CSI-free vs CSI-based multi-antenna WET schemes for massive low-power Internet of Things

05/29/2020 ∙ by Onel L. A. López, et al. ∙ 0

Wireless Energy Transfer (WET) is a promising solution for powering massive Internet of Things deployments. An important question is whether the costly Channel State Information (CSI) acquisition procedure is necessary for optimum performance. In this paper, we shed some light into this matter by evaluating CSI-based and CSI-free multi-antenna WET schemes in a setup with WET in the downlink, and periodic or Poisson-traffic Wireless Information Transfer (WIT) in the uplink. When CSI is available, we show that a maximum ratio transmission beamformer is close to optimum whenever the farthest node experiences at least 3 dB of power attenuation more than the remaining devices. On the other hand, although the adopted CSI-free mechanism is not capable of providing average harvesting gains, it does provide greater WET/WIT diversity with lower energy requirements when compared with the CSI-based scheme. Our numerical results evidence that the CSI-free scheme constitutes the optimum for most of the configurations; although its performance degrades significantly if the setup is not optimally configured in case of Poisson traffic. Finally, we show the prominent performance results when the uplink transmissions are periodic, while highlighting the need of a minimum mean square error equalizer rather than zero-forcing for information decoding.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 12

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

I Introduction

The Internet of Things (IoT) is a major technology trend that promises to interconnect everything towards building a data-driven society enabled by near-instant unlimited wireless connectivity [1, 2]. A key feature/challenge of the IoT is the massive connectivity since around 80 billion connected devices are foreseen to proliferate globally by 2025, thus resulting in a massive technology-led disruption across all industries [3].

The IoT ranges from cloud (e.g., data centers, super computers, internet core network) and fog (e.g., computers, smartphones, smart appliances) technologies, to edge (e.g., wearables, smart sensors, motes) and extreme edge (e.g., smart dust and zero-power sensors) technologies [4]. Energy efficiency and/or power consumption criteria become more critical as one descends over such layers. In fact, edge or extreme edge devices are usually powered by batteries or energy harvesters and are very limited in computing and storage capabilities to reduce costs and enlarge lifetime. Many types of energy harvesting (EH) technologies are under consideration, e.g., based on solar, piezoelectric energy sources; but those relying on wireless radio frequency (RF) signals are becoming more and more attractive. RF-EH provides key benefits such as [5, 6]: i) battery charging without physical connections, which significantly simplify the servicing and maintenance of battery-powered devices; ii) readily available service in the form of transmitted energy (TV/radio broadcasters, mobile base stations and handheld radios), iii) low cost and form factor reduction of the end devices; iv) increase of durability and reliability of end devices thanks to their contact-free design; and v) enhanced energy efficiency and network-wide reduction of emissions footprint.

RF-EH is a wide concept111Herein we focus on RF-EH networks where the RF signals are intentionally transmitted for powering the EH devices. Alternatively, the devices may opportunistically harvest energy from RF signals of different frequencies already in their surrounding environment and to which they are sensitive. The latter is known as ambient RF EH, and readers can refer to [7] for an overview. that encompasses two main scenarios when combined with Wireless Information Transfer (WIT), namely Wireless Powered Communication Network (WPCN) and Simultaneous Wireless Information and Power Transfer (SWIPT) [6]. In the first scenario, a Wireless Energy Transfer (WET) process occurs in the downlink in a first phase and WIT takes place in the second phase. Meanwhile, in the second scenario, WET and WIT occur simultaneously. An overview of the recent advances on both architectures can be found in [8], while herein the discussions will focus on WPCN and pure WET setups. Notice that WET may have a much more significant role than WIT in practical applications as highlighted in [6]. This is because WET’s duration is often required to be the largest i) in order to harvest usable amounts of energy, and/or ii) due to sporadic WIT rounds, e.g., event-driven traffic. Since SWIPT may happen just occasionally, WPCN use cases are often of much more practical interest.Therefore, enabling efficient WPCNs is mandatory [2, 6, 9], and constitutes the scope of this work.

