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Characterization of Space-Time Focusing in Time Reversed Random Fields Claude Oestges, Arnold D. Kim, George Papanicolaou, and Arogyaswami J. Paulraj
Abstract— This paper proposes various metrics to characterize space-time focusing resulting from application of time reversal techniques in richly scattering media. The concept and goals of time reversal are presented. Pertinent metrics describing both the time and space focusing effects are outlined. Two examples based on a model of discrete and continuous scattering media are used to illustrate how the proposed metrics vary as a function of various system and channel parameters, such as the bandwidth, delay and angle spreads, number of antennas, etc. Index Terms— Space-time focusing, random media, time reversal.
I. I NTRODUCTION
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N a time reversal (TR) experiment, a transducer captures the response received from an impulsive point source, and re-emits the time reversed version of this response into the propagation medium. For non-dissipative media the emitted signal back-propagates and focuses in both space and time at the original impulsive source [1]–[3]. For richly scattering media, this space-time compression can be very strong. This basic principle is well known in acoustics and has lead to remarkable applications in underwater sound [4]–[12] and ultra-sound [13]–[15]. The extension of TR techniques to radio electromagnetic propagation for wireless communications has yet to be investigated. However, the idea of exploiting scattering is not new. Indeed, there has been recently a tremendous activity in exploiting the richness of the scattering medium in spacetime communications by using multiple antennas at both transmit and receive ends (i.e. MIMO or Multiple-Input/MultipleOutput systems) [16]. However, current communication systems still rely on a fairly small communication bandwidth times channel delay-spread product B × τRMS . By using a large B × τRMS , it is believed that the transmitter can use an additional leverage of TR techniques to offer power gain and diversity gain together with space-time focusing. More precisely, TR techniques could be used to: Manuscript Draft: January 19, 2004 C. Oestges is a Post-Doctoral Fellow of the Belgian NSF - Fonds National de la Recherche Scientifique (FNRS), associated with the Microwave Laboratory, Universit´e catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (email:
[email protected]) A. D. Kim is a Postdoctoral Fellow at the Department of Mathematics of Stanford University, Stanford, CA 94305-2125, USA (email:
[email protected]) G. Papanicolaou is a Professor at the Department of Mathematics of Stanford University, Stanford, CA 94305-2125, USA (email:
[email protected]) A. J. Paulraj is a Professor at the Information Systems Laboratory, Stanford University, Stanford, CA 94305, USA (email:
[email protected])
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Reduce interference/intercept probability in view of secure communications. By selectively focusing the energy in both space and time at a target point, TR ensures that intercept receivers will have difficulty detecting or decoding the intended signal. Similarly, the co-channel interference can be strongly reduced in cellular networks. Shorten the temporal channel response. The use of TR can dramatically lower the effective delay-spread at the receiver, and lead to lower receiver complexity.
The goal of the paper is dual. First, characteristic metrics describing space-time focusing are presented in the context of time-reversed signals in wireless communications. They are estimated for two simplified models of random fields: a geometrical scattering model and a random-medium waveguide. Then we analyze the impact of system and channel parameters on the focusing by investigating the variation of the various metrics as a function of the channel properties (delay and angle spreads) and the system parameters (bandwidth, number of antennas, data rate, etc.). II. T IME R EVERSED R ANDOM F IELDS A. Channel Impulse Response Consider a transmission between transmit point P and receive point Q. The channel impulse response (CIR) is denoted as hB (τ, P → Q), where B is the bandwidth of the transmitted pulse. It is important to note that • •
the symmetry properties of usual transmission channels imply that hB (τ, P → Q) = hB (τ, Q → P) the CIR depends on the bandwidth of the transmitted pulse.
The so-called CIR is actually the convolution of the infinitebandwidth physical channel response and the filter impulse response. In this paper, the filter is implemented as a Nyquist filter with given roll-off factor. By default, the roll-off is taken as equal to zero, so the default filter is a rectangular window in the frequency domain. Depending on the scattering channel and the bandwidth, the CIR results in a temporal spreading of the initial pulse. Scatterers indeed create multipath mechanisms which, in turn, cause echoes to arrive at the receiver with different delays. However, the resolvability of the different delays depends upon the ratio of the inverse of the bandwidth to the physical channel spread (i.e. the interval between successive delays). The smaller 1/B is relative to the channel delay spread, the larger is the number of resolved paths.
IEEE TRANSCATIONS ON ANTENNAS AND PROPAGATION
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B. Time Reversal 1) Time domain relationships: Using the above formalism, the received signal at any point R for a Nyquist pulse emitted from R0 is hB (τ, R0 → R). At a particular point T, which we define as the transmitter, hB (τ, R0 → T) is captured. If the transmitter sends back the time reversed version of the captured signal, i.e. hB (−τ, R0 → T), then at any point R, the received signal can be expressed as sB (τ, R) = hB (τ, T → R) ? hB (−τ, R0 → T)
(1)
(2)
As a consequence of (2), the time-reversal operation causes the received signal at R0 to be focused in both time and space through constructive interference, meaning that all multipath signals add coherently at R0 , and incoherently elsewhere. 2) Frequency domain relationships: Models usually compute the channel transfer function HB (ω, R0 → T) over the system bandwidth (where ω is the angular frequency). It is the Fourier transform of hB (τ, R0 → T). Relationships (1) and (2) are naturally easily written in the frequency domain, since the time-reversal operation corresponds to a complex conjugation in frequency (denoted by the superscript *): ∗ SB (ω, R0 ) = HB (ω, T → R0 )HB (ω, R0 → T)
R0 D
where ? denotes the convolution product. Note that (1) assumes a perfect estimation of hB (τ, R0 → T). In practical settings, noise and interference considerations will cause this estimation to be biased. We shall not cover the impact of imperfect channel estimation in this paper. From now on, we define the point R = R0 as the focal or target point. Based on (1) and on the symmetry properties mentioned above, the signal received at R0 is sB (τ, R0 ) = hB (τ, R0 → T) ? hB (−τ, R0 → T)
limiting the aperture of the annular region to a given portion, i.e. specifying minimum and maximum angles, ϑm and ϑM (see Fig. 1). In this paper, we refer to ∆ϑ = ϑM − ϑm as the scattering angle-spread or simply the angle-spread.
(3)
According to (3), the time-reversal operation is equivalent to a perfect channel matching. We shall illustrate the characterization of sB (τ, R), i.e. the quality of space-time focusing in two different space-time random fields. The first simulated random field corresponds to a typical wireless radio channel at 2.5 GHz. The second one is a continuous heterogeneous medium consisting of a filled waveguide also operating at 2.5 GHz.
T
ρm
ϑM
Fig. 1.
ρM
ϑm
Geometrical representation of the propagation model.
We further simplify the channel description by choosing ρM = 2ρm and ρM
