Reversible Computation in Wireless Communications

Rev ersible Computation in Wireless Comm unications Harun Siljak 1 CONNECT Cen tre, T rinit y College, The Universit y of Dublin, Ireland harun.siljak@tcd.ie Abstract. This c hapter presents the pioneering w ork in applying re- v ersible computation paradigms to wireless communications. These ap- plications range from dev eloping rev ersible hardw are arc hitectures for underw ater acoustic communications to nov el distributed optimisation pro cedures in large radio-frequency antenna arra ys based on reversing P etri nets. Throughout the chapter, we discuss the rationale for intro- ducing rev ersible computation in the domain of wireless comm unications, exploring the inheren tly reversible properties of communication channels and systems formed by devices in a wireless net w ork. 1 In tro duction Wireless communication systems come in dierent shap es and sizes: from radio frequency (RF) systems we use in ev eryday life, to underw ater acoustic comm uni- cations (UAC) used where RF atten uation preven ts use of radio communications. These tw o examples are of interest to this case study , as we explored the p oten- tial role of rev ersible computation in impro ving modern wireless comm unications in RF and acoustic domain. In the RF context, w e examine the concept of distributed massive MIMO (m ultiple input multiple output) systems. The distributed massive MIMO paradigm will hav e an increasing relev ance in fth generation (5G) wireless systems and p ost-5G era, as it will allo w formerly centralised base stations to op erate as a group of hundreds (thousands) of small antennas distributed in space , serv- ing many users by b eamforming the signal to them, op erating using distributed algorithms hence pro viding reduced p ow er consumption and reduced computa- tional o verhead. Our aim is to explore the application of reversible computation paradigms in such systems to contribute in additional reduction of p ow er con- sumption, but also to help in fault recov ery and meaningful undoing of algorith- mic steps in con trol and optimisation of such systems. In the underwater acoustic context, we recognised the wa ve time rev ersal sc heme as a physical example of rev ersibility , a physical method w aiting for its reversible circuit implementation. The mechanism of wa v e time rev ersal is analoguous to reversible computation as we know it, and as suc h it admits elegan t and simple circuit implementation b enetting from all reversible computation adv an tages. With this inheren t reversibilit y in mind, we take the question of wa ve 2 Harun Siljak time reversal in underwater conditions a step further, and ask ab out realistic mo dels of suc h systems using rev ersible computation paradigms, and inv estigate the options of controlling the environmen t in which this pro cess is used for comm unication. Comm unication is inherently reversible: the communication channel changes direction all the time, with the transmitter and the receiver changing roles and transmitting through the same medium. Mo dulation and demo dulation, co ding and deco dingall these pro cesses aim for information conserv ation and rev ersibil- it y . Hence the motiv ation for this study is clear: can reversible computation help in achi eving goals of modern wireless comm unication: increasing access, decreas- ing latency and p o wer consumption, minimising information losses? In this chapter, we present results on optimisation schemes for massive MIMO based on rev ersing Petri nets, reversible hardw are for wa ve time reversal, and some preliminary though ts on our w ork in progress on mo delling and control of wa v e time reversal in rev ersible cellular automata, as well as control of these automata in general. 2 Rev ersing Petri nets and Massive MIMO 2.1 The Problem In the distributed massive MIMO system describ ed in the previous section, not all antennas need to b e active at all times. Selecting a subset of an tennas to op erate at a particular time instant allows the system to retain adv antages of a large an tenna array , including interference suppression, spatial multiplexing and diversit y [16] while reducing the num ber of radio frequency (RF) chains and num b er of antennas to p o wer [13]. The computational demand of optimal transmit antenna selection for large antenna arrays [11] makes it impractical, sug- gesting the necessity of sub optimal approac hes. T raditionally , these approaches w ere centralised and based on the knowledge