Field-Assisted Molecular Communication: Girsanov-Based Channel Modeling and Dynamic Waveform Optimization

1 Field-Assisted Molecular Communication: Girsano v-Based Channel Modeling and Dynamic W a v ef orm Op timiza tion Po-Chun Chou, Y en-Chi Lee, Member , IEEE, Chu n -An Y ang , Graduate Studen t Member , IEEE, Chia-Han L e e, Me m b er , IEEE, and Ping-Cheng Y e h , Membe r , IEEE Abstract —Analytical modeling of field-assisted mo lecular com- munication under dynamic electric fields is fundamentally challenging due to the coupling between stochastic transport and complex boundary geometries, which renders conv entional partial differential equation (PDE) approac hes intractable. In this work, we introduce a stochastic framew ork b ased on the Camero n–Martin–Girsanov theore m to address this challenge. By lev eraging a change-of-measure technique, we deriv e analyt- ically tractable chann el impulse response (CIR) expressions for both fu lly-absorbing and p assive sph erical receiv ers, where t h e latter serv es as an exact mathematical baseline to validate our framewo rk. Building upon these models, we establi sh a dynamic wa vef orm design framework fo r system optimization. Under a maximum a posteriori decision-feedb ack equalizer (MAP- D F E) framewo rk, we show that th e first-sl ot recei ved probability serv es as the p rimary determinant of the b i t error probability (BEP), while inter -symbol interference manifests as higher -order corr ec- tions. Expl oiting the monotonic response of the fully-absorbing architecture and usi n g the limitations of th e passive model to justify this strategic fo cus, we reform ulate BEP minimization into a distance-based optimization problem. W e propose a un ified, low-complexity Maximize Receiv ed Probability (MRP) algorithm, encompassing the Maximize Hitting Probability (MHP) and Maximize Sensin g Probability ( M SP) methods, to dynamically enhance d esired signals and su ppress inter-sy mbol interference. Numerical results validate th e accuracy of the proposed modeling approach and demonstrate near -optimal detection performance. Index T erms —Molecular communication, drift–diffusion envi- ronment, time-varying d rift, chan nel impu lse response, receiv er design. I . I N T R O D U C T I O N R ECENT ad vances in nanotechnolo gy have enab led the development of nan omachine s which, despite in divid- ual physical constraints, can achieve complex objectiv es throug h coop e rativ e nano networks [ 1 ], necessitating effectiv e This wo rk was supported by the National Scie nce and T e chnology Counc il of T aiwa n (NSTC 113-2115-M-008-013-MY3). (Correspondin g author : Y en- Chi Lee.) P . -C. Chou and P .-C. Y eh are with the Graduate Institute of Commu- nicat ion Engineerin g, Nationa l T aiwan Uni versit y , T aipei , T aiw an. (e-mail: d02942012@ntu. edu.tw, pcyeh@nt u.edu.tw ) Y .-C. Lee is with the Departme nt of Mat hematics, Nationa l Central Uni- versi ty , T aoyua n, T aiw an. (e-mail : yclee@ma th.ncu.edu.tw ) C.-A. Y ang is with the Department of Compute r Science, Nationa l Tsing Hua Uni versi ty , Hsinchu, T aiw an. (e-mail: s110065505@m11 0.nthu.edu.tw ) C.-H. Lee is with the Institute of Communicati ons E ngineering, Na- tional Y ang Ming Chia o Tung Uni versit y , Hsinchu, T aiwan. (e-mail: chiahan@nycu.e du.tw ) paradigm s such as m o lecular co mmunicatio n (MC) [ 2 ], [ 3 ]. Among various MC mech anisms—including dif fusion [ 4 ], [ 5 ], gap junctions [ 6 ], [ 7 ], and m o lecular or bacter ial m otors [ 8 ], [ 9 ]—molecular commu nication via diffusion (MCvD) is the most extensiv ely inv estigated and serves as the focus of th is work. T o en sure reliable, time-slotted infor mation exchang e, accurate end-to-e nd ch annel mod els encom p assing emission, diffusion, and re ception are essential [ 10 ]. In this stud y , we lev erage e xisting techniques to assum e per fect sy nchron ization between th e transmitter (Tx) and r eceiv er (Rx) [ 11 ], [ 12 ], focusing ou r analysis on the u nderlyin g chan nel dyn a m ics. A critical consequ ence of these diffusion-driven dyn amics is th e inheren t chann el m e mory , whe re residual molecules from pr evious intervals cause inter-symbol inter ference (ISI) and degrad e p erforma n ce. While various strategies ha ve been propo sed to mitigate I SI—ranging fr om adv anced detection filters [ 10 ], [ 13 ] and chemical-based solutio ns [ 14 ], [ 15 ] to en viron mental flow-based ap proache s [ 16 ], [ 17 ]—an other line of r esearch explor es using external assisti ve fields to actively regulate tran sport. For instance, constant electric [ 18 ] or mag - netic fields [ 19 ] can accelerate m olecular prop a gation. How- ev er, fixed-field designs are limited in time-slotted systems because ch a nnel memo ry effects vary across in tervals. This motiv ates th e design o f assisti ve fields as contro llable, time- varying f unctions that can b e dynam ically shaped to match the underly ing channel dynam ics. In biomedica l com m unication scenarios, information- bearing par ticles are often ions, wh ich natur a lly facilitates transport regulation via assisti ve electr ic fields [ 20 ]. I n this work, we inves tigate such electrically assis ted M Cv D systems by incor p orating two rep r esentativ e m odels [ 21 ]: the passive (P A) and the fu lly-absorb ing (F A) spherical receivers. The P A model, wh ich y ields an exact clo sed-form solutio n u n der ou r framework, is explicitly include d as a rig orous benchmar k to establish math ematical consistency and to theo retically justify why sub seq uent system optim izations must prior itize the F A architecture. While characterizing the channel imp u lse response (CIR) is e ssential for p e rforman ce ev aluation, the interplay betwee n spherical boun d ary c o nditions and time- varying electric fields—descr ibed by the Nernst–Plan ck equ a- tion—rend ers con ventional p artial-differential-equ ation (PDE) approa c h es analytically intractab le [ 22 ], [ 2 3 ]. This shift from simpler cu bic geometr ies [ 2 4 ] to physically r epresentative spherical models ind uces n ontrivial bound ary-value pro blems that create a sig n ificant analytical bo ttleneck. T o overcome 2 this, it is necessary to d eriv e fu n damentally new received probab ility expressions, which can then serve as the basis for robust symb ol r ecovery under detection frameworks such as the maximu m a posteriori (M AP) rule paire d with a Decision- Feedback Equ alizer (DFE) [ 2 5 ], [ 26 ]. T o overcome this ana ly tical bottleneck , we emp loy a stochastic framework that bypasses the direct solution o f complex PDEs by reweighting particle trajectories [ 27 ], [ 28 ]. Rather than tac kling the Nernst– Plan ck eq uation head-o n, this approa c h leverages the Cameron –Martin–Gir sanov theo r em [ 29 ] to transfo rm prob a bility measu r es, effecti vely cap turing the influe nce of time-varying ele c tr ic field s as a reweighting of free - diffusion paths. By applying this change-of -measure technique , we derive analytical tractab le CIR expressions for both P A an d F A spherical receivers. These expressions are then subseq u ently integrated into a time- slo tted MCvD system model, enablin g a rigorou s performance analysis under the MAP-DFE framework. Beyond perf ormance characteriza tio n, ou r analy sis iden- tifies the recei ved probab ility as the primar y determinant of system bit erro r p robability (BEP). This insight allows us to reform ulate th e complex BEP m inimization task into a more tr a ctable o bjective: m aximizing the received prob- ability throug h the optim a l design of th e time-varying as- sisti ve electric field . Specifically , we pro pose two receiver - specific strategies— Maximize Sen sing Pr o bability ( MSP) fo r P A receivers and Maximize Hitting Pr obab ility (MHP) for F A re ceiv ers—both