Time reversal (TR) communication in various configurations (single input, single output (SISO); multiple inputs, single output (MISO); or multiple inputs, multiple outputs (MIMO)) is studied. In particular, we report an experimental demonstration of time reversal focusing with electromagnetic waves in a SISO scheme. An antenna transmits a 1 μs electromagnetic pulse at a central frequency of 2.45 GHz in a high-Q cavity. Another antenna records the strongly reverberated signal. The time-reversed wave is built and transmitted back by the same antenna acting now as a time reversal mirror. The wave is found to converge to its initial source and is compressed in time. The quality of focusing is determined by the frequency bandwidth and the spectral correlations of the field within the cavity. A spatial focusing of the compressed pulse is also shown. This experiment is the first step for a communication scheme based on time reversal. It would be very interesting for ultrawideband communication in complex media since TR would permit compensation for delay spreading. MISO and MIMO TR communications are discussed on the basis of small-scale experiments with ultrasound. In particular, the binary error rate of the method is studied as a function of both data rate and external noise. A simple theoretical approach explains the results.
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 Nowadays, telecommunication systems demand higher and higher data transfer rates. In the recent past, intelligent coding [Proakis, 2000] led to the increase of the transfer rate. However, a further increase will require larger bandwidths [Paulraj et al., 2004]. A major problem is that, as the bandwidth is enlarged, electromagnetic waves undergo diffraction, reverberation and scattering, which results in fading and delay spreading. An initially short pulse that propagates in a complex environment spreads in time on a typical duration τ which can be much larger than the inverse of the bandwidth. Usually the time spreading is due to one or several wave reflections off obstacles inside buildings or in the streets. This phenomenon induces intersymbol interference (ISI) when the distance between symbols is shorter than the spreading time. ISI is an important problem to overcome, particularly in the field of ultrawideband (UWB) communications where the symbol length is potentially very short.
 A few years ago, an original technique based on time reversal [Fink, 1997] of acoustic waves was proposed for acoustic communications in shallow water reverberating environments [Edelmann et al., 2002]. Later the time reversal (TR) communication technique was generalized to MIMO schemes for ultrasonic (submillimeter wavelength) multiuser communication through a strongly multiple scattering medium [Derode et al., 2003]. In these experiments, the propagation medium consisted in a 40-cm large and 35-mm thick forest made of thousands of parallel steel rods. In such a complex medium, the “iterative time reversal” [Montaldo et al., 2004], an iterative algorithm that robustly converges to the inverse filter, was also successfully tested.
 More recently, the first time reversal experiment was reported for electromagnetic waves in the 2.45 GHz band [Lerosey et al., 2004]. This suggests that the techniques developed for ultrasound might be used for the electromagnetic case. It could be an interesting challenge because in many real environments (buildings or cities), microwaves with wavelengths between 10 and 30 cm are scattered off objects such as walls, desks, cars and so on, which produces a multitude of paths from the transmitter to the receiver. In such situations, a time reversal antenna should be able not only to compensate for these multipaths, but also to increase the information transfer rate thanks to the many reflections/reverberations. Indeed, it has been shown that reflections could be taken advantage of: in the case of multiple antenna arrays, reflections yield more independent propagation channels and therefore help to increase the capacity [Moustakas et al., 2000]. A basic experimental proof of such a principle, using phase conjugation, can be found in work by Henty and Stancil .
 From a theoretical point of view, time reversal of electromagnetic waves has been discussed in work by Cheney and Kristenson , Micolau et al. , Chambers and Berryman , and Oestges et al. . The potential interest of time reversal in real environments has been confirmed by two recent papers. Using broadband radio wireless measurements in an office space with many cubicles, Emami et al.  have computed the time-reversed field around a given receiver when a single transmitter is used in a time reversal mode. They retrieved the remarkable space-time focusing properties of time reversal. Nguyen et al.  as well as Kyritsi et al.  have studied the feasibility of applying TR techniques in multiple antenna wireless communication systems on the basis of outdoor measurements. They showed that TR in wireless communication mitigates the effect of channel dispersion and reduce the cochannel interference.
