, the data outage probability is expressed as { D,out αt β h T α T,η T,R T =r α T η T D } + η T E, α T η T log + h ĥ <R T, ĥ 9 that is the probability that the harvested energy is sufficient to engage in the pilots transmission and decode the received data, but the achieved downlink rate is smaller than R T.The following result provides a closed-form expression for 9 and concludes the study of the TDD case. However, before proceeding, let us denote b 3 = N R T 0 α T η T, b 4 = η T E + α T η T D α T β, b 5 = +η T T C E and b 6 = η T T C E, for the sake of the simplicity of the representation of the result. emma 4: When N R T 0 α T η T < η T E + α T η T D α T β, then the data outage probability for the TDD scheme, as in 9, can be computed as b5 D,out b 3 T = θ 3 =0 θ =0 I 0 θ θ 3 b 6 Conversely, when it can be computed as b5 D,out b 4 T = θ 3 =0 θ =0 Γ,b 5 b 4 θ θ 3 + Γ Γ R T α T η T θ 3 e θ + θ3 b 6 + θ 3 dθ dθ 3 + e θ + θ 3 b 6 + θ 3 dθ dθ η T E + α T η T D α T β, Γ,b 5 b 4 θ θ I0 θ 3 b 6 + Γ Γ θ 3 =0 e θ 3 θ 3 Q θ 3 b 6, b 5 b 4 Q θ 3 b 6, b 5 b 3 Γ dθ 3. 3 D. Data Outage robability for the FDD Scheme We conclude our study on the data outage probability by considering the FDD case. For given α F, η F, τ F, and a specific target downlink rate R F, the data outage probability can be stated mathematically as D,out F { α F,η F,τ F,R F α F β h =r τ F F ĥut + α F τ F D, } α F η F τ F log + h ĥ A <R F, 3 ĥa that is the probability that the harvested energy is sufficient to estimate the downlink channel, feed back its estimated version in the uplink, and decode the received data, but the achieved downlink rate is smaller than R F. The following result provides a closed-form expression of 3 and concludes the study of the FDD case. However, as before, let us introduce some new notation to further simplify representation of the results. Accordingly, we let σ = +η F T C, σ 3 = η F T C, σ 4 = τ F F T C, σ 5 = +σ 3σ 4 +σ 3 +σ 4, b 7 = N R F 0 α F η F τ F, b 8 = α F τ F D α F β, and b 9 = τ F F α F β. emma 5: The data outage probability for the FDD scheme, as in 3, can be computed as D,out F = θ 9 =0 θ 8 σ 5 Q, σ σ σ 3 I 0 θ5 θ 7 σ 5 + e + θ 9 =0 σ σ3 θ 7 σ 5 σ σ 3 θ 7 +θ 8 > b 7 b 8 b 9 σ 5 θ 7 +θ 8 b 7 b 8 b 9 σ 5 σ b 7 θ 5 =0 b 8 + b 9 θ 7 + θ 8 σ 5 θ I 7 θ 9 +σ 3 0 σ 4 + θ 5 +θ 7 +θ 8 +θ 9 + +σ 3 z θ8 θ9 ΓΓ θ 5 dθ 5 dθ 7 dθ 8 dθ 9 33 σ 4 σ b 8 + b 9 θ 7 +θ 8 σ 5 θ 5 =0 θ 8 σ 5 Q σ σ3, σ b 8 + b 9 θ 7 + θ 8 σ 5 θ 5 θ 5 θ 7 σ 5 I 0 σ σ3 I 0 θ 7 θ 9 + σ 3 θ8 θ9 σ 4 e θ 7 σ 5 σ σ 3 + θ 5 +θ 7 +θ 8 +θ 9 + +σ 3 θ 9 σ 4 Γ Γ + dθ 5 dθ 7 dθ 8 dθ 9 34 θ I 7 θ 9 +σ 3 0 σ 4 + θ 9 =0 θ 7 +θ 8 b 7 b 8 +σ3 b 9 σ 5 θ 9 σ + e θ 7 +θ 8 +θ 9 4 θ 7 σ 5 [Q σ σ3, σ b 8 + b 9 θ 7 + θ 8 σ 5 θ 7 σ 5 Q σ σ3, ] σ b 7 θ 8 θ9 Γ Γ dθ 7dθ 8 dθ roof: See Appendix C. roof: See Appendix D.