of which are executed thr o ugh a unified Maximize Receive d Pr obability (MRP) algo rithm. By map ping the o ptimization o nto a distance- based formu lation, the M RP algorithm achieves ne a r-optimal performan c e with remark ably low compu tational com plexity , offering a scalab le solution for various field-assisted molecular co mmunicatio n scen arios. The main contributions of this work are summar ized as follows. (i) Core Measure-Theoretic Framework under Dy na mic Drift: W e pr o pose a novel stochastic f ramew ork leverag- ing the Camer on–Martin –Girsanov theorem to circumvent the analytica l intractability of Nern st–Planck PDEs. By introdu c ing the Effective Drift Appr oximation (EDA) , we decoup le the time-varying comp osite drif t f rom com- plex spherical boun daries, der iving analytically tractable CIR expressions via effecti ve drif t mappin g for fu lly- absorbing receivers, and exact CIR expr essions for p as- si ve ones. (ii) Fundamental Insights into Receiver Dynamics: Build- ing on this framework, we furthe r show tha t a critical physical distinction exists between receiver architectu res. W e mathe m atically demo nstrate that the fully- a bsorbing receiver exh ib its a robust, mo notonic response to as- sisti ve field s, ensuring a stable optimization lan dscape. Con versely , the passi ve receiver suffers f rom a high-dr ift “overshoot” phenom enon. By contr a sting these beh aviors against ou r exact b aseline, we m athematically justify the strategic focus on a cti ve architectur es fo r reliab le nanon e twork design. (iii) Unified W avef orm O ptimization (MRP Algor ithm): Recognizing that the received probab ility domina tes the system BEP , we refor mulate the intrac ta b le BEP min- imization into a tractable distance- minimization prob- lem. W e propo se the MRP algorithm , which intelligently allocates the energy b udge t into a two-phase wa ve- form: a signa l-enhancin g acceleration phase and an ISI - suppressing cou nter-drift p hase. (iv) Near-Optimal Perf ormance with Lo w Complexity: Ex- tensiv e numerical ev aluations validate the accur a cy of our derived CIRs. Under a MAP-DFE d e tection f ramew ork, the ligh twe ight MRP algo rithm achieves near-optimal BEP perfo rmance comp a r able to exha u sti ve search meth- ods, maintaining r obustness even un der imp erfect channel state info rmation (CSI). The re m ainder of th is pap er is organized a s follows. Sec- tion II establishes the ph ysical system model and introduce s the measur e -theoretic stoch astic fram ew ork for character iz- ing par ticle dynam ics un der time-varying dr ift. Section I II derives th e closed -form analytical CIR expre ssion s for both fully-ab so rbing an d passive spherical receivers. Sectio n IV presents the discrete-time signal mod el and in vestigates the fundam ental phy sical pro perties of the chan nel co efficients, specifically hig hlighting the distinction betwee n receiv er archi- tectures. Sec tion V details the symbol detection rule an d the ev aluation o f BEP un d er the MAP-DFE fr amew ork. Section VI formu late s the wa veform optimizatio n problem an d introdu ces the unified M RP a lgorithm for assisted electric field d esign. Numerical results and v alidations are presented in Section VII, and finally , Section VIII concludes the p aper and su m marizes the key findings. I I . S Y S T E M M O D E L A N D S T O C H A S T I C F O R M U L AT I O N T o characterize the end -to-end be h avior of Field -Assisted Molecular Communica tio n (F AMC) systems, it is necessary to establish a tra ctable stochastic model that encapsula te s b oth the ph y sical r e ceiv er geometr y and th e particle dynamics under a tim e-varying drift. In th is section, we de fin e a n d set u p the p hysical environment an d intr oduce a measure-th e o retic framework that serves as the universal analytical engine for our ch a nnel deriv ations. A. Ph y sical System and Receiver Models W e con sider a m olecular co mmunicatio n system o perating in a three- d imensional (3-D) flu idic environment, as illustrated in Fig. 1 . The system comprises a po int Tx lo cated at x 0 and a spherical Rx o f r adius 𝑟 Rx centered at th e o rigin. The Eu clidean distance from the Tx to the Rx c e nter is denoted by 𝑑 0 = k x 0 k . Communica tio n is analyzed in a co ntinuou s-time fram ew ork starting fr om an initial emission. At 𝑡 = 0 , the Tx impu lsi vely releases a designated q u antity of 𝑀 c harged message particles (ions) into the flu idic environment. W e f ocus o ur analysis o n the underly in g tran sp ort dynamics and the resulting reception statistics, which form the f u ndamen tal basis for subsequent discrete-time system modeling . T o c o mprehe nsi vely characterize the re ception p r ocess, we in vestigate two distinct spherical Rx mod els, or dered by their biological relevance an d analytical c omplexity: 3 𝑥 1 -axis 𝑥 2 -axis 𝑥 3 -axis Origin Spherical Rx 𝑟 Rx Point Source Tx at x 0 Message Particles Composite Drif t 𝚽 ( 𝑡 ) 𝑑 0 = k x 0 k Fig. 1: System illustration o f MCvD with a time-varying assisted electric field. A p oint Tx located at x 0 releases ch arged particles, while a spher ical Rx centered at the o r igin captures the signal either p a ssi vely or via full absorption. The trajecto- ries ar e influenc ed by a composite dr ift field 𝚽 ( 𝑡 ) , combin ing periodic backgrou nd flow with an externally designed electric field align ed along th e 𝑥 1 -axis. • Fully-Absorbing (Active) Rx : Particles ar e perma nently removed fro m th e fluidic ch annel upon their first contact with the receiver surface, defined by R hit = { x | k x k = 𝑟 Rx } . This absorb ing boun d ary accurately mo dels th e ligan d- receptor b inding mec h anisms prevalent in targeted dr ug delivery and biological cells [ 30 ]. Due to its pra ctical significance in bio-n anomach ines, th is mode l co nstitutes the primar y focus of our system design. • Passiv e Rx: Message particles ca n fre e ly enter and exit the sensing volume R s = { x | k x k ≤ 𝑟 Rx } without un d ergoing ch emical reaction s o r phy sical ab - sorption [ 31 ]. While less represen tati ve o f biolog ical receptors, this “transparent” bo undary serves as a fun- damental theoretical b aseline and a mathem atical co nsis- tency check for our u nified fram ework. The pro p agation o f the charged particles is governed by a 3 -D compo site dr ift field. This field consists of a period ic backgr o und flow u ( 𝑡 ) [ 17 ] a n d an externally controlled, time- varying assisted electric field E ( 𝑡 ) [ 18 ]. Th e co mposite drift field is defined a s 𝚽 ( 𝑡 ) : = u ( 𝑡 ) + 𝑐 𝑒 E ( 𝑡 ) , (1) where 𝑐 𝑒 = 𝐷 𝑧 𝑒 𝑘 𝐵 𝑇 is th e mob ility constant, with 𝐷 being the d iffusion co efficient, 𝑧 the ion valence, 𝑒 the elemen- tary charge, 𝑘 𝐵 the Boltzmann constan t, and 𝑇 the absolute fluid temperature. In this g eneralized fr amew ork, th e spatially unifor m com posite d rift 𝚽 ( 𝑡 ) is tr eated as an arbitrar y time- varying vector field, enab ling th e c h aracterization of particle dynamics und er comp lex, no n-stationary environmental flows and assistiv e e le c tr ic fields. T ABLE I: Key Notations Symbol Description 𝑟 Rx , R s , R hit Rx radius, sensing region, and ab- sorbing bo u ndary . 𝐷 , 𝑐 𝑒 Diffusion co efficient and m obility constant. 𝚽 ( 𝑡 ) , E ( 𝑡 ) Composite drift and assisted electric field. 𝜉 , 𝛽 T otal energy budget and sup pression onset fractio n. P , Q , 𝑀 𝑡 Reference/ph ysical me asures and likelihood ratio. 𝚽 eff ( 𝑇 ) T ime-averaged effective drift field over [ 0 , 𝑇 ] . 𝑓 𝚽 hit ( 𝑡 ) , 𝑝 s ( 𝑇 ) FHT d ensity (ac tive) and sensing prob. (p assi ve). 𝑇 𝑏 , 𝑡 𝑠 , 𝑡 peak Symbol duration , sampling time, and peak time. 𝑝 R [ 𝑖 ] Receiv ed probab ility in the 𝑖 -th time slot. 