 In this paper, we discuss time reversal communication in various configurations: single input, single output (SISO); multiple inputs, single output (MISO); and multiple inputs, multiple outputs (MIMO). In section 2, we recall the principles of time reversal and its interest for communication purposes. Section 3 reports an experiment in which wideband electromagnetic waves are time reversed. It has been performed around 2.45 GHz in a reverberant chamber with a one-channel time reversal mirror (TRM), which is the first step of a real SISO communication based on TR of electromagnetic waves. Section 4 deals with MISO communication with ultrasound instead of electromagnetic waves. Indeed, as EM communication with multiple wideband antennas is still difficult to implement, we studied MISO time reversal telecommunication in a one hundredth scale model of an indoor environment with megahertz ultrasonic waves. The binary error rate (BER) of the method is studied as a function of both data rate and external noise. The communication channel is analyzed as a binary symmetrical communication channel and TR theory in random media is applied to explain the results. Section 5 briefly discusses the extension of the method to MIMO communication.
2. Basics of Time Reversal Communication
 In a propagation medium with reflecting surfaces, corners and so on, a wave can propagate between two points through multiple paths. In such a case, the response to a short pulse is spread in time because of the difference between travel times related to different paths (delay spread). One way to compensate for reverberations is to use time reversal. Basically the principle of the TR technique is the following: a transient wave e(t) is emitted by a source located at r. The wave propagates through a complex medium, which may be inhomogeneous, scattering or reverberating. The time dependence of the field is recorded by a set of M antennas located at ai and digitized. The signals are time-reversed in a computer memory. Then each antenna sends back the reanalogized time reversed version of the signal it had previously recorded. Because of reciprocity the wave back propagates through the medium and finally refocuses at the initial source position (spatial focusing) and recovers its original duration (time compression).
 Mathematically the TR communication process can be described in the following way: we assume that the propagation medium is reciprocal; that is, h(ai → r, t) = h(r → ai, t) where h(r1 → r2, t) is the impulse response between r1 and r2. Hence if e(t) is a Dirac pulse the signal at the user location is
 This signal consists of a peak of maximum amplitude at t = 0 surrounded by sidelobes which in the following will be referred to as “noise.” The more complex the propagation paths, the more ℛTR is peaked around time t = 0 [Derode et al., 2001; Blomgren et al., 2002; Draeger and Fink, 1997]. It has been shown that the peak-to-noise ratio (in amplitude) varies roughly as where Δν is the bandwidth of the initially emitted pulse and δν defines the correlation frequency of the reverberated field, that is, roughly the inverse of the spread time. The spread time is defined as the RMS delay spread of the signal, τ.
 The application of time reversal to telecommunications between a user located at point r and a base made of M antennas can be described as follows: in a preliminary step, the user sends a short pulse and the base records the M impulse responses between r and the antenna positions ai. Each impulse response is time reversed and convolved with the message s(t) before emission. For instance, in digital communication, the message s(t) may consist of a random series of bits with a rate 1/δt. After time reversal, the signal received by the user consists of a series of short pulses with the same rate, and little or no ISI, depending on the peak-to-noise ratio.
3. Time Reversal of Electromagnetic Waves
 From a technical point of view, the main difficulty to transpose this technique to the electromagnetic case lies in the very high sampling frequencies that are needed to digitize radio frequency (RF) signals. In this part, we propose a means to perform a time reversal experiment for electromagnetic waves using a sampling frequency much less than the carrier frequency. One way to perform time reversal for EM waves is to work with quasi-monochromatic signals, that is, to do a phase conjugation using the so-called three-wave or four-wave mixing in a nonlinear material in order to naturally produce the analogic phase-conjugated wave [Pepper, 1982; Chang et al., 1998]. Here, we want to perform a truly broadband time reversal for an electromagnetic pulse m(t)cos(2πν0t + ϕ(t)), or equivalently mI(t)cos(2πν0t) + mQ(t)sin(2πν0t). The carrier frequency is ν0, mI(t) and mQ(t) are the “baseband signals.” The time reversal operation can be directly performed on these baseband signals. The advantage is that the sampling frequency can be much lower than 2ν0.
