Time Reversal Technique Experiments with a Software-Defined Radio

Abstract

Time reversal techniques have been investigated for ultrasound and electromagnetic waves. They offer some advantages, particularly in cluttered and inhomogeneous environments, for point-to-point applications. The instrumentation usually employed for electromagnetic time reversal involves costly vector network analyzers, different interconnected generators and receivers, or a base station for mobile phones. This article explores the use of a low-cost commercial software-defined radio, in frequencies between 700 MHz and 2100 MHz, with indoor tests showing its performance and observed voltage gains for the received pulse.

1. Introduction

The time reversal (TR) technique for propagation was initially proposed for mechanical waves (ultrasound) [1] and later extended to electromagnetic fields [2], having been experimentally demonstrated in a reverberation chamber at a frequency of 2.45 GHz. When subject to a TR filter, waves are focused spatially and temporally on a specific point. Figure 1 shows an example of the complex environment where the TR technique can be of interest. The signal arriving at the receiver is a superposition of several attenuated and filtered copies of the transmitted pulse, in general, without a distinct line-of-sight (LOS), with the channel propagation following a Rayleigh model. TR helps focus the waves into the RX point, enabling a higher received amplitude at that specific local point. It is, therefore, aimed at point-to-point operations.

TR basically consists of two stages:

• Training: where a signal $u\left(t\right)$ occupying the desired bandwidth B is transmitted and captured by the receiver, which in response develops a signal $v\left(t\right)$ at its input, which is digitally recorded for the next step.
• TR: the received waveform $v\left(t\right)$ is phase conjugated in the frequency domain, $v_{TR}\left(t\right)$, and retransmitted by the TX antenna. In the receiver, a signal $y_{TR}\left(t\right)$ develops on its terminals, which is expected to have a larger energy than the original $v\left(t\right)$. The relation between the amplitude of both signals is the gain parameter which expresses the advantage of the TR technique.

The phase conjugate wave is equivalent, in the time domain, to reverse time, flipping the time series that represents the received pulse. That has the effect of focusing the wave into that specific point. It benefits from very complex wireless channels, because all multipath rays contribute to the overall response [3]. Mathematically, the TR operation is equivalent to a pulse compression on the channel impulse response (CIR) $h\left(t\right)$, and the TR output resembles the response to an exciting pulse $\delta \left(t\right)$ when $h\left(t\right)$ has a large complexity [4], which in propagation terms implies an inhomogeneous and cluttered scenario. The operation is similar to a matched filter [5], and it is therefore akin to a retrodirective antenna array [6], which retransmits the signal back to the direction where the source is located. Concerning applications involving electromagnetic waves, TR has been reported to be used in communications [7], increasing the efficiency of wireless power transmission [8,9] in Radio-Frequency Identification systems (RFID) [10], ground-penetrating radar enhancement [5], radar imaging [11], and also in Ultra-Wideband radar detection [4,12,13]. Specifically for RFID applications, a framework for simulating an environment using a 3D field solver was presented [14], where the whole virtual scenario was taken into account, including the RFID antennas. With the objective of alleviating the intersymbol interference (ISI) problem in a wireless system operating at 2.14 GHz, using a 10 MHz bandwidth and multiple-input single-output (MISO), a TR technique was used with better results in contrast to single-value decomposition (SVD), using a mobile phone base station as a transmitter [15].

A software-defined radio (SDR) is a radio communication system where components traditionally implemented in hardware—such as mixers, filters, amplifiers, modulators, and demodulators—are instead implemented through software on a programmable computing platform [16]. This architecture provides unprecedented flexibility, enabling a single hardware platform to support multiple communication standards, frequency bands, and modulation schemes through software reconfiguration alone. SDR technology has reduced development costs, accelerating the deployment of new protocols, and enabling cognitive radio applications that can dynamically adapt to spectral conditions. The fundamental principles were established by Mitola [17], who coined the term and articulated the vision of fully programmable radios capable of implementing arbitrary waveforms. Modern SDR implementations typically employ high-speed analog-to-digital converters (ADCs) positioned close to the antenna, with digital signal processing performed on field-programmable gate arrays (FPGAs), general-purpose processors (GPPs), or application-specific integrated circuits (ASICs).

