A Finite Input Alphabet Perspective on the Rate-Energy Tradeoff in SWIPT Over Parallel Gaussian Channels

Simultaneous wireless information and power transfer (SWIPT) has gained significant popularity in the recent past owing to its applications in a wide range of use cases. Although SWIPT has been fairly well investigated in the literature, the existing work has mainly focused on attaining the optimal rate energy (RE) tradeoff assuming Gaussian input alphabet. However, practical systems operate with finite input alphabets such as quadratic-amplitude modulation (QAM)/phase-shift keying. We characterize the attainable RE tradeoff in SWIPT systems employing finite input alphabet for transmission over parallel Gaussian channels of say orthogonal frequency-division multiplexing subcarriers or multiple-input multiple-output streams. Some of the key results in the literature that assume Gaussian input alphabet are shown to be special cases of our results. Furthermore, we provide insights into our results with the aid of graphical illustrations, which throw light on the optimal power allocation policy for various energy-harvesting constraints. Furthermore, we consider practically relevant time-sharing and power-splitting schemes operating with finite input alphabet and characterize their RE tradeoff. Their optimal solutions in the asymptotic regime are obtained, which serve as low-complexity solutions suitable for practical implementation. Our simulation studies have demonstrated that the Gaussian input assumption significantly over-estimates the attainable RE tradeoff, especially when the signal set employed is small. Furthermore, it is observed through numerical simulations that the proposed optimal power allocation performs significantly better than the power allocation based on the Gaussian input assumption. Specifically, as much as 30% rate improvement is observed when employing the classic 4-QAM signal set.


I. INTRODUCTION
The number of networked devices is expected to proliferate owing to a wide range of compelling applications [1], such as smart sensors in home applications, wearable devices, environmental sensors, warehouse management etc. Since most of these devices are battery operated, it is imperative that they have high energy-efficiency. While harvesting energy from the environment is well investigated, the recently emerged simultaneous wireless information and power transfer (SWIPT) philosophy [2]- [4] has gained significant popularity owing to its capability of extending the lifespan of the batteryoperated devices. SWIPT was first proposed in [5] and its various extensions were studied in [6]- [11]. Specifically, Grover and Sahai studied the SWIPT in frequency-selective additive white Gaussian noise (AWGN) channels [6]. Zhang and Ho characterised the rate-energy (RE) trade-off in multiple-input multiple-output (MIMO) SWIPT systems operating relying both on co-located as well as spatially separated energy harvesting (EH) and information decoding (ID) receivers [7]. Furthermore, Zhou et al. proposed dynamic power splitting aided SWIPT and characterised its RE trade-off [8]. They also studied the performance of SWIPT in the downlink of multiuser communication by considering both time sharing (TS) and power splitting (PS) receivers [9]. A detailed overview of various existing SWIPT schemes as well as the challenges and open issues of practical SWIPT systems can be found in [10], [11].
A key requirement of SWIPT systems is that they require channel state information (CSI) at the transmitter for energy/information beamforming. Hence, several beamforming schemes have been proposed for SWIPT systems [12]- [16], which focus on reducing the CSI feedback overhead while maximizing both the energy as well as the information efficiencies. Specifically, adaptive energy beamforming was proposed in [12] in conjunction with a CSI quantization scheme for improving the power vs. information transmission trade-off. In [13], Shi et al. studied the joint optimization of the transmit beamforming and the receive power splitting in a multiuser downlink communication scenario. As a further development, Park and Clerckx focused on reducing the CSI feedback overhead in MIMO interference channel with the aid of Geodesic energy beamforming [14], which requires partial CSI. Low-cost alternatives such as antenna selection and hybrid beamforming aided SWIPT schemes were investigated by Demir and Tuncer [15]. Recently, SWIPT was also studied by Yang et al. in the context of millimeter wave communication by considering lens antenna arrays [16].
However, the existing SWIPT solutions discussed so far were conceived by considering Gaussian input alphabet, which do not model the practical communication systems, where finite signal sets such as QAM/PSK are employed. It is widely recognized that when using a finite input alphabet, the achievable performance of transmission schemes under the Gaussian input assumption is significantly lower than that of the finite input alphabet. This phenomenon has been extensively studied both in scalar as well as vector Gaussian channels [17], [18] relying on transmit power allocation/precoding schemes [19]- [23]. Recently, the performance degradation due to Gaussian input assumption in SWIPT systems operating with finite input alphabet has been studied in [24]- [26]. Specifically, Zewde and Gursoy [24] have studied the RE trade-off in a point-to-point SWIPT system by optimizing the probabilities associated with the input constellation points. Furthermore, the authors of [25], [26] have characterised the RE trade-off in precoded MIMO systems employing finite input alphabet. Although [24]- [26] provide some insights, the implications of having finite input alphabets on the RE trade-off remain largely unexplored.
Against this background, the following are the contributions of this paper: 1) Orthogonal frequency division multiplexing (OFDM) [27], MIMO systems employing singular value decomposition (SVD) aided beamforming [28], etc. are some of the well-known scenarios of parallel Gaussian channels, whose RE trade-off remains unknown in case of finite input alphabet. In this paper, we first characterise the RE trade-off of SWIPT systems operating with finite input alphabet over parallel Gaussian channels, and show that the seminal results on co-located EH and ID receivers presented in [7] are special cases of our results. Furthermore, we provide insights into our results with the aid of graphical illustrations and show them to be a generalization of the well-known mercury-water pouring algorithm [19]. 2) We consider practically relevant TS as well as PS schemes and characterise their RE trade-off by considering finite input alphabets. Furthermore, we obtain asymptotic high signal-to-noise ratio (SNR) results on power allocation for these schemes, and compare them to their non-asymptotic counterparts. Since the SWIPT systems have to operate at high SNR for gleaning sufficient energy, our asymptotic high-SNR results are shown to be a simple solution for the practical implementation of the aforementioned schemes. Furthermore, we quantify the suboptimality of the Gaussian input assumption in terms of the attainable mutual information (MI) and harvested energy, and show how substantial it is when the signal set employed is small.
For the ease of reading, the paper provides tutorial insights interspersed with analytical and numerical results. The organization of the paper is as follows. The system model as well as the definitions and terminologies used in the paper are presented in Section II. Our main results on the RE tradeoff in SWIPT systems operating with finite input alphabet are presented in Section III. The graphical illustrations of our results are presented in Section IV. The TS and PS schemes are discussed in the context of finite input alphabet and their asymptotic behaviours are studied in Section V. Our conclusions are presented in Section VI.
Notations: R and C represent the field of real and complex numbers, respectively. The notation of | · | represents the magnitude of a complex quantity, or the cardinality of a given set. Expected value of a random variable z is denoted by E[z]. Given a random variable y, E[z|y] represents the conditional mean of z given y. A complex-valued circularly symmetric Gaussian distribution with a mean of µ and a variance of σ 2 is denoted by CN (µ, σ 2 ). The maximization problem is denoted by max v1,··· ,vN , where v 1 , . . . , v N are the optimization variables. The mutual information is measured in nats/channel use, unless stated otherwise.