𝑉 1 , 𝑉 2 , 𝑞 S , 𝑞 R Field magn itu des and ideal veloci- ties for signal/ISI. B. Sto chastic P article Dynamics un der T ime-V arying Drift The co ncentration evolution of the charged par ticles un- der the Nernst–Plan ck equ a tion can b e translated into th e stochastic trajectory of a single ion v ia a stochastic differential equation (SDE). The Itô process g overning th e position X 𝑡 of a single ion at time 𝑡 is g i ven by 𝑑 X 𝑡 = 𝚽 ( 𝑡 ) 𝑑 𝑡 + √ 2 𝐷 𝑑 B 𝑡 , (2) where B 𝑡 is a standard 3 -D Brownian mo tion defined on a probab ility space ( Ω , F , P ) with B 0 = 0 , and 𝚽 ( 𝑡 ) acts as the deterministic, time-varying drift. Evaluating the first-hittin g time ( FHT) density f o r the ab- sorbing bound ary un der a time-varying drift direc tly from the Fokker–Planck PDE is notor iously intractable. T o b y pass this barrier, we emp loy a measure-theor etic ap proach lever - aging the Cameron–M a rtin–Girsanov (CMG) th eorem [ 29 ], [ 32 ]. This fr amew ork allows us to analytically decou ple any arbitrary time-varying drift vector from the comp lex b ound a r y geometry by transfor ming th e pr obability measure. Specifically , rath e r than solvin g the PDE directly , we co n- sider the p rocess und er a referen ce measur e P where it behaves as a pu re diffusion (zero -drift) process, i.e., 𝑑 X 𝑡 = √ 2 𝐷 𝑑 B 𝑡 . Since the physical drift 𝚽 ( 𝑡 ) is d eterministic and boun ded, it satisfies the Novikov cond itio n [ 29 ]. Accordin g to th e CMG theorem, ther e exists an e q uiv alent p robability measu r e Q under which the process exhibits th e true time- varying drift. The transfo r mation between the p hysical m e a sure Q and the referenc e m easure P is governed by the Radon–Nikody m 4 deriv ativ e (i.e., the likeliho od ratio pro cess), 𝑀 𝑡 = 𝑑 Q 𝑑 P = exp   𝑡 0 𝚽 ( 𝛼 ) √ 2 𝐷 · 𝑑 B 𝛼 − 1 2  𝑡 0 k 𝚽 ( 𝛼 ) k 2 2 𝐷 𝑑 𝛼  . (3) Let 𝐴 deno te the arbitrar y event o f a pa rticle arriving at a target spatial state x at observation time 𝑇 . Utilizing th e ED A (see Append ix . A ) over the interval [ 0 , 𝑇 ] , wher e the effectiv e drift is defined as the time-average 𝚽 eff ( 𝑇 ) : = 1 𝑇  𝑇 0 𝚽 ( 𝛼 ) 𝑑 𝛼 , we can evaluate the prob ability of e vent 𝐴 und er the phy sical measure Q by evaluating the stoch astic integral using th e endpo in t X 𝑇 − x 0 = √ 2 𝐷 B 𝑇 , Q ( 𝐴 ) = E P [ 𝑀 𝑇 · 1 𝐴 ] ≈ exp  𝚽 eff ( 𝑇 ) · ( x − x 0 ) 2 𝐷 − k 𝚽 eff ( 𝑇 ) k 2 𝑇 4 𝐷  · P ( 𝐴 ) , (4 ) where P ( 𝐴 ) is the probability o f the e vent u n der p ure dif fusion (zero d rift), and the expo nential term serves as a geom etric weighting factor en capsulating the energy an d directio nality of the assisted electric field . T he ED A should be in terpreted as a path - av eraged dr ift app roximatio n . In particular, th e approx imation is m o st accurate when the v ariation of th e drift is slow relative to the characteristic hitting tim e scale. This is empirically validated in Fig. 4 . Impo rtantly , the ap proxim a tio n preserves the exponen tial tilting structure induced by the Girsanov transforma tion. Expressed in terms o f pro bability density functions, the joint spatial-tempor al distribution 𝑓 𝚽 ( x , 𝑇 ) u nder th e time-varying electric field can be seam lessly derived fro m the zer o-drift distribution 𝑓 ( x , 𝑇 ) via 𝑓 𝚽 ( x , 𝑇 ) = exp  𝚽 eff ( 𝑇 ) · ( x − x 0 ) 2 𝐷 − k 𝚽 eff ( 𝑇 ) k 2 𝑇 4 𝐷  𝑓 ( x , 𝑇 ) . (5) This exponential tilting relationship ( 5 ) serves as our univer - sal mathematical eng ine. In the subseq uent section , we demon - strate how this engine elegantly resolves the CIR for both th e formida b le fully- absorbin g boundary and the baseline passi ve sensing volume, pr ovid ing a general mapping regard less of the relative or ientation between th e drif t vector and the Tx– Rx axis. I I I . A NA L Y T I C A L C H A N N E L I M P U L S E R E S P O N S E S Building on the stochastic modelin g framework developed in Section II , this section evaluates the CIR fo r elec trically assisted MCv D system s. The CIR en capsulates th e pro bability that a single m essage particle, im pulsiv ely re le a sed at th e transmitter at time 𝑡 = 0 , is successfu lly captur ed or sensed at the r e c ei ver at ob ser vation time 𝑇 . W e first tackle the primary challeng e of this work: c h ar- acterizing the h itting density for a fu lly-absorb ing bou n dary under a time-varyin g electric field. Subsequ ently , we apply the ide n tical measure-th eoretic engine to the p a ssive receiver to verify the theore tical co nsistency of o ur u nified framework. A. CIR of the Fully-A bsorbing Receiver For the F A (active) receiver , the CIR is d efined by the FHT density . Unlike p assi ve observation, absorption is a path- depend ent pr ocess; if a par ticle touch es the boun dary R hit at any tim e 𝑡 ≤ 𝑇 , it is perman ently r e m oved from the fluidic channel. Recently , the exact analy tical CIR for a F A spherical receiver und er a consta n t uniform drift v with an arbitrary direction was established in [ 28 ]. Let 𝑓 ( v ) hit ( 𝑡 ) denote this zero-varying- drift CIR. By utilizing the joint time-lo c ation distribution and a spatial measure change, it was sho wn in [ 28 ] that 𝑓 ( v ) hit ( 𝑡 ) = e xp  − v · x 0 2 𝐷 − k v k 2 𝑡 4 𝐷  √ 4 𝐷 √ 𝜋 k v k 1 / 2 k x 0 k 1 / 2 × ∞  𝑚 = 0  𝑚 + 1 2  𝐼 𝑚 + 1 2  k v k 𝑟 Rx 2 𝐷  𝑃 𝑚 ( cos 𝜓 ) ×  ∞ 0 H 𝑚 ( 𝜆 ) 𝑒 − 1 2 𝜆 2 𝑡 𝑑 𝜆, (6) where 𝜓 = ∠ ( v , x 0 ) is the drif t ang le, 𝐼 𝜈 ( · ) is th e mo dified Bessel fu nction of the first kind, 𝑃 𝑚 ( · ) is the Legendre polyno mial, a n d H 𝑚 ( 𝜆 ) is the geometr y-depen dent spectral kernel comp osed of cross-prod ucts of Bessel function s (de- tailed definition s can be foun d in [ 28 ]). Howe ver , d irectly solving the Fokker–Planck eq uation or applying trad itional boun dary value methods for a time- varying compo site drift field 𝚽 ( 𝑡 ) is analytically intractab le. T o bridge this critical g ap, we le verage the un iversal Girsanov- based transf o rmation formu la te d in Section II . Under the ED A, the highly comp lex tempora l variations of the electric field over th e interval [ 0 , 𝑇 ] ar e integrated into the likelihood ratio pro cess ( 4 ). This e ssentially linearizes the path-dep endent weighting, allowing us to decouple the time-varying field from the spherical absorb ing boun dary . Mathematically , this maps the tim e-varying problem into an eq uiv alent constant- drift scenario, wh ere th e equ ivalent constant drift is precisely the time- av eraged effective drif t 𝚽 eff ( 𝑇 ) = 1 𝑇  𝑇 0 𝚽 ( 𝛼 ) 𝑑 𝛼 . Consequently , th e analytically tra c table CIR via effective drift map ping u nder the time-varying elec tr ic field, denoted by 𝑓 𝚽 hit ( 𝑇 ) , can be elegantly o btained by substituting the constant dr ift v in ( 6 ) w ith ou r effective d rift vector 𝚽 eff ( 𝑇 ) . This sub stitutio n inh erently con stitutes a quasi-static appro x- imation b a sed on th e ED A. By trea tin g the effective drift as a co nstant repr esenting th e time-averaged path weighting over the observation interval, this ap proach m aintains high computatio nal efficiency while pr eserving analytical accu r acy , particularly whe n th e time-varying field chang es relatively slowly compared to the charac teristic hitting time. Namely , 𝑓 𝚽 hit ( 𝑇 ) ≈ 𝑓 ( v ) hit ( 𝑇 )      v = 𝚽 eff ( 𝑇 ) = e xp  − 𝚽 eff ( 𝑇 ) · x 0 2 𝐷 − k 𝚽 eff ( 𝑇 ) k 2 𝑇 4 𝐷  × √ 4 𝐷 √ 𝜋 k 𝚽 eff ( 𝑇 ) k 1 / 2 k x 0 k 1 / 2 × ∞  𝑚 = 0  𝑚 + 1 2  𝐼 𝑚 + 1 2  k 𝚽 eff ( 𝑇 ) k 𝑟 Rx 2 𝐷  5 × 𝑃 𝑚 ( cos 𝜓 ( 𝚽 ) )  ∞ 0 H 𝑚 ( 𝜆 ) 𝑒 − 1 2 𝜆 2 𝑇 𝑑 𝜆, (7) where the effecti ve d rift an gle is 𝜓 ( 𝚽 ) = ∠ ( 𝚽 eff ( 𝑇 ) , x 0 ) . It is worth e m phasizing that ( 7 ) is a powerful and gener a l 3-D an alytical resu lt. The effecti ve dr ift 𝚽 eff ( 𝑇 ) can point in any direction relativ e to the Tx – Rx axis, with its ph y s- ical impact seamlessly captured throug h the tim e - av eraged magnitud e an d th e dy namic effecti ve drift an gle 𝜓 ( 𝚽 ) . This formu latio n pr oves th a t by u sing the Girsanov me a sure