9 IU et al.: SWIT UNDER DIFFERENT CSI ACQUISITION SCHEMES 99 V. N UMERICA RESUTS In this section we evaluate the performance of SWIT for MISO systems, to assess its merit under the transmit schemes considered in this work. The parameters used in our numerical results are as follows. We consider β =0.5, which is a good approximation of the performance delivered by state-of-the-art commercial products [5], and typically adopted value in the literature on this subject [], [], [7]. We consider = and T C = 000 for simplicity, and {3, 6}. We assume that the system operates in the industrial, scientific and medical ISM band, i.e., carrier frequency of.4 GHz. Accordingly, we set a distance between the A and the UT in the order of meters such that we can ensure that the latter is situated in the far-field region of the radiating A. Furthermore, we assume that the signals transmitted by both the A and the UT experience a generic path loss attenuation, with a path loss exponent equal to 3. As a consequence, we can safely let D = 000. The rationale for this is that by incorporating the propagation losses in D, we frame a more realistic scenario. Finally, we model the ratio between the power budgets available at the A and the UT following the same logic. We let F E = = 00, in accordance with the typical ratio between the available power budgets at both sides of the communication in modern networks, which is roughly 0 db [8]. Fig. 4. ζ T for analytic and numerical parameters, TDD and =3antennas. A. Downlink Rate Now, we focus on the ergodic downlink rate. First, we compute the optimal numerical performance of the system by numerically solving the problems in 8 and 5, by means of an exhaustive search whose complexity and time requirements are not suitable for realistic implementations. Subsequently, we evaluate the accuracy of our theoretical results by comparing them to the numerical performance results. Throughout this section, we will refer to the derived approximated parameters in emma and emma as analytic results, for the sake of clarity. Now, for the TDD scheme, let RT and η T be the optimal downlink rate and the optimal duration of the portion of the coherence time devoted to the channel training/estimation, computed by extensive Monte-Carlo simulations. For the sake of clarity, with a little abuse of notation, we denote ˆη T as the optimal parameter of interest for the TDD scheme, computed according to emma. For the FDD scheme, a similar notation is defined. Now, we define ζ T = R T α T,ˆη T R [0, ], T for TDD, and ζ F = R F α F,ˆη F,ˆτ F R [0, ], for FDD, as the F ratio between the downlink rate obtained with the analytic and optimal numerical results. 3 We let SNR [0, 30] db and compute ζ T and ζ F for both =3 and =6 in Fig Quantitatively, if we focus on the best performer for each of the considered SNR values, the gap between ζ T ζ F in the FDD case and is remarkably small. Thus, the accuracy of our derivations is confirmed. If we focus on the impact of on the two analytic results, we note that they confirm the intuitions provided in Section III-B and Section III-C. However, Fig. 5. ζ T for analytic and numerical parameters, TDD and =6antennas. 3 Note that, α T and α F are computed according to 6 and 3 respectively. Fig. 6. ζ F for analytic and numerical parameters, FDD and =3antennas.

