In this section, we provide an overall reasoning and mathematical justification behind the proposed design of the power signal.

Design constraints and objectives

As already mentioned above, the goal of this design is to superpose the power and the information signals in such a way that the high power WPT signal interferes as little as possible onto the information signal. To do so, and considering that the samples within the cyclic prefix can be discarded at the receiver, the power signal is sent as a pulse during the cyclic prefix duration only, repeated periodically at every cyclic prefix.

Looking back on the objectives of the waveform design listed in the introduction section, we can say that objectives 0.1 and 0.2 are already satisfied in this design. However, objectives 0.3 and 0.4 leave a room for pulse optimization such as its shape, duration, and other aspects. According to objective 0.3, the design should avoid degradation of WIT performance by reducing the risk of interference. Particularly, the wireless channel dispersion from the power head to the IoT device causes the pulse signal to spread longer in time (as illustrated in Fig. 4) which may lead to interference on the next OFDM symbols. Therefore, the pulse should end the furthest away possible from the end of the cyclic prefix to allow for some margin. In general, the pulse should have the smallest possible duration (in time) in order to avoid interference on the useful OFDM symbol. As for objective 0.4, the power signal should maximize the energy harvested at the receiver taking into account the nonlinear model of the rectifier. According to [30], high-PAPR signals usually increase WPT performance. The measurements in [39] confirm this claim that WPT is better with high-peak signals. Besides, from the transmitter point of view, it is important to note that the high-peak power signals increase the efficiency of the power amplifier (PA). On the other hand, the regulations impose limitation on both the total power transmitted by the signal and its PAPR, where the PAPR (Upsilon) is defined as the ratio between the peak power (P_{text{P}}) and the average power (P_{text{av}}) i.e., (Upsilon = frac{P_{text{P}}}{P_{text{av}}}). The regulations state that the peak power should not exceed 20 dB above the average power [40]. Hence, the three important elements that need to be considered for SWIPT pulse optimization based on the above discussion are as follows:

  1. 1

    The pulse should be designed to maximize the energy harvested according to the rectifier model.

  2. 2

    The pulse should respect the transmit power and peak power constraints.

  3. 3

    If possible, the pulse should have a short time duration to minimize the interference on the information part.

Considering the above points, it can be shown that the best possible waveform satisfying these conditions is the rectangular-shaped pulse (in equivalent baseband) where a limited average transmitted power (P_{text{av}}) is utilized with a maximum possible peak during the smallest duration. The mathematical justification is provided in the next subsection.

Fig. 4
figure 4

Effect of Wireless channel dispersion of the power signal onto the previous and successive useful OFDM symbol

Optimization of the power signal

Let us consider the design of a modulated power pulse (w_{text{p}}(t)) with a band limited baseband that is sent during the CP duration, of length (T_{text{cp}}). The power pulse is repeated every CP-OFDM symbol, i.e., every (T+T_{text{cp}}), where T is the useful OFDM data duration. The transmitted power signal in passband can be written as:

$$begin{aligned} w(t)=Re left[ w_{text{P}}(t) e^{jmath omega _{text{c}} t}right] =frac{1}{2}w_{text{P}}(t) e^{jmath omega _{text{c}} t}+frac{1}{2}w_{text{P}}(t)^* e^{-jmath omega _{text{c}} t}, end{aligned}$$

(2)

where (omega _{text{c}}) is the carrier frequency. We assume that the bandwidth of the power pulse is very small compared to the carrier frequency ((mathcal{{B}}<< omega _{text{c}})). We also assume that the channel is ideal so that the received signal is exactly w(t).

Considering the nonlinear model of the rectifier in [2], which is based on a simple single-diode rectifier (diode and RC circuit), the harvested current (i_{text{out}}) can be approximated by a Taylor series as

$$begin{aligned} i_{text{out}}=sum _{i=0}^{+infty } alpha _i {mathcal {E}}left( [w(t)]^iright) , end{aligned}$$

(3)

where (mathcal {E}left( .right)) is the time average, i is the summation index, and the coefficients (alpha _ige 0) depend on the rectifier characteristics. Since the pulse has a repetition period of (T+T_{text{CP}}), we can restrict the averaging on one period, giving

$$begin{aligned} {mathcal {E}}left( [w(t)]^iright) =frac{1}{T+T_{text{CP}}} int _{0}^{T_{text{CP}}} [w(t)]^i hbox {d}t, end{aligned}$$

(4)

where the upper bound of the integral is set to (T_{text{CP}}) since it is the maximal pulse duration. Since the modulated power signal is assumed to be band limited, the average of the real signal will be zero when i is odd (i.e., (mathcal {E}left( [w(t)]^iright) = 0) when i is odd). On the other hand, the even order terms do not cancel. For instance, if we compute the second-order time average

$$begin{aligned} {mathcal {E}}left( [w(t)]^2right)&=frac{1}{T+T_{text{CP}}} frac{1}{4} int _{0}^{T_{text{CP}}} left[ w_{text{P}}(t) e^{jmath omega _{text{c}} t}+w_{text{P}}(t)^* e^{-jmath omega _{text{c}} t}right] ^2 hbox {d}t. end{aligned}$$

