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On the Relationship between Transmission Power and Capacity of an Underwater Acoustic Communication Channel
Daniel E. Lucani, Milica Stojanovic, Muriel Médard
TL;DR
Because underwater acoustic path loss depends on both distance and frequency, power and useful bandwidth vary with transmission distance, and relating power, band, distance, and capacity is complicated. The paper derives closed-form approximations for power, band-edge frequency, and bandwidth as functions of distance and required capacity; numerical evaluation reports a good fit to empirical values across studied ranges and noise profiles.
Problem
Frequency-dependent path loss makes power consumption and useful bandwidth depend on transmission distance, while relating power, band, distance, and capacity is complicated.
Source: §I · p. 1Method
The paper develops closed-form approximate models for power, band-edge frequency, and bandwidth as functions of distance and required capacity, using an information-theoretic analysis of acoustic propagation loss and ambient noise.
Source: §I · p. 1, §VI · p. 6Results
Numerical evaluation found that the approximate models fit empirical values well across different distance and capacity ranges and noise profiles.
Source: §VI · p. 6Takeaways & Limitations
For a certain range of distance and capacity, the power model is nearly independent of shipping activity but depends markedly on wind speed, and the models can be used in network optimization for a required data rate.
Source: §VI · p. 6
Abstract
from arXiv · showhide
The underwater acoustic channel is characterized by a path loss that depends not only on the transmission distance, but also on the signal frequency. As a consequence, transmission bandwidth depends on the transmission distance, a feature that distinguishes an underwater acoustic system from a terrestrial radio system. The exact relationship between power, transmission band, distance and capacity for the Gaussian noise scenario is a complicated one. This work provides a closed-form approximate model for 1) power consumption, 2) band-edge frequency and 3) bandwidth as functions of distance and capacity required for a data link. This approximate model is obtained by numerical evaluation of analytical results which takes into account physical models of acoustic propagation loss and ambient noise. The closed-form approximations may become useful tools in the design and analysis of underwater acoustic networks.
I. INTRODUCTION
Frequency-dependent path loss makes both power consumption and useful bandwidth depend on transmission distance. The paper develops closed-form approximations for power and operating band as functions of distance and required capacity, extending earlier distance-based models from an SNR constraint to capacity.
I. INTRODUCTION
- Frequency-dependent path loss makes both power consumption and useful bandwidth depend on transmission distance in underwater acoustic links.This dependence distinguishes underwater acoustic systems from terrestrial radio systems.Source: §I · p. 1
- A simple expression linking transmission power to required capacity could support point-to-point link design and minimum-cost network optimization.The paper frames such expressions as useful from both theoretic and engineering standpoints.Source: §I · p. 1
- The paper presents closed-form approximations for power consumption and operating frequency band as functions of distance and capacity.The approximate model accounts for acoustic propagation loss and colored Gaussian ambient noise.Source: §I · p. 1
- Unlike earlier distance-based models whose coefficients depend on required SNR, this work uses desired link capacity as the figure of merit.The earlier model described power as P(l) = pl^γ and a similar relationship for operating bandwidth.Source: §I · p. 1
II. CHANNEL MODEL
The channel model combines distance- and frequency-dependent acoustic path loss with ambient noise, then treats capacity under a Gaussian-channel assumption. Its operating band and power depend on distance and target capacity, while changing the spreading factor leaves bandwidth unchanged and scales power.
II. CHANNEL MODEL
- Acoustic path loss depends on both link distance and signal frequency, with the spreading factor and absorption coefficient describing propagation.The spreading factor represents propagation geometry; examples include spherical spreading at k = 2, cylindrical at k = 1, and practical spreading at k = 1.5.Source: §II · p. 1
- The ambient-noise model includes turbulence, shipping, waves, and thermal noise, with shipping activity and wind speed among its parameters.The shipping activity factor ranges from 0 to 1, and wind speed is measured in m/s.Source: §II · p. 2
- Assuming a Gaussian channel, the model uses waterfilling to determine capacity and allows power and operating band to adjust to reach a target capacity.The optimum band may comprise non-overlapping intervals, each with lower- and higher-end frequencies.Source: §II · p. 2
- The analytical relations between distance, capacity, power, and band edges are complicated and provide little insight, motivating an approximate model over relevant ranges.The model is intended to allow flexibility in spreading factor, wind speed, and shipping activity.Source: §II · p. 2
- Changing the spreading factor leaves the transmission bandwidth unchanged, while the model extends to other spreading factors through a scaling of K(l, C).For the same distance and capacity, K(l, C, kj) = l^(kj−ki)K(l, C, ki).Source: §II · pp. 2–3
III. NUMERICAL EVALUATION PROCEDURE
A numerical procedure evaluates power and frequency-band quantities over distance and capacity values. For each distance, it raises a parameter until the target capacity is reached.
