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Underwater Acoustic Signal and Noise Modeling

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Underwater Acoustic Signal Processing

Part of the book series: Modern Acoustics and Signal Processing ((MASP))

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Abstract

In this chapter, the four dimensions in which underwater acoustic signals can be categorized are introduced: time, frequency, consistency from observation to observation, and knowledge of structure. Recalling the remote-sensing application, the impact of propagation through an underwater acoustic channel on source-signal characterization is described in terms of its effect on signal amplitude and phase. Various representations of bandpass signals are presented, including the analytic signal, complex envelope, envelope and instantaneous intensity. Statistical models for sampled time-series data are obtained for signals and noise to support derivation and analysis of detection and estimation algorithms. Reverberation in active systems is characterized as a random process in order to describe its autocorrelation function and power spectral density. The effect on reverberation arising from the motion of the sonar platform or reverberation-source scatterers, known as Doppler spreading, is introduced and approximated. In addition to the standard Gaussian noise model, a number of heavy-tailed distributions are described including the K distribution, Poisson-Rayleigh, and mixture distributions. Standard statistical models for signals and signals-plus-noise are presented along with techniques for evaluating or approximating the probability of detection and probability of false alarm.

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Notes

  1. 1.

    As noted in Sect. 7.4.1, ambient noise can be assumed stationary over short periods of time. However, as described in Sect. 7.4.2, a normalization process needs to occur in order to treat reverberation as a WSS random process.

  2. 2.

    The bandwidth of the complex envelope accounts for both the positive and negative frequencies.

  3. 3.

    As described in [14, Ch. 5] there exist broadband waveforms with significant Doppler sensitivity that are not subject to (7.7).

  4. 4.

    As shown in Sect. 4.4 or [24, Ch. 12], the Hilbert transform of s(t) is \(s_h(t)= \pi ^{-1}\int _{-\infty }^{\infty } s(\tau )/(\tau -t) \,\mathrm {d} \tau \). Note, however, that some texts define the Hilbert transform as the negative of that used here (i.e., with t − τ in the denominator of the definition rather than τ − t).

  5. 5.

    Recall from Sect. 5.5 that the ACF for complex WSS random processes is .

  6. 6.

    Recall from Sect. 5.5 that \(R_{\tilde {z}\tilde {z}}(\tau ) = 2R_{uu}(\tau )+j2R_{vv}(\tau )\) and R uv(τ) = −R uv(−τ) for a WSS proper complex random process.

  7. 7.

    Ergodicity in a WSS bandpass process with no zero-frequency content implies the ensemble mean is zero and that the temporally averaged power equals the variance.

  8. 8.

    Recall from the discussion in Sect. 5.3.6 that matrix–vector notation representing a vector as a lower-case bold letter (e.g., x) takes precedence over the mathematical-statistics notation of random variables taking an upper case letter and its observed value the lower case. Whether x = [X 1⋯X n]T or x = [x 1⋯x n]T must be discerned from context.

  9. 9.

    A trivial example of this can be found by letting θ ∼Unif(0, 4π).

  10. 10.

    Acknowledgement: CTBTO [29] with gratitude to Drs. G. Haralabus, M. Zampolli, P. Nielsen, and A. Brouwer for their assistance in identifying, accessing, and interpreting the data.

  11. 11.

    A distribution family is closed under a mathematical operation if the result of the operation is still within the family. For example, the Gaussian distribution is closed under translation, scale, and the addition of other Gaussian random variables.

  12. 12.

    Note that the approximations in Table 7.6 for the K distribution are accurate enough for use as a CRLB, but not accurate enough to obtain the off-diagonal term of the Fisher information matrix without adding terms in the 1∕α polynomial.

  13. 13.

    Note that this differs from that presented in [85] by not assuming the data have been normalized and by a factor of 2 so the average intensity is P o + λ rather than 2(P o + λ).

  14. 14.

    This technique has its origins in [96] and was applied in [97] to obtain the envelope PDF.

  15. 15.

    The computational routine from [98] has been implemented as a MATLAB® subroutine in [99]. Note that the definition of the Hankel transform in [99] differs from that used here, which is from [24, pg. 248], by absorbing the “x” in the integrand of (7.234) into the function g(x).

  16. 16.

    Note that the parameter variables used here differ from those in [53] to align with the (α, β) gamma-distribution convention: a g is the shape, b g the scale, and c g the location or shift parameter.

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    Douglas A. Abraham

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Abraham, D.A. (2019). Underwater Acoustic Signal and Noise Modeling. In: Underwater Acoustic Signal Processing. Modern Acoustics and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-92983-5_7

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