10 90 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 Fig. 7. ζ F for analytic and numerical parameters, FDD and =6antennas. the difference in terms of the best ζ T and ζ F between the two antenna configurations is rather small. This shows that the impact of the number of antennas at the A on the accuracy of the analytic results is not very significant. Furthermore, we see that ζ F ζ T, SNR [0, 30] db and {3, 6}. This is due to the two-step channel estimation process that is needed in the FDD scheme for the CSI acquisition at the A. As a consequence, a greater number of approximations is necessary. This reduces the accuracy of our closed-form representation of ηf and τ F.4 Focusing on the practical implementation, we note that the presence of the analytic results provides a twofold alternative for the A, depending on the system intrinsic constraints. When the time available for the optimization of the transmit parameters is small, the analytic results could be used to achieve a performance which is reasonably close to the optimal, without resorting to an exhaustive search. Conversely, if more time is available for the A, the analytic results can be used to improve the efficiency of the search for the optimal parameters. In this regard, we note that the downlink rate is a concave function of η, τ. Accordingly, in this case, the local optimum coincides with the global optimum. Now, assume that the results of emma and emma were adopted as a starting point for finding the numerically optimal parameters, by means of an exhaustive search inside a smaller set. Then, the necessary time to identify the global optimum could be significantly reduced w.r.t. a blind exhaustive search, due to the proximity of the analytic results and the actual global optimum. To conclude our analysis on the downlink rate, we investigate the advantages, if any, that the two duplexing schemes discussed so far can bring w.r.t. the non-csi case in terms of the downlink rate. We remark that our goal is to characterize the performance of the system under the realistic assumptions made in Section I. Thus, in the following study, all the system parameters discussed so far are set according to the analytic results derived in Section III. Moreover, for simplicity in the rep- 4 The interested reader may refer to Appendix B for further details. Fig. 8. Ratio between the ergodic downlink rate for the CSI acquisition schemes and the non-csi case. resentation, we let R = R T α T, ˆη T and R = R F α F, ˆη F, ˆτ F be the downlink rate for TDD and FDD, respectively, when the analytic results are adopted. The ratio between these rates R and their counterpart for the non-csi case i.e., R NC, with R NC as in 3 is represented in Fig. 8, for SNR [0, 30] db. Remarkably, both duplexing schemes clearly outperform the non-csi approach in terms of downlink rate. This shows that, despite the penalties incurred to acquire the CSI, evident downlink rate enhancements are experienced by the A, thanks to presence of the CSI, however imperfect the latter might be. The result in Fig. 8 is even more remarkable, considering that therein the two duplexing schemes always outperform the non- CSI approach, regardless of the antenna configuration and the SNR value. Furthermore, the largest advantage over the non- CSI performance is obtained in the low-to-mid SNR regime. In this regard, we first focus on the TDD case. In both cases, i.e., R R NC =3and =6, is a monotonically decreasing function of the SNR, confirming that the MF performs better for low than for high SNR values [5], [9]. In particular, this shows that the availability of the CSI at the A, albeit imperfect, is sufficient to achieve a much larger downlink rate as compared to the non-csi approach. Furthermore, the performance for =6 is strictly larger than for =3, showing that, as in the case of traditional wireless communications, the SWIT can effectively exploit the transmit diversity gain delivered by a MISO system as grows. Interestingly, the same is true for the FDD scheme. The CSI acquisition procedure in this case is more complex and prone to a higher uncertainty, R R NC especially at low SNR. This impacts the behavior of that presents a maximum at SNR =0dB, for both the considered antenna configurations. On one hand, the gain brought by the FDD scheme over the non-csi approach is dominated by the power gain at the UT, brought by a more accurate beamformer design at the A, for SNR 0 db. On the other hand, the reduction of the multiplexing gain due to the increasing impact that both the channel estimation and feedback phases have on available time for information transfer, as the quality of the CSI increases, determines the decreasing behavior of R R NC for

11 IU et al.: SWIT UNDER DIFFERENT CSI ACQUISITION SCHEMES 9 Fig. 9. Energy shortage probability, non-csi and =3antennas. SNR > 0 db. Finally, we note that the difference at high SNR R between the values of R NC for TDD and FDD is very low, but increases with the. In fact, when the SNR is high, the channel estimation/feedback phases are very short, thus the difference in the amount of time available for the information transfer in both cases is small. Nevertheless, a bigger entails a larger τ F thus α F and, in turn, increases the difference between the values of for TDD and FDD at high SNR as well. R R NC Fig. 0. Energy shortage probability, TDD and =3antennas. B. Outage robability We switch our focus to the analysis of the energy shortage and the data outage probability. A set of Monte-Carlo simulations is performed to obtain the numerically computed probabilities. Subsequently, we set the values of η T, η F, and τ F according to emma and emma and compute the exact value of both metrics by means of the analytic results in Section IV. At this stage, we only consider the case = 3 owing to space economy. In the previous subsection, we verified that the impact of change in the number of antennas on the accuracy of the analytic results on the downlink rate is rather small. Accordingly, a robustness of the accuracy of our results to a change in the number of antennas could be conjectured. For the sake of clarity we let p E,out and p D,out be the energy shortage probability and the data outage probability when no energy shortage occurs, respectively. Furthermore, we let R NC = R T = R F =6 bit/s/hz 5 be the target rate for the considered system. Finally, we depict p E,out for SNR [0, 30] db in Fig. 9, Fig. 0, and Fig. and p D,out for SNR [0, 5] db in Fig.. As shown in these figures, the numerical results perfectly match the analytic results derived in Section IV for all the three schemes. This perfect match verifies the correctness of our derivations. We start by noting that the energy shortage probability strongly depends on the considered parameters. Thus, a comparison between schemes could have limited interested w.r.t. a Fig.. Energy shortage probability, FDD and =3antennas. 5 Referring to Section IV-B, IV-C, and IV-D, we note that R NC, R T,and R F are specified values. Fig.. Data outage probability when no energy shortage occurs, =3 antennas.

