Reversibility in space, time, and computation: the case of underwater acoustic communications
RReversibility in space, time, and computation:the case of underwater acoustic communications (cid:63)
Work in Progress Report
Harun Siljak [0000 − − − CONNECT Centre, Trinity College Dublin, Ireland [email protected]
Abstract.
Time reversal of waves has been successfully used in commu-nications, sensing and imaging for decades. The application in underwa-ter acoustic communications is of our special interest, as it puts togethera reversible process (allowing a reversible software or hardware realisa-tion) and a reversible medium (allowing a reversible model of the envi-ronment). This work in progress report addresses the issues of modelling,analysis and implementation of acoustic time reversal from the reversiblecomputation perspective. We show the potential of using reversible cellu-lar automata for modelling and quantification of reversibility in the timereversal communication process. Then we present an implementation oftime reversal hardware based on reversible circuits.
Keywords:
Acoustic time reversal · Digital signal processing · Latticegas · Reversible cellular automata · Reversible circuits.
The idea of wave time reversal has been considered for decades: among otherreferences, we can find an early mention in Rolf Landauer’s work [5]. The the-ory and practice of modern time reversal of waves stems from Mathias Fink’sidea of time reversal mirrors [3]. While the first theoretical and practical re-sults came from the case of sound waves (acoustics), the concept was translatedin electromagnetic domain as well, through applications in optics [11] and ra-dio technology [7]. The particular scenario we are considering here is the caseof underwater acoustic communications (UAC), an application where the poorelectromagnetic wave propagation makes sound waves the best solution.Fluid dynamics (motion of liquids and gases) using reversible cellular au-tomata (RCA) has been discussed extensively in the past [4,9]. However, this (cid:63)
This publication has emanated from research supported in part by a research grantfrom Science Foundation Ireland (SFI) and is co-funded under the European Re-gional Development Fund under Grant Number 13/RC/2077. The project has re-ceived funding from the European Unions Horizon 2020 research and innovationprogramme under the Marie Skodowska-Curie grant agreement No 713567 and waspartially supported by the COST Action IC1405. a r X i v : . [ n li n . C G ] J un H. Siljak century has seen only a few applications of RCA to macro-scale engineeringproblems such as acoustic underwater sensing [10]. Similarly, proofs of conceptfor hardware and software implementations of reversible digital signal processingexist [1], and we yet have to see it applied. This work will combine these under-utilised methods of modelling and hardware implementation and relate them totime reversal, another physical form of reversibility. The work is fully practicalin nature, as it is conducted on the UAC use case.The style of exposition is tailored to give the first introduction to time re-versal of waves to reversible computation community. Linking the two fields inboth analytic (RCA) and synthetic sense (reversible hardware) is the main con-tribution of this paper: in future publications we will present the results. Thewide approach taken, ranging from physical phenomena to cellular automataand reversible hardware is intended to show the role of different aspects of re-versibility in a single practical application. We first introduce time reversal andits application in UAC, followed by a section motivating the use of RCA inmodelling and quantification of the time reversal process. Second, we present areversible hardware solution for the time reversal processing chain and a briefset of conclusions and future work pointers.
The concept of (acoustic) time reversal is illustrated in Fig. 1 [3]. If we place asound source in a heterogeneous medium within a cavity and let it emit a pulse,this pulse will travel through the medium and reach an array of transducers placed on the cavity walls (Fig. 1(a)). If we emit what the transducers havereceived in a reversed time order (Fig. 1(b)), we will get a sound wave resemblingan echo. However, unlike an echo, this sound wave will not disperse in the cavity,but be focused instead, converging at the point where the original wave wasgenerated, at the acoustic source.Covering the whole cavity with transceivers is not feasible: not only does itask for a large number of transceivers, but sometimes the system is deployed in(partially) open space, not a cavity. Hence, the option of a localised time reversalmirror (TRM) has to be considered: only a few transceivers co-located in a singleposition that use the multipath effects resulting from multiple reflections of theemitted sound wave on the scatterers in the environment. As it turns out, it ispossible to have the time reversal effect and a coherent pulse at the original sourceif the environment is complex enough (Fig. 2(a) illustrates such an experiment).This counter-intuitive result relies on the effect of ergodicity inherent to ray-chaotic systems [2]: the wave will pass through every point in space eventuallyand collect all the environment information on the way to the mirror. From the electronic point of view, these elements are piezo transducers capableof converting mechanical to electrical energy while operating as receivers and theopposite while operating as transmitters. From the communications standpoint, theyare transceivers, and from the everyday standpoint they are microphones/speakers.eversibility and Underwater Acoustic Communications 3
Fig. 1.