change, the impact of arbitrary time-varying assistiv e electric fields can be in c orpora ted into the CIR as an explicit reweighting o f the geometric modes, preser ving an alytical tractability f or the subsequen t system ev aluation. B. CIR of the P assive Rece iv e r T o d e monstrate the universality of o ur measure- theoretic approa c h , we apply the identical tr ansformatio n engin e to the passi ve r eceiv er ca se. Un like the active receiver , passive sensing evaluates the snapshot pro bability of a particle residin g within the transpar e nt sensing volume R s at ob servation time 𝑇 . For mathem atical conv enience, we interchan ge the coor di- nate system suc h that the Tx is at th e origin and the Rx is centered at x 0 . Under zero drift, the co n centration p rofile is the f u ndamen tal solu tio n of Fick’ s law [ 33 ], 𝑐 ( x ; 𝑇 ) = 1 ( 4 𝜋 𝐷 𝑇 ) 3 2 e xp  − k x k 2 4 𝐷 𝑇  . (8) Applying the transform a tion relation ( 5 ), the co ncentration profile under th e time-varying drift 𝚽 ( 𝑡 ) is re weighted by the Girsanov density , 𝑐 𝚽 ( x ; 𝑇 ) = e xp  𝚽 eff ( 𝑇 ) · x 2 𝐷 − k 𝚽 eff ( 𝑇 ) k 2 𝑇 4 𝐷  𝑐 ( x ; 𝑇 ) . (9) By completing th e square in th e expon ent and r ecognizin g that 𝚽 eff ( 𝑇 ) 𝑇 =  𝑇 0 𝚽 ( 𝛼 ) 𝑑 𝛼 , th e density seamlessly con - denses into a sh ifted Gaussian fo rm, 𝑐 𝚽 ( x ; 𝑇 ) = 1 ( 4 𝜋 𝐷 𝑇 ) 3 2 e xp  − k x − 𝚽 ( 𝑇 ) k 2 4 𝐷 𝑇  , (10) where 𝚽 ( 𝑇 ) =  𝑇 0 𝚽 ( 𝛼 ) 𝑑 𝛼 d enotes the cumulativ e drif t vector representin g the total sp atial displacemen t, which is related to 𝚽 eff ( 𝑇 ) via 𝚽 ( 𝑇 ) = 𝑇 𝚽 eff ( 𝑇 ) . Remark 1 (Universal Consistency): It is crucial to note that for the p a ssive observation case, our Girsanov-based effecti ve drift mapping inheren tly degenerates to an e xact character- ization. Th e endpo int distribution ( 10 ) aligns perfectly with classical diffusion theo ries governed b y a varying dr if t [ 17 ]. This proves that ou r un ified framework not only ap p roxi- mates comp lex b ounda r y condition s with high accur acy but also preserves strict mathematical consistency in fun damental scenarios. The CIR (i.e. , sensing p robability ) of the P A spherica l Rx is finally obtained by integrating ( 10 ) over the transparen t volume R s , 𝑝 s ( 𝑇 ) : =  R s 𝑐 𝚽 ( x ; 𝑇 ) 𝑑 x = 1 2  er f  𝑟 Rx − 𝑑 0 √ 4 𝐷 𝑇  + er f  𝑟 Rx + 𝑑 0 √ 4 𝐷 𝑇   − √ 𝐷𝑇 𝑑 0 √ 𝜋  e xp  − ( 𝑟 Rx − 𝑑 0 ) 2 4 𝐷 𝑇  − exp  − ( 𝑟 Rx + 𝑑 0 ) 2 4 𝐷 𝑇   , (11) where 𝑑 0 : =   x 0 − 𝚽 ( 𝑇 )   represents th e effectiv e Tx– Rx separation. This distance imp licitly ac c o unts for the total 3- D spatial displacem ent ind uced by the time-varying field, providing a r obust, geometry-in depend ent validation of our analytical fr a mew ork. I V . D I S C R E T E - T I M E S I G NA L I N G A N D S Y S T E M P R O P E RT I E S W ith the analytica l CIRs estab lish e d , we now tr ansition from the contin u ous-time physical diffusion model to a discrete-time commu nication system. I n this section, we f or- mulate the en d-to-end signal mo d el and ana ly ze the phy sical proper ties of th e channel coefficients. Cru cially , we identify a fundam ental trade-off between receiver geo metries th at moti- vates o ur subseq uent optimization strategy . A. T ime-Slotted T ransmission Mo del T o enab le successiv e symbo l transmissions, we adopt a time-slotted MCvD framework [ 34 ] with a symb ol dur ation 𝑇 𝑏 . W e consider On- Off K eying (OOK) mod ulation (see [ 35 ]), where the transmitted bit 𝑏 𝑖 ∈ { 0 , 1 } in slot 𝑖 d ic ta tes the emission pr ocess: if 𝑏 𝑖 = 1 , an impulsiv e release of 𝑀 particles occurs at the star t of the slo t; if 𝑏 𝑖 = 0 , no particles are released. For efficient transpor t and tractable waveform design, we hencefo rth assum e that the assistive electric field E ( 𝑡 ) is aligned with the Tx–Rx axis. Under this configur ation, the composite drift vector can be characterized by its scalar compon ent Φ ( 𝑡 ) along the transmission axis. The received probab ility 𝑝 R [ 𝑖 ] in the 𝑖 - th time slot, origin ating from a sing le emission at 𝑡 = 0 , is given by 𝑝 R [ 𝑖 ] =   𝑖𝑇 𝑏 ( 𝑖 − 1 ) 𝑇 𝑏 𝑓 Φ hit ( 𝑡 ) 𝑑 𝑡 , Fully-Absor b ing Rx , 𝑝 s  𝑡 𝑠 + ( 𝑖 − 1 ) 𝑇 𝑏  , Passi ve Rx , (12) where 𝑡 𝑠 ∈ ( 0 , 𝑇 𝑏 ] is the samp ling time for the P A receiver . Assuming a sufficiently large 𝑀 ( correspon ding to d ilute- solution regime), the total received ions 𝑦 [ 𝑖 ] in slot 𝑖 follows a Gaussian d istribution due to the linear superp osition o f the current signal an d the ISI fro m previous slots [ 36 ], 𝑦 [ 𝑖 ] = 𝑀 𝑏 𝑖 𝑝 R [ 1 ] + 𝑖 − 1  𝑘 = 1 𝑀 𝑏 𝑖 − 𝑘 𝑝 R [ 𝑘 + 1 ] + 𝑁 total [ 𝑖 ] , (13) where 𝑁 total [ 𝑖 ] encapsulates the signal-de p endent c o unting noise from all relev ant slots. Equation ( 13 ) high lights that the system perfo rmance is dictate d b y the target signal stre ngth 𝑝 R [ 1 ] r elativ e to the interfe r ence weights 𝑝 R [ 𝑘 + 1 ] fo r 𝑘 ≥ 1 . 6 B. Ph y sical Pr operties of the Chann el Coefficients T o m inimize the BEP , the time-varying field 𝐸 ( 𝑡 ) must be designed to max im ize 𝑝 R [ 1 ] wh ile sup pressing ISI. Howe ver , the feasibility of th is o ptimization stric tly depend s on the receiver’ s physical nature. Remark 2 (On the Robustness and Monotonicity of the T arget Functio n): While our un ified Girsan ov fram ew ork analytically c a ptures both architectu res, their respo n ses to a directed d rift field are fun damentally dif feren t. • Fully-Absorbing Rx (Monoto nicity): As an active cap - ture mechanism, the hitting p robability is a cu m ulative process. Th is follows from the cumulative natur e of absorption , where any acceleration of particle trajector ie s strictly increa ses the total h itting prob ability over time . I n- creasing the drift field streng th strictly acceler ates par ticle arriv als, cau sing 𝑝 R [ 1 ] to m onoton ically increase to wards saturation. This prop e rty ensur es a well-behaved, co ncave optimization land scape for gr a dient-based designs. • Passiv e Rx (Overshoot ): Sensing is a snap sh ot mea- surement within a transpare n t v olume. Exc e ssively strong drift fields can force the molecu la r cloud to traverse an d exit the sensing volume befor e the samp ling time 𝑡 𝑠 . This overshoot phenomeno n results in a non-mo n otonic “b ell- shaped” r esponse, intr oducing n on-con cav e regions and se vere num erical sensiti vity into the o ptimization. Based on this p rofou nd ph ysical distinction, the F A receiver not only m ore accur a tely reflects bio lo gical ligand-rece p tor mechanisms [ 30 ], but it also fund amentally gu arantees th e mathematical ro bustness of o ptimal wa veform desig n s. Con- sequently , to ensur e a r e liable link contro l framework, o ur subsequen t optimiza tion will focus exclusively o n the F A spherical rec e i ver . V . D E T E C T I O N R U L E A N D B E P A N A L Y S I S This section ev aluates the end-to- end system pe r forman ce using BEP as the p r imary metric. T o system a tica lly an alyze the impa c t of th e time-varying ele c tr ic field, we adopt a MAP detection rule combined with DFE. For analytical trac tability , we assum e a perfect decision-f eedback im plementation where all p r evious bits b [ 𝑖 − 1 ] = [ 𝑏 1 , . . . , 𝑏 𝑖 − 1 ] are correctly decod ed and available at the r e ceiv er . Recalling the statisti cal framework from our pre vious work [ 24 ], the MAP d e tection rule is form ulated as 𝑝 ( 𝑦 [ 𝑖 ] | 𝑏 𝑖 = 1 , b [ 𝑖 − 1 ] ) 𝑝 ( 𝑦 [ 𝑖 ] | 𝑏 𝑖 = 0 , b [ 𝑖 − 1 ] ) 𝐻 1 ≷ 𝐻 0 𝑝 ( 𝑏 𝑖 = 0 ) 𝑝 ( 𝑏 𝑖 = 1 ) , (14) where hy potheses 𝐻 1 and 𝐻 0 correspo n d to the tran smission of bit ‘1’ an d ‘0’, respe cti vely . T o ch aracterize the d etection statistics, we introdu ce the superscript [ 𝑖 , 𝑗 ] fo r th e 𝑖 - th symbol c o nditioned o n the 𝑗 - th candidate sequence of preced in g bits. As established in [ 24 ], let 𝜇 [ 𝑖 , 𝑗 ] 0 be the expected re c ei ved molecule co unt un der 𝐻 0 , which accounts