I-a Related Work

Over the past few years, the analysis and optimization of WPCNs has evolved from the simple Harvest-then-Transmit (HTT) protocol [10, 11, 12] towards more evolved alternatives that are capable of boosting the system performance either via cooperation [13], power control [14], rate allocation schemes [15] and/or retransmissions [16]. However, most of the works so far are concerned with rather optimistic setups where either i) most of the power consumption sources at the EH devices are ignored, ii) Channel State Information (CSI) procedures are assumed cost free, and iii) only one or few EH devices are powered. Regarding the latter, the number of EH devices is often not greater than the number of powering antennas such that full gain from energy beamforming (EB) is attained in the WET phase, e.g., [12, 17, 18]. For instance, a setup where a multi-antenna hybrid access point (HAP) transfers power to the devices via EB, followed by the devices sending their data simultaneously by consuming the harvested energy, is investigated in [17]. The authors cast a max-min rate optimization problem with practical non-linear EH and solve it via several iterative optimization methods. However, no other power consumption sources besides transmissions are considered, and Zero Forcing (ZF) equalization is used for information decoding at the HAP without analysing the CSI acquisition costs. Meanwhile, the authors in [18] do consider the CSI acquisition costs when optimizing the HAP pilots power and the power allocated to the energy transmission, while the EH devices are under the effect of several power consumption sources. Yet, the imposition of having more antennas than devices may be strong towards future low-power massive IoT networks. Finally, the lack of a traffic source model for data transmissions is also a strong limitation for most of the works, which intrinsically assume full-buffer EH devices, e.g., [10, 11, 12, 13, 14, 15, 16, 17, 18].

One important observation is that the gains from EB decrease quickly as the number of EH IoT devices increases [6]. This holds even without accounting for the considerable energy resources demanded by CSI acquisition. Therefore, in massive deployment scenarios, the broadcast nature of wireless transmissions should be intelligently exploited for powering simultaneously a massive number of IoT devices with minimum or no CSI. To that end, the authors in [19] propose a new form of signal design for WET relying on phase sweeping transmit diversity, which forces the multiple antennas to induce fast fluctuations of the wireless channel and does not rely on any form of CSI. This is accomplished by exploiting the non-linearity of the EH circuitry. Meanwhile, several multi-antenna CSI-free WET solutions have been recently proposed and analyzed in [20, 21] to improve the statistics of the RF energy availability at the input of the EH circuitry of a massive set of energy harvesters:

  • One Antenna (), under which the power beacon (PB) uses only one antenna transmitting with full power;

  • All Antennas transmitting the Same Signal (), under which the PB transmits the same signal simultaneously with all antennas but with equal, hence proportionally reduced, power at each;

  • All Antennas transmitting Independent Signals (), under which the PB transmits power signals independently generated across the antennas; and

  • Switching Antenna (), under which the PB transmits with full power by one antenna at a time such that all antennas are used during a coherence block.

Notice that i) is the simplest scheme since it does not take advantage of the multiple spatial resources, while ii) may reach considerable gains in terms of average harvested energy under Line of Sight (LOS) but it is highly sensitive to the different mean phases of the LOS channel component, and iii) , do not improve the average energy availability but do provide transmit diversity. It was demonstrated in [21] that devices closer to the PB benefit more from , while those that are far, and more likely to operate near their sensitivity level, benefit more from the . All these CSI-free WET schemes have been considered without the information communication component typical of a WPCN, and consequently, their influence on the overall system performance is so far unclear.

I-B Contributions and Organization of the Paper

This paper aims at analyzing for the first time the gains from operating with/without CSI for powering massive low-power IoT deployments with uplink transmission requirements. Specifically, we consider a WPCN where a massive set of IoT nodes require occasional uplink information transmissions to a HAP, which in turn is constantly transferring RF energy to them in the downlink. Herein, we adopt the strategy [20, 21] as the CSI-free WET scheme, which, besides the benefits aforementioned, allows a better coupling to the co-located information transmission processes. The latter is because only one antenna is used for WET at any time, while the remaining antennas stay silent, thus such idle antennas may be used for uplink information decoding in WPCN setups. The main contributions of this work are listed as follows:

  • We investigate and analyze a WPCN setup under CSI-based and CSI-free powering schemes. We are concerned with the overall outage probability, which encompasses both WET and WIT processes’ failures. The performance is evaluated in terms of the worst node’s performance such that we can assure Quality of Service (QoS) guarantees for all nodes in the network. We consider the power consumption from several sources, e.g., transmission, circuitry, and CSI-acquisition procedures;

  • We decouple WET and WIT processes and cast a max-min WET optimization problem when CSI is available at the HAP. We provide analytical bounds on the performance of the CSI-based WET beamforming by relying on Maximum Ratio Transmission (MRT). We show that the MRT is near the fairest EB, e.g., the EB that provides max-min performance guarantees, even in a massive deployment, if the farthest EH node experiences at least 3 dB of power attenuation more than the remaining devices;

