2. Principles of TR

Time reversal is a digital signal processing technique. If applied to waveform design, it can be used as a beamforming technique. According to Fink's time reversal cavity (TRC) in [6], the time reversal technique uses the detection signal to obtain all the information of the spatial channel so that the source signal can be perfectly reconstructed in the original position. Considering the actual situation, Fink uses the Huygens principle to transform the three-dimensional TRC into a two-dimensional and finite number of time reversal mirrors (TRM). The basic idea of time reversal technique is to reverse the signal in the time domain or phaseconjugate the signal in the frequency domain. The process of TR contains two stages as shown in Figure 1. The first stage is detection stage. The source transmits the

Figure 1. The process of TR (a) detection stage (b) TR stage.

Long-Distance Wireless Power Transfer Based on Time Reversal Technique DOI: http://dx.doi.org/10.5772/intechopen.89078

Figure 2.

temporal effect and reduce the pollution by the spatial effect. In [2], it conducted a

TR can improve the transmission efficiency, and we also consider the microwave and DC conversion efficiency. The conversion efficiency depends on rectenna. It is not only a function of rectenna design but also a function of input waveform, i.e., input power and shape. This results in that the conversion efficiency is not a constant, but a nonlinear function of the input waveform [5]. Therefore, we consider a nonlinear rectenna circuit model in our system model with TR technique. Then we propose a sequential convex programming (SCP) to get the optimal fre-

The potential of TR and SCP to handle efficiency issues for WPT systems is investigated in this chapter. We introduce the background of WPT and TR in section 1. Then we detail the principles of TR in section 2. We describe the TR-SCP-WPT system model in section 3. Finally, the energy efficiency performance of the

Time reversal is a digital signal processing technique. If applied to waveform design, it can be used as a beamforming technique. According to Fink's time reversal cavity (TRC) in [6], the time reversal technique uses the detection signal to obtain all the information of the spatial channel so that the source signal can be perfectly reconstructed in the original position. Considering the actual situation, Fink uses the Huygens principle to transform the three-dimensional TRC into a two-dimensional and finite number of time reversal mirrors (TRM). The basic idea of time reversal technique is to reverse the signal in the time domain or phaseconjugate the signal in the frequency domain. The process of TR contains two stages as shown in Figure 1. The first stage is detection stage. The source transmits the

TR indoor experiment with a nanosecond pulse with a carrier frequency of 2.45 GHz. It is proven that TR can achieve an energy gain of 30 dB, and when continuous wave is used, the TR scheme can avoid indoor fading phenomenon. In [3], it explores the spatial profile of TR-WPT for energy reception at the vertical direction of the focus point in a metal cavity and gives a closed-form expression. In

[4], it uses the temporal–spatial focusing effect of TR, combined with coils containing metamaterials, to illuminate centimeter-level LED lamps, thus demon-

strating the precise control of near-field electromagnetic waves.

quency amplitude for the better conversion efficiency.

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WPT system is proposed.

2. Principles of TR

Figure 1.

94

The process of TR (a) detection stage (b) TR stage.

An illustration of TR (a) the channel probing phase and (b) the data transmission and focusing phase [7].

detection signal. In the second stage, TR stage, TRC records the signal with the channel impulse response (CIR) and then reverses it. The time reversal signal will arrive at the same position of the source through the same CIR.

The CIR goes through by the multipaths. To more detail, let us imagine that there are two points A and B in the space of the metal box like in Figure 2. When A transmits radio signals, its radio waves bounce back and forth in the box. Some of them bounce back and forth through B. After a period of time, the energy level decreases and can no longer be observed. Meanwhile, B can record the multipath distribution of arrival wave as time distribution. Then, such multipath distribution is inverted (and conjugated) by B and emitted accordingly, the last one and the last one. Through channel reciprocity, all waves following the original path will arrive at A at the same specific time and add up in a perfect and constructive way. This is called focusing effect [7] (Figure 2).

The specific operation of TR is to reverse the time-domain signal on the time axis or to adopt phase conjugation for the complex frequency domain signal. In our research, we choose the operation in time domain.