for the residu al ISI from all p r evious emissions, we have 𝜇 [ 𝑖 , 𝑗 ] 0 = 𝑀 𝑖 − 1  𝑘 = 2 𝑏 [ 𝑗 ] 𝑖 − 𝑘 𝑝 R [ 𝑘 ] . (15) Under 𝐻 1 , the expe cted count includes the cur r ent signal com- ponen t, such th a t 𝜇 [ 𝑖 , 𝑗 ] 1 = 𝑀 𝑝 R [ 1 ] + 𝜇 [ 𝑖 , 𝑗 ] 0 . The co rrespon d ing condition al v ariances are given by 𝜎 [ 𝑖 , 𝑗 ] 0 = (  𝑖 − 1 𝑘 = 2 𝑏 [ 𝑗 ] 𝑖 − 𝑘 𝜎 2 𝑘 ) 1 / 2 and 𝜎 [ 𝑖 , 𝑗 ] 1 = ( 𝜎 2 1 + ( 𝜎 [ 𝑖 , 𝑗 ] 0 ) 2 ) 1 / 2 , wh e re 𝜎 2 𝑘 = 𝑀 𝑝 R [ 𝑘 ] ( 1 − 𝑝 R [ 𝑘 ] ) repr esents the signal-d ependen t coun ting noise. Based on th e Gaussian ap proxima tio n, th e observation 𝑦 [ 𝑖 ] follows the cond itio nal distributions N ( 𝜇 [ 𝑖 , 𝑗 ] 1 , ( 𝜎 [ 𝑖 , 𝑗 ] 1 ) 2 ) and N ( 𝜇 [ 𝑖 , 𝑗 ] 0 , ( 𝜎 [ 𝑖 , 𝑗 ] 0 ) 2 ) . By d etermining the op timal MAP deci- sion thresho ld 𝛾 [ 𝑖 , 𝑗 ] , we recall fr om [ 24 ] that the cond itional false alarm pr o bability 𝑃 [ 𝑖 , 𝑗 ] F A and detectio n pr obability 𝑃 [ 𝑖 , 𝑗 ] D are form ulated as 𝑃 [ 𝑖 , 𝑗 ] F A = 𝑄  𝛾 [ 𝑖 , 𝑗 ] − 𝜇 [ 𝑖 , 𝑗 ] 0 𝜎 [ 𝑖 , 𝑗 ] 0  , 𝑃 [ 𝑖 , 𝑗 ] D = 𝑄  𝛾 [ 𝑖 , 𝑗 ] − 𝜇 [ 𝑖 , 𝑗 ] 1 𝜎 [ 𝑖 , 𝑗 ] 1  , (16) where 𝑄 ( ·) is the stand ard 𝑄 - function. Detailed deriv ations of the exact optima l threshold 𝛾 [ 𝑖 , 𝑗 ] can b e foun d in [ 24 ]. The a verage BEP 𝑃 [ 𝑖 ] 𝑒 is th en o btained b y marginalizing over all 2 𝑖 − 1 possible ISI sequenc es, 𝑃 [ 𝑖 ] 𝑒 = 2 𝑖 − 1  𝑗 = 1 𝑝 ( b [ 𝑗 ] [ 𝑖 − 1 ] )  𝑝 ( 𝑏 𝑖 = 0 ) 𝑃 [ 𝑖 , 𝑗 ] F A + 𝑝 ( 𝑏 𝑖 = 1 ) ( 1 − 𝑃 [ 𝑖 , 𝑗 ] D )  . (17) Remark 3 (Surrogate Objective for System Optimizat io n): Directly minim izin g th e av erage BEP in ( 17 ) with respect to the time-varying elec tric field E ( 𝑡 ) is analytically prohib iti ve due to the hig hly n on-linear de pendenc e of the thresh old on th e entire channel coefficient sequen ce { 𝑝 R [ 𝑘 ] } . In prior work [ 24 ], such intractability necessitated co mputation ally expensiv e exhaustive searches or complex g radient descent approx imations (e.g. , Min EP and MaxSIR algorithms) to find suboptimal field wa veforms. T o circu mvent this bottleneck in our generalized f r amew ork, we ana ly ze th e system in an ideal ISI-f r ee regime where 𝑝 R [ 𝑘 ] ≈ 0 fo r 𝑘 ≥ 2 . I n this cond ition, 𝑃 [ 𝑖 ] F A → 0 and the detection probability is purely dominated by the target signal, 𝑃 [ 𝑖 ] D ≈ 1 − 𝑄   𝑀 𝑝 R [ 1 ]  . (18) Equation ( 18 ) reveals a crucial ph ysical in sight: the BEP monoto nically d ecreases as the first-slot received prob ability 𝑝 R [ 1 ] increases. While this ap proxim ation is universal, its practical utility as an op timization objec ti ve strictly depe n ds on the receiver architectu re. As hig hlighted in Remar k 1, unlike passive sensing wh ich suffers f r om hig h-drift overshoot, the F A receiver g uarantees a strictly monoto nic an d concave response to the d irected electric field. Consequently , for the F A ar chitecture, maximizing 𝑝 R [ 1 ] serves as a ro bust, math ematically gu aranteed surr o gate ob- jectiv e for BEP min imization. Guided by this insight, we bypass the intracta b le exact BEP form ulation and ado pt a computatio nally efficient two-phase sequ ential strategy: first designing the field wa veform to aggressively maximize 𝑝 R [ 1 ] , 7 𝑡 peak 𝑇 𝑏 Phase I Phase II 𝑉 1 𝑉 2 𝛽𝑇 𝑏 · · · · · · Deterministic term for 𝑠 ( 𝑡 ) Assisted electric-field wa veform 𝐸 1 ( 𝑡 ) (a) Fully- absorbin g (activ e) spher ical Rx 𝑡 𝑠 𝑇 𝑏 Phase I Phase II 𝑉 1 𝑉 2 · · · · · · Deterministic term for 𝑠 ( 𝑡 ) Assisted electric-field wa veform 𝐸 1 ( 𝑡 ) (b) Passi ve spherical Rx Fig. 2: Illu stration of the optimiz e d assisted electric-field wav eform s 𝐸 1 ( 𝑡 ) fo r the two receiver architec tures. (a) Fully- absorbing Rx : The wa veform is d e signed to maximiz e the hitting density at 𝑡 peak throug h the MHP method , with a strategic suppression ph ase 𝑉 2 to d ivert residu al par ticles. (b) Passive Rx: The waveform op timizes the sen sing pro bability at the sampling time 𝑡 𝑠 using the MSP m ethod. Both d esigns are g enerated by the unified MRP engine under a total energy constra int 𝜉 . and subsequen tly utilizing the residual energy to su p press the leading I SI term 𝑝 R [ 2 ] . V I . W A V E F O R M O P T I M I Z A T I O N A N D S Y S T E M D E S I G N Driv en by the analytical insight fro m Section V that the first-slot received pro bability 𝑝 R [ 1 ] dictates the sy stem BEP , we now formulate the assisted electric field d esign as a tractable op timization prob lem. In alig nment with the ro bust- ness rationale established in Rema r k 1, th is section exclu si vely focuses o n the F A sph erical receiver . W e propose a tw o-ph ase wa veform optimizatio n stra tegy , termed the Maximize Hitting Pr o bability method , which sequen tially maximizes the target signal and suppresses the sub seq uent ISI u nder a strict energy budget. A. Pr oblem F o rmulation and Ener gy Constraint T o activ ely shape the chan nel impulse response, we design a per iodic, time-varying assisted electric field E ( 𝑡 ) . For design tractability and to maximize transp o rt efficiency along the transmission link, we orien t the composite field along the 𝑥 1 - axis, coincid in g with the Tx –Rx tr a jectory . T his allows us to simplify the d esign variable to a scalar e le c tr ic field co mponen t 𝐸 1 ( 𝑡 ) . The molecular propagatio n is simultaneously influenced b y a time-varying back groun d flow 𝑢 1 ( 𝑡 ) acting alon g the Tx–Rx axis. Th e backg r ound flo w is selected as in [ 37 ], 𝑢 1 ( 𝑡 ) = 𝜇 𝐹 0 cos 2  𝜋 𝑡 𝑇 𝑑  , (19) where 𝜇 is the molecu lar mo bility , 𝐹 0 is the peak flow f o rce amplitude, and 𝑇 𝑑 is the flo w period . The transmitter is located at x 0 = [ 𝑥 0 , 0 , 0 ] where 𝑥 0 = − 𝑑 0 . For an alytical simplicity and without loss o f generality , the initial p hase of the b ackgro und flow is set to z ero. The system operates under a fundam ental energy co n straint 𝜉 , which bou nds the kinetic energy imparte d by the external field over a symbo l period 𝑇 𝑏 . The n ormalized energy co n- straint is defined as  𝑇 𝑏 0 | 𝐸 1 ( 𝜏 ) | 2 𝑑 𝜏 ≤ 𝜉 . (20) W e decou ple the wa veform 𝐸 1 ( 𝑡 ) into two sequ ential phases: a sign al-enhan cing field 𝐸 S 1 ( 𝑡 ) to maximize 𝑝 R [ 1 ] , followed by an ISI-sup pressing field 𝐸 R 1 ( 𝑡 ) utilizing the resid u al energy 𝜉 res . B. Ph a se I: Sign al Enhanc e m e nt via MHP Appr oximation For the active receiver , max imizing 𝑝 R [ 1 ] =  𝑇 𝑏 0 𝑓 Φ hit ( 𝑡 ) 𝑑 𝑡 is co mputation ally pro h ibitiv e due to the complex temp oral integral of Bessel f unctions. T o establish a tractab le su r rogate, we maximize the instantaneo u s hitting density at the peak - receiving lo cation x peak = [ − 𝑟 Rx , 0 , 0 ] at a target peak time 𝑡 peak . Based on the universal engine in ( 5 ), the contro l variable 𝐸 S 1 ( 𝑡 ) influences the hitting density thr o ugh the expo nential Girsanov weight, Ψ ( 𝑇 ) = e xp  − 𝑇 4 𝐷    𝚽 eff ( 𝑇 ) − x peak − x 0 𝑇    2 + 𝐶 x  , (21) where 𝐶 x is a con stan t indep endent of th e electric field. T his reform u lates the max imization into a distance-minimizatio n pr oblem : ( P sig ) argmin 𝐸 S 1     𝚽 eff ( 𝑡 peak ) − x peak − x 0 𝑡 peak     2 . ( 22) 8 Algorithm 1 Maximize Received Probability (MRP) E n gine Require: T arget displacemen ts 𝑞 S , 𝑞 R ; durations 𝑇 𝑝 1 , 𝑇 𝑝 2 ; budget 𝜉 Ensure: Optimal electr ic field m a gnitudes ( 𝑉 ∗ 1 , 𝑉 ∗ 2 ) 1: if | 𝑞 S | ≥  𝜉 / 𝑇 𝑝 1 then ⊲ Insufficient energy to reach optimal pe a k 2: 𝑉 ∗ 1 ← sgn ( 𝑞 S ) ·  𝜉 / 𝑇 𝑝 1 3: 𝑉 ∗ 2 ← 0 ⊲ No residu a l energy for ISI suppression 4: else ⊲ Unconstra in ed optimum achieved 5: 𝑉 ∗ 1 ← 𝑞 S 6: if 𝑇 𝑝 1 = 𝑇 𝑏 then 7: 𝑉 ∗ 2 ← 0 8: else 9: 𝜉 res ← 𝜉 − ( 𝑉 ∗ 1 ) 2 · 𝑇 𝑝 1 ⊲ Comp ute residua l energy 10: 𝑉 ∗ 2 ← − sgn ( 𝑞 R ) ·  𝜉 res / 𝑇 𝑝 2 ⊲ Apply maxim um suppression 11: end if 12: end if The g lobal minimum is ach iev ed when the effecti ve drift completely bridge s the spatial gap with in th e peak time. Solving th is cond ition yields the ideal