  • We consider two types of information traffic sources: i) periodic traffic, such that the network is perfectly synchronized; and ii) Poisson traffic, which is uncoordinated and random. The overall performance is analyzed for both traffic profiles. Our results not only evidence that the system performance deteriorates under Poisson random access when compared to deterministic traffic, but also that it is more challenging to optimally configure the network. We cast an optimization problem to determine the optimum pilot reuse factor such that the collision probability keeps below a certain limit. A solution algorithm is provided and shown to converge in few iterations;

  • For information decoding in the uplink, the HAP implements either ZF or the Minimum Mean Square Error (MMSE) equalization. We show that the MMSE provides large performance gains for the WPCN under consideration when compared to ZF, mainly because of the low-rate low-power transmissions, which are typical in the analyzed scenario;

  • The impact of the CSI-based and CSI-free scheme on the WET performance is analytically analyzed and several trade-offs are identified. It is shown that the CSI-free scheme is preferable as the number of IoT devices increases and/or the CSI acquisition costs increase. In terms of overall performance, the CSI-free scheme is shown to be the optimum for most of the configurations; although its performance may degrade significantly if the setup is not optimally configured in case of Poisson traffic.

Next, Section II presents the system model and assumptions, Section III discusses the energy outage performance under the CSI-based and CSI-free WET schemes, while Section IV addresses the information outage performance under ZF and MMSE decoding schemes. Section V presents and discusses numerical results. Finally, Section VI concludes the paper.

Notation:

Boldface lowercase letters denote column vectors, while boldface uppercase letters denote matrices. For instance,

where is the -th element of vector ; while where is the -th row -th column element of matrix . By

we denote the identity matrix, and by

we denote a vector of ones. Superscripts and denote the transpose and conjugate transpose operations, while and denote the trace operator and a diagonal matrix with elements , respectively. , and are the set of complex, real and non-negative integer numbers, respectively; while is the imaginary unit and denotes the imaginary part of . The absolute/cardinality operations in case of scalars/sets is denoted as , while denotes the euclidean norm of vector . Additionally, and are the floor and ceiling functions, respectively, while and are the supremum and infimum notations. The curled inequality symbol is used to indicate positive definiteness of a matrix, while is the big-O notation.

denotes expectation with respect to random variable (RV)

, which is characterized by a Probability Density Function (PDF)

and Cumulative Distribution Function (CDF)

, while is the probability of event . Also, denotes the sum of RVs distributed as . is a circularly-symmetric Gaussian complex random vector, with mean vector and covariance matrix , while is a non-central chi-squared RV with degrees of freedom and parameter such that [22]

(1)

where denotes the Marcum Q-function.

Ii System model

We consider the scenario depicted in Fig. 1. In the downlink, a HAP wirelessly powers a large set of single-antenna EH sensor nodes located nearby. Such low-power devices require in turn to sporadically send some short data messages of bits/Hz over time blocks of seconds in the uplink. The HAP is equipped with antennas, of which are used for downlink energy transmission, and the remaining for information decoding in the uplink. We assume that the coherence time is sufficiently large such that for any feasible . On the one hand, notice that since the RF-EH devices are extremely-low-power nodes, they are foreseen to be mostly static devices, thus, the coherence time is large. On the other hand, such devices are expected to transmit for short times due to intrinsically small data payloads, low-latency requirements, and/or lack of energy resources to support longer transmissions [23]. Then, by limiting for instance the analysis of this work to we can set , although extending any of our analyses for any other smaller would be straightforward.

Fig. 1: System model: a HAP equipped with antennas powers wirelessly in the downlink a set of single-antenna sensor nodes located nearby, while it receives information from a subset of them in the uplink.

Ii-a Channel model

The average channel gain between the HAP and is denoted as , e.g. the path loss is , while the small-scale fading channel coefficient between the HAP’s antennas and (downlink) is denoted as , and the channel between and HAP’s antennas (uplink) is denoted as . Notice that even when the network is configured to operate over the same frequency band in uplink and downlink, the channel reciprocity is difficult to hold in this kind of setup since devices at both ends are extremely different [24], hence we assume fully independent uplink and downlink channels222Even when certain dependence may exist, this does not affect significantly our results. This is because WIT phases in WPCNs are mostly sporadic, then, the aggregated harvested energy between consecutive WIT phases is much less dependent on the fading experienced in a particular coherence block..