II. SYSTEM MODEL
Consider a single-user SWIPT system consisting of a transmitter and a receiver having co-located EH and ID blocks. Both the energy as well as the information are transmitted to the receiver over N parallel channels. Explicitly, the N channels may correspond to the N orthogonal subcarriers in case of an OFDM system, or N parallel data streams in case of an N × N MIMO system employing beamforming [28]. We consider a generic system model so that the results derived can be readily applied to various scenarios. Let the N parallel Gaussian channels be modeled as where y i ∈ C is the signal received over the i th channel, h i ∈ C is the i th channel coefficient, n i ∼ CN (0, 1) ∀ i, and x i is the symbol transmitted over the i th channel. The transmitted symbol x i satisfies the power constraint and it is modeled as and s i is the equiprobable information bearing symbol chosen from a unit-energy constellation S i . The SNR over the i th channel is represented by ρ i = p i γ i , where γ i = |h i | 2 P . Without loss of generality, we assume γ 1 ≥ γ 2 ≥ · · · ≥ γ N .
Definition 1: The total achievable MI over the N parallel channels for a given power allocation {p 1 , . . . , p N } is given by where the achievable MI over the i th channel at an SNR ρ i is given by where The total power received over the N parallel channels is given byQ where η denotes the energy harvesting efficiency [7], which is taken to be one for the ease of presentation. Definition 3: The minimum mean-squared error (MMSE) in the estimation of the i th symbol s i at an SNR ρ i is given by Closed-form expressions of MMSE i (ρ i ) for various signal sets such as BPSK, QPSK, 16-QAM etc. can be found in [19]. Furthermore, the inverse function of MMSE i (ρ i ), which is required in our subsequent analysis for various signal sets, can be stored in a look-up table.
The following proposition proves instrumental in characterising the RE trade-off in Section III.
Proposition 1: [17] The MI and the MMSE associated with the i th channel satisfy