uncon strained field velocity 𝑞 A S . If the energy r equired exceed s the budget 𝜉 , th e wa veform is trun cated to 𝑉 ∗ 1 = sgn ( 𝑞 A S )  𝜉 / 𝑡 peak . C. Ph ase II: I SI Suppression a n d the MRP Algo rithm The residual energy 𝜉 res = 𝜉 − ( 𝑉 ∗ 1 ) 2 𝑡 peak is deployed to suppress 𝑝 R [ 2 ] . Since th e F A recei ver continuo usly in tegrate s arriv als, we de p loy an o pposing drift field late in the symbo l period to deviate residual pa r ticles. W e defin e a fraction 𝛽 ∈ [ 0 , 1 ] for the supp r ession onset, applyin g a co n stant field 𝑉 2 during 𝑡 ∈ [ 𝛽𝑇 𝑏 , 𝑇 𝑏 ] . The velocity allocatio n is co nsolidated into the Maximize Received Pr obability alg orithm, p resented in Alg orithm 1 , where the temporal durations for the fully-absor bing archi- tecture map to 𝑇 𝑝 1 = 𝑡 peak and 𝑇 𝑝 2 = ( 1 − 𝛽 ) 𝑇 𝑏 . The fina l composite electric field is comp actly expressed as 𝐸 1 ( 𝑡 ) =          𝑉 ∗ 1 , ( 𝑖 − 1 ) 𝑇 𝑏 ≤ 𝑡 < 𝑡 peak + ( 𝑖 − 1 ) 𝑇 𝑏 , 𝑉 ∗ 2 , 𝛽𝑇 𝑏 + ( 𝑖 − 1 ) 𝑇 𝑏 ≤ 𝑡 < 𝑖𝑇 𝑏 , 0 , otherwise . (23) Remark 4 (Universality of t he MRP Engine): While we fo - cus on the F A receiver (i.e., MHP), th e logic in Algo rithm 1 is inherently u niversal. By substitutin g x peak with th e volumetric center, the en gine gener ates th e Maximize Se n sing Pr obability solution fo r P A receivers (see Appen dix C ), un derscorin g the cohesiveness of our fr amew ork. V I I . N U M E R I C A L V A L I DA T I O N A N D P E R F O R M A N C E E V A L UAT I O N In this sectio n, we p rovide a compreh ensiv e n umerical ev aluation of the p roposed F AMC framework. T o ensur e the full repro ducibility of our resu lts, we first d etail the ph ys- ical parameters and simulatio n environment. Subsequen tly , we validate th e accur acy of the Girsanov-based analytical 0 0 . 5 1 1 . 5 2 0 0.5 1 1.5 × 10 − 3 Sampling T ime (sec.) Amplitude ( cm/s) W ave form fi eld on 𝑥 1 -axis W ave form fi eld on 𝑥 2 -axis W ave form fi eld on 𝑥 3 -axis Fig. 3: Arbitrary time- varying drift r ealizations 𝚽 ( 𝑡 ) . CIR expressions v ia particle-based Monte Carlo simulations. Finally , we e valuate th e system-level perf o rmance, demo nstrat- ing the efficac y of the proposed MRP wa veform op timization algorithm s in ter ms of sign al-to-ISI enhan c ement and BEP reduction . A. Simu lation Setup Unless oth erwise sp e cified, the system parame te r s fo r the numerical ev aluations ar e configur ed as fo llows. T he spherical receiver has a radiu s of 𝑟 Rx = 10 𝜇 m , and the d istance fro m the point transmitter to the receiver center is 𝑑 0 = 30 𝜇 m . The message pa r ticles ar e calcium ions ( Ca 2 + ) with a valence of 𝑧 = 2 and a d iffusion coefficient of 𝐷 = 7 . 5 × 1 0 − 6 cm 2 / s in a fluidic e nvironmen t at an ab solute te m perature of 𝑇 = 300 K [ 37 ]. F or the p assi ve rece iver , the sam p ling time is 𝑡 𝑠 = 0 . 1 s , whereas for the fully-ab so rbing receiver , the target p eak time is set to 𝑡 peak = 0 . 1 s . The com munication symbol duratio n is strictly defined as 𝑇 𝑏 = 2 s . T o explicitly evaluate the im pact of th e I SI suppression tim ing in th e p roposed MHP method, we investigate two supp ression onset fraction s: 𝛽 = 0 . 5 and 𝛽 = 0 . 8 . Furth e rmore, to accurately capture the ISI dyn amics, the BEP evaluation accoun ts for the residual mo lecules from the p receding two sym bol intervals (i.e. , evaluating up to the third symbol). The backg round flow is characterized b y a drift duration of 𝑇 d = 1 s , a mobility factor of 𝜇 = 1 . 7 7 × 10 11 s / kg , and a fo rce amp litude of 𝐹 0 = 4 . 18 × 1 0 − 15 N , w h ich align with typical micro- fluidic modeling [ 37 ]. B. CIR V erification with P article-Based Simula tion T o validate the accur a cy of the proposed Girsanov-b ased analytical CIR expressions, we compar e the theoretical distri- butions against p article-based Monte Carlo simulation s tracing three-dim ensional Brownian ran dom walks [ 33 ]. T o ensure high statistical fidelity , the Mo nte Carlo simulatio ns indepen - dently track 𝑁 = 1 0 6 message particles for each scenario. Furthermo re, we con struct arbitrary time-varying comp osite drift field s 𝚽 ( 𝑡 ) , gener ated as 3-D piecewise co nstant vector fields, and ob serve the re c eption statistics fo r bo th the active and passiv e receiv er a rchitectures. For exact reprod ucibility and a consistent co mparison between the two ar chitectures, 9 Arrival Time (sec.) 0 0.5 1 1.5 2 Hitting Probability Density 0 0.2 0.4 0.6 Particle-Based Simulation Theoretical Value Sampling Time (sec.) 0 0.5 1 1.5 2 Sensing Probability 0 0.004 0.008 0.012 Particle-Based Simulation Theoretical Value Fig. 4: V e r ification o f the analytical CIR for the fu lly-absorbin g spherical receiver and the passive sph erical r eceiver . Th e left p anel compares the analytical hitting pr obability den sity (dashed lines, co mputed v ia ( 7 )) with particle-based Monte Carlo simulations. The right panel compares the analytical sen sing probab ility (dashed lines, compu ted via ( 11 )) with particle-based Monte Carlo simulations. the drift field realization shown in Fig. 3 is g enerated by initializing the pseud o-rand om numb er generator with a fixed seed (i. e., seed = 1). As illustrated in the left and rig ht pan e ls of Fig. 4 , the hitting probab ility den sity in ( 7 ) for the fully -absorbin g rec ei ver and the analytical sen sin g prob ability in ( 11 ) for the pas- si ve receiver p erfectly track the simu lated p a rticle behaviors. This excellent agree m ent confir m s that the EDA successfully captures the p a th-depen dent we ig hting indu c ed by the time- varying electr ic field, thereb y preservin g th e math ematical tractability req uired for subsequ ent system op timization. C. I mp act of En er gy Constraints and W aveform Design Subsequen tly , we in vestigate how th e optimal allo cation of the assisted electric field affects the system perfor mance u nder varying energy budgets 𝜉 . Specifically , we ev aluate the signal- to-ISI d ifference, d efined as 𝑝 R [ 1 ] − 𝑝 R [ 2 ] , alon gside the overall BEP for energy constraints up to 𝜉 = 200 V 2 s / m 2 . The propo sed MHP an d MSP methods are executed via the un ified MRP en gine (Algo rithm 1 ), utilizing the ana lytical par ameters for the passiv e architecture d e r iv ed in Append ix C . These propo sed designs are sy stematically comp a red again st a pure diffusion baseline (No Drift) and a co n stant-field appr oach (Undesign ) that consum es th e energy budget evenly over the symbol d uration 𝑇 𝑏 . The nu merical re sults pre sen ted in Fig. 5 provide a profo u nd physical validation of th e r o bustness rationale established in Remark 2 . For the fully-ab so rbing receiver (left pan e ls), the MHP method exhib its strict mono tonicity . As 𝜉 increases, the algorithm efficiently allocates energy to accelera te par ticles, steadily m aximizing th e target signa l 𝑝 R [ 1 ] while effecti vely utilizing the residual energy to de viate tra ilin g particles, thereby suppressing 𝑝 R [ 2 ] . Consequen tly , the signal-to-I SI difference expands continu o usly , leading to a r a pid an d robust decay in BEP . Furtherm ore, comp aring the two o nset fr actions, a delay ed suppression ph a se ( 𝛽 = 0 . 8 ) y ie ld s superior ISI mitigation and a strictly lower BEP compared to an earlier onset ( 𝛽 = 0 . 5 ) when sufficient e nergy is av ailable. Con versely , th e passi ve receiver (right panels) is fun damen- tally limited b y its snap shot sen sin g mech anism. Although the signal-to-I SI difference continu es to imp rove as 𝜉 increa ses tow ards 200 V 2 s / m 2 , the abso lute magnitud e of th is gap remains se verely bou nded comp ared to the active architecture. Specifically , the p assi ve receiver on ly achiev es a maximu m difference of approxim ately 0 . 11 , whereas the fully-absorbin g receiver rapidly ap proache s 0 . 