The antenna elements are sufficiently separated such that the fading seen at each antenna can be assumed independent. We assume quasi-static channels undergoing Rician fading, i.e., , which is a very general assumption that allows modeling a wide variety of channels by tuning the Rician factor [25, Ch.2], e.g., when the channel envelope is Rayleigh distributed, while when there is a fully deterministic LOS channel.

Ii-B Transmission model

We assume homogeneous (in terms of hardware, supported services and traffic characterization) IoT devices which are harvesting RF energy from HAP’s transmissions. They require

power units to keep active, otherwise they are in outage. Such a value obviously depends on their circuitry but also on the services they require to support. Additionally, the EH devices need to report their data to the HAP at some moments, so they interrupt briefly (during

seconds) their EH process to send it. We model such transmission activation in two different ways, by considering:

  • periodic traffic, such that the network is perfectly synchronized and every EH device has a predefined slot allocated for transmission. If the periodicity is , then there are slots available. If , then each device operates alone in the channel; otherwise there will be up to concurrent transmissions eventually;

  • Poisson traffic, such that the network traffic is uncoordinated. Let us take as the mean number of messages per coherence time that are required to be transmitted by each EH device. Notice that it is evident that needs to hold according to our previous discussions.

It is worth noting that neither the periodic nor the Poisson model are suitable for mimicking bursty traffic, for which other more suitable models are recommended, e.g., [26]. However, a WPCN implementation is not suitable in scenarios requiring bursty transmissions mostly due to its inherent and strict energy limitations, thus we resorted to the above simple but effective models covering two extreme ends. Additionally, note that the multiple antennas at the HAP require to be exploited for spatially separating the concurrent transmissions with high reliability. We delve into the specific details in Section IV.

Finally, denotes the fixed transmit power of , while represent the energy resources333Notice that the time resource for is limited by , while for is limited by the overall transmission duration . In fact, we assume in our analyses that the uplink pilot training phase is much shorter than the actual data transmission and ignore its impact on the information outage performance in Section IV. (power time) utilized by such EH node to let the HAP know the uplink and donwlink CSI, respectively. Notice that since channel reciprocity does not hold, it is expected that as transmissions from the EH devices are requried in both downlink and uplink (pilot transmissions in uplink, feedback in downlink), but decoding/processing the pilots sent by the HAP is also required in the downlink.

Ii-C Performance evaluation

We adopt the outage probability formulation as the main performance metric. We say is in outage when: i) the harvested energy was insufficient for supporting its operation and consequently no uplink data transmission occurred: energy outage , or ii) uplink transmission occurred but the transmitted message could not be decoded at the HAP: information outage . Since donwlink and uplink channels are independent and transmit powers are fixed we have that ’s outage probability is given by

(2)

Finally, the network performance is evaluated in terms of the worst node’s performance by computing the network outage probability as

(3)

Then, we can assure that every EH device in the network performs reliably at least the of time. Notice that for , the term dominates (2). Consequently, and since is required in practical scenarios, we can examine independently the bounds on and .

Iii Wireless Energy Transfer

In Subsection III-A, we first propose a CSI-based precoding scheme for optimizing the WET process. Then, we address the CSI-free WET alternative in Subsection III-B. The energy outage performance under both CSI-based and CSI-free WET schemes is also analyzed therein.

Iii-a CSI-based WET

In each coherence block time, the HAP sends pilot signals that are used by the EH devices to estimate the donwlink channels. Then, such information is fedback to the HAP through the uplink channels in an ordered way. As commented before, in such processes, the EH devices spend

energy units each time, which is approximately given as

(4)

where denotes the energy required for decoding, processing and sending back to the HAP the information related to the pilot signals coming from each antenna. As we will show later in Subsection III-A3, very often, the HAP only requires the WET-CSI from a small set of EH devices, and therefore it is expected that their CSI feedback can be scheduled without overlapping.

As there are transmit antennas, the HAP is able to transmit energy beams to broadcast energy to all sensors in . Then, the incident RF power at is given by

(5)

where is the HAP’s transmit power, , denotes the precoding vector for generating the th energy beam, and is its normalized energy carrying signal, i.e., , which is independently generated across the antennas, i.e., .