This article describes a TR setup using a software-defined radio as the core instrument, which is innovative in the current literature. Real-world implementations of TR have used vector network analyzers (VNAs) [2,13], where the instrument’s high sensitivity and large bandwidth were exploited, and the measured signals sampled in the frequency domain were subjected to the TR mathematical operations elsewhere. Besides VNAs, a setup with an oscilloscope, arbitrary waveform generator (AWG), and antennas [8,10] has been employed. The different instruments demand unique software control, usually Labview [18], to synchronize their operation and articulate the mathematical processing routines. SDRs, in turn, offer some benefits in relation to the other hardware items as follows:

• Lower costs when compared with complex instrumentation like AWGs and VNAs.
• Open-source solutions to software control and integration.
• The software control interface has several built-in tools that help the TR flow design, such as filters, fast Fourier transform (FFT), decimators, etc.
• The seamless integration of software and hardware that enables a quick deployment of modifications, for instance, testing different frequencies, amplifier gains, and transmitted power amplitudes.
• In case SDRs with two MIMO ports are employed, a complex signal is able to be retrieved (IQ samples), with information on the phase.
• Lighter and with smaller dimensions.

As a downside, SDRs in general have narrower bandwidths in contrast to VNAs and AWGs, which limits the TR pulse width in the frequency domain and therefore the overall scheme efficiency. SDRs operating with large instantaneous bandwidths (in excess of 100 MHz) are still costly.

This article describes tests with an SDR performing TR in an indoor environment. The used hardware (Section 2.1) and the software (Section 2.2) are described, with the procedures synthesized in the latter. Section 3 checks if the pulses delivered in time and frequency domains by the SDR are consistent using a loopback configuration. Two indoor experiments are carried out in Section 4, with the respective gain results achieved by the TR technique.

2. Materials and Methods

The next subsections will describe both the hardware and software applications that implemented the focusing technique. Since the focusing uses point-to-point communication, the first step is the correct antenna positioning and power levels adjustments, according to Figure 2. The original pulse is synthesized, following the bandwidth and center frequency. In cases where the receiving level is too small, a proper gain can be set in the SDR. After this initial checking, the software applications can be run; first, the training stage is run, where the receiving pulse is recorded and processed, followed by the learning stage, where the actual processed pulse is transmitted and the gain can be computed.

2.1. Hardware

The software-defined radio NI USRP B210 [19] was employed for the tests. It covers the frequency range of 70 MHz to 6 GHz, and has four RF ports, which can be configured as multiple-input and multiple-output (MIMO). Each channel has a 12-bit analog-to-digital (ADC) and digital-to-analog converter (DAC), enabling a theoretical 74 dB signal-to-noise ratio (SNR). An internal Xilinx Spartan 6 XC6SLX150 FPGA controls the RF block, and it communicates with the host PC by means of a USB channel. For the RF part, a 15 cm telescopic-type wire monopole was used for the transmission, and a planar log-periodic antenna (LPDA) as the receiver. The latter is directive whereas the former is omnidirectional; this choice justified by the possibility to orient the receiver antenna in directions other than facing the transmitter, so that a more complex propagation channel can be enforced. The SDR always operated with its maximum output power (16 dBm), confirmed after measurements with a spectrum analyzer. The B210, being MIMO-capable, has its ports locked into the same phase reference, making IQ detection possible, where a complex signal is retrieved. In case one port-only SDRs were used (e.g., Hack RF One, capable of operating as either a transmitter or receiver, but in half-duplex mode), the measured signal would contain both IQ samples, but the phase part would be subjected to variations since the two devices (RX and TX) have their oscillators running freely and independently, without lock. Figure 3 shows a block diagram of the used hardware—the actual SDR and the laptop—along with an example of its use in an indoor environment. Two of the four SDR ports are used, while the other ones were left unconnected. The receiving antenna can be placed at distances up to 10 m away from the transmitter by means of a coaxial cable.

Figure 2. Steps for the TR technique based in SDR.

Figure 2. Steps for the TR technique based in SDR.

Figure 3. Block diagram of the hardware and depiction of its application. The different colors of the arrows mean the multiple paths reaching the receiver.

Figure 3. Block diagram of the hardware and depiction of its application. The different colors of the arrows mean the multiple paths reaching the receiver.

In the USB port the SDR delivers a time domain stream of complex numbers, occupying the baseband and alternating I and Q samples. The used application interface has to acquire these data and proceed with further tasks such as visualization and signal processing.

