Exact 3-D Channel Impulse Response for Spherical Receivers with Arbitrary Drift Directions

TL;DR

The paper tackles the lack of an exact 3-D CIR for a fully absorbing spherical receiver under arbitrary drift in molecular MIMO channels. It introduces a Girsanov-based measure transformation to map zero-drift hitting-time statistics to the drifted case, yielding a closed-form, exact CIR expressed as a series with Bessel and Gegenbauer components. The authors validate the formula against extensive Monte Carlo simulations (NRMSE < 4%) and demonstrate substantial speedups in evaluating peak metrics and performing parameter sweeps. This work enables precise, physics-based channel modeling for 3-D spherical receivers and lays the groundwork for efficient MIMO design in advective molecular communications.

Abstract

Accurate channel modeling for spherical absorbing receivers is fundamental to the design of realistic molecular multiple-input multiple-output (MIMO) systems. While advanced modulation schemes have been proposed to mitigate interference, determining the channel impulse response (CIR) under arbitrary flow directions remains a challenge; existing exact solutions are restricted to either 1-D/no-drift scenarios or planar receiver geometries. Addressing this gap, we derive the first exact analytical CIR for a spherical receiver in a 3-D molecular communication system with uniform drift in an arbitrary direction. Unlike prior approximations that ignore the angle between the drift and the transmission axis, our approach utilizes the Girsanov theorem to analytically transform the hitting-time distribution from a stationary medium to a drifted one. The proposed closed-form expression not only eliminates modeling errors inherent in previous approximations for off-axis receivers but also enables efficient parameter-space exploration of critical system metrics (e.g., peak time and amplitude), a task that would be computationally costly with pure simulation-based approaches.

Figures (4)

• Figure 1: System model of the 3-D molecular communication channel. A point transmitter is located at $\mathbf{x}_0$, and a fully absorbing spherical receiver of radius $r$ is at the origin. The medium is subject to a uniform drift $\mathbf{v}$ forming an angle $\psi = \angle(\mathbf{v}, \mathbf{x}_0)$ with the position vector $\mathbf{x}_0$.
• Figure 2: Derivation flow of the proposed method. We start from Yin's joint distribution Yin:2009 and apply the Girsanov theorem to derive the general solution.
• Figure 3: Validation of analytical CIR against Monte Carlo simulations. Top Row: Moderate drift ($|\mathbf{v}|=5 \, \mu\text{m/s}$). Bottom Row: Strong drift ($|\mathbf{v}|=10 \, \mu\text{m/s}$). Columns correspond to positive ($\psi=180^\circ$), perpendicular ($\psi=90^\circ$), and negative ($\psi=0^\circ$) drift directions.
• Figure 4: Analytical characterization of peak amplitude (top panels) and peak time (bottom panels). (a) Dependency on drift velocity magnitude. (b) Dependency on receiver radius. The continuous curves demonstrate the model's capability to efficiently sweep parameter spaces, revealing distinct sensitivities for different drift angles $\psi$.

Theorems & Definitions (2)

• Lemma 1
• Theorem 1: Exact CIR for Arbitrary Drift