#### 2.1 Temporal focusing effect of TR

The temporal–spatial focusing effect of TR utilizes the principle of channel reciprocity, which means that in the two stages, the channel information is time invariant. Channel reciprocity requires a high correlation between the CIR of the forward link and the backward link, while channel stability requires that the CIR be stationary for at least one of the detection and TR stages. Experiments in the laboratory area show that the correlation of CIR between forward and backward links is as high as 0.98 in [8]. Therefore, TR can play an important role. The temporal focusing effect of TR refers to that the signal originally arriving last will be transmitted first while the signal originally arriving first will be transmitted last, and finally the signal of all paths arrive at the same moment.

We consider a long distance and wireless environment, namely, rich multipath channel can be represented as

$$h(t) = \sum\_{l=0}^{L-1} a\_l \delta(t - \tau\_l) \tag{1}$$

where L is the whole number of multipath, al is the amplitude of the path, τ<sup>l</sup> is the delay of the path, and δðÞ is the Dirichlet function.

According to the process of TR, the TR signal containing the reversed CIR will go through the same CIR. Therefore, there is an equivalent channel of TR, such as

$$\begin{split} h\_{eq}(t) &= h(-t) \otimes h(t) \\ &= \sum\_{l=0}^{L-1} a\_l \delta(-t - \tau\_l) \otimes \sum\_{k=0}^{L-1} a\_k \delta(t - \tau\_k) \\ &= \sum\_{l=0}^{L-1} \sum\_{k=0}^{L-1} a\_l a\_k \delta(t - (\tau\_l - \tau\_k)) \\ &= \sum\_{l=0}^{L-1} a\_l^2 \delta(t) + \sum\_{l=0}^{L-1} \sum\_{k=0, \ k \neq l}^{L-1} a\_l a\_k \delta(t - (\tau\_l - \tau\_k)) \\ &= A\_{\delta \text{vac}} + A\_{\text{sideloke}} \end{split} \tag{2}$$

(200, 0, and 0 mm), TRM2 (�200, 0, and 0 mm), TRM3 (0, 200, and 0 mm), TRM4 (0, �200, and 0 mm). The distance between the source node A and the TRM array elements is 200 mm, which is in the far field( L > λ, λ is the component of the

The basic TR process in this reverberant metal cavity proceeds as follows: first, the detection signal x tð Þ is injected into the cavity at the source node A, and the received signal at point B of the TRM array element is y tð Þ after passing through the

where ⊗ represents the convolution. Second, the received signal y tð Þ is time reversed at point B and injected into the cavity again. The received signal at point A

¼ xð Þ �t ⊗ hABð Þ �t ⊗ hBAð Þt

where heqð Þt is an autocorrelation function of hABð Þt , which can be regarded as

When the channel impulse response between point C (at a certain distance from

¼ xð Þ �t ⊗ hABð Þ �t ⊗ hBCð Þt

Because of the maximum value range of coherent superposition of multipath signals, the spatial focusing effect of TR can make the multipath signals superpose coherently at the source node A at a certain time, thus enhancing the electric field intensity at the source node to produce a peak. As shown in Figure 5, it can be seen that TR processing results in a lower signal amplitude in an area other than the location of the source node and the recovery of x tð Þ occurs at the primary source node A, and the peak value of RAð Þt is higher than the peak value of RCð Þt . Because hABð Þ �t and hBCð Þt are non-autocorrelation, there will be coherent cancelation in the signal. The obtained RCð Þt and x tð Þ signals are completely different. The peak power ratio of the signal that the source node and the non-source node can receive is determined by the total multipath gain of the channel and the correlation

RAðÞ¼ t yð Þ �t ⊗ hBAð Þt

¼ xð Þ �t ⊗ heqð Þt

point source node A) and point B of TRM array unit is recorded as hBCð Þt , the expression of received signal at non-source node can be obtained as follows:

RCð Þ¼t yð Þ �t ⊗ hBCð Þt

y tðÞ¼ x tð Þ ⊗ hABð Þt (3)

(4)

(5)

signal included the shortest wavelength).

DOI: http://dx.doi.org/10.5772/intechopen.89078

A reverberant metal cavity with one source node and four TRM nodes.