of the communication channel b e- t ween every user and every an tenna in the array; one widely used algorithm is the greedy algorithm [12] whic h op erates iteratively by adding the an tenna that increases the sum rate the most when joined with the set of already selected an tennas. In decen tralised algorithms similar pro cedures are conducted on m uch smaller subsets of an tennas [21], leading to similar results in o v erall performance. Our approach here is decentralised, and it relies on Reversing P etri nets (RPN) [17] as the underlying paradigm. As this chapter fo c uses on applications, the reader in terested in details ab out reversing Petri nets used in this example is advised to see [18]. The presen tation here is based on [22]. The optimisation problem we are solving is downlink (transmit) antenna selection of N T S an tennas at the distributed massive MIMO base station with N T an tennas, in presence of N R single an tenna users. W e maximise the sum- capacit y C = max P , H c log 2 det  I + ρ N R N T S H c PH H c  (1) Rev ersible Computation in Wireless Communications 3 where ρ is the signal to noise ratio (SNR), I a N T S × N T S iden tity matrix, P a diagonal N R × N R p o wer distribution matrix. H c is the N T S × N R c hannel submatrix for a selected subset of antennas from the N T × N R c hannel matrix H [10]. In the case of receiver antenna selection, addition of any antenna to the set of selected antennas improv es the o verall sum-capacity , as its e quiv alen t of equation (1) do es not inv olv e scaling by the num b er of selected antennas (i.e. there is not a p o wer budget to b e distributed ov er an tennas in the receive case). This problem is submo dular and has a guaranteed (sub optimal) p erformance b ound for the previously describ ed greedy algorithm. Greedy algorithm do es not hav ep erformance b ound for the transmitter antenna selection, as the case describ ed b y equation (1) do es not fullfil the submo dularity condition [24]; the addition of an an tenna to the already selected set of antennas can decrease c hannel capacity . As done in [24,21], we optimise (1) with tw o v ariables, the subset of selected an tennas and the optimal p ow er distribution o ver them succesiv ely: first, P is fixed to having all diagonal elements equal to 1 / N R (total p ow er is equal to ρN R / N T S ), and after the antenna selection P is optimised by water filling for zero forcing. Fig. 1 illustrates the prop osed algorithm based on RPN: the antennas are P etri net plac es (circles A-G), with the token (bright circle) in a place indicates that the curren t state of the algorithm asks for that place (that antenna) to b e on. The places are divided into o verlapping neighb ourho o ds ( N 1 and N 2 in our toy example) and each t wo adjacent places hav e a common neigh b ourho o d. T r ansitions b etw een places mov e tokens around based on the sum capacity cal- culations, with rules describ ed b elow: 1. T ransition is p ossible if there is a token in exactly one of the tw o places (e.g. B and G in Fig. 1) it connects. Otherwise (e.g. A and B, or E and F) it is not p ossible. 2. The enabled transition will o ccur if the sum capacity (1) calculated for all an tennas with a token in the neigh b ourho o d shared b y the tw o places (for B and G, that is neigh b ourho o d N 1 ) is less than the sum capacity calculated for the same neigh b ourho o d, but with the token mov ed to the empt y place (in case of B-G transition, this means C AB < C AG ). Otherwise, it do es not o ccur. 3. In case of several p ossible transitions from one place (A-E, A-D, A-C) the one with the greatest sum-capacity difference (i.e. improv ement) has the priorit y . 4. There is no designated order in transition execution, and they are performed un til a stable state is reached. The algorithm starts from a conguration of n tokens in random places and con verges to a stable final conguration in a small n umber (in our experiments, up to fiv e) of iterations (passes) through the whole netw ork. As the RPN conserv es the num b er of tokens in the netw ork, and our rules allow at most one token p er 4 Harun Siljak place, the algorithm results in n selected antennas. Executing the algorithm on sev eral RPNs in parallel (in our exp eriments, up to five) allows tokens to tra v erse all parts of the netw ork and find go o d congurations even with a relatively small n umber of antennas and users. The conv erged state of the RPN b ecomes the ph ysical state of antennas: an tennas with