Shallow-water underwater acoustic (UWA) orthogonal frequency-division multiplexing (OFDM) systems suffer Doppler-induced inter-channel interference (ICI), and analytical studies quantify it through the channel's time-correlation function, itself obtained by Fourier conversion from a Doppler power spectrum (Jakes, uniform, or two-path). That conversion presupposes wide-sense stationarity, while shallow UWA channels are documented to be non-stationary; the source study (Ha, Nguyen, and Nguyen, 2021) derives ICI-aware per-subcarrier signal-to-interference-plus-noise ratio (SINR) and capacity from a geometry-based simulation model, explicitly noting that the conversion between Doppler spectrum and time-correlation function 'should be accompanied by certain conditions' that are not available. The open problem is to establish those explicit conditions — including the corrected time-varying spectra-to-correlation relation with a quantified approximation error over an OFDM symbol — and, on that basis, to derive corrected ICI-aware SINR and capacity expressions for shallow-water UWA OFDM that clarify how far the source's numerical conclusions extend beyond its single simulated scenario (carrier frequency 30 kHz, receiver SNR 20 dB, uniformly distributed scatterers). No later work — whether non-stationary geometry-based channel models, stationarity-testing methods, or ICI-mitigation schemes — states the conversion conditions or validates the ICI-aware capacity analysis, leaving the problem open.

Background

Underwater acoustic (UWA) communication is the only practical way to transmit data wirelessly over useful distances in the ocean, and its channel is among the hardest propagation media in communications: sound travels at only about 1500 m/s, so modest platform or sea-surface motion produces severe Doppler distortion; reflections from the sea surface and seabed in shallow water create long multipath delay spreads; and absorption loss grows quickly with frequency and distance, confining useful systems to bandwidths of a few tens of kilohertz. Orthogonal frequency-division multiplexing (OFDM), which transmits on many narrow, mutually orthogonal subcarriers so that a long channel delay spread appears as flat fading per subcarrier, is therefore attractive — but orthogonality survives only if the channel stays effectively constant over one OFDM symbol. Time variation destroys orthogonality and leaks power between subcarriers, an effect called inter-channel interference (ICI, also intercarrier interference). Because the total usable bandwidth of a UWA system is small, the subcarrier spacing is narrow and even modest Doppler-induced spectral broadening produces ICI that materially reduces achievable rate.

A standard analytical route quantifies ICI through second-order channel statistics. Under the wide-sense stationary uncorrelated scattering (WSSUS) assumption — the channel impulse response has time-invariant second-order statistics, and contributions from different path delays are uncorrelated — the time-correlation function and the Doppler power spectrum form a Fourier-transform pair (Bello's classical characterization of randomly time-variant linear channels), so one may specify, say, a Jakes U-shaped Doppler spectrum and read off the time correlation. Classical OFDM analyses, e.g., Li and Cimini (2001), compute ICI power from exactly such a time-correlation function, and per-subcarrier signal-to-interference-plus-noise ratio (SINR) and capacity expressions follow. In underwater acoustics this route is questionable twice over. First, Doppler in UWA systems is not a small common frequency shift: the signal is wideband relative to its carrier, so each propagation path experiences a Doppler scaling and the distortion differs from subcarrier to subcarrier (Li, Zhou, Stojanovic, Freitag, and Willett, 2008). Second, and centrally here, shallow UWA channels are widely documented to be non-stationary — their statistics drift with moving scatterers, sea state, and platform motion — so the Fourier conversion from a Doppler spectrum to a single time-correlation function is not automatically valid. van Walree (2013) observes that the WSSUS assumption is rarely tested on UWA data and that no typical, canonical UWA channel exists; experimental statistical characterizations (Radosevic, Proakis, and Stojanovic, 2009; Qarabaqi and Stojanovic, 2013) instead build the channel from physical paths, each carrying its own Doppler scale.

The source study sits in this setting. Using a geometry-based stochastic channel model (GBSM — a model whose multipath is generated from randomly placed scatterers on the rough sea surface and seabed, due to Naderi, Pätzold, and Zajic, 2014) with scattering points uniformly distributed between transmitter and receiver, Ha, Nguyen, and Nguyen (2021) derived the time-variant channel impulse response, the per-subcarrier SINR including ICI, and the corresponding channel capacity for a shallow-water UWA OFDM system, evaluated numerically at a carrier frequency of 30 kHz and receiver SNR of 20 dB. The paper explicitly acknowledges the gap this problem tracks: prior ICI analyses convert a Doppler spectrum (Jakes, uniform, or two-path) into a time-correlation function, an assumption that 'may not be applicable to UWA channels due to the complicated time varying characteristics of the shallow UWA propagation environments'; since UWA channels are non-stationary, 'the conversion between the Doppler spectrum and the time correlation function should be accompanied by certain conditions' — conditions the paper does not provide. The same group had earlier modeled UWA Doppler power spectra (Ha, Nguyen, and Nguyen, 2017; Naderi, Ha, Nguyen, and Pätzold, 2017, the latter explicitly for non-stationary channels from measurements), and the general mathematics for non-WSSUS channels — time-frequency-dependent correlation functions and local scattering functions (Matz, 2005) — exists in wireless channel theory. Neither line, however, has been instantiated into explicit conversion conditions or into corrected ICI/SINR/capacity formulas for shallow UWA OFDM. Meanwhile, non-stationary GBSMs for shallow UWA (Zhu, Wang, and Ma, 2021; Yang, Ma, Chang, and Wang, 2025) produce time-varying correlation functions and Doppler spectra directly from geometry, yet their own authors flag measurement-based validation as incomplete, and they are not connected to OFDM ICI analysis.