8 . Because the powerful electr ic field forces the mo lecular cloud to traverse the sensing volume rapidly , th e peak concentration captu red at any single sampling instant is fun damentally lo wer than the cu mulative n umber of molec u les absor bed over a time in terval. This intrin sic physical limitation inh erently bottlen ecks the overall BEP reduction , preventing th e passiv e receiver fr om reaching th e ultra-reliab le per forman c e levels of the active architectu re. This stark co ntrast confirms tha t the cumu lativ e ab sorption mechanism f undame n tally guar antees superio r optimizatio n potential, validating our pr imary design f ocus. D. BEP P erformance Evalu a tion Finally , we evaluate the overall system reliability across dif- ferent symb o l durations 𝑇 𝑏 . Fig . 6 pre sen ts b oth the signal-to- ISI difference ( 𝑝 R [ 1 ] − 𝑝 R [ 2 ] ) and the BEP u nder a m o derate energy budget of 𝜉 = 25 V 2 s / m 2 . The tran smission inv olves 𝑀 = 100 and 𝑀 = 100 0 particles fo r th e absorb ing an d passive receivers, respe c tively . As ob ser ved in the top panels of Fig. 6 , extending the symbol dur ation 𝑇 𝑏 provides an expande d tempo ral w in dow for th e electr ic field to acti vely shap e the molecular tr ansport, thereby in creasing the separatio n between the target signal and the leading ISI. Th is dir ect improvement in th e phy sical channel response translates into a significant reductio n in BEP , as demo nstrated in the bottom p a n els. The pro p osed two-phase wa veform designs (M HP and MSP) consistently ou tperform both the pure diffusion (No Drift) and the con stant-field (Undesign ) baselines across all ev aluated sy m bol duration s. For th e active receiver operating under the MHP strategy , em ploying a d e layed suppression onset ( 𝛽 = 0 . 8 ) yie ld s a more r o bust signal- to -ISI gap and 10 ξ (V 2 s/m 2 ) 0 50 100 150 200 p R [1] - p R [2] 0.2 0.4 0.6 0.8 1 No Drift Undesign MHP ( β = 0.5) MHP ( β = 0.8) ξ (V 2 s/m 2 ) 0 50 100 150 200 p R [1] - p R [2] 0 0.03 0.06 0.09 0.12 No Drift Undesign MSP 0 50 100 150 200 ξ (V 2 s/m 2 ) 10 -48 10 -36 10 -24 10 -12 10 0 BEP No Drift Undesign MHP ( β = 0.5) MHP ( β = 0.8) 0 50 100 150 200 ξ (V 2 s/m 2 ) 10 -28 10 -21 10 -14 10 -7 10 0 BEP No Drift Undesign MSP Fig. 5: Perfo rmance metrics versus energy constraint 𝜉 u nder different electric-field design metho ds. (T op Row) Th e signal-to- ISI d ifference 𝑝 R [ 1 ] − 𝑝 R [ 2 ] for F A (left) a n d P A (righ t) receivers. (Bot tom Row) Th e cor respondin g BEP for F A (left) an d P A (right) receivers. The fully- a bsorbing arch itecture (lef t) guaran tees robust op timization an d mono tonic improvement, whereas the p assi ve receiver (rig ht) faces perfo rmance saturation at h igher energy lev els du e to the inheren t overshoot ph enomen o n. a lower BEP compar e d to an earlier onset ( 𝛽 = 0 . 5 ), furth er highligh ting the critical im p ortance of timely ISI mitigation. By efficiently allocating the constrained energy budget, the lightweight MRP engin e gu arantees h ighly reliable detection perfor mance, establishing it as a highly practical solution for low-complexity b io-nano receivers. V I I I . C O N C L U S I O N This pap er has established a co mprehe nsiv e analytical and optimization f ramew ork fo r m olecular com munication systems operating und er time-varying comp o site dr ift fields. By h ar- nessing the CMG th eorem, we successfully decou pled the temporal variations of th e assisti ve electric field from the complex spatial boundar ies, deriving analytically tractable CIRs via effecti ve drift ma p ping fo r F A r eceiv ers, and exact CIR expression s fo r P A spherical rece ivers. T hese analytical models were rigorou sly validated via particle-based simula - tions, con fir ming that the EDA capture s the p a th-depen dent weighting of dynamic fields with h igh fidelity . Building upon th is foun dation, we id e n tified th e first-slot received p r obability as the prima ry deter minant o f BEP perf o r- mance in decision- f eedback detection architectu res. Recogniz- ing the math ematical vu lnerability of passiv e sensing to high- drift overshoot, we strategically fo cused o ur o ptimization on the F A recei ver , which exhibits a robust m onoton ic respon se to assisti ve fields. W e pro posed the Maximize Received Pr obabil- ity algorithm, which intellig ently shapes the electric field into a signal- enhancin g a c celeration p h ase and an ISI-su ppressing counter-drift phase. Num erical evaluations demonstrate that our design ac h iev es near-optimal perfo rmance u n der strict en- ergy co nstraints with low com putationa l comp lexity , offering a scalable and r eliable control para d igm for advanced field- assisted bio- nanonetwork s. A P P E N D I X A E FF E C T I V E D R I F T A P P ROX I M AT I O N B A S E D O N M E A S U R E - C H A N G E F R A M E WO R K The fu ndamen ta l fr amew ork for e valuating r eception statis- tics via the m e asure-chan ge framework was established in o ur previous work [ 28 ] for chan n els with a constan t unifo rm drift. T o rigoro u sly justify the EDA u tilized in ( 4 ) for time-varyin g fields, we exten d this Gir san ov-based ap proach . W e co nsider th e ev ent of in te r est defined as an arriv al within a small spatial-tempo ral window , 𝐴 : =  𝑡 ∈ [ 𝑇 , 𝑇 + Δ 𝑇 ] , X 𝑡 ∈ [ x , x + Δ x ] , 𝑇 < ∞  . (24) Recalling the app lication o f the Radon –Nikodym theorem from [ 28 ], th e likelihood ratio 𝑀 𝑡 transform s th e zero -drift referenc e m easure P to the dr ifted ph ysical measur e Q . The probab ility of event 𝐴 u nder the true time- varying dr ift is ev aluated b y taking the expectation un d er the p ure diffusion 11 T b (sec.) 0.5 1 1.5 2 2.5 3 p R [1] - p R [2] 0 0.2 0.4 0.6 0.8 No Drift Undesign MHP ( β = 0.5) MHP ( β = 0.8) T b (sec.) 0.5 1 1.5 2 2.5 3 p R [1] - p R [2] 0 0.01 0.02 0.03 0.04 No Drift Undesign MSP 0.5 1 1.5 2 2.5 3 T b (sec.) 10 -12 10 -8 10 -4 10 0 10 4 BEP No Drift Undesign MHP ( β = 0.5) MHP ( β = 0.8) 0.5 1 1.5 2 2.5 3 T b (sec.) 10 -8 10 -6 10 -4 10 -2 10 0 BEP No Drift Undesign MSP Fig. 6: Per forman ce m e trics versus symbol duratio n 𝑇 𝑏 under an en ergy budge t of 𝜉 = 2 5 V 2 s / m 2 . (T op Row) The sign al-to-ISI difference 𝑝 R [ 1 ] − 𝑝 R [ 2 ] fo r F A (le f t) and P A ( right) receivers. ( Bottom Row) The correspo nding BEP for F A (left) and P A (right) r eceiv ers. The prop osed MHP and MSP methods sign ificantly and consistently outp e rform th e constant-field b aseline (Undesign ) acr oss all ev aluated symb o l duration s. measure, Q ( 𝐴 ) = E P [ 𝑀 𝑡 · 1 𝐴 ] =  Ω 𝑀 𝑡 · 1 𝐴 𝑑 P =  Ω e xp   𝑡 0 𝚽 ( 𝛼 ) √ 2 𝐷 · 𝑑 B 𝛼 − 1 2  𝑡 0 k 𝚽 ( 𝛼 ) k 2 2 𝐷 𝑑 𝛼  · 1 𝐴 𝑑 P . (25) Unlike the co nstant drift scenario in [ 28 ] where the stochas- tic integral simplifies exactly , the arbitrary time-varying field 𝚽 ( 𝛼 ) render s the Itô integral an alytically intractable. T o resolve this, we approximate the composite drift by its time-average over the interval [ 0 , 𝑡 ] , deno ted as 𝚽 eff ( 𝑡 ) : = 1 𝑡  𝑡 0 𝚽 ( 𝛼 ) 𝑑 𝛼 . Substituting this effective drift allows u s to extract it fro m the stoch astic integral,  𝑡 0 𝚽 ( 𝛼 ) √ 2 𝐷 · 𝑑 B 𝛼 ≈ 𝚽 eff ( 𝑡 ) √ 2 𝐷 ·  𝑡 0 𝑑 B 𝛼 = 𝚽 eff ( 𝑡 ) √ 2 𝐷 · B 𝑡 . (26) Follo wing the Brownian motion substitution B 𝑡 = X 𝑡 − x 0 √ 2 𝐷 under the pu r e d iffusion ref erence measure P , we substitute this d y namic relation back into the simplified Radon–Nikod ym deriv ativ e to yield 𝑀 𝑡 ≈ exp  𝚽 eff ( 𝑡 ) · ( X 𝑡 − x 0 ) 2 𝐷 − k 𝚽 eff ( 𝑡 ) k 2 𝑡 4 𝐷  . (27) Finally , th e ind icator fun ction 1 𝐴 tightly restricts the spatial and temporal variables to X 𝑡 ≈ x and 𝑡 ≈ 𝑇 . Consequen tly , the exponential term becomes a determin istic weighting factor that can be factored out o f the integral, Q ( 𝐴 ) ≈ exp  𝚽 eff ( 𝑇 ) · ( x − x 0 ) 2 𝐷 − k 𝚽 eff ( 𝑇 ) k 2 𝑇 4 𝐷  × P ( 𝐴 ) . (28) This explicit