Time Reversal Signal Processing
Encyclopedia
Time Reversal Signal Processing
is a technique for focusing wave
s. 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, as they are usually arrays
of transducers, but they do not have to be arrays. TRM are known and used for decades in the optical domain, and are used in the ultrasonic domain by Mathias Fink
at the École supérieure de physique et de chimie industrielles de la ville de Paris
.
Yet another variation is to use a single transducer
and an ergodic
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.
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
waves in human bodies, and electromagnetic waves
in free space. The medium must also be approximately linear
.
Time reversal techniques can be modeled as a matched filter
. If a delta function is the original signal, then the received signal at the TRM is the impulse response
of the channel. The TRM sends the reversed version of the impuse response back through the same channel, effectively autocorrelating it. This autocorrelation function 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.
of École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris. 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. Interestingly, the spatial width was about 6 times narrower with the scatterers than without. Moreover, the spatial width was less than the diffraction limit as determined by the size of the TRM with the scatterers. This is possible because the scatterers increased the effective aperture 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 is currently leading a research team working to extend the principles of Time Reversal to electromagnetic waves, 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
systems, and trying to improve detection and imaging schemes in highly cluttered environments, where Time Reversal techniques seem to provide the greatest benefit.
.
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 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.
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
is a technique for focusing wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
s. 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, as they are usually arrays
Array processing
Array processing is signal processing of the outputs of an array of sensors to:* Enhance the signal-to-interference-plus-noise ratio compared to that of a single sensor using conventional or adaptive beamforming....
of transducers, but they do not have to be arrays. TRM are known and used for decades in the optical domain, and are used in the ultrasonic domain by Mathias Fink
Mathias Fink
Mathias Fink, born in 1945 in Grenoble, is a French physicist, professor at the École supérieure de physique et de chimie industrielles de la ville de Paris, member of the French Academy of Sciences and professor at the Collège de France....
at the École supérieure de physique et de chimie industrielles de la ville de Paris
École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris
The école supérieure de physique et de chimie industrielles de la ville de Paris or ESPCI ParisTech is a chemistry and physics engineering college run by the city of Paris, France and a member of ParisTech...
.
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
Transducer
A transducer is a device that converts one type of energy to another. Energy types include electrical, mechanical, electromagnetic , chemical, acoustic or thermal energy. While the term transducer commonly implies the use of a sensor/detector, any device which converts energy can be considered a...
and an ergodic
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
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 equationWave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
known as reciprocity
T-symmetry
T Symmetry is the symmetry of physical laws under a time reversal transformation: T: t \mapsto -t.Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics.Time asymmetries...
: 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
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...
waves in human bodies, and electromagnetic waves
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
in free space. The medium must also be approximately linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
.
Time reversal techniques can be modeled as a matched filter
Matched filter
In telecommunications, a matched filter is obtained by correlating a known 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 with a conjugated time-reversed version of the template...
. If a delta function is the original signal, then the received signal at the TRM is the impulse response
Impulse response
In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...
of the channel. The TRM sends the reversed version of the impuse response back through the same channel, effectively autocorrelating it. This autocorrelation function 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 FinkMathias Fink
Mathias Fink, born in 1945 in Grenoble, is a French physicist, professor at the École supérieure de physique et de chimie industrielles de la ville de Paris, member of the French Academy of Sciences and professor at the Collège de France....
of École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris. 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. Interestingly, the spatial width was about 6 times narrower with the scatterers than without. Moreover, the spatial width was less than the diffraction limit as determined by the size of the TRM with the scatterers. This is possible because the scatterers increased the effective aperture 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 is currently leading a research team working to extend the principles of Time Reversal to electromagnetic waves, 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
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
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 inhomogenous mediaHomogeneity (physics)
In general, homogeneity is defined as the quality or state of being homogeneous . For instance, a uniform electric field would be compatible with homogeneity...
.
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 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.