III. RE TRADEOFF IN SWIPT SYSTEMS OPERATING WITH FINITE INPUT ALPHABET
In order to characterise the RE trade-off, we first consider the two extreme cases, where only either the EH receiver or the ID receiver is present, followed by the case where both the EH and ID receivers operate concurrently. Considering only the EH receiver to be present, the objective is to maximize the harvested energyQ = N i=1 p i γ i . Thus, we have the following optimization problem:

Proposition 2:
The solution to (P1) is given by p 1 = N and p j = 0 for 2 ≤ j ≤ N .
Proof: The proof directly follows from the fact that γ 1 ≥ γ j for 2 ≤ j ≤ N , which is essentially the popular energy beamforming solution [10].
The maximum harvested energy in this case is given bȳ Q max = N γ 1 . If the ID receiver were to operate concurrently with the EH receiver, the attainable MI is given by R EH = I 1 (N γ 1 ) ≤ log(M 1 ). If the system were to operate with the idealized Gaussian input alphabet, we have R EH = log(1 + N γ 1 ) = log(1 + N |h 1 | 2 P ), which grows unbounded with P . By contrast, in the practical finite input alphabet case R EH is upperbounded by log(M 1 ). This key difference plays an important role in optimally exploiting the available power for striking the most appropriate information vs. energy transmission trade-off as discussed in the subsequent sections.
Assuming only the ID receiver to be present, the objective is to maximize the attainable rate R = N i=1 I i (p i γ i ). Thus, we have the following optimization problem:

Proposition 3:
The solution to (P2) is given by the wellknown mercury-water pouring algorithm [19]: where µ is the solution to The maximum attainable MI in this case is given by If the EH receiver was to operate concurrently with the ID receiver, the attainable energy transmission would be given byQ Thus, the two extremities of the RE region are given by (R EH ,Q max ) and (R max ,Q ID ). Note that while attainingQ max , any rate obeying R < R EH can be achieved by reducing the coding rate employed by the transmitter.
In order to characterize all the legitimate RE pairs in the region we now consider R EH < R < R max andQ ID <Q <Q max , which leads us to the following optimization problem:

Proposition 4:
The solution to (P3) is given by for i = 1, . . . , N , where µ and λ are solutions to and µ > λγ 1 . The power allocation in (16) is referred to as the mercury-gallium pouring 1 algorithm. Proof: The objective function in (P3) is concave over the constraint set, since each of the MI functions is strictly concave over its input power. The Lagrangian [29] of (P3) is given by and the Lagrange dual function of (P3) is given by Since the solution to (P3) can be obtained by solving min µ≥0,λ≥0 G(λ, µ), we first solve the problem in (19) for a fixed λ and µ. We have and the first-order necessary conditions for optimality [29] are given by It is readily inferred from (20) that it is necessary to have µ > λγ 1 in order to have a bounded solution. From Proposition 1, we have d dp and hence the optimal power allocation should satisfy Since the domain of The optimal µ and λ for a given set of {p 1 , . . . , p N } can be obtained by employing the well-known subgradient techniques [29]. The solution to (P3) can be obtained by iterating between the primal and dual solutions until convergence. Specifically, for a given µ ≥ 0, λ ≥ 0 the sub-gradient of the dual metric is computed: which is used for updating (µ, λ) based on the ellipsoid method. From the new set of (µ, λ), p i is updated based on (27). This procedure is repeated until µ and λ converge to a prescribed accuracy. This concludes the proof.
The following corollary provides the optimal power allocation for the Gaussian input assumption, which was studied in [7].
Let us now compare the attainable RE trade-off in a SWIPT system employing either Gaussian or finite input alphabets. Fig. 1 compares the RE trade-off attained by various constellations in a SWIPT system having N = 2, γ 1 = 4 and γ 2 = 1.2. It is evident from Fig. 1 that the Gaussian input assumption results in a significant over-estimation of the RE trade-off, especially when the constellation employed is small. Furthermore, it can be seen from Fig. 1 that the discrepancy between the pair of trade-offs reduces, as the constellation size is increased. This is expected, since a high-order constellation is a better approximation of the Gaussian input.
So far, we considered co-located EH and ID receivers, which is practically more relevant than the case, where the pair of receivers are separated. Nonetheless, our results can be readily extended to the case of spatially separated receivers as well, which is briefly discussed below.
represent the set of channel gains over the N parallel channels spanning from the transmitter to the ID and EH receivers, respectively. Furthermore, Proceeding along the lines of Proposition 3 and Proposition 4, we have The optimal power allocation for the case of spatially separated EH and ID receivers is given by the following proposition.
Proposition 5: The optimal power allocation for the case of separated EH and ID receivers is given by for i = 1, . . . , N , where µ and λ are solutions to and µ > λγ ′ max . The proof is similar to that of Proposition 4, hence it is omitted due to space constraints. Note that since EH receivers are not capable of sophisticated signal processing, the channel gains {γ ′ i } N i=1 may not be made available to the transmitter for employing the optimal power allocation presented in Proposition 5. In this paper, we focus only on SWIPT systems having co-located EH and ID receivers.
IV. GRAPHICAL ILLUSTRATIONS In this section, we provide further insights into the solution of (P3) presented in Section III with the aid of graphical illustrations.