formulatio n confir ms that under the EDA, th e time-varying effecti ve d rift acts as a g e o metric en ergy weigh t applied to the un-d rifted pro bability P ( 𝐴 ) . Strictly speak ing, approx imating the true Itô integral with the effecti ve dr ift p rod- uct introduce s a slight second-o rder variance mismatch, since the variance of the tr ue stochastic integral  𝑇 0 k 𝚽 ( 𝛼 ) k 2 𝑑 𝛼 differs fro m that of the app r oximation k 𝚽 eff ( 𝑇 ) k 2 𝑇 . How- ev er, this d ifference primarily affects h igher-order fluctuatio n statistics. The E D A should be interpr e ted as a path-averaged drift appro ximation that becomes accura te when the drift variation over the hittin g time scale is m o derate, mean in g the first-o rder geome tric weighting de fin iti vely dom inates the path measure chang e. Th is is empirically validated in Fig. 4 . Impor tan tly , the ap proxim a tion preserves the exponential tilt- ing structur e in duced by the Gir sanov transform ation. Th is directly g eneralizes the static analytical mapping established in 12 [ 28 ] to dynamic en vironmen ts, the r eby v alidating the universal mathematical engin e employed in Sectio n II . A P P E N D I X B D E R I V A T I O N O F T H E S E N S I N G P RO B A B I L I T Y F O R P A S S I V E S P H E R I C A L R E C E I V E R S This appen dix p rovides a rigoro us d eriv ation of th e sens- ing probability 𝑝 𝑠 ( 𝑇 ) for a P A spherical r eceiv er in a 3-D en viron ment under the referen ce measure 𝑃 (zero -drift case). It is worth n oting that while the exact expected nu mber of molecules fo r a spherical re c ei ver was p reviously for mulated by Noel et al. [ 38 , Ap pendix] in a dimensionle ss context, ou r deriv ation pr esented here of fers three distinct contributions. First, we adop t a fundam entally different mathematical ap- proach ; rather than rely ing on complex Bessel functio n iden- tities, we employ a direct integration meth od over spherical coordin ates. Second , our formu lation is explicitly dimen sional, retaining physical p arameters ( e.g., 𝐷 , 𝑇 , 𝑑 0 , an d 𝑟 Rx ) to seam- lessly integrate with the system and signal models established in our earlier sections. Finally , our step-b y-step de riv ation is physics-in f ormed; by utilizing the geo metric interactio n between the molecular cloud and the sensing v olume , it provides clear physical interpretation s at each stage of the integration. In the absenc e o f drift, the con centration profile resulting from a point sou r ce release at x 0 is g i ven by 𝑐 ( x , 𝑇 ) = 1 ( 4 𝜋 𝐷 𝑇 ) 3 / 2 e xp  − k x − x 0 k 2 4 𝐷 𝑇  . (29) The sensing prob ability is defined as the integral of 𝑐 ( x , 𝑇 ) over the sensing region R s = { x | k x k ≤ 𝑟 Rx } . By e stab - lishing a spherical co ordinate sy stem ( 𝑟 , 𝜃 , 𝜙 ) centered at the origin—wh ere 𝑟 denotes th e radial distance, 𝜃 ∈ [ 0 , 2 𝜋 ) is th e azimuthal ang le, and 𝜙 ∈ [ 0 , 𝜋 ] is the polar ang le m easured from th e 𝑥 1 -axis—and a ligning the Tx at distance 𝑑 0 along the 𝑥 1 -axis, the distance term is expressed a s k x − x 0 k 2 = 𝑟 2 + 𝑑 2 0 − 2 𝑟 𝑑 0 cos 𝜙 . The volume in tegral is form ulated as 𝑝 𝑠 ( 𝑇 ) = 1 ( 4 𝜋 𝐷 𝑇 ) 3 / 2  𝑟 Rx 0  𝜋 0  2 𝜋 0 𝑟 2 sin 𝜙 × e xp  − 𝑟 2 + 𝑑 2 0 − 2 𝑟 𝑑 0 cos 𝜙 4 𝐷 𝑇  𝑑 𝜃 𝑑 𝜙 𝑑 𝑟 . (30) W e first ev aluate the angular integrals. Th e integral over the azimuthal an gle 𝜃 yields 2 𝜋 . For the pola r angle 𝜙 , letting 𝑢 = cos 𝜙 gives  1 − 1 e xp ( 𝑟 𝑑 0 2 𝐷 𝑇 𝑢 ) 𝑑 𝑢 = 2 𝐷 𝑇 𝑟 𝑑 0  𝑒 𝑟 𝑑 0 2 𝐷 𝑇 − 𝑒 − 𝑟 𝑑 0 2 𝐷 𝑇  . Substituting this back into ( 30 ) yields a radial in tegral: 𝑝 s ( 𝑇 ) = 1 √ 4 𝜋 𝐷 𝑇 𝑑 0  𝑟 Rx 0 𝑟  𝑒 − ( 𝑟 − 𝑑 0 ) 2 4 𝐷 𝑇 − 𝑒 − ( 𝑟 + 𝑑 0 ) 2 4 𝐷 𝑇  𝑑𝑟 . (31) T o solve th e Gaussian-like integrals, we app ly integration b y parts. For the first term , we let 𝑣 = 𝑟 − 𝑑 0 √ 4 𝐷 𝑇 , yield ing  𝑟 Rx 0 𝑟 𝑒 − ( 𝑟 − 𝑑 0 ) 2 4 𝐷 𝑇 𝑑𝑟 = − 2 𝐷𝑇  𝑒 − ( 𝑟 Rx − 𝑑 0 ) 2 4 𝐷 𝑇 − 𝑒 − 𝑑 2 0 4 𝐷 𝑇  + 𝑑 0 √ 𝜋 𝐷 𝑇  er f  𝑟 Rx − 𝑑 0 √ 4 𝐷 𝑇  + e r f  𝑑 0 √ 4 𝐷 𝑇   . A similar pr ocedur e is applied to th e seco nd term inv olving ( 𝑟 + 𝑑 0 ) . Upo n co mbining the terms, the error f unction com - ponen ts containing only 𝑑 0 cancel each other out. The final expression for the sensing pro bability is der iv ed as 𝑝 s ( 𝑇 ) = 1 2  er f  𝑟 Rx − 𝑑 0 √ 4 𝐷 𝑇  + e r f  𝑟 Rx + 𝑑 0 √ 4 𝐷 𝑇   − √ 𝐷𝑇 𝑑 0 √ 𝜋  e xp  − ( 𝑟 Rx − 𝑑 0 ) 2 4 𝐷 𝑇  − e xp  − ( 𝑟 Rx + 𝑑 0 ) 2 4 𝐷 𝑇   , (32) which matches th e r esult presented in Section III an d en- sures physical co n sistency regard ing d imensions and the n on- monoto nic nature of passive sensing. Sensing Region R 𝑠 (Radius 𝑟 𝑅 𝑥 ) 𝑥 1 -axis 𝑑 0 𝑟 | | 𝑥 − 𝑥 0 | | Origin (Rx) Tx ( 𝑥 0 ) Point 𝑥 𝜙 Fig. 7: Geometric illustration o f the P A spherical r eceiv er relativ e to the point Tx. T o align with th e system mode l in Fig. 1 , the T x is positioned on th e negati ve 𝑥 1 -axis. T o facilitate the volume in tegration over the sensing region R 𝑠 , the coordin ate origin is set at the center of the re c e i ver . By the law of cosin es, the squared distan c e from an arbitrar y point 𝑥 to the Tx at 𝑥 0 is evaluated as | | 𝑥 − 𝑥 0 | | 2 = 𝑟 2 + 𝑑 2 0 − 2 𝑟 𝑑 0 cos 𝜙 , whic h establishes th e sp a tial relationship u tilized f or the de r iv ation in Appen dix B . A P P E N D I X C F I E L D D E S I G N F O R P A S S I V E R E C E I V E R S V I A S E N S I N G P R O B A B I L I T Y M A X I M I Z A T I O N This append ix d etails th e Maximize Sensing Pr obab ility wa veform design f or the passi ve spherical receiv er , dem on- strating the universality of th e MRP engine presented in Algorithm 1 . For the pa ssi ve receiver , the r eceiv ed pro bability in th e first slot is d etermined by th e sensing p robability at the sampling time 𝑡 𝑠 , i.e., 𝑝 R [ 1 ] = 𝑝 s ( 𝑡 𝑠 ) . From ( 11 ), it can be sh own that 𝑝 s ( 𝑡 𝑠 ) is a concave, monoto nically d ecreasing function of the effecti ve distance 𝑑 0 ( 𝑡 𝑠 ) . The maximu m sensin g probability is achieved whe n the effecti ve distance is n ullified, 𝑑 0 ( 𝑡 𝑠 ) = 0 , wh ich correspo nds to p hysically shifting the center of the particle clou d to the origin of the Rx sphe r e. T o maximize 𝑝 s ( 𝑡 𝑠 ) using the Phase I signal-enhan cing field 𝐸 S 1 ( 𝑡 ) = 𝑉 1 applied durin g 𝑡 ∈ [ 0 , 𝑡 𝑠 ] , we fo rmulate the distance-min im ization p r oblem: ( P pas ) argmin 𝑉 1   𝑥 0 + 𝑐 𝑒 𝑉 1 𝑡 𝑠 + 𝑈 1 ( 𝑡 𝑠 )   2 s . t . 𝑉 2 1 𝑡 𝑠 ≤ 𝜉 , (33) 13 where 𝑈 1 ( 𝑡 𝑠 ) =  𝑡 𝑠 0 𝑢 1 ( 𝛼 ) 𝑑 𝛼 is the b a c kgrou n d flow dis- placement. Solving fo r the root y ie ld s the ideal unconstrained velocity for Phase I a s 𝑞 P S = − ( 𝑥 0 + 𝑈 1 ( 𝑡 𝑠 ) ) / ( 𝑐 𝑒 𝑡 𝑠 ) . If residu al en ergy 𝜉 res = 𝜉 − ( 𝑉 ∗ 1 ) 2 𝑡 𝑠 > 0 rem ains, it is u tilized in Phase II to app ly an oppo sing field 𝑉 2 during 𝑡 ∈ [ 𝑡 𝑠 , 𝑇 𝑏 ] to suppress th e leading ISI at the n ext sampling instance 𝑡 𝑠 + 𝑇 𝑏 . Since the field is period ic, th e total displacement at 𝑡 𝑠 + 𝑇 𝑏 includes the effects of 𝑉 ∗ 1 applied twice (in both slots) a n d 𝑉 2 applied once . T o ag gressiv ely deviate th e particles away from the RX, we determine the optima l 𝑉 2 that maximizes the distance: argmax 𝑉 2   𝑥 0 + 2 𝑐 𝑒 𝑉 ∗ 1 𝑡 𝑠 + 𝑈 1 ( 𝑡 𝑠 + 𝑇 𝑏 ) + 𝑐 𝑒 𝑉 2 ( 𝑇 𝑏 − 𝑡 𝑠 )   2 s . t . 𝑉 2 2 ( 𝑇 𝑏 − 𝑡 𝑠 ) ≤ 𝜉 res . (34) The target unc o nstrained veloc ity to perfe c tly bridge the displacement back to the o rigin (from which we app ly the maximum o p posite sign) is 𝑞 P R = −  𝑥 0 + 2 𝑐 𝑒 𝑉 ∗ 1 𝑡 𝑠 + 𝑈 1 ( 𝑡 𝑠 + 𝑇 𝑏 )  /  𝑐 𝑒 ( 𝑇 𝑏 − 𝑡 𝑠 )  . T o ensure structural symmetry with the fully- a bsorbing design case, by directly inputting the parameters 𝑞 S = 𝑞 P S , 𝑞 R = 𝑞 P R , and explicitly map ping the temporal phases as 𝑇 𝑝 1 = 𝑡 𝑠 , a n d 𝑇 𝑝 2 = 𝑇 𝑏 − 𝑡 𝑠 into the un iv ersal MRP Engine (Algorithm 1 ), the optimal field m agnitudes ( 𝑉 ∗ 1 , 𝑉 ∗ 2 ) fo r the passiv e receiver ar e efficiently generated . R E F E R E N C E S [1] I. F . Akyildiz, F . Brunetti, and C. 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