Receiver-Surface Hit Patterns via Legendre Approximation for Molecular Signal Detection

I. INTRODUCTION

• Count-only observations discard directional surface patterns that can distinguish transmitter-induced arrivals from background or unrelated molecular activity.The detection test compares the angular-time distribution under noise alone with that under noise plus an active signal.Source: §I · p. 1
• Finite Legendre expansions approximate the axially symmetric angular hit density using the cosine similarity between transmitter and absorption directions.Exact surface-distribution characterization usually requires diffusion equations with absorbing boundary conditions, which can be difficult to use directly in likelihood-based detection.Source: §I · p. 1
• The proposed detectors combine molecule counts, surface locations, and arrival times in memoryless and Viterbi-based schemes.The Legendre-Viterbi detector uses the Legendre-based angular-time likelihood within sequence detection.Source: §I · p. 1
• Incorporating receiver-surface angular information improves detection over count-only approaches, particularly in low-observation regimes.The detectors offer different performance-complexity tradeoffs.Source: §I · p. 1

II. SYSTEM MODEL

• The modeled system uses BCSK, a single point transmitter, and a perfectly absorbing spherical receiver in an unbounded three-dimensional environment.Molecules propagate by Brownian motion, and absorbed molecules are removed from the environment.Source: §II · pp. 1–2
• The figure depicts early arrivals concentrated near the transmitter-facing side and later ISI arrivals becoming more spread and uniform over the receiver surface.Its right panel plots angular density against θ, represented through z = cos θ.Source: Figure 1 · p. 2
• Each observation includes the absorbed molecules’ surface locations and arrival-time bins, alongside their count.The surface locations provide spatial structure in addition to the capture probability that determines expected absorbed counts.Source: §II · p. 2
• The noise component can include environmental background, unrelated molecular activity, and intersymbol interference from previous symbols.Under H0 the per-molecule likelihood is the normalized noise angular-time density qN, and the count model is centered around λN.Source: §II · p. 2
• Under H1, the current symbol adds a structured signal with expected count λS and normalized angular-time density pS to the noise component.The active-symbol per-molecule likelihood is a weighted superposition, with the count centered at λN + λS.Source: §II · p. 2

• Axial symmetry reduces the surface-density model to the cosine of the angle between transmitter and absorption directions, a one-dimensional variable on [−1, 1].Legendre polynomials form an orthogonal basis on this interval.Source: §III · p. 2
• The conditional angular density h(z | d, b) is approximated by a finite Legendre expansion whose coefficients are orthogonal projections when the exact density is available.In practice, the coefficients are fit to particle-based empirical angular histograms with a ridge penalty.Source: §III · pp. 2–3
• The truncated expansion is floored and renormalized before being converted into a surface density with respect to solid angle.This handles small negative values in the truncated expansion.Source: §III · p. 3
• When the time-bin masses sum to one, pS is a normalized joint angular-time density, while molecule counts are modeled separately through λS.The expected signal count is λS = NTxFhit(Ts, d, r).Source: §III · p. 3
• A particle-based simulator records absorption directions and arrival-time bins across transmitter distances and symbol taps to build the reference library used by the detectors.Experiments use L = 10 and B = 100; environmental noise is absent, so the noise component is ISI only.Source: §III · p. 3

• The detector compares the expected-noise model under H0 with the noise-plus-current-signal model under H1 using a likelihood ratio.The expected current-signal count is λS = λ0 = NTxp(0).Source: A. Legendre Detection Rule · p. 3
• Because past bits are unavailable to the memoryless detector, it replaces each unknown previous bit with its prior mean when forming expected ISI.The resulting noise model weights each previous-symbol angular profile by its expected mass; in the experiments, environmental noise is zero.Source: A. Legendre Detection Rule · p. 3
• The log-likelihood ratio combines count and shape terms, and the decision threshold is selected to meet a desired BER or false-alarm probability.The same count/shape decomposition is reused by the hybrid detector.Source: A. Legendre Detection Rule · p. 3

• Sequence detection replaces prior-averaged ISI with candidate bit histories that specify which past symbols were active.The Viterbi state stores the previous K −1 candidate bits, and each library tap supplies an angular-time intensity for a past emission.Source: B. Legendre-Viterbi Sequence Detection · p. 3
• Each Viterbi branch is scored using the angular-time intensity of the absorbed molecules for that candidate history.The branch observations are modeled as an inhomogeneous Poisson point process.Source: B. Legendre-Viterbi Sequence Detection · p. 4

C. Hybrid Legendre-Viterbi Detector

• Because the prior-averaged expected-ISI background is independent of bit history, the shape term is computed once per symbol and reused across branches.The shape log-likelihood excludes Poisson count factors.Source: C. Hybrid Legendre-Viterbi Detector · p. 4
• The hybrid branch score retains state-conditioned counts while crediting the current active bit with a precomputed Legendre shape term.Unlike the full detector, it does not recompute the angular-time intensity for each candidate branch.Source: C. Hybrid Legendre-Viterbi Detector · p. 4
• Per-symbol costs are O(1) Fixed threshold, O(NRxL) Legendre, O(2K−1) Count Viterbi, O(2K−1 + NRxL) Hybrid Legendre-Viterbi, and O(2KNRxK) Full Legendre-Viterbi.Table I summarizes detector computational complexity per symbol.Source: C. Hybrid Legendre-Viterbi Detector · p. 4, Table I · p. 4

V. PERFORMANCE EVALUATION

• As NTx increases at Ts = 0.2 s, BER falls for all detectors; Legendre-Viterbi further improves on Count Viterbi by adding angular-time likelihood to sequence memory.The memoryless Legendre detector achieves BER comparable to Count Viterbi.Source: §V · p. 4
• At fixed NTx = 100, longer Ts improves BER, while Legendre-based detectors show their advantage most clearly at small Ts.Shorter symbol durations yield fewer absorbed molecules and less reliable count statistics.Source: §V · p. 4, Figure 3 · p. 5
• Legendre-Viterbi achieves the lowest BER, with the angular-pattern advantage most visible at small Ts or small NTx.The low-count settings make count-only information less discriminative.Source: §V · pp. 4–5, Figure 3 · p. 5
• In the P0|1-versus-P1|0 comparison, the lower-left operating points indicate better performance, and the Legendre detector improves on the fixed-threshold detector.The fixed-threshold detector relies only on molecule counts.Source: §V · p. 5, Figure 4 · p. 5

VI. CONCLUSION

• The proposed detector models molecule-absorption angles with Legendre coefficients, retaining surface information discarded by count-only methods.It compares observed angular patterns under physically meaningful models; a true bit-1 shifts the angular-time pattern toward the Legendre profile around direction a.Source: §VI · p. 5
• The full Legendre-Viterbi detector achieves the best BER by using state-dependent angular-time likelihoods with sequence memory.Its gains are especially visible in short Ts regimes, where count statistics become less reliable.Source: §VI · p. 5
• The memoryless Legendre detector performs comparably to count-based Viterbi without Viterbi’s exponential sequence complexity.Source: §VI · p. 5
• The hybrid Legendre-Viterbi detector combines state-conditioned count likelihoods with a memoryless Legendre shape term.Source: §VI · p. 5