For our setup and performance evaluation criterion, the optimum precoder is the one that minimizes . However, since the set is known by the HAP after the CSI acquisition procedures, the problem translates to maximize subject to . The previous objective function is not concave and therefore the problem is not convex. However, it can still be optimally solved by rewriting it as a semi-definite programming (SDP) problem [27], as shown next.

Iii-A1 Energy beamforming

First, define , while in (III-A) can be rewritten as

(6)

where and . Second, notice that is a Hermitian matrix (with maximum rank ) that can be found by solving

(7a)
subject to (7b)
(7c)
(7d)

which is an SDP problem. Notice that (7c) corresponds to the power budget constraint. Finally, the beamforming vectors

match the eigenvectors of

but normalized by their corresponding eigenvalues’ square roots such that

. This procedure allows finding the optimum precoding vectors, and hereinafter it is referred to as CSI-based beamforming.

Interior point methods are mostly adopted to efficiently solve SDP problems. Since consists of a linear function, linear constraints, one positive semi-definite constraint, and the more challenging optimization variable has size , interior point methods will take iterations, with each iteration requiring at most arithmetic operations [28], where represents the solution accuracy at the algorithm’s termination. In addition, an eigendecomposition of , which has complexity , is required in order to derive the set of beamforming vectors. Consequently, the SDP solution becomes computationally costly as the number of PB’s antennas and/or the number of EH devices increases.

Iii-A2 Energy outage lower bound

Notice that

(8)

where is the sensor under the greatest path loss: , e.g., the farthest sensor. The above expression strictly holds as long as we consider the same energy requirements for all devices, e.g., homogeneous devices with the same transmit power . However, (8) should also hold when intelligent power allocation polices are utilized.

In the best possible scenario, where the HAP requires compensating only the channel impairments of since the remaining nodes are under much better favorable channel/propagation conditions, a MRT precoding will be the optimum. Such MRT precoding is indistinctly and equivalently given by

(9)

for which in each coherence interval becomes

(10)

which comes from using [20, Eq. (45)] and setting .

We assume that the energy harvested between consecutive uplink transmissions requires to be enough for powering the circuits, performing the CSI acquisition procedures, and sending an uplink information message, while the remaining (if any) energy is used in other tasks, e.g., sensing, signal processing, etc. Therefore,

  • for periodic traffic, the total energy harvested by between its uplink transmissions is at most given by

    (11)

    where in , denotes the energy conversion efficiency, and the summation is over independent RVs of the form of , comes from using (10), while follows after using the definition of a non-central chi-square RV. Notice that although we conveniently used to take advantage of a finite summation, the last expression holds without such a constraint.

    Meanwhile, the energy requirements under periodic traffic are given by

    (12)

    thus (8) becomes

    (13)

    which comes from using the CDF of a non-central chi-square RV; while

  • for Poisson traffic, the messages arrive with an exponential inter-arrival random time with mean (given in coherence intervals). For analytical tractability, let us assume that transmissions also occur in an slotted fashion, where slots are of duration . Then, devices with a ready-to-send message wait for the next time slot for transmission.

    Let us denote the inter-arrival RV by , which is now discrete and with PMF given by

    (14)

    for . Now, becomes a random sum of RVs, i.e., , while the energy requirements to make the uplink transmissions take place are random as well, and can be written as

    (15)

    Then

    (16)

    where comes from averaging the outage events conditioned on a given and using (10), while follows by taking the summation of

    non-central chi-squared RVs, which obeys a non-central chi-squared distribution as well, but with

    times the number of degrees of freedom and non-centrality parameter, and using its CDF by taking advantage of (1), while follows after taking the expectation with respect to by using (14). Notice that we avoided using an infinite notation in the last step, and instead considered only the first summands, hence (16) is, in general, an approximation that becomes exact as . However, since is a discrete exponential-like random variable characterized in (14), setting such that is enough for a good accuracy. Notice that

    (17)

    which follows from realizing that computing is equivalent to evaluate into the transform of the sequence , which is .