3.1. Experimental Setup
 The experimental setup is the following (see Figure 1). We use two antennas working around 2.45 GHz manufactured by Astron Inc. and two circuit boards (RD0310 from RFMD, effective bandwidth 5 MHz) implementing the physical layer of a wireless local area network (WLAN). On the transmit side, it enables us to transform a baseband signal coded on two analog inputs I and Q, its in-phase component (cos) and its quadrature component (sin), to RF. On the receive side, it enables to go from the RF signal back to analog baseband I/Q.
 The experiment takes place in a strongly reverberant cavity with dimensions 3.08 m × 1.84 m × 2.44 m. Using an arbitrary waveform generator, we deliver a short pulse mI(t) (central frequency 3 MHz, −6 dB bandwidth 2 MHz) to the I analog input (cf. Figure 2a) of the transmit board. No signal is delivered to the Q analog input (Figure 2a, top plot). A mixer upconverts this signal to the GHz band and delivers e(t) = mI(t)cos(2πν0t). Then the waveform e(t) is transmitted by antenna A. After propagation, the signal e′(t) = m′I(t)cos(2πν0t) + m′Q(t)sin(2πν0t) is recorded by antenna B and downconverted to produce the I and Q components of the output signal m′I(t) and m′Q(t) that can be observed at the oscilloscope (Figure 2b). The received signals last more than 8 μs, that is, eight times longer than the initial baseband pulse, or 19600 periods of the RF wave. It means that the RF wave has traveled more than 2 km in the 14-m3 cavity, that is, undergone a few thousand reflections.
3.2. Baseband Time Reversal
 Our goal is now to time reverse the received radio signal e′(t) = m′I(t)cos(2πν0t) + m′Q(t)sin(2πν0t). Thus the baseband I and Q signals m′I and m′Q are digitized by the oscilloscope at a 40-MHz sampling rate, sent to a computer and time reversed. The wave carrier is conjugated as well. The following step consists in reanalogizing the time-reversed I and Q signals, and encoding them on the phase-conjugated wave carrier: the resulting RF signal writes m′I(−t)cos(2πν0t) − m′Q(−t)sin(2πν0t) = e′(−t). It is then transmitted back by antenna B. After propagation, the RF signal received at antenna A is downconverted to baseband.
3.3. Temporal Compression
 As can be seen in Figure 3a, the received signal on channel I is compressed in time and recovers its initial duration. Note that the acquisition window has been arbitrarily chosen in order to center the focused pulse within the Figure 3.
 Actually, since the reverberated wave field has been captured by a single antenna, the time reversal operation is not perfect. The waveform that is recreated is not the exact replica of the initial pulse: there are sidelobes around the peak on channel I, and a signal is measured on channel Q (Figure 3a, top plot) although nothing was sent on that channel. As mentioned in section 1, the peak-to-noise ratio in a one-channel time reversal experiment varies as . Therefore one could expect an even stronger pulse compression if the bandwidth was larger, or the correlation frequency smaller. Given the dimensions of the cavity, the Heisenberg time (i.e., the inverse of the mean distance between two successive eigenmodes of the cavity) is tH ≈ 80 μs. But the characteristic absorption time is ta = 3.6 μs because of the attenuation due to the skin effect on the walls; therefore the modes are not resolved and the correlation frequency δν is determined by ta rather than tH. Taking Δν = 2 MHz and δν = 1/ta ≈ 280 kHz leads to a predicted peak-to-noise ratio roughly equal to 3, comparable to our experimental results.
 A high-Q cavity with such a large reverberation time seems to be very far from a realistic environment. However, it should be noted that the relevant parameter is not the absolute value of the reverberation time, but rather the “spread factor,” that is, the ratio of the reverberation time to the pulse duration (or equivalently, the ratio between the bandwidth and the correlation frequency). Here we used a very reverberant chamber, but commercial transceivers with a relatively small bandwidth. The emergence of new UWB devices used in realistic indoor environment will lead to situations in which the reverberation time is not as large as in our cavity, but the bandwidth being much larger, the spreading factor will be comparable or even higher. Thus the concepts we have developed here will be transposable to realistic situations.
