2454 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 11, NOVEMBER 2001 All-Purpose and Plug-In Power-Law Detectors for Transient Signals Zhen Wang and Peter K. Willett, Senior Member, IEEE Abstract—Recently, a power-law statistic operating on discrete Fourier transform (DFT) data has emerged as a basis for a remark- ably robust detector of transient signals having unknown structure, location, and strength. In this paper, we offer a number of im- provements to Nuttall’s original power-law detector. Specifically, the power-law detector requires that its data be prenormalized and spectrally white; a constant false-alarm rate (CFAR) and self- whitening version is developed and analyzed. Further, it is noted that transient signals tend to be contiguous both in temporal and frequency senses, and consequently, new power-law detectors in the frequency and the wavelet domains are given. The resulting detectors offer exceptional performance and are extremely easy to implement. There are no parameters to tune. They may be consid- ered “plug-in” solutions to the transient detection problem and are “all-purpose” in that they make minimal assumptions on the struc- ture of the transient signal, save of some degree of agglomeration of energy in time and/or frequency. Index Terms—Crack detection, nonlinear detection, signal de- tection, sonar detection. I. INTRODUCTION AND CONTEXT A. Background I T IS often of considerable interest to identify short-duration nonstationarities in observed signals. Applications include surveillance (e.g., [6]) in which an acoustic “transient” may in- dicate the presence of a threat, industrial monitoring (e.g., [16]), in which the number and severity of transients reflects machine health, and medicine (e.g., [2]). Naturally, the problem is com- paratively simple if the signal to be detected is known—the only uncertainty is the time of occurrence, but knowledge of the tran- sient is usually not available or dependable; of interest here is to detect transient signals with unknown form, location, and strength. The hypothesis test is naturally composite, with any structure open to challenge. Basically, the detector is tasked to determine whether all observations belong to a known stationary probability distribution or whether they do not. Now, if there were nothing whatever that could be assumed about a transient signal, the detection task would be more or less hopeless. There are, fortunately, two rather qualitative proper- ties that most transient signals possess. The first is the obvious Manuscript received August 11, 2000; revised June 11, 2001. This work was supported by the Naval Undersea Warfare Center under Contract N66604-99-1-5021 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 Dr. Vikram Krishnamurthy. The authors are with the Department of Electrical and Systems Engi- neering, University of Connecticut, Storrs, CT 06269-2157 USA (e-mail: willett@engr.uconn.edu). Publisher Item Identifier S 1053-587X(01)07771-6. temporal contiguity: A transient signal is often couched as a lo- calized burst (or bursts) in time, although the duration of such a burst is unknown in most applications. The second is a tendency for most transient signals to be bandpass, that is, it is reasonable to expect most of a transient signal’s energy to be contained in contiguous frequency observations, although again, there is usu- ally little to be said about which frequencies. To exploit only the former, and considering a transient event as a two-sided change (at some unknown time, the observations switches from having pdf to having pdf , and at a later time, there is a return to ), Page’s test has been explored and found to be quite useful [1], [5]. Very similar to this, Nuttall couched a transient as a contiguous burst of bins in time, where is known, and developed the “maximum” detector [13]. To ex- ploit only the latter, there are detectors that begin their work on frequency domain data (usually DFT bins). Via (maximum likelihood) estimation of unknown signal parameters via the es- timation–maximization (EM) algorithm, a GLRT approach is presented [21]. Of greatest interest here is Nuttall’s frequency domain “power-law” detector [12], which will be introduced shortly. It is natural to use both kinds of contiguities, and for this, we have, for example, the Gaussian-mixture time-spectrogram model in [17] and the GLRT approaches arising from linear data transformations (either time-frequency or time-scale) [3], [10], [11]. These transforms are directed toward signal representation and classification, trying to distinguish signals in the transform domain. In [23], an attempt was made to compare the performances of a number of the above transient detection approaches on a fairly wide variety of signals. Those using time contiguity alone (Page and “maximum”) were perhaps the sturdiest performers overall but suffer from the need that certain parameters (signal strength or length) be prespecified and that data be prewhitened and prenormalized. Among the others, it was surprising that the most robust performance came from the simplest processor: Nuttall’s frequency-domain power-law statistic. It is a very good detector indeed, and in this paper, we show a number of ways to make it better still. B. Nuttall’s Power-Law Statistic There has been significant recent attention to Nuttall’s power-law detector [12], [14] due to its simple implementation and good performance. The test is based on the following formulation. Under the signal-absent hypothesis ( )—that the time-domain data is complex white Gaussian noise—prepro- cessing by the magnitude-square DFT yields independent and identically distributed (iid) exponential random variates. Under 1053–587X/01$10.00 © 2001 IEEE