tokens are turned on for the duration of the coherence interv al. At the next up date of the channel state information, algorithm pro ceeds from the curren t state. The computational fo otprint of the describ ed algorithm is very small: tw o small matrix multiplications and determinan t calculations are p erformed at a no de which contains a token in a small n umber of iterations. As such, this algorithm is significantly faster and computationally less demanding than the cen tralised greedy approach whic h is a low-complexit y representativ e of global optimisation algorithms in antenna selection [11]. The worst case complexity of RPN based approac h is O ( N ω /a T ) (here, N T denotes the n umber of antennas, and ω , 2 < ω < 3 is the exp onent in the emplo yed matrix multiplication algorithm complexit y) if neighbourho o d of N 1 /a T , a > 1 suffices for RPN algorithm (as √ N T suffices in our case, we wen t for a = 2). The constant factor m ultiplying the complexit y is small because of few computing nodes (only those with tokens) and few iterations. Fig. 1: A to y mo del of antenna selection on a reversing Petri net 2.2 Results and Discussion The algorithm was tested using raytracing Matlab to ol Ilmprop [9] on a system comp osed by 64 omnidirectional antennas randomly distributed in space shown in Fig. 2(a). In all computations, channel state information (CSI) in matrix H w as normalised to unit av erage energy ov er all an tennas, users and sub carri- ers,follo wing the practice from [10]. 75 randomly distributed scatterers and one Rev ersible Computation in Wireless Communications 5 (a) Randomly distributed antennas (b) The mapping to RPN top ology Fig. 2: An tennas in physical and computational domain 6 Harun Siljak large obstacle are placed in the area with the distributed base station. The num- b er of (randomly distributed) users with omnidirectional antennas v aried from 4 to 16, and w e used 300 OFDM sub carriers, SNR ρ = -5 dB, 2.6 GHz carrier fre- quency , 20 MHz bandwidth. Antennas are computationally arranged in an 4 × 16 arra y folded into a toroid, creating a contin uous infinite netw ork, as shown in Fig. 2(b), e.g. antenna 1 is direct neigh b our of an tennas 2, 16, 17 and 49. Imme- diate V on Neumann (top, do wn, left, righ t) neigh b ours can exc hange tok ens, and o verlapping 8-antenna neighbourho o ds are placed on the grid: e.g. for antenna 1, transitions to 16 and 17 are decided up on within the neighbourho o d { 16, 32, 48, 64, 1, 17, 33, 49 } and the transitions to 2 and 49 are in { 1, 17, 33, 49, 2, 18, 34, 50 } . In Fig. 3 we compare greedy and random selection with tw o v ariants of our RPN approach: the av erage of five concurrently running RPNs, and the p erformance of the b est RPN out of those five. The p erformance is comparable in all cases, and b oth v arian ts of our prop osed algorithm tend to outp erform the cen tralised approach as the n umber of users grows. This in practice means that a single RPN suffices for netw orks with a relatively large exp ected n umber of users. The inherent reversibilit y of this problem and its solution generalises to the common problem of resource allo cation in wireless netw orks, and sharing any p o ol of resources (p ow er, frequency , etc) can b e handled b etw een an tennas (and an tenna clusters) o ver a Reversing Petri Net. At the same time, such a solution w ould b e robust to changes in the environmen t, p oten tial faults, sudden changes in the mo de of op eration, and could op erate on reversible hardware. Fig. 3: Achiev ed sum rates for 4-16 users using the proposed algorithm vs random and cen tralised greedy selection Rev ersible Computation in Wireless Communications 7 Fig. 4: The effects of imp erfect CSI and random selection of sub carriers on opti- misation In [21], it has b een sho wn that the distributed algorithms are resistant to errors in CSI and that they p erform well ev en with just a (randomly selected) subset of sub carriers used for optimisation. Results in Fig. 4 in the case of 12 users confirm this for the RPN algorithm as well. 3 Rev ersible Hardware for Time Reversal The technique called w av e time reversal [6] has b een introduced in acoustics almost three decades ago, and has since b een applied to other wa ves as well– optical and RF. In our w ork, w e fo cused on acoustic time reversal, thinking of its applications in acoustic underwater communications. How ever, it is w orth not- ing that wa v e time rev ersal plays a significant role in RF communications as w ell–conjugate b eamforming for MIMO systems is based on it. In the remainder of this section, we introduce the concept of wa ve time rev ersal and explain our prop osed solution for its reversible hardware implementation. The presen tation here follo ws the one in [20]. 