Iii-A3 On the optimality of the MRT beamforming

Let us assume that the HAP is using the MRT beamformer to power the farthest node . One question arises: How such beamformer impacts the wireless powering of the remaining devices? To shed some light into that matter, let us focus on the performance in terms of average incident RF power in a certain device . By using (III-A) and the second MRT beamfomer given in (9), we have that

(18)

Unfortunately,

follows a cumbersome projected normal distribution when

[29], which makes the analysis of the distribution of already very complicated even for such a simplified scenario. Meanwhile, decoupling the expression as shown in the last line of (18) does not solve the problem since numerator and denominator are correlated. We resorted to simulation and standard fitting procedures, and found out that

(19)

matches (18) accurately, which is corroborated in Fig. 2. Now, notice that based on (10), we have that

Fig. 2: vs for . Comparison between the Monte Carlo-based (18) and the analytical approximation (19).
Schemes No. Tx. Antennas Average EH Gain EH Diversity No. Rx. Antennas Energy Requirements
CSI-based ModerateHigh
CSI-free (SA) LowModerate
TABLE I: Main system performance characteristics under the considered CSI-based and CSI-free WET schemes
(20)

thus,

(21)

When , we have that even when the HAP uses only the CSI statistics referred to , the remaining devices harvest more energy at least on average. In such scenario the MRT beamformer is most of the time the optimum from a system perspective. Now notice that if we consider the large-LOS scenario, (21) simplifies to

(22)

which basically tells us that when undergoes a path-loss at least 3dB greater than the experienced by the remaining EH nodes, the optimum energy beamformer is given by (9) most of the time.

Iii-B CSI-free WET

Several CSI-free powering schemes have been recently proposed and analyzed in [20, 21], e.g., , , and . In Section I, we highlighted their main characteristics and argued why we adopt the scheme as our CSI-free scheme in this paper. Summarizing, the reasons are three-fold: i) among the schemes taking advantages of the spatial resources, exhibits a homogeneous performance over the space, which is not sensitive to the different mean phases of the LOS channel component; ii) it is more suitable than the scheme for powering devices far from the HAP; and iii) it allows a better coupling to the co-located information transmission processes since only one antenna may be transferring energy in the donwlink while the remaining may be receiving uplink information.

In our setup, the adoption of the SA scheme implies that each transmit antenna is active during seconds, while the remaining antennas function as receive antennas, i.e., . The incident RF power at is then given by

(23)

which comes from exploring the connection to (10) and defining since all antennas transmit during a coherence block, although not simultaneously. Consequently, by taking , and the energy outage bounds given in (13) and (16) for periodic and Poisson traffic, respectively, hold here as well. Notice that under the CSI-based scheme, the average harvested energy can be up to times greater than under scheme, for which , however, the diversity gain of is greater since all antennas contribute.

A summary on the system performance characteristics under the CSI-based and CSI-free WET schemes is presented in Table I. Notice that the average EH gain is counted as , while the energy requirement field accounts for all energy consumption sources including the uplink CSI-acquisition procedure which is required for both analyzed schemes.

Iv Wireless Information Transmission

As commented in Section II, at some points, the EH devices require sending short data messages of bits/Hz over blocks of seconds to the HAP. The HAP utilizes antennas to decode the arriving messages and resolve possible simultaneous transmissions. Uplink CSI is required to implement the ZF or MMSE linear decoding schemes which we adopt here, while the analysis under non-coherent decoding schemes is left for future work.

Notice that we consider an uninterrupted downlink WET, while now and then a subset of the devices interrupt their harvesting process to send uplink data. The self-interfering powering signals, traveling through the channels between the transmit antennas and receive antennas are assumed to be perfectly canceled via Successive Interference Cancellation (SIC) techniques, which may include analog and digital processes. Such SIC techniques can benefit from the fact that the powering signals can be chosen deterministically.

Finally, under the SA scheme we assume that the transmit slots are scheduled such that no antenna switching occurs during an actual uplink transmission, which would complicate the information decoding procedures. Next, we analyze the WIT performance under the considered traffic profiles.

Iv-a WIT under periodic traffic

As commented in Subsection II-B the maximum number of concurrent transmissions is deterministically , thus, the HAP requires that same number of orthogonal pilot signals and consequently pilot symbols. Then, under periodic traffic, can be broken approximately into

(24)

where is the per-symbol pilot energy. Next, we investigate the outage performance of the data transmission phase.