A. MMSE-Power Charts
The optimal power allocation in (21), which is equivalent to (27), can be obtained by plotting γ i MMSE i (p i γ i ) as a function of p i ∈ [0, N ] and finding its intercept with µ * − λ * γ i for a givenQ. For instance, whenQ =Q ID we have λ * = 0 and the intercept of γ 1 MMSE 1 (p 1 γ 1 ) and γ 2 MMSE 2 (p 2 γ 2 ) with µ * yields the optimal power allocations p * i . Fig. 2(a) illustrates this case in a SWIPT system having N = 2, γ 1 = 4, γ 2 = 1.2 and using a 4-QAM signal set. WhenQ >Q ID , we have λ * > 0 and the intercepting lines µ * − λ * γ i differ for each of the channels depending on γ i . Fig. 2(b)-(c) illustrate this case for three values ofQ ∈ (Q ID ,Q max ), which correspond to 10%, 50% and 90% of ∆Q =Q max −Q ID . It is evident from Fig. 2(d) that asQ →Q max , we arrive at p * 1 → N , which is in accordance with Proposition 2. Fig. 3 illustrates the optimal power allocation for various Q in the aforementioned SWIPT system for the 16-QAM case. The key difference between the 4-QAM and the 16-QAM case is that the stronger channel receives more power in the latter case, whereas in case of 4-QAM, the weaker channel may receive more power depending on the value of Q. Note that the larger the size of the QAM constellation, closer it is to the Gaussian alphabet. Since a higher-gain channel receives more power in the Gaussian alphabet case, the power allocation in case of 16-QAM is quite similar. This phenomenon can be observed by comparing Fig. 2(a)-(b) to Fig. 3(a)-(b). Specifically, whenQ =Q ID , we have p * 2 = 1.17 and p * 1 = 0.83 in case of 4-QAM, whereas in case of 16-QAM we have p * 2 = 0.8 and p * 1 = 1.2. Note that in case of BPSK, this phenomenon occurs even for larger values ofQ. We have omitted the illustration of this scenario owing to the space constraint.

B. Mercury-Gallium Pouring
In this section, we provide an alternative interpretation of the mercury-gallium pouring algorithm, which is essentially a generalization of the mercury-water pouring algorithm presented in [19].
Let us define the following function, which measures the discrepancy w.r.t. the Gaussian input assumption: Note that the power allocation solution should satisfy When G i (µ, λ) = 1, we have 1 γi + p i = 1 µ−λγi , which corresponds to the power allocation associated with the Gaussian input assumption. Thus, G i (µ, λ) is a measure of discrepancy w.r.t. the Gaussian input assumption.
The power allocation solution presented in Proposition 4 can be interpreted as follows: 1) Set up a vessel having unit-base, which is solid up to the height 1/γ i for the i th channel.   4) The height of the gallium over the mercury in each of the vessels gives the optimal power allocation p MGP i . Fig. 4(a) provides a pictorial depiction of the above interpretation of the mercury-gallium pouring algorithm. When λ = 0, which corresponds to the no energy harvesting constraint, the solution in (16) reduces to that of mercury-water pouring, as depicted in Fig. 4(b). It is readily seen that when λ = 0, the height 1/(µ − λγ i ) = 1/µ up to which gallium has to be filled becomes fixed for all the channels.