Long-Distance Wireless Power Transfer Based on Time Reversal Technique

the equivalent channel impulse response of TR.

channel hABð Þt :

97

Figure 4.

of source node is as follows:

According to Eq. (2), the equivalent channel is focused to zero moment and is superposed by L paths. In addition, this will generate a lot of sidelobe information as Asidelobe. Since the multipath delay range is large and the difference of ð Þ τ<sup>l</sup> � τ<sup>k</sup> is the same, the equivalent channel has many smaller side lobes on the time axis. We prove the temporal focusing effect of TR by the indoor channel, IEEE802.15.3a in [9] like in Figure 3.

According to Eq. (2), at zero-focus moment, the effect of delay is just offset, and the L multipath components add up, resulting in an increase in peak value at that time. Figure 3a shows the normalized indoor channel, and the TR equivalent channel of Figure 3b is time reversed by the channel of Figure 3a. It can be seen from the peaks of the two that the TR equivalent channel has an order of amplitude higher than the peak, which is consistent with Eq. (2). Multipath delays can cause phase shifts in the signal. Therefore, for the transmitted signal, it is equivalent to the fact that at the time of focusing, the signal information is superimposed in phase, and a focus peak appears. This is the temporal focusing effect of the TR.

Figure 3. Temporal focusing effect of TR (a) CIR (b) TR channel.

#### 2.2 Spatial focusing effect of TR

The spatial focusing effect is illustrated by the time reversal simulation experiment in a reverberant metal cavity through the XFDTD. As shown in Figure 4, there is the source node A at the center of reverberant metal cavity (0, 0, and 0 mm) and four time reversal mirror (TRM) array elements namely TRM1

Long-Distance Wireless Power Transfer Based on Time Reversal Technique DOI: http://dx.doi.org/10.5772/intechopen.89078

Figure 4. A reverberant metal cavity with one source node and four TRM nodes.

(200, 0, and 0 mm), TRM2 (�200, 0, and 0 mm), TRM3 (0, 200, and 0 mm), TRM4 (0, �200, and 0 mm). The distance between the source node A and the TRM array elements is 200 mm, which is in the far field( L > λ, λ is the component of the signal included the shortest wavelength).

The basic TR process in this reverberant metal cavity proceeds as follows: first, the detection signal x tð Þ is injected into the cavity at the source node A, and the received signal at point B of the TRM array element is y tð Þ after passing through the channel hABð Þt :

$$\mathbf{y}(t) = \mathbf{x}(t) \otimes h\_{AB}(t) \tag{3}$$

where ⊗ represents the convolution. Second, the received signal y tð Þ is time reversed at point B and injected into the cavity again. The received signal at point A of source node is as follows:

$$\begin{aligned} \mathcal{R}\_A(t) &= \mathbf{y}(-t) \otimes h\_{BA}(t) \\ &= \mathbf{x}(-t) \otimes h\_{AB}(-t) \otimes h\_{BA}(t) \\ &= \mathbf{x}(-t) \otimes h\_{eq}(t) \end{aligned} \tag{4}$$

where heqð Þt is an autocorrelation function of hABð Þt , which can be regarded as the equivalent channel impulse response of TR.

When the channel impulse response between point C (at a certain distance from point source node A) and point B of TRM array unit is recorded as hBCð Þt , the expression of received signal at non-source node can be obtained as follows:

$$\begin{split} R\_C(t) &= \mathcal{y}(-t) \otimes h\_{BC}(t) \\ &= \mathcal{x}(-t) \otimes h\_{AB}(-t) \otimes h\_{BC}(t) \end{split} \tag{5}$$

Because of the maximum value range of coherent superposition of multipath signals, the spatial focusing effect of TR can make the multipath signals superpose coherently at the source node A at a certain time, thus enhancing the electric field intensity at the source node to produce a peak. As shown in Figure 5, it can be seen that TR processing results in a lower signal amplitude in an area other than the location of the source node and the recovery of x tð Þ occurs at the primary source node A, and the peak value of RAð Þt is higher than the peak value of RCð Þt . Because hABð Þ �t and hBCð Þt are non-autocorrelation, there will be coherent cancelation in the signal. The obtained RCð Þt and x tð Þ signals are completely different. The peak power ratio of the signal that the source node and the non-source node can receive is determined by the total multipath gain of the channel and the correlation

According to the process of TR, the TR signal containing the reversed CIR will go

