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A Performance Study of Some Transient Detectors Zhen Wang and Peter Willett Abstract—We present a simulation study of several different statistics applied to the detection of unknown transient signals in white Gaussian noise. The results suggest that relatively unsophisticated tests based on tem- poral localization of power, such as the Page test and a test based on a new statistic due to Nuttall, give reliable results. Index Terms—Transient detection. I. INTRODUCTION In many applications, it is desired to identify blocks of data that con- tain, in addition to noise, a signal of short duration. For example, in the passive sonar situation, a threat may betray itself by a single accidental report, and in the very different application of process-monitoring by acoustic emission (AE), transients may be due to the sudden release of stress (cracking). At any rate, were the transient signal known, the problem would be trivial. The interest, of course, is to develop an ap- proach useful for signals regardless of their form, length, or location. There are a number of techniques proposed for the detection of tran- sients, some with simple implementation (a Page test), some more com- plex (for example, techniques based on the Gabor transformation), and some quite numerically-intensive indeed and, at present, not well suited to real-time application. In this correspondence, we compare numeri- cally the performances of several of the first two types of schemes on a variety of simulated transients, each added to white Gaussian noise. We do not wish to represent that this study is exhaustive, either in terms of the transients used or the detectors tested—we have tried to make the former representative, and as to the latter, we apologize for omissions. We also have made no attempt at “tuning” beyond what has been sug- gested in the open literature; on the contrary, we wish to interrogate the various algorithms in situations to which they are not particularly well matched. The detectors are as follows. Gb: GLRT Based on the Gabor Transformation: The signal model is , in which denotes the observations arranged Manuscript received February 22, 1999; revised April 19, 2000. This work was supported by the National Science Foundation under Con- tract DMI-9634859 and by the Office of Naval Research under Contract N00014-98-1-0049. The associate editor coordinating the review of this paper and approving it for publication was Prof. S. M. Jesus. The authors are with the Department of Electrical and Systems Engi- neering, University of Connecticut, Storrs, CT 06268-2157 USA (e-mail: willett@engr.uconn.edu). Publisher Item Identifier S 1053-587X(00)06676-9. in an -dimensional column vector, is a pre-chosen matrix of di- mensions whose columns consist a basis for the signal sub- space, the vector is unknown and is called the “signal descriptor,” the mismatch describes the residual, and is white and Gaussian, with zero mean and unity variance (for convenience). Assume the signal can have frequencies in the range , and let the sampling rate be and the observation interval be seconds. We have and . We let the transform matrix be the left inverse of and use instead, if necessary, to make orthonormal. The generalized likelihood ratio test (GLRT) statistic is given by [4]. (1) in which the type of basis determines . Here, the basis is the Gabor transformation [9], which is most appropriate for decaying narrowband signals. Wd: GLRT Based on the Wavelet Transform: The GLRT in (1) is based on the wavelet transformation [4], [1], [2]. The rows of are the sampled versions of the wavelet transform operator. Since we do not wish our results to be specific to any type of transient, we choose the Daubechies (order 4) basis. We scale to six levels. P2: Nuttall’s Power-Law Detector: A detector attracting interest uses the “power-law” [6] (2) where the are the magnitude-squared FFT bins corresponding to the observations . The choice has been shown to have good robustness properties, and we refer to it as P2. Mx: Nuttall’s “Maximum” Detector: In the time domain, when both the signal duration and the average signal power are known, the processor (3) is optimal under the assumption that a white Gaussian transient’s start- point is uniformly distributed [7]. Clearly, the signal power must be known precisely; the “maximum” processor (4) is introduced [7] as an approximation to the previous when only the duration is known, and in our experience, little is lost by its use. In our simulation, we use , which, admittedly, is well matched to the signals of interest. EM: EM Detector: In [10] it is proposed to model DFT data as ei- ther a single population of independent exponential random vaxiables (the transient-free situation) or as two populations. Under the addi- tional assumption that the membership of the DFT data in these two populations is i.i.d. Bernoulli, the parameters (in fact, a mean and a Bernoulli probability since it is assumed in this correspondence that the transient-free noise power is known) may be efficiently estimated via the EM algorithm. Insertion of the result to a GLRT is straightfor- ward. Pg: Page Detector: The Page test [8] is the comparison of the cusum statistic (5) 1053–587X/00$10.00 © 2000 IEEE