V. TIME SHARING AND POWER SPLITTING SCHEMES OPERATING WITH FINITE INPUT ALPHABET
In this section, we discuss practically feasible time sharing and power splitting aided SWIPT schemes operating with finite input alphabet and characterise their RE trade-off.

A. Time Sharing
Let us consider a SWIPT system in which the energy and the information transmission take place over independent timeslots. During the information transmission phase, only the ID  receiver would be active, while in the energy transmission phase, only the EH receiver would be active. Let 0 ≤ α ≤ 1 denote the percentage of time allocated for EH. Let {p and {p (2) i } N i=1 represent the power allocations during the ID and EH phases, respectively. Under the fixed power constraint [7] for both EH and ID phases, we have where TS 1 refers to the TS scheme having a fixed power constraint. The boundary of E TS1 (P ) is essentially the line connecting the two extremities (R max , 0) and (0,Q max ).
Under the flexible power constraint [7], where the transmission power averaged over the EH and ID phases is bounded, where TS 2 refers to the flexible power constraint. The following proposition provides a simplified RE region for the case of flexible power constraint, which is analogous to Proposition 4.1 of [7].
Proposition 6: Under the flexible power constraint, all the boundary points of E TS2 (P ) except the two extremities (R max , 0) and (0,Q max ) can be attained, when α → 0. Under this limiting condition, we have The proof follows similar lines to that of Proposition 4.1 [7] and it is omitted due to space constraints. Note that for anyQ <Q max , the boundary point of E TS2 (P ) in (36) is given by the mercury-water pouring algorithm described in Proposition 3. Furthermore, the limiting condition of α → 0 essentially assumes having infinite transmit power capability in the EH phase, which is practically infeasible. Under the finite transmit power condition, the optimal α that achieves the boundary points of E TS2 (P ) can be shown to be α =Q N γ1kp , where k p = P p /P and P ≤ P p < ∞ is the peak transmit power. Unless stated otherwise, TS 2 refers to the time sharing scheme with P p = ∞.

B. Power Splitting
Let us now consider a SWIPT system in which both the energy as well as the information transmission take place simultaneously. The receiver is assumed to have a power splitter that splits the received signal and simultaneously feeds both the EH as well as the ID receivers. Assuming uniform power splitting across all the N parallel channels, the achievable RE region is given by where β decides the fraction of the received power allocated for EH. The boundary points of the RE region given in (37) are obtained by the mercury-water pouring algorithm described in Proposition 3. A practical application of the above scheme arises, for instance, in the filter bank multi-carrier system [30], where each of the receive filters in the filterbank can be equipped with a power splitter that feeds both the EH and ID blocks.

C. High-SNR Asymptotics
A key requirement of the SWIPT systems is that they need a high SNR owing to the high energy threshold of the EH receiver. As a result, it is important to understand the behaviour of the various SWIPT schemes discussed earlier, when P → ∞. Specifically, we study the asymptotic behaviour of TS 2 and UPS schemes.
Proposition 7: When employing TS 2 , the optimal power allocation under P → ∞ is given by where S i is assumed to be the same for all the N parallel channels andq = lim P →∞Q /γ 1 . Proposition 8: When employing UPS, the optimal power allocation under P → ∞ is given by where S i is assumed to be the same for all the N parallel channels.
The proofs of both Proposition 7 and Proposition 8 follow from Theorem 5 [19] and are omitted due to space constraints. Note that the asymptotic power allocation in (38) depends on the harvested energy throughq, whereas that in (39) does not depend on the harvested energy. Furthermore, the power allocation in (39) is the same as that of the system operating without energy harvesting [19]. Considering the lowcomplexity nature of TS 2 and its explicit dependence onQ, we restrict our attention to TS 2 while evaluating the asymptotic high-SNR performance.