3.1 W a ve Time Rev ersal Time reversal mirrors (TRMs) [6] are based on emitter–receptor an tennas p osi- tioned on an arbitrary enclosing surface. The wa ve is recorded, digitized, stored, time-rev ersed and rebroadcasted b y the same antenna array . If the array on the b oundary intercepts the en tire forward wa v e with a go o d spatial sampling, it generates a p erfect backw ard-propagating cop y . The pro cedure b egins when the 8 Harun Siljak Fig. 5: A closed surface is filled with transducer elements. The w av efront dis- torted by heterogeneities comes from a p oint source and is recorded on the ca vity elements. The recorded signals are time- reversed and re-emitted b y the elemen ts. The time-reversed field bac k-propagates and refo cuses exactly on the initial source. [7] Fig. 6: Time-rev ersal exp eriment through a diffusive medium [7] Rev ersible Computation in Wireless Communications 9 source radiates a wa ve inside a volume surrounded by a tw o-dimensional surface with sensors (microphones) along the surface which record field and its normal deriv ativ e until the field disapp ears (Fig. 5). When this recording is emitted back, it created the time-reversed field which lo oks like a con vergen t wa vefield until it reaches the original source, but from that p oin t it propagates as a div erging w av efield. This can b e comp ensated b y an active source at the fo cusing p oint canceling the field, or a passiv e sink as a p erfect absorb er. [3] This description asks for the whole surface to b e cov ered with the TRM transceiv ers, and for both the signal and the deriv ativ e to b e stored: for practical purp oses, less hardware-demanding solutions are needed. First, we note that the normal deriv ative of the field is prop ortional to the field in case the TRM is in the far field, halving the necessity for signal recording. Second, we note that a TRM can use complex environmen ts to app ear as an antenna wider than it is, resulting in a refo cusing qualit y that do es not dep end on the TRM ap erture. [4] Hence, it can b e implemented with just a subset of transceiv ers lo cated in one part of the b oundary , as seen in Fig. 6. (a) (b) Fig. 7: (a) The three realms of reversibilit y , (b) The classical (top) and the re- v ersible solution (b ottom) for the classical time reversal chain 3.2 The Design Fig. 7 illustrates the challenge of designing reversible hardware solution for a TRM: 1. The environmen t is reversible to an extent (we will return to this question later in this c hapter). The physics of w av e propagation in w ater is reversible, but the issues arise as w e lose information in the pro cess. 2. The analog computation part of the TRM loses information due to filtering and analog-to-digital/digital-to-analog conv ersion (ADC/D AC), amplifiers accompan ying the filters and the conv erters themselves, at the transition to the digital domain. 3. Finally , the digital computation part of the TRM is reversible and no increase in entrop y is necessary: writing in memory and unwriting, in the fashion of 10 Harun Siljak Bennett’s trick, enabling reuse of memory for the next incoming wa v e, while not increasing the en tropy . Analog pro cessing The real amplifier is an imp e rfect device with a limited bandwidth, hence prone to losing signal information. By definition, it tak es addi- tional energy for the signal, so it asks for an additional pow er source. At the same time, the analog to digital and digital to analog con verters both lose information b ecause of the finite resolution in time and amplitude, preven ting full reversibil- it y . How ev er, a single device can b e b oth an ADC and a DA C dep ending on the direction [14]. In this solution, we assume put bi-directional conv erters together with bi-directional amplifiers [14]. The conv ersion is additionally simplified in the one-bit solution [5] where the receiv ers at the mirror register only the sign of the wa veform and the transmitters emit the rev ersed version based on this information. It is a sp ecial case of analog-to-digital and digital-to-analog conv er- sion with single bit con verters. The reduction in discretisation lev els also means simplification of the pro cessing chain and making its reversal (bi-directivity) ev en simpler. The question of the information loss is not straightforw ard: while the information ab out the incoming w av e is lost in the con version pro cess (and the loss is maximal due to minimal resolution), spatial and temp oral resolution are not significantly degraded. This sc heme can also b e called ”one-trit” rev ersal: there are three p ossible states in the practical implementation: p ositive pressure, negativ e pressure, and ”off”. Digital pro cessing The first, straightforw ard wa y of p erforming time rev ersal of a digitally sampled wa v e is storing it in memory and reading the samples in the reverse order (last in, first out, LIFO), analoguous to storing the samples on the stac k. The design of registers in reversible logic is a w ell-explored topic [15] and b oth serial and parallel reading/writing can be implemented. Design of latches in reversible logic is a w ell-studied problem with known solutions; a com bination of latches makes a flip-flop, and a series of flip-flops makes a register (and a reversible address counter). In the case of wa ve time reversal, the recording of data is a large register b eing loaded serially with wa v e data. m bits from the ADC are memorised at the conv erter’s sample rate inside a k × m bit register matrix (where k is the num b er of samples to b e stored for time reversal). In the receiving pro cess, the bits are stored, in the transmission pro cess they are unstored, returning the memory into the blank state it started from (uncomputation). W e utilise Bennett’s trick and lose information without the en tropic p enalty: the information is kept as long as it is relev ant. When additional signal pro cessing, e.g. filtering or mo dulation is p erformed, it is conv enient to reverse w av es in frequency domain: there, time domain reversal is ac hiev ed b y phase conjugation, i.e. changing the sign of the signal’s phase. The transition from time to frequency domain (and vice versa) in digital domain is p erformed by the F ast F ourier T ransform (FFT) and its inv erse coun terpart, whic h are rev ersibly implementable [23]. The necessary phase conjugation is an arithmetic op eration of sign reversal, again reversible. Any additional signal Rev ersible Computation in Wireless Communications 11 pro cessing can b e reversible as well: e.g. filter banks and wa velet transforms. These processes remain reversible with preserv ation of all comp onents of signals [2]. Fig. 8(a) gives a comparison of the bit erasures in different implementa- tions of the digital circuitry: frequency domain (FFT) and time domain rev ersal p erformed by irreversible circuits, compared to reversible implementations. The n umber of erasures c hanges dep ending on tw o parameters: bit resolution of the ADC and the waiting time–the length of the in terv al in which samples are col- lected b efore rev ersal starts, equiv alent to the num b er of digitised samples. The increase in b oth means additional memory lo cations and additional dissipation for irreversible circuits. The irreversible FFT implementation has an additional information loss caused b y additional irreversible circuitry compared to the irre- v ersible time domain implementation. Our implementation has no bit erasures whatso ev er. The price that is paid reflects in the larger n umber of gates used in the circuit: the num b er of gates has only spatial consequences, information- related energy dissipation is zero thanks to information conserv ation. On the other hand, Fig. 8(b) sho ws the information loss in the analog part of the system, and we differen tiate tw o typical environmen ts, the chaotic cavit y and the complex (multiple scattering) medium. The chaotic ca vity is an ergo dic space with sensitive dep endence on initial conditions for w av es. In suc h an en- vironmen t there is little to no loss in the information if the waiting time is long enough and the ADC resolution is high enough. In the complex media, the dif- ference is caused by some of the w av e comp onents being reflected backw ards by the scattering environmen t, hence not reac hing the TRM. Again, more informa- tion is retained with the increase in the ADC resolution. How ever, as rep orted in [5], the information loss from low-resolution ADC use do es not affect the p er- formance of the algorithm. The analog part of the scheme remains a topic of our future w ork, as it leav es space for improv emen ts of the scheme. 