2.2. Software

GNU Radio [20] was used to interface with the SDR. It is an open-source framework, operating under the General Public License, and is used for interfacing with the SDR’s incoming data and performing signal processing and visualization functions. The final application, encapsulated as a Python file, can be run in a stand-alone fashion, or using the GNU Radio Companion, where the programming actually takes place with a block-oriented format. The same control and interface functions hereby used could have been implemented with Matlab [21] or Labview [18], although they require paid licenses. Using the same B210 SDR with GNU Radio, a Digital TV 3.0-compatible ISDB-T_B receiver was implemented, based on resources such as Low-Density Parity Check (LDPC) codes, bit interleaving, and Layer Division Multiplexing (LDM) [22]. In another reported example, different waveforms, e.g., orthogonal frequency division multiplexing (OFDM), triangular, and sawtooth, were tested to check which one gave a better performance for wireless power transfer [23].

The block diagram in Figure 4 shows the final solution employed to integrate all the necessary TR steps. The pulse $u\left(t\right)$ is formed with a Python application 3.8.2 from two user inputs: the SDR sample rate and the desired pulse bandwidth B. A real-valued sequence of numbers, representing the pulse, is then passed to the GNU Radio Companion program, which is then transmitted by the TX antenna and received by the RX port, $v\left(t\right)$, after passing by the wireless channel. It is saved as a binary file. Once the training stage is performed, it is followed by the focusing stage. The stored train of received $v\left(t\right)$ pulses is displayed by a Python code and the user selects two potential candidates to be used as the channel frequency response, based on their shape and amplitude. The Python code then outputs the time reversal version of the channel response, the so-called $v_{TR}\left(t\right)$ pulse, which is radiated by the SDR port. For the sake of future comparisons, the original $u\left(t\right)$ is also transmitted, so that the processing gain can be measured.

Preliminary tests were performed by integrating more tasks in GNU Radio, instead of using external Python programs. For instance, the pulse $u\left(t\right)$ was first programmed using a block called Python snippet within the GNU Radio Companion. However, due to hardware constraints imposed by the available laptop, it was decided to keep the GNU Radio application as lean as possible. Also, Python offers better possibilities than GNU Radio to extract figures and debug the code. Both flowgraphs are shown in Figure 5, where the training program consists of the USRP reading a vector source and transmitting it, and the receiving channel receiving the data from the antenna and saving it in a file (block named File Sink in the flowgraph). For the test, performance evaluations between the pulses were carried out by either transmitting one vector or another, which are input to the USRP sink as a vector source, imported from Python code.

The pulse to be transmitted occupies a bandwidth B and is subjected to an SDR $samp\text{\_}rate$. It is interesting to note that the $samp\text{\_}rate$ variable also describes the RF band centered on the central frequency $f_c$. The pulse $u\left(t\right)$ is then formed as

$$u\left(t\right)=sinc\left(Bt\right)\times h\left({\displaystyle \frac{Bt}{5}}\right)exp\left(j2\pi f_{c}t\right).$$

(1)

where h is the Hamming window, $f_c$ is the carrier frequency, and the variable $t_k$ is defined as the function of the SDR internal sample rate as follows:

$$t_k={\displaystyle \frac{k}{samp\text{\_}rate}},k=0,1,\dots ,N.$$

(2)

Following [10], the Hamming window multiplies the sinc function to limit out-of-band emissions. The SDR operated with a sample rate of 25 MHz, and the pulse bandwidth was chosen to be 5 MHz and 10 MHz. The B210 unit reaches a maximum of 56 MHz, but this data rate has to be divided by two channels, including 28 MHz. However, tests showed that a too-fast sample rate produced lost samples in the GNU Radio application, due to hardware (USB ports and processor overload) constraints. The pulse bandwidth B was then set to be slightly smaller than half of the sample rate parameter, so that the pulses can be adequately represented in the time domain. Nevertheless, it has been observed in the literature that the TR gain improves with the pulse bandwidth B [24].

Once the pulse $u\left(t\right)$ is synthesized by the Python code, it is transmitted as a train (every 20 $\mathsf{\mu }\mathrm{s}$) and received by the antenna, $v\left(t\right)$; the process is carried out in the Training GNU Radio program. It is recorded and processed to generate the TR response according to Figure 6. The next steps are followed in the Python code: First, two pulses (named A and B) are manually selected in the pulse train; they allow the choice of which one has less noise and a better performance. It also allows the user to compare their frequency domain shape to see if they are similar. As alternatives to this selection, a method based on the SNR or correlation analysis could be implemented, where the analytical pulse could be used as a noise-free comparison. Then, a low-pass filter is applied to both pulses (Butterworth, order 12), to attenuate noise and out-of-band signals. Later, the two time series (pulse A and B) are flipped and conjugated to form the TR signal $v_{TR}\left(t\right)$. One of them is chosen to be transmitted, based on its shape, amplitude, and SNR.