(5)

When developing the square, two of the terms correspond to the Fourier transform of (w_{text{P}}(t)^2) evaluated at (pm 2omega _{text{c}}) and can be ignored given the low pass nature of (w_{text{P}}(t)). Only the cross-terms remain, leading to

$$begin{aligned} {mathcal {E}}left( [w(t)]^2right) =frac{1}{T+T_{text{CP}}} frac{1}{2} int _{0}^{T_{text{CP}}} |w_{text{P}}(t)|^2 hbox {d}t. end{aligned}$$

(6)

Similarly, if we compute the fourth time average, the only remaining cross-term is

$$begin{aligned} {mathcal {E}}left( [w(t)]^4right) =frac{6}{T+T_{text{CP}}} frac{1}{2^4} int _{0}^{T_{text{CP}}} |w_{text{P}}(t)|^4 hbox {d}t. end{aligned}$$

(7)

We can generally induce that ({mathcal {E}}left( [w(t)]^{2i}right) propto int _{0}^{T_{text{CP}}} |w_{text{P}}(t)|^{2i} hbox {d}t), and the output harvested current can be written as

$$begin{aligned} i_{text{out}} =sum _{i=0}^{+infty }beta _i int _{0}^{T_{text{CP}}} |w_{text{P}}(t)|^{2i} hbox {d}t =sum _{i=0}^{+infty }beta _i int _{0}^{T_{text{CP}}} P(t)^i hbox {d}t, end{aligned}$$

(8)

where (P(t)=|w_{text{P}}(t)|^2) representing the power of the power pulse signal.

To simplify the optimization, we transform these expressions to a discrete time and replace the integrals by sums. The samples of (w_{text{P}}(t)) and P(t) are denoted by (w_{text{P}}[m]) and p[m], respectively.Footnote 1 The objective is to find the optimal values of ({p[0],ldots ,p[L_{text{cp}}-1]}) that maximize the output harvested current subject to the average power constraint, the PAPR constraint (i.e., a peak constraint), and a positivity constraint. Thus, the optimization problem can be formulated as

$$begin{aligned}&max _{p[0],ldots ,p[L_{text{cp}}-1]} i_{text{out}} approx sum _{i=0}^{+infty }beta _i sum _{m=0}^{L_{text{cp}}-1} p[m]^i, nonumber \&text {subject to} end{aligned}$$

(9)

$$begin{aligned}&frac{1}{L_{text{cp}}+N}sum _{m=0}^{L_{text{cp}}-1} p[m] = P_{text{av}},nonumber \&p[m]le p_{text{max}},nonumber \&p[m]ge 0 . end{aligned}$$

(10)

The objective function is a polynomial function, which is convex given the positiveness of (p[0],ldots ,p[L_{text{cp}}-1]). Moreover, the constraints correspond to a convex set of solution. The solution of the maximization of a convex function lies on the extreme of the feasible sets.

One possible solution for the above formalized problem which satisfies the Karush–Kuhn–Tucker (KKT) conditions is an almost rectangular pulse, where we can define

$$begin{aligned} L=Big lfloor {frac{(L_{text{cp}}+N)P_{text{av}}}{p_{text{max}}}}Big rfloor . end{aligned}$$

(11)

For the constraint to make sense and recalling that the power pulse samples has to be within the cyclic prefix, the average power is assumed to be smaller than the cyclic prefix multiplied by (p_{text{max}}) ((P_{text{av}}<L_{text{cp}} p_{text{max}})) and therefore the value of L is always less than the cyclic prefix (L<L_{text{cp}}). A solution is obtained by setting (p[m]=p_{text{max}}) for (m=0,ldots ,L-1) and the remaining power (p[L]=P_{text{av}}(L_{text{cp}}+N)-Lp_{text{max}}).

Technically, implementing the last sample p[L] is not really practical. From design point of view, it makes more sense to only consider the samples with full (p_{text{max}}) amplitudes and discard the remaining one p[L]. Hence, we can safely assume that we want a signal that is purely rectangular within the cyclic prefix duration (Fig. 5).

Fig. 5
figure 5

Illustration of rectangular power signal parameters. Refer to Sect. 3.3 for the explicit definition of these parameters

Mathematically, one could argue that the nonzero samples with amplitude (p_{text{max}}) could be spread within the cyclic prefix rather than creating a rectangular shape. However, from both implementation and interference point of view, it is better to group these samples together in time as far as possible from the start of the following useful OFDM part rather than spreading them.

In order to compare the rectangular shape with other signal shapes, we provide an example illustrated in Fig. 6. In this example, we use the nonlinear model of the rectifier in equation 20 in [2] to compute the harvested energy for these various signal shapes. These signals are assumed to be within the cyclic prefix duration, and for the fairness of comparison, they all have equal average power. Besides, for this particular example, we assume that the PAPR constraint imposes that the instantaneous amplitude of any sample can not exceed 0.7. The results show that the rectangular shape signal X2 with the highest possible peak during the shortest possible duration provides the greatest amount of energy harvested.