12 9 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 comparison between the results obtained for each scheme, as the duration of the energy transfer phase varies. Accordingly, we restrain our focus to the latter aspect. As expected, the energy shortage probability is independent of the SNR, regardless of the value of α N. However, for both the TDD and the FDD scheme, the energy shortage probability decreases with the SNR, regardless of the value of α T and α F. In these cases, a larger SNR reduces the optimal time for both devices to perform the operations intrinsic to the CSI acquisition and achieve accurate channel estimations. In other words, the channel estimation accuracy increases with the SNR value, thus the CSI acquisition requires less time. Therefore, the energy consumption at the UT is lower when the SNR is large. Now, if the duration of the energy transfer phase is doubled or tenfold, a reduction of the energy shortage probability from almost one to three orders of magnitude is observed, depending on the considered scheme. In practice, if the coherence time is long enough, even a rather small increase of the duration of the energy transfer phase can positively impact the energy shortage probability. We now switch our focus to the data outage probability illustrated in Fig.. We start by noting that, to compute the numerical data outage probability in this case, the duration of the energy transfer phase for the three considered schemes, i.e., α N, α T, and α F, is chosen at each iteration of the simulations such that the harvested energy at the UT is sufficient to perform the receiver operations intrinsic to each scheme. As a matter of fact, the obtained quantitative results for a study of this kind are not extremely relevant, in fact they clearly depend on the selected target rate. In practice, their qualitative behavior is definitely more interesting. In this regard, the lowest data outage probability is experienced by the considered system in the case of the TDD scheme. This could have been expected after our findings on the downlink rate in the previous section, in which the TDD scheme resulted as the best performer out of the three considered cases. VI. CONCUSION In this work, we have examined the efficacy of SWIT in a MISO system consisting of an A and a single UT. In particular, the latter is not equipped with any local power source, but instead harvests the necessary energy for its operations from the received RF signals. The performance of the considered system has been analyzed under realistic and practically relevant system assumptions. Three practical cases have been considered: a absence of CSI at the A, b imperfect CSI at the A acquired by means of pilots estimation TDD, c imperfect CSI at both the UT and the A acquired by means of analog CSI feedback in the uplink FDD. We have compared the considered scenarios by means of three performance metrics of interest, i.e, the ergodic downlink rate, the energy shortage probability, and the data outage probability. Accordingly, we have derived closed-form expressions for each metric, and for the ergodically optimal duration of both the WT and the channel training/feedback phases, to maximize the downlink rate in all the three scenarios. The accuracy of our derivations has been verified by an extensive numerical analysis. First, it is worth noting that TDD has consistently been the best performer for each considered metric, confirming the potential of this duplexing scheme for the future advancements in modern networks. More specifically, concerning the downlink rate, our findings show that CSI knowledge at the A is always beneficial for the information transfer in SWIT systems, despite the resources devoted to the channel estimation/feedback procedures and the presence of estimation errors. In a follow-up of this work, we will study both strategies to maximize the efficiency of the WT, in the case of the availability of CSI knowledge at the A prior to the WT phase or part of