Problem Statement

The problem, as left open by Ha, Nguyen, and Nguyen (2021), has two linked parts serving one objective — a scientifically justified characterization of ICI in shallow-water UWA OFDM under channel non-stationarity.

(1) Conversion conditions. Establish explicit conditions under which, for non-stationary shallow underwater acoustic channels, a Doppler power spectrum may legitimately be converted into the time-correlation function used in ICI-power, per-subcarrier SINR, and capacity analyses of UWA OFDM. This requires formulating the correct relation for genuinely non-stationary channels (for example, an instant- or window-dependent Doppler spectrum together with a time- and instant-dependent correlation function) and proving sufficient — ideally also necessary — conditions under which the classical stationary Fourier conversion approximates it, with an error bound expressed in identifiable channel and system parameters, for example the stationarity interval (the time span over which the channel's statistics can be treated as wide-sense stationary) versus the OFDM symbol duration, Doppler spread and the rate of change of channel statistics, subcarrier spacing, number of subcarriers, and signal bandwidth.

(2) Consequences for ICI-aware capacity analysis. Using those conditions, derive the corresponding corrected (or provably approximate) per-subcarrier SINR and capacity expressions for shallow-water UWA OFDM. In the stationary limit — when the channel's second-order statistics become wide-sense stationary, so that the time-correlation function and the Doppler spectrum form a Fourier-transform pair — these must recover the per-subcarrier SINR and capacity that the source evaluates from the time-variant channel transfer function of its geometry-based model. Through those expressions, determine how far the source's numerical conclusions (for example, that ICI is negligible for bandwidths above 10 kHz at Doppler frequencies of 1-2 Hz, and its recommended subcarrier counts, symbol lengths, and bandwidths) depend on the specific simulation assumptions: uniformly distributed scatterers, a single carrier frequency of 30 kHz, a fixed receiver SNR of 20 dB, and one family of scattering statistics.

An answer that supplies only mitigating receiver algorithms (equalization, Doppler compensation, or alternative waveforms) does not resolve this problem: the question concerns the validity of the analytical channel-statistics-to-ICI-prediction chain itself.

Both parts are jointly required, and every sub-task listed within them must be addressed: the correct non-stationary relation, the conditions and error bound for the classical conversion, the corrected per-subcarrier SINR and capacity expressions, their recovery of the source's stationary-limit values, and the assessment of how far the source's numerical conclusions extend; an answer that establishes only the conversion conditions, or that omits the corrected SINR and capacity expressions or that assessment, does not resolve the problem.