1) RE Tradeoff:
Let us now compare the RE trade-off attained by the various schemes discussed so far. Consider a SWIPT system having N = 2, γ 1 = 4 and γ 2 = 1.2. Fig. 5 compares the RE trade-off using the 4-QAM signal set over both the channels. The outer bound along with the RE regions of TS 1 , and of TS 2 using P p = 2P as well as P p = ∞ transmit peak power constraints and that of the UPS scheme are depicted in Fig. 5. It can be seen from Fig. 5 that E TS1 (P ) ⊆ E TS2 (P ). The RE region of TS 2 with finite P p as well as that of UPS are observed to lie between E TS 1 (P ) and E TS 2 (P ), as in the Gaussian alphabet case [7]. In case of UPS, when β is large, a large fraction of the received power is allocated to energy harvesting. As a result, a small fraction of power is available for information transmission. In this case, only the channel having the highest gain receives all the available power. When β is decreased, more power becomes available for information transmission, hence multiple channels may receive non-zero power allocation. This is evident from Fig. 5, where the tradeoff curve associated with the UPS shows two different trends. Fig. 6 compares the RE trade-off in the aforementioned system for the 16-QAM signal set. Note that all the observations made from Fig. 5 hold for this case as well. Thus, we can conclude from Fig. 5 and Fig. 6 that the relative performance trends of the various schemes under finite input alphabet are similar to those of Gaussian input alphabet (cf. Section IV-D [7]).
2) Optimal vs. Asymptotic Power Allocation: Let us compare the performance of the asymptotic power allocation of (38) to that of the optimal power allocation discussed in Section V-A. Fig. 7 compares the attainable RE trade-off when employing both optimal and asymptotic power allocations for various values of P . The system is assumed to have N = 2, |h 1 | 2 = 4, |h 2 | 2 = 1.2 and using the 2-PAM signal set over both the channels. It is evident from Fig. 7 that whenQ is negligible, the asymptotic and optimal power allocations attain nearly the same performance. Furthermore, when the operating rate is much lower than R max , we see that both the asymptotic and optimal power allocations give nearly the same performance. Since most of the practical systems employ channel codes whose rates are much less than one, the asymptotic power allocation would be a suitable lowcomplexity solution.  Fig. 7. Comparison of the achievable RE trade-off in a SWIPT system employing TS 2 with optimal and asymptotic power allocations. The system parameters are assumed to be N = 2, |h 1 | 2 = 4, |h 2 | 2 = 1.2 and 2-PAM signal set over both the channels. Fig. 8 and Fig. 9 compare the attained RE trade-off by optimal and asymptotic power allocations in case of the 4-QAM and 16-QAM signal sets, respectively. It is evident from Fig. 7, Fig. 8 and Fig. 9 that the performance loss with respect to the optimal power allocation increases, when the size of the signal set is increased. However, when the operating datarate is lower than R max , we can still employ the asymptotic power allocation without incurring a significant performance loss.

3) Optimal vs. Gaussian Power Allocation [7]:
A key difference in the performance of the Gaussian and finite input alphabets such as QAM/PSK is that in the latter case the associated MI saturates with the SNR, hence resulting in no further incentive for any additional power allocation, which is in contrast to the Gaussian alphabet case, where the associated MI does not saturate with the SNR. This key difference renders the SWIPT schemes that assume Gaussian input alphabets suboptimal for practical systems, which employ practical QAM/PSK constellations. We quantify the penalty of the Gaussian input assumption both in terms of the attainable MI as well as the harvested energy, and show that the penalty imposed by this assumption can be severe, when the modulation alphabet is small. In the following, the Gaussian power allocation refers to the power allocation policy assuming Gaussian input alphabet [7]. Fig. 10 compares the RE trade-off in a SWIPT system employing TS 2 using both optimal as well as Gaussian power allocations. The signal sets considered are 2-PAM, 4-QAM and 16-QAM, and the transmission powers considered are given by P ∈ {5, 15, 25}. The system is assumed to have N = 2 and the channel powers are assumed to be |h 1 | 2 = 4 and |h 2 | 2 = 1.2, as earlier. It is evident from Fig. 10 that the Gaussian power allocation is outperformed by the optimal power allocation for all the signal sets considered, especially in the Pareto frontier region. Table I quantifies the rate-penalty imposed by the Gaussian power allocation.
So far, we studied the performance of both the Gaussian and optimal power allocations by considering fixed channel gains. Let us now compare their performance as a function of the channel powers. This is accomplished in Fig. 11 by studying the achievable rate as a function of γ2 γ1 between -20 dB to 0 dB. The signal set employed over both the channels is 2-PAM and the transmit power is P = 25. It is evident from Fig. 11 that as the difference between γ 2 and γ 1 increases, the rate-penalty of the Gaussian power allocation becomes more  pronounced. WhenQ = 175 and γ2 γ1 < −15 dB, we observe that the Gaussian power allocation does not allocate any power to the 2 nd channel, hence the achievable rate saturates at 1 bits/channel use. By contrast, the optimal power allocation ensures that the weaker channel gleans sufficient power, while the stronger channel does not receive more power than what is optimal for the 2-PAM signal set. Similar observations hold for the 4-QAM case depicted in Fig. 12. It is evident from Fig. 12 that asQ increases, the achievable rate becomes more sensitive to γ2 γ1 . This is indeed expected, since the power available for information transmission reduces whenQ increases, where the specific power allocation policy employed plays an influential role. Comparison of achievable rate for variousQ with optimal and Gaussian power allocations (2-PAM).
Optimal power alloc.
Gaussian power alloc. Fig. 11. Comparison of the achievable rate as a function of " " in a SWIPT system employing TS 2 with optimal and Gaussian power allocations. The system is assumed to operate with 2-PAM signal set over both the channels. byQ n =Q γ1 (0 ≤Q n ≤ N ) for characterising the RE trade-off. Consider a 2 × 2 MIMO system employing SVD aided beamforming and 4-QAM signal set over both the channels. Fig. 13 compares the attainable RE trade-off in the aforementioned system when employing Gaussian and optimal power allocations for P = 5 and P = 15. It is evident from Fig. 13 that the the optimal power allocation significantly outperforms the Gaussian power allocation. Specifically, when Q n = 1 a rate improvements of 0.3 and 0.35 bits/channel use are observed when P = 5 and P = 15, respectively. Fig. 14 compares the attainable RE trade-off in a 3 × 3 MIMO system employing SVD aided beamforming and 2-PAM signal set over all the channels. It is evident from Fig. 14 that the Gaussian power allocation suffers from performance loss compared to the optimal power allocation as in the 4-QAM case (cf. Fig. 13). Specifically, the optimal power  Fig. 13. Comparison of the achievable RE trade-off in a SWIPT system employing TS 2 and operating over 2 × 2 Rayleigh fading channel with the aid of beamforming. The system is assumed to operate with 4-QAM signal set over both the channels.
allocation attains a power gain of about 26% when operating at 2.75 bits/channel use and P = 5, and about 40% when operating at 2.9 bits/channel use and P = 15, respectively.