III. NUMERICAL EVALUATION PROCEDURE
- The procedure computes P(l, C), f̂ini(l, C), and f̂end(l, C) over a region of distance and capacity values.Source: §III · p. 3
- Figure 4 identifies a1 and a2 for the bandwidth approximation over 0–10 km and 0–2 kbps.Source: Figure 4 · p. 3
- For each distance, initialize K(l, C) to the minimum of A(l, f)N(f); the minimizing frequency f0 is the optimal frequency.Source: §III · p. 3
- Increase K(l, C) iteratively until the target capacity is met, then repeat the process for each capacity in the range of interest.Source: §III · p. 3
- At each iteration, define B(n)(l, C) by frequencies satisfying A(l, f)N(f) ≤ K(n)(l, C), compute capacity, and stop once it reaches the target.Source: §III · p. 3
IV. APPROXIMATE MODELS
The paper presents approximate models for transmission power, high-end frequency, and bandwidth. The power models cover two distance-capacity ranges, and similar models fit the frequency and bandwidth quantities.
IV. APPROXIMATE MODELS
- Figure 5 identifies a1 and a2 for the power model over 0–100 km and 0–100 kbps.Source: Figure 5 · p. 4
- Power consumption is approximated for 0–10 km and 0–2 kbps, and for 0–100 km and 0–100 kbps, with k = 1.5.Source: §IV · pp. 3–4
- Similar models provide a good fit for the high-end frequency f̂end(l, C) and bandwidth B̂(l, C) = f̂end(l, C) − f̂ini(l, C).Source: §IV · p. 4
V. NUMERICAL RESULTS
The proposed models fit the numerical results well across the two examined distance-capacity ranges. For the low-range, low-rate case, the power-model parameters depend more on wind speed than on shipping activity, allowing a wind-only approximation.
V. NUMERICAL RESULTS
- The approximation tables report a1 and a2 parameters for power, band-edge frequency, and bandwidth in the 0–10 km, 0–2 kbps and 0–100 km, 0–100 kbps ranges.The tables also report mean square error relative to the actual parameters.Source: Table I · p. 4, Table II · p. 5, Table III · p. 5
- For the 0–100 km, 0–100 kbps case, the reported figures show a1 and a2 parameters for power, band-edge frequency, and bandwidth.This case uses k = 1.5, s = 0.5, and w = 0 m/s.Source: §V · pp. 4–5, Figure 6 · p. 4, Figure 7 · p. 5
- For the 0–10 km, 0–2 kbps case, the a2(C) parameter for P(l, C) can use a linear approximation instead of a quadratic model.The figures for the cases use k = 1.5, s = 0.5, and w = 0 m/s.Source: §V · pp. 4–5
- For the low-range, low-rate case, the power-model parameters change little with shipping activity but depend more strongly on wind speed.The passage relates this difference to the modeled noise dependencies: shipping activity scales as O(f^-3.4), while wind speed scales as O(f^-2).Source: §V · p. 5
- For that low-range case, the approximation can use wind speed alone rather than both shipping activity and wind speed.The paper reports that α and β can be treated as functions of w only.Source: §V · pp. 4–5
VI. CONCLUSION
The paper presents closed-form approximate models for power, bandwidth, and band-edge frequency in a time-invariant Gaussian acoustic channel. Numerical evaluations show good fit across examined ranges and noise profiles; band-edge frequency and bandwidth are invariant to spreading factor, while power scales with distance and spreading-factor change.
VI. CONCLUSION
- The closed-form approximate models for power, bandwidth, and band-edge frequency fit numerical values well across examined distances, capacities, and shipping-activity and wind-speed noise profiles.The models assume a time-invariant channel with Gaussian noise and account for acoustic path loss and ambient noise.Source: §VI · p. 6
- The band-edge frequency and bandwidth are invariant to spreading factor k, while power scales as P(l, C, k′) = l^(k′−k)P(l, C, k).Source: §VI · p. 6
- For a certain range of distance and capacity, the power model is nearly independent of shipping activity but depends markedly on wind speed, with a smooth dependence that can be approximated simply.Source: §VI · p. 6
- The models can be used in network optimization to determine optimal power consumption for a required data rate.Source: §VI · p. 6
- Future work will study convexity properties of the P(l, C) model and its use in network optimization.Source: §VI · p. 6