L�1

αkδð Þ t � τ<sup>k</sup>

αlαkδð Þ t � ð Þ τ<sup>l</sup> � τ<sup>k</sup>

(2)

k¼0

X L�1

<sup>k</sup>¼0, <sup>k</sup>6¼<sup>l</sup>

According to Eq. (2), the equivalent channel is focused to zero moment and is superposed by L paths. In addition, this will generate a lot of sidelobe information as Asidelobe. Since the multipath delay range is large and the difference of ð Þ τ<sup>l</sup> � τ<sup>k</sup> is the same, the equivalent channel has many smaller side lobes on the time axis. We prove the temporal focusing effect of TR by the indoor channel, IEEE802.15.3a in

According to Eq. (2), at zero-focus moment, the effect of delay is just offset, and the L multipath components add up, resulting in an increase in peak value at that time. Figure 3a shows the normalized indoor channel, and the TR equivalent channel of Figure 3b is time reversed by the channel of Figure 3a. It can be seen from the peaks of the two that the TR equivalent channel has an order of amplitude higher than the peak, which is consistent with Eq. (2). Multipath delays can cause phase shifts in the signal. Therefore, for the transmitted signal, it is equivalent to the fact that at the time of focusing, the signal information is superimposed in phase,

The spatial focusing effect is illustrated by the time reversal simulation experiment in a reverberant metal cavity through the XFDTD. As shown in Figure 4, there is the source node A at the center of reverberant metal cavity (0, 0, and 0 mm) and four time reversal mirror (TRM) array elements namely TRM1

αlαkδð Þ t � ð Þ τ<sup>l</sup> � τ<sup>k</sup>

through the same CIR. Therefore, there is an equivalent channel of TR, such as

<sup>α</sup>lδð Þ �<sup>t</sup> � <sup>τ</sup><sup>l</sup> <sup>⊗</sup> <sup>X</sup>

<sup>l</sup> <sup>δ</sup>ðÞþ<sup>t</sup> <sup>X</sup> L�1

and a focus peak appears. This is the temporal focusing effect of the TR.

l¼0

heqðÞ¼ t hð Þ �t ⊗ h tð Þ

<sup>¼</sup> <sup>X</sup> L�1

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<sup>¼</sup> <sup>X</sup> L�1

<sup>¼</sup> <sup>X</sup> L�1

[9] like in Figure 3.

2.2 Spatial focusing effect of TR

Temporal focusing effect of TR (a) CIR (b) TR channel.

Figure 3.

96

l¼0

l¼0

l¼0 α2

X L�1

k¼0

¼ Afocuse þ Asidelobe

#### Figure 5.

Spatial focusing effect of time reversal electromagnetic wave: Electric field distribution Z = 0 plane of the source node (a) in NO-TR system (b) in TR-WPT system.

3.2 Simulation

TR-SCP-WPT system model.

Figure 6.

(VNA) [13].

numbers (2–8).

Figure 7. The metal cavity.

99

3.3 Result and analysis

Firstly, we consider a multipath channel response based on a metal cavity like in Figure 7, with the measurement in frequency domain by Vector Network Analyzer

In the simulation, we set the carrier frequency as 5.8GHz and the emitting power as 36 dBm. Then we combine the measured channel with Matlab module like in Figure 6 to perform the experiment. As in Figure 9a, the TR-SCP and TR systems perform much better than the DT system because the TR technique can take

In [14], it proposes that the larger the bandwidth, the more the number of multipath like in Figure 8. In the above analysis and Eq. (2), the performance of TR is related to the multipath. In addition, the factor of SCP is frequency point numbers. Therefore, we compare three system models, namely, direct transmission (DT), TR, and TR-SCP, in different bandwidth (20–200 MHz) and frequency point

Long-Distance Wireless Power Transfer Based on Time Reversal Technique

DOI: http://dx.doi.org/10.5772/intechopen.89078

coefficient between the channels. Even if the multipath gain of the source node and the non-source node is the same or slightly higher, the ratio of peak power will still have a large gap when the channel correlation coefficient is very small. In the scattering-rich environment, as long as the distance is enough, the correlation of impulse response of multipath channel decreases to a very low level. The spatial focusing effect of TR is adaptive and can reduce electromagnetic radiation pollution.