E. Future Directions
In this paper, we have studied the performance of SWIPT systems considering realistic signal constellations such as BPSK/QAM. These classic signal sets are known to be less energy-efficient than multi-dimensional signal sets such as spatial modulation (SM) [31], [32]. The SM is capable of striking more beneficial design trade-offs owing to using a low number of RF chains and low-complexity maximum likelihood decoding complexity [33]. It would be interesting to study the RE trade-off in SWIPT systems employing SM. Furthermore, the existing literature mainly focuses on SWIPT in multicarrier systems [9], [34]. However, multicarrier systems tend to require complex power-hungry signal processing and hence they are less suitable for battery-operated devices. By contrast, single carrier transmission schemes [35] are a suitable choice. It would be interesting to study the attainable RE trade-off in single carrier schemes employing SM and classic signal sets, which has hitherto not been studied. There has been a significant interest in SWIPT-enabled full-duplex systems [36], Comparison of the achievable RE trade-off in a SWIPT system employing TS 2 and operating over 3 × 3 Rayleigh fading channel with the aid of beamforming. The system is assumed to operate with 2-PAM signal set over all the channels. [37]. It would be worthwhile studying the attainable RE tradeoff in these systems considering finite input alphabets.

VI. CONCLUSIONS
We have characterised the RE trade-off in SWIPT systems operating with finite input alphabet over parallel Gaussian channels, which has hitherto not been studied in the literature. The RE trade-off of the SWIPT system operating with Gaussian input alphabet was shown to be a special case of our result (cf. Corollary 1). The Gaussian input assumption was observed to significantly overestimate the actual attainable RE trade-off with finite input alphabets. Further insights into our results were provided with the aid of graphical illustrations: a) MMSE-power charts and b) Mercury-Gallium pouring algorithm. Furthermore, the proposed mercury-gallium pouring algorithm was shown to be a generalization of the well-known mercury-water pouring algorithm. Practically relevant time sharing and power splitting schemes were discussed and their RE trade-off was characterised considering finite input alphabet. Furthermore, their asymptotic power allocation solutions were obtained, which were shown to be optimal except for the Pareto frontier region. Furthermore, the attainable RE tradeoff with Gaussian and finite input alphabet were studied by considering various signal sets and transmit powers. The rate attained by the optimal power allocation was observed to be significantly better than that of Gaussian power allocation.