Iv-A1 Signal model

At the HAP, the data signal received after each transmission is given by

(25)

where the th column of is and consequently such matrix has dimension , , is the normalized vector of the normal signals transmitted by the devices, and is the Additive White Gaussian Noise (AWGN) vector at the antennas. If is not active in a given transmission slot of duration , we consider that the respective entries in , and are zero. Consequently the number of non-zero columns of is at most , which matches also the maximum number of non-zero rows and columns of , and the number of non-zero elements of . Finally, the equalizer at the receiver decouples the transmitted data streams such that its output is given by

(26)

Iv-A2 Zf

The ZF equalizer is

(27)

and by substituting it into (26) yields

(28)

Then, the instantaneous Signal to Interference-plus-Noise Ratio (SINR) of the output stream corresponding to the one transmitted by is given by

(29)

where with . For Rayleigh fading, i.e., , has the central inverse Wishart distribution, which for the case of greater than the number of data streams , yields to . Meanwhile, the analysis under Rician fading is encumbered by the noncentrality of the Wishart distribution of . The usual approach in such case lies in approximating the noncentral Wishart distribution by the virtual central Wishart distribution as summarized in [30]. In any case, the analysis is cumbersome, specially for the general scenario where does not need to necessarily hold, thus, we take no further steps to characterize the distribution of . Whenever the statistics of are required, we use a Monte Carlo-based approach.

Iv-A3 Mmse

The MMSE equalizer is

(30)

while the corresponding component for decoding the th data stream is given by

(31)

Then, the corresponding instantaneous SINR is given by

(32)

where . Even for the simplest scenario with Rayleigh fading, equal per-user SNR, and , the distribution of is cumbersome as corroborated in [31, 32]. This, and the fact that for a more general scenario there is no closed-form expression for the PDF and CDF of , we rely again on Monte Carlo simulations whenever its statistics are required.

Iv-A4 Information outage performance

For the sake of fairness, we assume that those devices with the most similar path losses are scheduled for concurrent transmission. This is possible under periodic traffic, which is deterministic by nature. Let us sort the devices according to their path loss such that is the device with the smallest attenuation, while is the device under the greatest path loss. Now, we evaluate the information outage performance at in order to get a bound on the performance of any node in the network444

Such bound is expected to hold under the assumption of equal devices’ transmit power, or a power allocation such that a greater attenuation implies a smaller transmit power. While the latter seems odd at first sight since the farthest node is usually allowed to transmit with greater power in traditional cellular networks, it is not the case in WPCNs where the farthest device harvests also less energy.

. Thus, we have that , while (25) and subsequent derivations can be compacted by eliminating the zero-rows/columns of , e.g. , , and reducing the dimension of vector , i.e., . Then,

(33)

which comes from using (29) and (32) such that .

Iv-B WIT under Poisson traffic

Concurrent transmissions happen randomly under Poisson traffic. Therefore, there is no way of completely avoiding the pilot collisions unless all devices are allocated orthogonal pilot sequence. However, this can be extremely energy-costly for large since , where was defined in the previous subsection. To overcome this, we herein allow collisions to occur with a probability not greater than , which is a system parameter to be efficiently designed.

Iv-B1 Collision probability

The probability that a given device is active at a certain time slot is given by

(34)

where denotes the set of active devices in such a time slot, and the last equality comes from using (17). Notice that the subset of active devices is random under Poisson traffic, and also its cardinality . Meanwhile, is a Binomial RV with parameters and such that

and , which represents the average number of concurrent transmissions.

Now, let us take as the number of orthogonal pilot sequences/symbols such that is the pilot reuse factor, and denote as , where , the set of devices using the same pilot signal, then

(35)

and similarly is a Binomial RV with parameters and . Consequently, represents the average number of concurrent transmissions of devices using the same pilot signals. As done previously, we focus our attention to the performance of . Assuming such a device is already active, its associated collision probability is

(36)

Then, we must choose such that . However, notice that if such is greater than , it is preferable to deterministically assign one unique pilot sequence to each user, thus, avoiding the collisions completely. Therefore, the optimum given is given by

(37)

where

(38)

In the Appendix, we illustrate a simple procedure for solving (38). Finally,

(39)

Next, we investigate the outage performance of the data transmission phase.

Iv-B2 Information outage performance

Herein, we need to consider the pilot collision events and the outages due to decoding errors. Since takes into account the events related to the collided ’s transmissions, we are now interested on the event where operates without collision while the remaining IoT sensors in may or may not be colliding. Consequently, we now have that

(40)

The latter term in both and can be easily evaluated by

  1. generating a sample conditioned on ;

  2. drawing elements from to conform the set of interfering devices;

  3. evaluating (33) for such configuration;

  4. averaging (33) over many possible realizations of .

Finally, notice that setting the collision target probability for optimum system perform