3.4. Spatial Focusing
 The experiment shows that time reversal is able to compensate for multiple reverberation and recreate a short electromagnetic pulse at the source. But there is more to it: we have also verified that the amplitude of the re-created signal is stronger at antenna A than anywhere else in the cavity (Figure 3b); that is, the time-reversed wave is spatially focused. Here once again the ratio between the peak and the spatial sidelobes is governed by the ratio [Derode et al., 2001]. Nevertheless, the actual shape of a focal spot when achieving a time reversal experiment with electromagnetic waves is still under study.
 The extension of the same TR process to multiple antenna is possible. In a MISO configuration, using M antennas would enhance the peak-to-noise ratio by a factor and further reduce the ISI. However, MISO or MIMO UWB communication experiments in reverberant media waves are difficult to implement. In section 4, we present a MISO communication experiment with ultrasonic waves propagating in a small-scale model mimicking indoor propagation of an UWB electromagnetic pulse.
4. MISO UWB Time Reversal Communication in a Downscaled Indoor Environment Using Ultrasound
 In this section, we study time reversal communications in the context of ultrasound in a downscaled indoor environment. Our aim is to underline some features of MISO and MIMO time reversal communication. Binary phase shift keying is used to transmit digital messages and the binary error rate (BER) is measured as a function of both data rate and external noise.
4.1. Experimental Setup
 All experiments were conducted with ultrasound in a water tank. In order to mimic multipath fading and delay spread in indoor environments we have built a centimetric scale model represented in Figure 4. The four walls of each room are made of 3-mm thick Plexiglas cubes (100 × 100 × 100 mm3). The corridors are 40-mm large. Decimetric electromagnetic wavelengths are replaced by millimetric ultrasonic ones. Needle-like piezoelectric transducers “simulate” electromagnetic antennas. The central frequency of the transducers is about 1.1 MHz with a 30% bandwidth, which would correspond to an ultrawideband electromagnetic system with a carrier frequency of about 2 GHz and a bandwidth of 700 MHz. A base station made of eight antennas is placed on the right side of the model. Eight other antennas that represent eight possible positions for the user are randomly distributed within the model (Figure 4). An experimental impulse response is obtained by sending a 3 μs-long pulse by one of the base antennas and recording the resulting wave field at one of the receiving antennas. Thus 64 impulse responses are recorded, corresponding to each base antenna/user antenna pair. The sampling rate is 32 MHz. A typical impulse response is plotted in Figure 5. In this setup we have evaluated the spread time (i.e., the RMS delay spread): τ ≈ 315 μs.
 Depending on how the user is placed relatively to the base antenna, the received energy may vary significantly: a line-of-sight path gives rise to a huge and short response, whereas multipathing decreases the signal amplitude. Therefore the set of 64 impulse responses is normalized; that is, each impulse response h(t) is divided by its total energy ∫h2(t)dt. Time compressions after TR are presented in Figures 5b and 5c. As expected, the peak-to-noise ratio increases when the number of active base station antennas is increased. Nevertheless, some secondary lobes still remain because the TR is imperfect. Indeed, a perfect time reversal with no secondary lobes would require a lossless medium on the one hand, and a base made of a very large amount of antennas [Fink, 1997] completely surrounding the medium on the other hand. Here, these two conditions are not fulfilled. These secondary lobes result in intersymbol interference (ISI), which is an intrinsic limitation of the imperfect TR approach.
 In the following, we use TR to transmit pseudo random sequences of bits and study the bit error rate (BER). Because of the limitations of our electronics, we cannot actually transmit thousands of bits within a reasonable time frame. Therefore, instead of actually performing the communication, we compute ℛTR from the set of measured impulse responses using equation (1), and infer the BER as follows. A 1000-bit pseudo random sequence is generated. This sequence is coded using binary phase shift keying (BPSK) to form the signal s(t). The received message is numerically computed convolving s(t) with the signal ℛTR given by equation (1), then demodulated and decoded by minimization of the quadratic distance on a constellation diagram. This operation is repeated until the number of wrong bits is large enough for the BER to be statistically significant. In addition, when the number N of transmitting antennas is less than eight, the BER is averaged upon all possible combinations of N antennas picked at random among the eight antennas of the base. Finally, the BER is also averaged on the eight BERs corresponding to the eight possible positions of the receiver.
 First, we study the BER versus the data rate. Then, we investigate the robustness of the TR communication as external noise is added. In these experiments the total emitted energy remains constant regardless of the number of antennas in the base.