it, and their impact on the energy shortage probability. Additional subject of future investigation will be the extension of the considered set-up to a multi-user scenario. AENDIX A ROOF OF EMMA To evaluate 8, we use the law of iterated expectations, i.e., E h, w [ ] =E h [E w [ h]]. Furthermore, we neglect D in 8 since, in practice, D generally [8]. We proceed by first computing the following expression: η T η ] T E β h E w [log + h ĥ, 36 ĥ for a given channel realization h. To compute 36, the following straightforward results can be derived: h ĥ = h Ψ and η T T C ĥ = Ψ, 37 E η T T C E ηt T C E h ηt T C E h. where Ψ χ and Ψ χ We break up the subsequent analysis into two cases, namely, the high SNR and low SNR cases. First, we consider the analysis at high SNR. In this case, applying the approximation, log + SNR log SNR when SNR 0, and 37 to 36, we can derive [ ] E w log + [ h ĥ h ] Ψ E w log. ĥ Ψ 38 Subsequently, using the Taylor series expansion of log Ψ and log Ψ at their respectively mean values i.e. +λand + λ respectively, where λ = η T T C E h, we have h 38 = log +log N e [ 0 E Ψ,Ψ ln+λ+ Ψ λ Ψ λ +λ + λ + ] ln+λ Ψ λ + Ψ λ +λ + λ 39 h + λ =log +log N e ln + λ 0 + λ + +λ ln + λ+ + λ 40 a h λ log + λ =log h log N +, 4 0 λ

13 IU et al.: SWIT UNDER DIFFERENT CSI ACQUISITION SCHEMES 93 where a in 4 is derived by noting that λ is large at high SNR, and hence we can neglect the higher order fractional terms in 40. Moreover, in the ln + λ term is neglected owing to λ. Further, we use 4 in 36 and take the expectation over h. Moreover, since /λ is small at high SNR, we use the approximation, log + x x log e when x 0, in the derivation. By some straightforward computations, we can rewrite 8 as η T argmax ηt B η T B log e η T T C E, 4 where B = E h [+ E log h ], and B is a constant value. To derive ηt, we differentiate 4 with respect η T and set it equal to 0. By some straightforward computations, we can derive the result 9. We now move to the analysis at low SNR. Before deriving, we first note that E[X] =E[expln X]. Subsequently, using the approximation, log + SNR SNRlog e when SNR 0, and Jensen s inequality, i.e., E[expln X] expe[ln X], wehave 36 η T η T E log e β h N [ 0 ] h exp E ĥ w ln. 43 ĥ β h Once again, ] we apply Taylor series expansion to E w [ln h ĥ, i.e., steps In this case, since λ is small ĥ at low SNR, we approximate the higher order terms, such as +λ +λ in 40, by a constant value κ. Using this, we can rewrite 43 as κ η T η T E β h h log e +λ + λ. 44 Finally, we take the expectation over h. By using Jensen s inequality [similar approaches in 43] and applying Taylor series expansion for the logarithm term at the mean value h [similar approaches in 39 4], we rewrite 8 as κ + η T T C E ηt η T η T E β arg max ηt + η T T C E where κ is a constant. To derive ηt, we differentiate the above formula with respect η T and set it equal to 0. By some straightforward computations, we can derive the result 0. AENDIX B ROOF OF EMMA To evaluate 5, we use the similar approaches as in 36. Hence, we first compute [ Eŵ τ F τ F F ĥut β h η F ] log + h ĥ A. 45 ĥa To compute 45, the following results can be derived details omitted for lack of space: h ĥ A = h T C η F + Φ, 46 τ F F ĥa = N 0 T C η F + Φ, 47 τ F F ĥut = η F T C Φ 3, 48 where Φ χ χ T C h T C h, Φ τ F F ηf T C h η F + Before proceeding, we note that the pre-log term and the term τ F F η F +, and Φ 3 χ inside the logarithm in 45 are correlated, due to the presence of ŵ UT in both terms. However, at high SNR, the variance of ŵ UT will be small. Thus, we assume that the pre-log term and term inside the logarithm are approximately independent [and hence we can take the expectations of these two terms in 45 separately]. For the term inside the logarithm, using and by Taylor series expansion [similar approaches in 39 4], we obtain Eŵ [log where λ = [ Eŵ + h ĥ A ĥa T C h η F + ] τ F τ F F ĥut β h η F. log h λ + λ, 49. For the pre-log term, we have τ F F ] τ F τ F F β η F. 50 The approximation in 50 is derived