The sequence of numbers is fed to the transmitter (represented by the USRP Sink in the diagram) by the vector source blocks. They are copied and pasted from text files saved from the Python program. When they are fed as the input to the USRP Sink block (Figure 5, Training), the waveform modulates the RF carrier, $f_c$, set by the user between 900 MHz and 2100 MHz, which are constraints set by the used antennas. Besides this, in the test program, both the TR pulse $v_{TR}\left(t\right)$ and the training pulse $u\left(t\right)$ are output, one at a time, to evaluate their amplitudes squared on the received channel. It is important to stress that the energy of all pulses is normalized prior to transmission to enable fair amplitude comparisons at the receiving end. This helps identify the received responses and measure their respective amplitudes, thereby computing the gain. Its user interface is shown in Figure 7.

Other parameters such as the $f_c$ frequency, the internal amplifier gain, and the transmitted power were set by the user within the GNU Radio application. As a final note, a better performance was observed when running the GNU Radio flowgraphs on an 8 GBytes RAM laptop with an Ubuntu operational system (version 20.04.6) in comparison to a 16 GBytes RAM laptop with Windows 10 in terms of stability (GNU Radio crashed on the Win PC when the program was terminated). More modern and powerful PC hardware would make the operation more stable regardless of the GNU Radio parameters and the used operational system.

3. SDR Output Investigation

A preliminary test was carried out to check if the SDR hardware was able to output the desired pulse, since it might be too fast for its circuits and therefore transmitted with a deformed shape. In addition to an isolated pulse, all pulses within the train should maintain the same expected waveform, with a stable and predictable performance. Figure 8 contains the results for two received pulses randomly selected from the sequence, in contrast to the expected computed waveform, in both the frequency and time domains. It can be seen that the curves in time comply reasonably well, though their power spectra shows that out-of-band emissions are larger than that expected from the theory. It justifies the use of a low-pass filter as shown in Section 2.2. The inset contains its basic measurement setup, with the SDR operated with its maximum output power and the RF input protected by means of a 30 dB attenuator.

It is interesting to stress that GNU Radio presents the plots in the frequency domain with its own unit (dBr—relative dB) and not using the usual dBm, so a calibration is needed if actual quantities are to be shown. This procedure was performed using the SDR and GNU Radio connected to an RF generator and later compared to a spectrum analyzer, so a relation between dBr and dBm could be determined. During these evaluations it was also observed that the SDR internal amplifier gain (LNA) was not constant throughout the used frequency range, so this parameter had to be kept fixed during comparisons with different pulses and conditions.

4. Results

Different sets of experiments were carried out in order to address the effectiveness of the TR technique using an SDR. One involved an indoor domestic environment, whereas other tests took place in a research laboratory, with larger distances.

4.1. Domestic Environment: Sweeping Frequency

The first evaluation was performed in a domestic environment, whose main dimensions are shown in Figure 9, with obstacles from furniture, domestic appliances, desktop computers, etc. The measured processing gain is defined following the model in [8] as follows:

$$G={\displaystyle \frac{\max |y_{TR}{\left(t\right)|}^2}{{\max \left|y\left(t\right)\right|}^2}}.$$

(3)

where both $y_{TR}\left(t\right)$ and $y\left(t\right)$ are the time domain received pulses of the transmitted TR and training signals, respectively, as shown pictorially in Figure 7. The magnitude squared is justified by the presence of complex signals, given the SDR IQ demodulation.

Since the received signal is actually a train of pulses, whose peak amplitudes vary in time, an average is computed in the QT GUI Number sink block, seen in Figure 5, in the bottom-right corner. The test phase analyzes the received signal for both u and $v_{TR}$, and picks the maximum value after an observation time of 4 s.

For the test, both RX and TX antennas were kept fixed at the same positions throughout the whole experiment, while the central frequency $f_c$ was swept to analyze its effect on the gain. Two different pulse bandwidths were tested, 10 MHz and 5 MHz, using a sample rate of 25 MHz. Each discrete frequency required its own training phase, given the dependency of the channel impulse response on the frequency. The results are shown in Figure 10. It is possible to see that gains are observed in all tested frequencies, with better performance around 1000 to 1200 MHz, for both pulse bandwidths. This peak can be ascribed to a better sensitivity, due to a combination of antennas, SDR frequency response, and the channel itself.