4.2.1. BER Versus Data Rate
 The measurements of the BER versus the data rate are simply performed by changing the time lag δt between two consecutive symbols in the message s(t). Each time-reversed impulse response is convolved by s(t). Because of the linearity and time translation invariance of the propagation medium, the symbols received by the user are separated by the same time lag δt. Obviously the data rate equals the inverse of δt. The BER is estimated for time lag ranging from 0.5 μs (2 Mbits/s) to 10 μs (100 kbits/s).
 As shown in Figure 6, increasing the number of antennas drastically improves the BER. Indeed as the number of base antennas is increased, the symbols' amplitude grows faster than the sidelobe level for data rates less than 1 Mbits/s.
4.2.2. BER Versus External Noise
 We also study the corruption of the transmission due to the presence of external noise. The SNR is defined as the ratio of the symbol amplitude to the standard deviation of the external noise; it will be expressed in dB. We assume a stationary additive white Gaussian (AWG) external noise. The computation works as previously, but here we keep the data rate at 188 kbits/s. For a base station made of M antennas, the standard deviation of the AWGN is expressed as a fraction of the symbol amplitude obtained with the same number of antennas but without external noise. Figure 7 displays the experimental results for a one-, four-, and eight-antenna base station.
 When the number of antennas is increased, the BER decreases drastically as soon as the SNR is higher than 6 dB. Indeed, for small SNR, the external noise dominates the BER, whereas the intersymbol interference is the main contribution to the BER when the SNR is high. Increasing the number of base antennas significantly improves the BER only if we operate in an environment with an external noise lower than the “internal noise” (sidelobes) induced by the intersymbol interferences.
 The logarithm of the BER seems to vary fairly linearly with the time lag δt between the bits, that is, the inverse of the data rate (see Figure 8). The slope has the dimensions of a frequency, and it also varies linearly with the number of antennas, M (see Figure 9). Hence the logarithm of the BER decreases linearly with both M and δt. These experimental results are confirmed by a simple theoretical approach. In the following, we assume that the statistical fluctuations of the BER are weak enough for the BER to be considered as a good estimate of the probability of error Pe. In the case of BPSK, it is well known that the probability of error is given by
where Eb is the squared amplitude of one symbol (i.e., the energy per bit), N0/2 is the noise power and Q is the complementary error function Q(x) = (/2dt)/. The noise power N0/2 is simply the addition of a “deterministic” contribution due to the intersymbol interference (sidelobes) and external noise. If 2Eb/N0 is larger than unity, a good approximation of log(Pe) is
 The amplitude of a symbol increases as M because it results from a coherent addition of the signals emitted by all the antennas of the base whereas the total sidelobe level between two symbols increases proportionally to because of the incoherent sum of the intersymbol sidelobes. Here it is assumed that the paths between the user and each antenna of the base are uncorrelated. In summary, log(Pe) is given by −2MEb(1)/N0(1), with Eb(1)/N0(1) the signal-to-noise ratio for a single antenna.
 When a short pulse propagates through a complex medium, scattering or reverberation spreads the initial pulse over a characteristic time τ. If the scattered signal is time reversed through the same reciprocal medium, it can been shown [Derode et al., 2001] that the amplitude of the refocused pulse is proportional to τΔν, while the standard deviation of the sidelobes surrounding the peak is proportional to (see Figure 10).
 Now when a sequence of bits is transmitted, while the amplitude of each symbol is still τΔν, each symbol is “contaminated” by the sidelobes of the neighboring symbols. More precisely, the ISI results from the incoherent sum of the sidelobes due to roughly τ/δt symbols, so we have N0(1) ∝ τ2Δν/δt. Finally we have
 Using the approximate expression for the probability of error, we finally obtain
 The experimental slope deduced from Figure 9 is in fairly good agreement with equation (5): the value of 0.25 MHz deduced from the linear fit corresponds roughly to the 0.35 MHz experimental bandwidth (Δν).