using Eŵ[ ĥut ]= h + η F T C h N since at high SNR, 0 η F T C is small enough to be neglected. Finally, we substitute 49 and 50 into 45 and take the expectation over h. Using the approximation log + x x log e when x 0 and by some straightforward computations, we can rewrite 5 as ηf,τf argmax ηf,τ F τ F τ F F β η F B 5 log e T C η F +, 5 τ F F [ where B 5 = E h log h ]. To find ηf and τ F,wedifferentiate 5 with respect η F and τ F and equate them to 0. By some straightforward computations, we can derive the results 6 and 7. We now turn to the analysis at low SNR. Herein, we follow the similar derivations as in Appendix A. First we use the same approaches as in 43. Further, we apply and use the Taylor series expansion [similar to the approaches in 44]. Thus, we can derive 45 η F τ F β h τ F F η F T C + λ κ 3 log e + λ β + λ, 5

14 94 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 where λ = η F T C h, and κ 3 is a constant value. Finally, as before we take the expectation over h. By using Jensen s inequality [similar approaches in 43] and applying Taylor series expansion of the logarithm term at the mean value h [similar approaches in 39 4], we can rewrite the expected value of 5 over h as + κ 4 η F τ F τ F F β + T C η F + τ F F τ F F η F T C β T C η F + τ F F 53 where κ 4 is a constant value. Now, we differentiate 53 with respect η F and τ F and equate them to 0. Using some straightforward computations, we obtain the result 8. When x 0, +x + x. By utilizing this approximation in 8 since is large at low SNR, we can derive τf η F T C β F. Substituting the latter approximated τf into the differentiated equation 5 η F =0, we can also derive the result 9. AENDIX C ROOF OF EMMA 4 We now proceed with the proof. In equation 9, we have two random variables, i.e., h ĥ and h. To evaluate 9, ĥ we first express h as the sum of h ĥ and another independent random variable see the steps ĥ below. This step simplifies the evaluation as shown in the following. To do so, first we project{ the row vector } h onto an orthonormal set of vectors Ω= ĥ, ĥ g,, g, where g,, g are chosen arbitrarily such that the vectors in Ω span the complex dimensional space. Recall that h ĥ CN +b 6 ĥ, b 5 I. Since the distribution of a Complex Gaussian random vector with distribution CN0, I projected onto an orthonormal h set remains unchanged [9], we can conclude that ĥ ĥ ĥ CN ĥ +b 6,b 5 and h g l ĥ CN0,b 5, l =,...,. Additionally, the h g l ĥ l =,..., are independent random variables. Thus we can conclude the following: h ĥ ĥ ĥ = b 5 Θ, 54 h g + + h g ĥ = b 5 Θ, 55 b 5 b 6 where Θ χ, ĥ and Θ χ. Furthermore, Θ and Θ are independent and Θ +Θ =b 5 h.now, applying the same approach as in 4 to 9 and using 54 and 55, we can derive the following: D,out T = Eĥ[r{Θ < b 5 b 3, Θ +Θ b 5 b 4 }]. et us focus on the relationship between b 3 and b 4 and consider two possible cases, i.e., b 3 <b 4 and b 3 b 4. We start from the former. In this case, denoting the DF of Θ i as f Θi θ i for i {, }, wehave, D,out T = Eĥ b5 b 3 θ =0 Γ,b 5 b 4 θ I0 ĥ θ b 5 b 6 dθ. Γ e θ + ĥ b 5 b 6 56 Since ĥ CN0, + b 6 I, it follows that ĥ = +b 6 Θ 3, where Θ 3 χ. Substituting the DF of ĥ into 56, we obtain our result 30. In the second case b 3 b 4, following the same approaches as in 56, we obtain our result 3 and conclude the proof. AENDIX D ROOF OF EMMA 5 Adopting a similar approach to what has been adopted in 4 and denoting the conditional distribution of ĥut given ĥ A as fĥut ĥa, the analytic expression for the outage probability, i.e., D,out F in 3, can be written as EĥA h ĥ A r <b 7, h b 8 ĥa + b 9 ĥut ĥa, ĥut ] } f ĥut ĥa dĥut. 57 To compute 57, we use the projection approach that we have used in Appendix C. First, we { project the row vector } h onto ĥ A an orthonormal set of vector, ĥa g,, g, such that these vectors span the dimensional complex space. Since CN h ĥa,ĥut +σ 3 ĥ UT, σ I, following the same approach as in Appendix C, we obtain h ĥ A = ĥa Θ 5, 58 σ ĥa,ĥut h g + h g = Θ 6, 59 ĥa,ĥut σ where Θ 5 χ ĥ σ σ3 UTĥA and Θ ĥa 6 χ ĥut. Moreover, Θ 5 σ σ 3 ĥ UTĥA ĥa and Θ 6 are independent and Θ 5 +Θ 6 =σ h. Using 58 and 59, we rewrite 57 as [ EĥA r {Θ 5 +Θ 6 σ b 8 + b 9 ĥut, ] Θ 5 < σ b 7 }f ĥut ĥa dĥut. 60