References

  1. Do Viet Ha, Tien Hoa Nguyen, and Van Duc Nguyen, 'Impact of Inter-Channel Interference on Shallow Underwater Acoustic OFDM Systems', 2021, arXiv:2101.02089, DOI 10.48550/arXiv.2101.02089, https://arxiv.org/abs/2101.02089; journal version: Nguyen Tien Hoa, Do Viet Ha, and Nguyen Van Duc, Journal of Research and Development on Information and Communication Technology, vol. 2020, no. 1, pp. 42-49, 2020, DOI 10.32913/mic-ict-research.v2020.n1.929, https://doi.org/10.32913/mic-ict-research.v2020.n1.929 (source paper)
  2. P. A. Bello, 'Characterization of Randomly Time-Variant Linear Channels', 1963, DOI 10.1109/tcom.1963.1088793, https://doi.org/10.1109/tcom.1963.1088793
  3. Ye Li and L. J. Cimini, 'Bounds on the interchannel interference of OFDM in time-varying impairments', 2001, DOI 10.1109/26.911445, https://doi.org/10.1109/26.911445
  4. B. Li, S. Zhou, M. Stojanovic, L. Freitag, and P. Willett, 'Multicarrier Communication Over Underwater Acoustic Channels With Nonuniform Doppler Shifts', 2008, DOI 10.1109/joe.2008.920471, https://doi.org/10.1109/JOE.2008.920471
  5. A. Radosevic, J. G. Proakis, and M. Stojanovic, 'Statistical characterization and capacity of shallow water acoustic channels', OCEANS 2009 - Europe, 2009, DOI 10.1109/oceanse.2009.5278349, https://doi.org/10.1109/oceanse.2009.5278349
  6. G. Matz, 'On non-WSSUS wireless fading channels', 2005, DOI 10.1109/twc.2005.853905, https://doi.org/10.1109/TWC.2005.853905
  7. P. Qarabaqi and M. Stojanovic, 'Statistical Characterization and Computationally Efficient Modeling of a Class of Underwater Acoustic Communication Channels', 2013, DOI 10.1109/joe.2013.2278787, https://doi.org/10.1109/JOE.2013.2278787
  8. P. A. van Walree, 'Propagation and Scattering Effects in Underwater Acoustic Communication Channels', 2013, DOI 10.1109/joe.2013.2278913, https://doi.org/10.1109/JOE.2013.2278913
  9. E. Baktash, M. J. Dehghani, M. R. Nasab, and M. Karimi, 'Shallow Water Acoustic Channel Modeling Based on Analytical Second Order Statistics for Moving Transmitter/Receiver', 2015, DOI 10.1109/tsp.2015.2411219, https://doi.org/10.1109/TSP.2015.2411219
  10. Meisam Naderi, Matthias Pätzold, and Alenka G. Zajic, 'A Geometry-Based Channel Model for Shallow Underwater Acoustic Channels Under Rough Surface and Bottom Scattering Conditions', 2014, DOI 10.1109/cce.2014.6916689, https://doi.org/10.1109/CCE.2014.6916689
  11. P.-J. Socheleau, C. Laot, and J.-M. Passerieux, 'Concise Derivation of Scattering Function from Channel Entropy Maximization', 2010, DOI 10.1109/tcomm.2010.091310.090247, https://doi.org/10.1109/TCOMM.2010.091310.090247
  12. D. V. Ha, V. D. Nguyen, and Q. K. Nguyen, 'Modeling of Doppler power spectrum for underwater acoustic channels', Journal of Communications and Networks, 2017, DOI 10.1109/jcn.2017.000044, https://doi.org/10.1109/JCN.2017.000044
  13. M. Naderi, D. V. Ha, V. D. Nguyen, and M. Pätzold, 'Modelling the Doppler power spectrum of non-stationary underwater acoustic channels based on Doppler measurements', OCEANS 2017 - Aberdeen, 2017, DOI 10.1109/oceanse.2017.8084993, https://doi.org/10.1109/OCEANSE.2017.8084993
  14. P. A. van Walree, P.-J. Socheleau, R. Otnes, and T. Jenserud, 'The Watermark Benchmark for Underwater Acoustic Modulation Schemes', 2017, DOI 10.1109/joe.2017.2699078, https://doi.org/10.1109/JOE.2017.2699078
  15. A. Radosevic, R. Ahmed, T. M. Duman, J. G. Proakis, and M. Stojanovic, 'Adaptive OFDM Modulation for Underwater Acoustic Communications: Design Considerations and Experimental Results', 2014, DOI 10.1109/joe.2013.2253212, https://doi.org/10.1109/JOE.2013.2253212
  16. Iwona Kochanska, 'Assessment of Wide-Sense Stationarity of an Underwater Acoustic Channel Based on a Pseudo-Random Binary Sequence Probe Signal', Applied Sciences, 2020, DOI 10.3390/app10041221, https://doi.org/10.3390/app10041221
  17. Do Viet Ha and Nguyen Van Duc, 'Analytical method of parameter determination for OFDM system over measurement-based underwater acoustic channel modeling', Physical Communication, 2020, DOI 10.1016/j.phycom.2020.101045, https://doi.org/10.1016/j.phycom.2020.101045
  18. Xiumeng Zhu, Cheng-Xiang Wang, and Ruofei Ma, 'A 2D Non-Stationary Channel Model for Underwater Acoustic Communication Systems', 2021, arXiv:2108.06513, DOI 10.48550/arXiv.2108.06513, https://arxiv.org/abs/2108.06513; also Proc. IEEE 93rd Vehicular Technology Conference (VTC2021-Spring), 2021, DOI 10.1109/VTC2021-Spring51267.2021.9448976, https://doi.org/10.1109/VTC2021-Spring51267.2021.9448976
  19. Xianpeng Li, Yupeng Tai, Haibin Wang, Jun Wang, Shuo Jia, Yonglin Zhang, and Weiming Gan, 'A metric to quantify the time-varying characteristics of underwater acoustic communication channels', JASA Express Letters, 2024, DOI 10.1121/10.0026601, https://doi.org/10.1121/10.0026601
  20. Yuxuan Yang, Yilin Ma, Hengtai Chang, and Cheng-Xiang Wang, 'A Quasi-Deterministic Channel Model for Underwater Acoustic Communication Systems', 2025, arXiv:2501.04238, DOI 10.48550/arXiv.2501.04238, https://arxiv.org/abs/2501.04238; conference version: Proc. IEEE 99th Vehicular Technology Conference (VTC2024-Spring), 2024, DOI 10.1109/VTC2024-Spring62846.2024.10683411, https://doi.org/10.1109/VTC2024-Spring62846.2024.10683411