In conclusion, the result of this optimization can be interpreted as having the smallest number of samples with the highest possible value instead of having a constant lower value all over the cyclic prefix. This concentration is better both in terms of harvested energy as well as from the point of view of interference. The model and parameters of this power signal are discussed in detail in the following subsection.

Fig. 6
figure 6

Comparing different shapes of signals in terms of harvested energy using the nonlinear model of the rectifier

Power signal model and parameters

As mentioned above, a power signal is generated from a power head using a modulated rectangular pulse signal that is sent only during the cyclic prefix duration (T_{text{cp}}). Without loss of generality, we assume that the width and position of the rectangular pulse are set as multiples of the sampling time ((T_{text{s}} = frac{T}{N})) of the signal. Then, the baseband discrete-time signal representation of the power signal generated by the power head can be written as

$$begin{aligned} w_{text{P}}[m] = {left{ begin{array}{ll} V_{text{p}} &{} text {if } bigg |m-bigg (frac{W_{text{p}}}{2} + L_{text{o}}bigg )bigg | le frac{W_{text{p}}}{2} \ 0 &{} text {otherwise} , end{array}right. } end{aligned}$$

(12)

which is sent in a synchronized fashion during the duration (0 le m le N + L_{text{cp}}). The signal parameters are defined as follows and as shown in Fig. 5:

  • (L_{text{o}}): represents the offset between the start of the cyclic prefix and the beginning of the rectangular pulse defined in terms of samples (the corresponding time is given by: (T_{text{o}} = L_{text{o}} T_{text{s}})).

  • (W_{text{p}}): represents the width of the rectangular pulse defined in terms of samples (the corresponding time is given by: (T_{text{w}} = W_{text{p}} T_{text{s}})).

  • (V_{text{p}}): represents the amplitude of the rectangular signal.

The discrete-time representation of the total transmitted power signal generated by the power head can be represented as:

$$begin{aligned} x_{text{P}}[m] = sum _{b=-infty }^{+infty } w_{text{P}}[m-b(N+L_{text{cp}})] , end{aligned}$$

(13)

where b is the index of the OFDM symbols transmitted. To avoid interference with the useful symbol duration, we restrict the rectangular pulse to be within the cyclic prefix duration such that (0< L_{text{o}} < L_{text{cp}}-W_{text{p}}).

One important indicator is the distance between the end of the peak and the end of cyclic prefix duration (L_{text{cp}}) that we refer to as “CP margin” and can be defined as

$$begin{aligned} Delta _{text{cp}} = L_{text{cp}} – (L_{text{o}} + W _{text{p}}). end{aligned}$$

(14)

The CP margin (Delta _{text{cp}}) , which is defined in samples, indicates how much the pulse is near or far from the next useful OFDM symbol. In practice, (Delta _{text{cp}}) should be positive otherwise the rectangular pulse overlaps with the useful OFDM symbol duration creating strong interference.

Fig. 7
figure 7

Block Diagram representing the interference caused by the power signal (x_{text{p}}[m]) due to the channel (h_{text{p}}[m]) onto the information decoding chain at the IoT receiver

The choice of waveform parameters is influenced by some regulations and limitations. One of them is the limitation on PAPR where the regulation imposes that the peak power should not exceed 20 dB above the average power [40]. We can infer that the average transmitted power over one OFDM symbol can be expressed as:

$$begin{aligned} P_{text{av}}=frac{P_{text{P}} W_{text{p}}}{N + L_{text{cp}}}. end{aligned}$$

(15)

where the peak power is defined as (P_{text{P}} = frac{V_{text{p}}^2}{R}) and we assume that (R = 1). Based on the maximal allowed PAPR, the minimal value of pulse width is set to

$$begin{aligned} frac{P_{text{peak}}}{P_{text{av}}} le Upsilon _{text{max}} leftrightarrow W_{text{p}} ge W_{mathrm{p, min}} = frac{N + L_{text{cp}}}{Upsilon _{text{max}}} . end{aligned}$$

(16)

For example, if we consider the maximum PAPR (Upsilon _{text{max}}) to be 18 dB higher than the average power and we use the fact that the normal cyclic prefix in LTE standard is around 7% of the useful OFDM duration T. Then, the minimum width of the rectangular pulse will be (W_{text{p,min}} = frac{N+L_{text{cp}}}{63} approx frac{1.07 N}{63} approx frac{N}{58.87}).

On the other hand, based on the current network setup, there might exist another limitation due to time travel difference between both transmitters. In many cases, the power head might be located closer to the IoT device than the base station. This is due to the difference in sensitivities of both systems, where energy harvesting requires higher power levels at the receiver. Due to the difference in travel time, the power signal could potentially arrive before the information signal which will create interference onto the previous useful OFDM symbol (see Fig. 4). So the rectangular peak offset (L_{text{o}}) should be adjusted correctly to accommodate for such problem and should be greater than a minimum offset (L_{text{o,min}}).

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