To show the channel variation with frequency, it was measured with a vector network analyzer, keeping the same antennas and respective positions; the result is shown in Figure 11. Its frequency response is not flat, with multiple peaks in the S21 parameter, which is characteristic of a multipath environment.

Ambient noise (Figure 12) was picked up in the majority of frequencies used in the test, since it was carried out in an urban area. Analyzing the train of $v\left(t\right)$ pulses, the 900 MHz case contains a larger noise energy in contrast to 1300 MHz, and the former peak amplitudes have a wider amplitude fluctuation. This means that the choice of an adequate $v\left(t\right)$ pulse within the train is harder, due to the larger SNR.

Concerning the output power level and the internal LNA gain, it was observed that for a better performance, in terms of stability and repeatability, operating with maximum gain and maximum output power was preferred; otherwise, fluctuations observed along the received train amplitude as well as the higher noise content jeopardized computing the gain.

4.2. Laboratory Environment: Computing the Gain for Different Positions

The other tests were carried out in a larger, more cluttered laboratory environment. Figure 13 shows the main dimensions of the area and its overall picture. The measurement results are summarized in

Table 1. Gains for different positions, defined according to Figure 13.

Position	Gain
1	2.43
2	3.82
3	2.00
4	1.69
5	1.91
6	2.08
7	2.07
8	2.18
9	3.32
10	2.17

. For each position, the transmitter was kept fixed and the receiving site was varied; with these conditions, the training was performed and later the processing gain computed. The frequency of 1300 MHz was chosen to be used, given its good observed performance in the domestic environment. The sample rate was set to 25 MHz and the pulse bandwidth B to 10 MHz. Points (9) and (10) are in direct line-of-sight, with distances of, respectively, 5 and 2 m from the transmitter.

Figure 12. Train of received $v\left(t\right)$ pulses for two different measured center frequencies.

Figure 12. Train of received $v\left(t\right)$ pulses for two different measured center frequencies.

Figure 13. Indoor dimensions (meters) with the measurement points, along with a picture of the actual environment, with the visible points marked in red.

Figure 13. Indoor dimensions (meters) with the measurement points, along with a picture of the actual environment, with the visible points marked in red.

From the results, one can see two interesting points: the largest gain was observed at point (2), where the receiver was placed right below a piece of metallic equipment, in a very complex and harsh channel, as well as (8), where there was a wall between both spaces and a metallic table below the antenna. Also, the two LOS points had the situation of higher gain for the larger distance, where multiple reflections are likely to arise. Figure 14 shows the positions of these aforementioned spots.

The results confirm that the TR performs better when the filter is applied to a highly cluttered environment, with marginal gains for more homogeneous channels.

4.3. Gain for One Position and Addressing the Focusing at Other Untrained Points

The goal of this test was see how well the focusing took place for a specific point in space. This evaluation had the receiving and transmitter antennas trained for one spatial point, named the target (chosen to be at position 8, with the metallic shield plates), and then the same gain was computed for untrained receiving positions other than the target. Throughout the whole procedure, the transmitter antenna was kept fixed. The receiving antennas were positioned on the same spots as depicted in Figure 13, and additional points were measured from the target, to see how the focusing behaves near the training spot. Also, two linearly spaced LOS spots were used. The results are presented in

Table 2. Gain for positions other than the target, defined according to Figure 13.

Position	Gain
1	1.32
2	1.02
3	0.97
4	0.90
5	1.00
6	1.01
7	1.22
8 (trained target)	2.10
8 (target) but without obstacles	1.46
1 m from 8 (target)	1.16
2 m from 8 (target)	1.57
LOS 3 m away	0.49
LOS 5 m away	1.52

.

The results show, again, that whenever the propagation is clean, TR does not provide relevant advantages. Off-target (i.e., untrained) points provided moderate gains, and as the measured point moved away from the target, its gain decreased, even reaching situations where the TR provided pulses with less energy than the original u. The target was also compared without the metallic plates at the same position but in a less complex propagation environment, and the gain also decreased, indicating that the presence of the obstacles imposes a large difference in the channel impulse response, though the relative TX and RX positions were kept the same.

5. Conclusions

Realizing a TR experiment can be an involved task, since it deals with both RF and signal processing, as well as recording received signals to be later post processed. This work presented the use of a commercial software-defined radio with open-source applications, namely Python for signal processing and GNU Radio for hardware control. This combination resulted in a seamless and versatile TR system, which was easily reconfigured and debugged. The results originated from tests in two different indoor environments and proved that the time reversal indeed provided energy gains and focusing, with better results for more complicated and inhomogeneous wireless channels.