4 Rev ersible Environmen t Mo dels and Con trol Time reversal describ ed in the previous section is an example of a reversible pro cess in a nominally reversible environmen t. While dynamics of water sub ject to wa v es are inherently reversible, most of the sources of the water dynamics do not reverse naturally: e.g. the Gulf stream or a motion of a school of fish. Hence, even though it would rarely b e completely reversed, the mo del for UA C should b e reversible. W e discuss the questions of rev ersible mo dels follo wing the exp osition in [19], and the w ork in progress on control of reversible cellular automata (R CA). R CA lattice gas mo dels are cellular automata ob eying the laws of fluid dy- namics describ ed by the Na vier-Stokes equation. One such mo del, FHP (F risch- Hasslac her-Pomeau) lattice gas [8] is simple and y et following the Navier-Stok es equations exactly . It is defined on a hexagonal grid with the rules of particle collision shown in Fig. 9. The FHP lattice gas provides us a tw o-dimensional 12 Harun Siljak (a) (b) Fig. 8: Information loss in (a) digital and (b) analog part of the system. Units are omitted as the particular asp ects of implementation are not relev an t for the illustration of effects. Plot (a) is obtained by coun ting op erations, plot (b) by sim ulation of back-scattering. Fig. 9: FHP rules Rev ersible Computation in Wireless Communications 13 mo del for UAC, easily implementable in softw are and capturing the neces sary prop erties of the rev ersible medium. F ollowing the exp osition in the previous section, w e observ e a model with an original source (transmitter) which causes the spread of an acoustic wa ve, the original sink (receiver) waiting for the wa v e to reach it, as well as scatterers and constan t flows (streams) in the environmen t. The constant stream and the loss of information caused by some wa ve comp onents never reaching the sink will result in an imp erfect reversal at the original source. The measure of returned p o wer giv es us a directivity pattern (fo cal p oint). The amplitude of the p eak will fluctuate based on the lo cation of the original source and may is a measure of reversibilit y , akin to fidelity or Loschmidt Echo. F or us, it is a measure of the quality of communication, but in a more general context it can measure rev ersibility of a cellular automaton. F rom the control viewp oint, it is in teresting to ask the following: if a certain part of the environmen t is controllable (i.e. a num b er of cells of the RCA do es not ob ey the rules of the RCA but allows external mo dification), how can it b e used to achiev e b etter time reversal? This is a comp ensation approach where w e engineer the environmen t to comp ensate for effects caused by sources of disturbance out of our con trol. The approac h we take is one of control of cellular automata [1], and it will b e shown that RCA are easier to control than regular CA, with easier searc h strategies and the ability to calculate control sequences. 5 Conclusions In this chapter, w e provided an o verview of results obtained in the case study on rev ersible computation in wireless comm unications. Some of the presen ted work, suc h as optimisation in massive MIMO and reversible hardware for wa v e time rev ersal is finished and sub ject to further extensions and generalisations; other w ork, mainly the parts fo cused on RCA and mo delling of rev ersible physics of comm unication, is still ongoing and more results are to come. This has b een a pioneering study into reversibilit y in communications, and the results obtained promise a lot of space for improv emen t and applications in the future, and we hop e these efforts will serve as an inspiration and a trigger for the developmen t of this field of researc h. 6 Ac kno wledgements The work presented in this c hapter was supp orted by the COST Asso ciation through the IC1405 Action on Reversible Computation, as well as a gran t from Science F oundation Ireland (SFI) co-funded under the Europ ean Regional De- v elopment F und under Grant Num b er 13/RC/2077 and Europ ean Union’s Hori- zon 2020 programme under the Marie Sk ao dowsk a-Curie grant agreement No 713567. I am grateful to my collab orators, Prof Anna Philipp ou, Kyriaki Psara, Dr Julien de Rosny , Prof Mathias Fink, and Dr F ranco Bagnoli for making this in terdisciplinary research p ossible, and to K. Popovic for the inspiring ideas. 14 Harun Siljak References 1. F ranco Bagnoli, Ra ´ ul Rech tman, and Samira El Y acoubi. Control of cellular au- tomata. Physic al R eview E , 86(6):066201, 2012. 