The time reversal mechanism in a cavity (from [6])
Fig. 2. (a) Localised time reversal mirror with a complex propagation medium (b) Asimplified reversal scheme with a 3-D focal spot visualisation (from [6])
The application of this concept to communications is straightforward: thewave, when returned to the original sender may convey information from thereceiver (TRM) and it will be focused only at the location of the original source(preventing both eavesdropping and interference at other locations). While thereare other applications as well (e.g. localisation, imaging) we focus on the com-munications aspect as it allows us to introduce multiple transmitters and re-ceivers (multiple inputs, multiple outputs, MIMO) and analyse the effects of(irreversible) signal interference in the (reversible) model.
Time reversal in the UAC setting is an example of a reversible process in a nom-inally reversible environment. While dynamics of water (or any fluid for that
H. Siljak
Fig. 3.
Collision rules for FHP gas purpose) subject to sound waves, streams, waves and other motions are inher-ently reversible, most of the sources of the water dynamics cannot be reversed:e.g. we cannot reverse the Gulf stream or a school of fish even though their mo-tion and the effect on water is in fact reversible. Hence, even though it wouldrarely be completely reversed, the model for UAC should be reversible.RCA give us such an option through the lattice gas models [13]: cellularautomata obeying the laws of fluid dynamics described by the Navier-Stokesequation. One such model, the celebrated FHP (Frisch-Hasslacher-Pomeau) lat-tice gas [4] has had several improvements after its original statement in 1986[17], but its basic form is simple and yet following the Navier-Stokes equationsexactly. This is a model defined on a hexagonal grid through a set of rules ofparticle collision shown in Fig. 3. The model can be interpreted as an RCAvia partitioning approaches [16], but the randomness of transitions when colli-sions include more possible outcomes (as seen in the figure) has to be taken intoaccount.The FHP lattice gas provides us a two-dimensional model for UAC, easily im-plementable in software and capturing the necessary properties of the reversiblemedium. It is not a novel idea to use a lattice gas to model water, but neitheracoustic underwater communications or time reversal of waves have been ob-served through this lens before. As already noted, however, time reversal is notgoing to be conducted by running the cellular automaton backwards in time, asthat would reverse parts of the environmental flow we usually have no influenceover. The acoustic time reversal is performed the same way as in real systems,by time reversing the signal received at the time reversal mirror.The model we observe consists of the original source (transmitter) whichcauses the spread of an acoustic wave, the original sink (receiver) waiting forthe wave to reach it, as well as scatterers and constant flows (streams) in theenvironment. The constant stream and the loss of information caused by somewave components never reaching the sink will result in an imperfect reversal at Partitioning of cellular automata is an approach rules are applied to blocks of cellsand the blocks change in successive time steps. Different approaches exist, dependingon the grid shape, e.g. Margolus neighbourhood for square grids, and Star of Davidand Q*Bert neighbourhoods for hexagonal grids.eversibility and Underwater Acoustic Communications 5
Fig. 4.
The reversible hardware scheme for acoustic time reversal the original source when the roles are switched (i.e. when the time reversal mirrorreturns the wave). If we measure the power returned, we will have a directivitypattern (focal point) similar to the one in Fig. 2(b).The amplitude of the peak will fluctuate based on the location of the originalsource and may serve as a metric: a measure of reversibility. If we move thesource over the whole surface of the model and measure this metric (whoseanalogue in quantum reversibility studies is fidelity or Loschmidt Echo [14]) weobtain a heatmap of the surface with respect to the quality of time reversal. Inthe context of time reversal studies, it is used as a measure of the quality ofcommunication, but in a more general context it can measure reversibility of acellular automaton. The functionality of the model increases if we observe severaltransceivers distributed over the area (e.g. underwater vehicles communicatingwith a central communication node) and/or allow motion of transceivers. Thecomplexity of the model increases as well, and the reversibility metrics becomea measure of interference. This is the first part of our ongoing work, as weinvestigate the effects contributing to communication quality loss in the FHPmodel for UAC.