15 IU et al.: SWIT UNDER DIFFERENT CSI ACQUISITION SCHEMES 95 To compute the double integration over θ 5 and θ 6 in 60, we have two two cases, i.e., b 7 <b 8 + b 9 ĥut and b 7 b 8 + b 9 ĥut. In the first case, the probability term in 60 can be computed as σ b 7 θ 5 =0 Q ĥut σ σ3 σ b 8 + b 9 ĥut θ 5 e θ 5 + σ σ 3 ĥ UT ĥa ĥa I 0 ĥ UTĥA ĥa, θ 5 ĥ UTĥA σ σ3 ĥa dθ 5. 6 Since ĥut ĥa CN σ5 σ 4 ĥ A,σ 5 I, using the projection approach once again, we can derive ĥ UTĥA = h A 5 Θ 7, ĥa 6 ĥut ĥ UTĥA h A = σ 5 Θ 8, ĥa 63 where Θ 7 χ σ5 σ4 ĥa and Θ 8 χ. Furthermore, Θ 7 and Θ 8 are independent and Θ 7 +Θ 8 = σ 5 ĥut. astly, we note that since b 7 <b 8 + b 9 ĥut,wehaveb 7 < b 8 + b 9 Θ 7+Θ 8 σ and hence Θ 7 +Θ 8 > b 7 b 8 b 9 σ 5. Now, applying 6 63 to 60, the integral of 60 can be computed as σ b 7 I 0 σ 5 θ 7 ĥa θ 8 + θ 7 +θ 8 > b 7 b 8 b 9 σ 5 θ 5 =0 θ 8 σ 5 Q σ σ3, σ b 8 + b 9 θ 7 + θ 8 σ 5 θ 5 Γ e I θ5 θ 7 σ 5 0 σ σ3 θ 5 +θ 7 +θ 8 + θ 7 σ 5 σ σ 3 σ 4 + σ 5 ĥa σ 4 dθ 5 dθ 7 dθ We note that, since ĥa CN0, + σ 3 + σ 4 I,wehave that ĥa = +σ 3+σ 4 Θ 9, where Θ 9 is χ. Using this fact and 64 in 60, we obtain 33 in emma 5. We now consider the second aforementioned case, i.e., b 7 b 8 + b 9 ĥut. Following similar steps as in the first case, we obtain 34 and 35 in emma 5 the detailed steps have been omitted for matters of space economy. At this stage, the outage probability in 3 is obtained as the sum of 33 35, and this concludes the proof. REFERENCES []. Xie, Y. Shi, Y. Hou, and A. ou, Wireless power transfer and applications to sensor networks, IEEE Trans. Wireless Commun., vol. 0, no. 4, pp , Aug. 03. [] A. 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16 96 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 4, NO. 4, ARI 05 [6] D. Samardzija and N. Mandayam, Unquantized and uncoded channel state information feedback in multiple-antenna multiuser systems, IEEE Trans. Commun., vol. 54, no. 7, pp , Jul [7]. iu, R. Zhang, and K.-C. Chua, Wireless information and power transfer: A dynamic power splitting approach, IEEE Trans. Commun., vol. 6, no. 9, pp , Sep. 03. [8] 3G, TR 36.84, further advancements for E-UTRA physical layer aspects, v.9.0.0, Cedex, France, Tech. Rep., Mar. 00. [9] D. Tse and. Viswanath, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge Univ. ress, Jun. 005, ch. A..3, pp Chen-Feng iu received the B.S. degree from National Tsing Hua University, Hsinchu, Taiwan, in 009 and the M.S. degree in communications engineering from National Chiao Tung University, Hsinchu, Taiwan, in 0. In 0, he joined Academia Sinica as a Research Assistant. In July 03 and from February 04 to June 04, he was also a Visiting Researcher at the Singapore University of Technology and Design. His research interests focus on wireless communications and simultaneous wireless information and power transfer. Marco Maso S 08 M 4 received the bachelor s degree in 005 and the M.Sc. degree in telecommunications engineering in 008, both from University of adova, Italy. He received both the h.d. degree in information engineering I.C.T. from University of adova and the h.d. degree in telecommunications ICST from Supélec, France, in 03. He has been engaged in research on practical implementations of OFDM packet synchronization in 005/06, DVB-T systems implementation in 008/09, signal processing algorithms for high speed coherent optical communications in 0/3 and dynamic spectrum management strategies for mobile ad-hoc networks in 03/04. He was a Research Engineer at Supélec in 0/3 and a ost-doctoral Research Fellow at the Singapore University of Technology and Design in 03/4. Since September 04, he is a Researcher with the Mathematical and Algorithmic Sciences ab of Huawei France Research Center. His research interests broadly span the areas of wireless communications and signal processing