2. Ying-Jui Chen and Kevin S Amaratunga. M-channel lifting factorization of p er- fect reconstruction filter banks and rev ersible m-band wa v elet transforms. IEEE T ransactions on Cir cuits and Systems II: Analo g and Digital Signal Pr o c essing , 50(12):963–976, 2003. 3. Julien de Rosn y and Mathias Fink. Ov ercoming the diffraction limit in wa v e ph ysics using a time-reversal mirror and a no vel acoustic sink. Physic al r eview letters , 89(12):124301, 2002. 4. Arnaud Derode, Philipp e Roux, and Mathias Fink. Robust acoustic time rev ersal with high-order m ultiple scattering. Physic al r eview letters , 75(23):4206, 1995. 5. Arnaud Derode, Arnaud T ourin, and Mathias Fink. Ultrasonic pulse compression with one-bit time rev ersal through m ultiple scattering. Journal of applie d physics , 85(9):6343–6352, 1999. 6. Mathias Fink. Time reversal of ultrasonic fields. i. basic principles. IEEE tr ans- actions on ultr asonics, ferr o ele ctrics, and fr e quency c ontr ol , 39(5):555–566, 1992. 7. Mathias Fink. F rom losc hmidt daemons to time-reversed wa v es. Philosophic al T ransactions of the R oyal So ciety A: Mathematic al, Physic al and Engine ering Sci- enc es , 374(2069):20150156, 2016. 8. Uriel F risch, Brosl Hasslacher, and Yves P omeau. Lattice-gas automata for the na vier-stokes equation. Physic al r eview letters , 56(14):1505, 1986. 9. G Del Galdo, Martin Haardt, and Christian Schneider. Geometry-based channel mo delling of mimo channels in comparison with channel sounder measurements. A dvanc es in R adio Scienc e , 2(BC):117–126, 2005. 10. Xiang Gao, Ove Edfors, F redrik T ufv esson, and Erik G Larsson. Massiv e mimo in real propagation environmen ts: Do all antennas contribute equally? IEEE T r ans- actions on Communic ations , 63(11):3917–3928, 2015. 11. Y uan Gao, Han Vinck, and Thomas Kaiser. Massiv e mimo antenna selection: Switc hing architectures, capacity b ounds, and optimal antenna selection algo- rithms. IEEE T r ansactions on Signal Pr o c essing , 66(5):1346–1360, 2017. 12. Mohammad Gharavi-Alkhansari and Alex B Gershman. F ast antenna subset se- lection in mimo systems. IEEE tr ansactions on signal pr o c essing , 52(2):339–347, 2004. 13. Jak ob Hoydis, Stephan T en Brink, and M ´ erouane Debbah. Massiv e mimo in the ul/dl of cellular netw orks: How many an tennas do we need? 2013. 14. Omid Mirmotahari and Yngv ar Berg. Pseudo floating-gate and reverse signal flow. In R e c ent A dvanc es in T e chnolo gies . Intec hOp en, 2009. 15. No or Muhammed Nay eem, Md Adnan Hossain, Lafifa Jamal, and Hafiz Md Hasan Babu. Efficien t design of shift registers using rev ersible logic. In 2009 International Confer enc e on Signal Pr o c essing Systems , pages 474–478. IEEE, 2009. 16. Ayfer Ozgur, Olivier L´ evˆ eque, and David Tse. Spatial degrees of freedom of large distributed mimo systems and wireless ad hoc netw orks. IEEE Journal on Sele cte d Ar eas in Communications , 31(2):202–214, 2013. 17. Anna Philippou and Kyriaki Psara. Reversible computation in p etri nets. In International Confer enc e on R eversible Computation , pages 84–101. Springer, 2018. 18. Anna Philipp ou, Kyriaki Psara, and Harun Siljak. Controlling reversibilit y in rev ersing p etri nets with application to wireless communications. In International Confer enc e on R eversible Computation , pages 238–245. Springer, 2019. Rev ersible Computation in Wireless Communications 15 19. Harun Siljak. Reversibilit y in space, time, and computation: the case of underw ater acoustic comm unications. In International Conferenc e on R eversible Computation , pages 346–352. Springer, 2018. 20. Harun Siljak, Julien de Rosn y , and Mathias Fink. Reversible hardware for acoustic comm unications. IEEE Communic ations Magazine , 2020. 21. Harun Siljak, Irene Macaluso, and Nicola Marchetti. Distributing complexity: A new approac h to antenna selection for distributed massive mimo. IEEE Wir eless Communic ations L etters , 7(6):902–905, 2018. 22. Harun Siljak, Kyriaki Psara, and Anna Philipp ou. Distributed antenna selection for massiv e mimo using reversing p etri nets. IEEE Wir eless Communic ations L etters , 2019. 23. Mariusz Skoneczn y , Yv an V an Rentergem, and Alexis De V os. Reversible fourier transform chip. In 2008 15th International Confer enc e on Mixe d Design of Inte- gr ate d Cir cuits and Systems , pages 281–286. IEEE, 2008. 24. Rah ul V aze and Harish Ganapathy . Sub-mo dularit y and an tenna selection in mimo systems. IEEE Communic ations L etters , 16(9):1446–1449, 2012.