 For a binary symmetrical channel (BSC), the channel capacity is given by [Proakis, 2000]
with p the transition probability p = P(0∣1) = P(1∣0). The maximum capacity of a BSC is 1 bit per channel use, it is reached when p = 0 (no noise) or p = 1 (every bit is surely inverted during the transmission). In a strict sense, the capacity is an ideal limit; its value should not depend on the type of modulation or on the technique of transmission. Therefore it is difficult to apprehend it from an experimental point of view. However, we can use the BER as an estimated probability of error for a given technique (TR) and a given modulation (BPSK), and from equation (6) estimate the maximum error-free data rate that we can hope to achieve in our experiment for the chosen modulation. Hence this transfer rate can be expressed as a function of the number M of antennas, the bandwidth Δν and the time lag δt between symbols. In the following, the time lag is fixed such that Δνδt = 1. When there is no external noise, the error-free transfer rate is only limited by internal noise, that is, the intersymbol interference (see Table 1). Figure 11 plots the number of antennas required to obtain a maximum transfer rate of 0.95 bit per channel use (notice that the maximum rate is 1 bit) versus the signal-to-external noise ratio. When the number of antennas becomes very large, the maximum transfer rate is intrinsically limited by the sidelobes (ISI) because of the imperfections of the time reversal. From Figure 11 we conclude that a 0.95-bit transfer rate cannot be reached for less than M = 1.6 antennas (two antennas in practice!) in the base.
Table 1. Maximum Transfer Rate D Obtained Without External Noise for M = 1 to M = 4 Antennas in the Base
Number of Antennas
5. From MISO to MIMO
 Another fundamental advantage of communications based on TR in a complex environment is that the message is spatially focused onto the receiver, with a spatial resolution that is much finer than in free space [Derode et al., 1995, 2001]. Besides, one can show that the typical amplitude of the spatial sidelobes also grows as while the amplitude of the refocused pulse at the source is proportional to τΔν. This is particularly interesting for MIMO communications between M transmitting and K receiving antennas, connected to one (MIMO single user) or different users (MIMO multiple user). Whatever the configuration, because of spatial focusing it is possible to focus different bitstreams to different receivers simultaneously within the same frequency band, the degree of “overlapping” between two receivers (spatial ISI) being once again determined by [Derode et al., 2003]. In such communication schemes, the amplitude of the ISI level on each receiver will be multiplied by and the error-free transfer rate can be inferred again. In that respect, the reverberating or “complex” character of the medium is essential since the number of receivers that can be simultaneously addressed depends on the number of independent paths within the medium.
 In this paper, we demonstrated the feasibility of a one-channel TR experiment for electromagnetic waves around 2.45 GHz. This first experiment opens interesting perspectives for SISO communications in reverberant media since TR enables us to compensate for reverberation and scattering. The use of the cavity enabled us to use commercial transceivers with a low bandwidth and nevertheless to obtain a significative pulse compression (the smaller the bandwidth, the smaller the spread factor for a given environment). The emergence of new UWB devices will lead to realistic indoor situations where though there are not as many reverberations as in our cavity, the spreading factor ratio Δν/δν will be at least as high as in our experiment. Thus the concepts we have developed will be transposable. TR MISO and MIMO communications have also been studied on the basis of small-scale experiments with ultrasound. In particular in a MISO communication, we have shown that TR reduces the level of intersymbol interference by a factor of compared to a SISO communication, M being the number of antennas on the base station. This is intrinsically related to the fact that in a TR communication scheme, the transmitter has the knowledge of the channel. When the user (and not the transmitter) has the knowledge of the channel, the capacity is not improved when using a MISO scheme instead of a SISO one. This is consistent with the predictions of information theory applied to MISO communications [Paulraj et al., 2003, p. 70]. Finally, a TR MIMO communication is well suited to take benefit of disorder and send independent messages to different users simultaneously. Indeed, the messages are naturally spatially focused onto the users, which limits the cross talking between them.
 The authors wish to acknowledge the Département de Recherche en Electromagnétisme, Laboratoire Signaux et Systèmes, Supelec, Gif-sur-Yvette, France (www.lss.supelec.fr) and particularly A. Azoulay and V. Monebhurrun for having let us use their reverberant chamber, as well as the Conseil Régional d'Ile de France and the Conseil Départemental de l'Essonne, who financed it. This work is a part of the research projects developed within the Groupement de Recherches ImCoDe (GDR 2253, CNRS).