for the physical layer, with focus on heterogeneous networks, self organizing networks, MIMO systems, interference management and suppression techniques, wireless power transfer and cognitive radios. He has been actively involved in organizing and chairing sessions, and has served as a member of the Technical rogram Committee in a number of international. Chia-Han ee received the B.S. degree from National Taiwan University, Taipei, Taiwan, in 999, the M.S. degree from the University of Michigan, Ann Arbor, MI, USA, in 003, and the h.d. degree from rinceton University in 008, all in electrical engineering. From 999 to 00, he served in the R.O.C. army as a Missile Operations Officer. From 008 to 009, he was a ostdoctoral Research Associate at the University of Notre Dame, USA. In 00, he joined Academia Sinica as an Assistant Research Fellow and since 04, is an Associate Research Fellow. His research interests include wireless communications and networks. He serves as an Editor for the IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS and IEEE COMMUNICATIONS ETTERS. Tony Q. S. Quek S 98 M 08 SM received the B.E. and M.E. degrees in electrical and electronics engineering from Tokyo Institute of Technology, Tokyo, Japan, respectively, and the h.d. degree in electrical engineering and computer science from the Massachusetts Institute of Technology. Currently, he is an Assistant rofessor with the Information Systems Technology and Design illar, Singapore University of Technology and Design SUTD. He is also a Scientist with the Institute for Infocomm Research. His main research interests are the application of mathematical, optimization, and statistical theories to communication, networking, signal processing, and resource allocation problems. Specific current research topics include sensor networks, heterogeneous networks, green communications, smart grid, wireless security, compressed sensing, big data processing, and cognitive radio. Dr. Quek has been actively involved in organizing and chairing sessions, and has served as a member of the Technical rogram Committee as well as Symposium Chair for a number of international conferences. He is serving as the Co-chair for the HY & Fundamentals Track for IEEE WCNC in 05, the Communication Theory Symposium for IEEE ICC in 05, and the HY & Fundamentals Track for IEEE EuCNC in 05. He is currently an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE WIREESS COMMUNICATIONS ETTERS, and an Executive Editorial Committee Member for the IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS. Hewas Guest Editor for the IEEE Signal rocessing Magazine Special Issue on Signal rocessing for the 5G Revolution in 04, and the IEEE Wireless Communications Magazine Special Issue on Heterogeneous Cloud Radio Access Networks in 05. Dr. Quek was honored with the 008 hilip Yeo rize for Outstanding Achievement in Research, the IEEE Globecom 00 Best aper Award, the CAS Fellowship for Young International Scientists in 0, the 0 IEEE William R. Bennett rize, the IEEE SAWC 03 Best Student aper Award, and the IEEE WCS 04 Best aper Award. Subhash akshminarayana S 07 M received the M.S. degree in electrical and computer engineering from The Ohio State University in 009, and the h.d. degree from the Alcatel ucent Chair on Flexible Radio and the Department of Telecommunications at Supélec, France in 0. Currently, he is with the Singapore University of Technology and Design SUTD. Simultaneously, he has been holding a visiting research appointment at rinceton University since August 03. His research interests broadly spans wireless communication and signal processing with emphasis on small cell networks SCNs, cross-layer design wireless networks, MIMO systems, stochastic network optimization, energy harvesting and smart grid systems. He has received nomination for the Best of Globecom award at Globecom 04. He has served as a member of Technical rogram Committee for WiOpt 05, IEEE WCNC 04, IEEE IMRC 04, IEEE VTC 04, and IEEE WCNC 04.