The reversibility in time of the communication scheme we use and the reversibil-ity in space of the medium both suggest that the reversibility in computationshould exist as well. Fig. 4 gives an overview of a reversible architecture we areproposing, which consists of speakers/microphones, AD/DA (analogue to digi-tal/digital to analogue) converters, Fast Fourier Transform (FFT) blocks and aphase conjugation block. This architecture is already the one used in wave timereversal–here we interpret it in terms of reversible hardware.From the electromechanical point of view, a microphone and a speaker arethe same device, running on the same physical principle, which makes the twoends of the scheme equivalent. The next element, the AD converter on one andDA converter on the other end are traditionally made in an irreversible fashionas the signal in traditional circuits flows unidirectionally. However, one of thefirst categories in the international patent classification of AD/DA converters isH03M1/02: Reversible analogue/digital converters. There has been a significant
H. Siljak number of designs proposed to allow bidirectional AD/DA conversion. With thatin mind, we may consider this step to be reversible as well.The signal received is manipulated in the Fourier (frequency) domain byconjugation (change of the sign of the complex image’s phase) as conjugation infrequency domain results in time reversal in time domain. This asks for a chainof transform, manipulation and inverse transform so the new time domain signalcan be emitted. All elements in this chain are inherently reversible. Reversibilityof the Fourier transform has been long utilised, and reversible software andcircuit implementations of its commonly used computational scheme, FFT havebeen proposed [8,12,18]. Hence, the FFT block can be considered reversible,and the Inverse Fast Fourier Transfom (IFFT) is just the FFT block with thereversed flow. Finally, the phase reversal is simply changing the sign of thehalf of the outputs coming from the FFT block, as the whole set of outputscomprises of phase and amplitude of the signal in frequency domain. Changingthe sign is the straightforwardly reversible action of subtraction from zero orsimple complementation of the number, and as such has been solved already inthe study of reversible arithmetical logic unit [15].Once we have determined the reversibility of the scheme, we note its sym-metry as well. If we fold the structure in the middle (at the conjugation block),the same hardware can be used both to propagate the inputs and the outputs.While the particular details of circuit implementation are left for future work,where details of the additional circuitry will be addressed as well, we have herepresented this scheme as a proof of concept, a reversible signal processing schemeconvenient for an implementation in reversible hardware. That is the second partof our ongoing work.
In this work in progress report, we have presented the potential of reversiblecomputation for time reversal in UAC. Future work will focus on both the mod-elling prospects using RCA and the reversible circuit implementation of the timereversal hardware. While going into more detail to cover all the practical issuesof it, future work also needs to address the appropriateness of the same or similarapproach to the question of time reversal in optics and radio wave domain. Theintegration of reversible computation with physical time reversal in this con-text opens a general discussion on the relationship of different interpretations ofreversibility and new venues for reversible computation.
References
1. De Vos, A., Burignat, S., Thomsen, M.: Reversible implementation of a discreteinteger linear transformation. In: 2nd Workshop on Reversible Computation (RC2010). pp. 107–110. Universit¨at Bremen (2010)2. Draeger, C., Aime, J.C., Fink, M.: One-channel time-reversal in chaotic cavities:Experimental results. The Journal of the Acoustical Society of America (2),618–625 (1999)eversibility and Underwater Acoustic Communications 73. Fink, M.: Time reversal of ultrasonic fields. i. basic principles. IEEE transactionson ultrasonics, ferroelectrics, and frequency control (5), 555–566 (1992)4. Frisch, U., Hasslacher, B., Pomeau, Y.: Lattice-gas automata for the navier-stokesequation. Physical review letters (14), 1505 (1986)5. Landauer, R.: Parametric standing wave amplifiers. Proceedings of the Institute ofRadio Engineers (7), 1328–1329 (1960)6. Lemoult, F., Ourir, A., de Rosny, J., Tourin, A., Fink, M., Lerosey, G.: Timereversal in subwavelength-scaled resonant media: beating the diffraction limit. In-ternational Journal of Microwave Science and Technology (2011)7. Lerosey, G., De Rosny, J., Tourin, A., Derode, A., Montaldo, G., Fink, M.: Timereversal of electromagnetic waves. Physical review letters (19), 193904 (2004)8. Li, J.: Reversible fft and mdct via matrix lifting. In: Acoustics, Speech, and SignalProcessing, 2004. Proceedings.(ICASSP’04). IEEE International Conference on.vol. 4, pp. iv–iv. IEEE (2004)9. Margolus, N., Toffoli, T., Vichniac, G.: Cellular-automata supercomputers for fluid-dynamics modeling. Physical Review Letters (16), 1694 (1986)10. McKerrow, P.J., Zhu, S.M., New, S.: Simulating ultrasonic sensing with the latticegas model. IEEE Transactions on robotics and automation (2), 202–208 (2001)11. Popoff, S.M., Aubry, A., Lerosey, G., Fink, M., Boccara, A.C., Gigan, S.: Exploitingthe time-reversal operator for adaptive optics, selective focusing, and scatteringpattern analysis. Physical review letters (26), 263901 (2011)12. Skoneczny, M., Van Rentergem, Y., De Vos, A.: Reversible fourier transform chip.In: Mixed Design of Integrated Circuits and Systems, 2008. MIXDES 2008. 15thInternational Conference on. pp. 281–286. IEEE (2008)13. Succi, S.: The lattice Boltzmann equation: for fluid dynamics and beyond. Oxforduniversity press (2001)14. Taddese, B., Johnson, M., Hart, J., Antonsen Jr, T., Ott, E., Anlage, S.: Chaotictime-reversed acoustics: Sensitivity of the loschmidt echo to perturbations. ActaPhysica Polonica, A. (5) (2009)15. Thomsen, M.K., Gl¨uck, R., Axelsen, H.B.: Reversible arithmetic logic unit forquantum arithmetic. Journal of Physics A: Mathematical and Theoretical43