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Detecting Signals with Unknown Duration and/or Starting Time: Sequential Detectors

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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 most applications of remote sensing, signals have an unknown arrival time (e.g., arising from an unknown range in active sensing) and often an unknown duration. Detectors for such signals, which sequentially incorporate and test data as it is measured, are derived and evaluated in this chapter. Sliding incoherent sum and sliding M-of-N (binary integration) detectors are presented for cases where the signal duration is known but the starting time is not. For the opposite scenario, where the starting time is known and the signal duration is not, the sequential probability ratio test (SPRT) is used. When neither the starting time nor signal duration is known, Page’s test is shown to arise as a generalized likelihood ratio detector. For each of these detectors, the probability of false alarm is one because they will eventually declare a detection when left to run for an infinitely long time. As such, the average time between false alarms is introduced and used as a performance metric in addition to the probability of detection and average delay before detection. Various techniques are presented for evaluating the performance measures including approximations and a quantization approach. The chapter is concluded with a design example applying Page’s test to data from a cell-averaging normalizer.

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Notes

  1. 1.

    In some texts the stopping time is described using the infimum (“inf”) rather than minimum so as to include the limiting case of K →∞.

  2. 2.

    The SNR s here is equivalent to the S d used in earlier chapters. However, owing to the applicability of sequential detectors to other scenarios a generic SNR term is used.

  3. 3.

    Note that in most texts on sequential detection and hypothesis testing the variable N is used to describe the stopping time. In this text the variable K is used so as to avoid confusion with the N in M-of-N detectors.

  4. 4.

    This can be shown using the results of [20] which dictate that if the data are iid and , then for some c > 0 and r ∈ (0, 1) with the stopping time K as in (10.42).

  5. 5.

    Note that in Wald’s book [1] and some others on sequential analysis, A and B are the opposite of the variables used here.

  6. 6.

    As an arcane side note, g(x) in (10.77) is not specifically a log-likelihood ratio of the data model in (10.74) because it does not differentiate whether signal is present or not based on the time index.

  7. 7.

    Note that if E 0[g(X)] = 0 and E s[g(X)] > 0, the relationship between D and F is quadratic (F ∼ D 2); clearly this should be avoided.

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  1. Ellicott City, MD, USA

    Douglas A. Abraham

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Abraham, D.A. (2019). Detecting Signals with Unknown Duration and/or Starting Time: Sequential Detectors. In: Underwater Acoustic Signal Processing. Modern Acoustics and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-92983-5_10

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  • DOI: https://doi.org/10.1007/978-3-319-92983-5_10

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