Time Reversal
Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
has three main uses: creating an optimal carrier signal for communication, reconstructing a source event, and focusing high-energy
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...
s to a point in space. A Time Reversal Mirror (TRM) is a device that can focus waves using the time reversal method. TRMs are also known as time reversal mirror arrays since they are usually
arrays
An array is a systematic arrangement of similar objects, usually in rows and columns.
Things called an array include:
{{TOC right
Music
* In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
of transducers. TRM are well-known and have been used for decades in the optical domain. They are also used in the ultrasonic domain.
Overview
If the source is passive, i.e. some type of isolated reflector, an iterative technique can be used to focus energy on it. The TRM transmits a plane wave which travels toward the target and is reflected off it. The reflected wave returns to the TRM, where it looks as if the target has emitted a (weak) signal. The TRM reverses and retransmits the signal as usual, and a more focused wave travels toward the target. As the process is repeated, the waves become more and more focused on the target.
Yet another variation is to use a single
transducer
A transducer is a device that converts energy from one form to another. Usually a transducer converts a signal in one form of energy to a signal in another.
Transducers are often employed at the boundaries of automation, measurement, and cont ...
and an
ergodic
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies t ...
cavity. Intuitively, an ergodic cavity is one that will allow a wave originating at any point to reach any other point. An example of an ergodic cavity is an irregularly shaped swimming pool: if someone dives in, eventually the entire surface will be rippling with no clear pattern. If the propagation medium is lossless and the boundaries are perfect reflectors, a wave starting at any point will reach all other points an infinite number of times. This property can be exploited by using a single transducer and recording for a long time to get as many reflections as possible.
Theory
The time reversal technique is based upon a feature of the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
known as
reciprocity: given a solution to the wave equation, then the time reversal (using a negative time) of that solution is also a solution. This occurs because the standard wave equation only contains even order derivatives. Some media are not reciprocal (e.g. very lossy or noisy media), but many very useful ones are approximately so, including sound waves in water or air,
ultrasonic
Ultrasound is sound waves with frequencies higher than the upper audible limit of human hearing. Ultrasound is not different from "normal" (audible) sound in its physical properties, except that humans cannot hear it. This limit varies f ...
waves in human bodies, and
electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
in free space. The medium must also be approximately
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
.
Time reversal techniques can be modeled as a
matched filter
In signal processing, a matched filter is obtained by correlating a known delayed signal, or ''template'', with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal w ...
. If a
delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
is the original signal, then the received signal at the TRM is the
impulse response
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
of the channel. The TRM sends the reversed version of the impulse response back through the same channel, effectively autocorrelating it. This
autocorrelation function
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as ...
has a peak at the origin, where the original source was. It is important to realize that the signal is concentrated in both space and time (in many applications, autocorrelation functions are functions of time only).
Another way to think of a time reversal experiment is that the TRM is a "channel sampler". The TRM measures the channel during the recording phase, and uses that information in the transmission phase to optimally focus the wave back to the source.
Experiments
A notable researcher is
Mathias Fink
Mathias Fink, born in 1945 in Grenoble, is a French physicist, professor at ESPCI Paris and member of the French Academy of Sciences.
Life and career
Mathias Fink received a M.S. degree in mathematics from Paris University, and the Ph.D. deg ...
of
. His team has done numerous experiments with ultrasonic TRMs. An interesting experiment involved a single source transducer, a 96-element TRM, and 2000 thin steel rods located between the source and the array. The source sent a 1 μs pulse both with and without the steel scatterers. The source point was measured for both time width and spatial width in the retransmission step. The spatial width was about 6 times narrower with the scatterers than without. Moreover, the spatial width was less than the
diffraction limit
The resolution of an optical imaging system a microscope, telescope, or camera can be limited by factors such as imperfections in the lenses or misalignment. However, there is a principal limit to the resolution of any optical system, due to t ...
as determined by the size of the TRM with the scatterers. This is possible because the scatterers increased the
effective aperture
In electromagnetics and antenna theory, the aperture of an antenna is defined as "A surface, near or on an antenna, on which it is convenient to make
assumptions regarding the field values for the purpose of computing fields at external points. T ...
of the array. Even when the scatterers were moved slightly (on the order of a wavelength) in between the receive and transmit steps, the focusing was still quite good, showing that time reversal techniques can be robust in the face of a changing medium.
In addition,
José M. F. Moura of Carnegie Mellon University has led a research team working to extend the principles of Time Reversal to electromagnetic waves,
José M. F. Moura, Yuanwei Jin. "Detection by Time Reversal: Single Antenna",IEEE Transactions on Signal Processing, 55:1, pp. 187-201, January 2007
/ref> and they have achieved resolution in excess of the Rayleigh resolution limit, proving the efficacy of Time Reversal techniques. Their efforts are focused on radar
Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
systems, and trying to improve detection and imaging schemes in highly cluttered environments, where Time Reversal techniques seem to provide the greatest benefit.
Applications
The beauty of time reversal signal processing is that one need not know any details of the channel. The step of sending a wave through the channel effectively measures it, and the retransmission step uses this data to focus the wave. Thus one doesn't have to solve the wave equation to optimize the system, one only needs to know that the medium is reciprocal. Time reversal is therefore suited to applications with inhomogeneous media.
An attractive aspect of time reversal signal processing is the fact that it makes use of multipath propagation. Many wireless communication systems must compensate and correct for multipath effects. Time reversal techniques use multipath to their advantage by using the energy from all paths.
Fink imagines a cryptographic
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
application based on the ergodic cavity configuration. The key would be composed of the locations of two transducers. One plays the message, the other records waves after they have bounced throughout the cavity; this recording will look like noise. When the recorded message is time reversed and played back, there is only one location to launch the waves from in order for them to focus. Given that the playback location is correct, only one other location will exhibit the focused message wave; all other locations should look noisy.
See also
* Phase conjugation Phase conjugation is a physical transformation of a wave field where the resulting field has a reversed propagation direction but keeps its amplitudes and phases.
Description
It is distinguished from Time Reversal Signal Processing by the fact th ...
References
External links
Mathias Fink. Time Reversal of Ultrasonic Fields--Part 1: Basic Principles. IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control. 39(5):pp 555--566. September 1992.
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Signal processing