IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 2004 713 Detection of Transient Signals With Unknown Localization Francisco M. Garcia, Member, IEEE, and Isabel M. G. Lourtie, Senior Member, IEEE Abstract—In the context of real-time detection of transient sig- nals, a likelihood ratio (LR) test is evaluated at every sampling in- terval. Performing the LR tests at a lower rate reduces significantly the computational complexity of the detection algorithm. However, in general, this simplification also leads to a strong degradation of the detector performance. For example, with a small shift error, an arriving transient may be in quadrature with its model. This degra- dation is particularly noticeable when the signals to detect are de- terministic and sampled at a frequency close to the Nyquist rate. This letter proposes a method to overcome this limitation by using locally stationary models of the signals to detect. The resulting de- tectors are robust to shift errors and computationally efficient. Index Terms—Bandpass signals, detection, local stationarity, nonstationary processes, real-time processing, transient signals. I. INTRODUCTION D ETECTION of transient signals is an issue of major con- cern in many applications such as wireless communica- tions, sonar and radar. Recent works in this field include [1] and [2] where, respectively, the gap metric and order statistics are used to detect signals with unknown parameters, namely delays or unknown arrival times. In a similar line of thought, in this letter, we propose a method to develop detectors for transient signals with random amplitude that are robust to shift errors, and exhibit low computational complexity. This work addresses the problem of detecting a bandpass transient signal with unknown localization. For example, con- sider a surveillance environment where a certain signal is ex- pected to arrive at the receiver with unknown arrival time. The generalized likelihood ratio test (GLRT) is the classical pro- cessor to solve this problem. It consists of an estimation/de- tection scheme, where the likelihood ratio is evaluated continu- ously along the time axis and its maximum value compared to a threshold. In general, the observation process itself is filtered and sampled with a sampling interval , and the likelihood ratio (LR) is evaluated at the same rate. The GLRT is computation- ally consuming because it requires high sampling frequencies to avoid performance degradation due to time-shift errors which are, at most, of length . To reduce the computational cost (number of operations per time unity) of the processor, it would be desirable to use sampling rates close to the Nyquist frequency of the signal to detect, and Manuscript received October 9, 2003; revised February 10, 2004. This work was supported by the FCT, POSI, and FEDER under Project POSI/32708/CPS/1999. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Marcelo G. S. Bruno. The authors are with the ISR-Instituto de Sistemas e Robótica, IST-Insti- tuto Superior Técnico, 1049-001 Lisboa, Portugal (e-mail: fmg@isr.ist.utl.pt; iml@isr.ist.utl.pt). Digital Object Identifier 10.1109/LSP.2004.833448 to evaluate the LR at every time intervals, with and . When the signals to detect are Gaussian stationary processes, the corresponding optimal quadratic processors are insensitive to time shifts. This is not the case for nonstationary processes (see [3]). In this letter, we consider a particular class of nonstationary processes, where the signals are known up to a random amplitude. As shown later, the 2-D autocorrelation function (ACF) of such signal is not locally stationary (NLS). In this case, a small shift error can strongly degrade the processor performance since the signal arriving at the receiver may be in quadrature with its model. In order to reduce the performance loss due to shift errors, we use a locally stationary (LS), rather than NLS, signal model. This leads to a robust detector that allows larger LR time intervals, thus with smaller computational complexity. Our framework is derived from second-order characteristics of stochastic signals, related to the sampling theorem [4] and to the local stationarity of nonstationary processes [3], [5]–[7]. It is shown in [3] and [7] that, due to the positive definiteness of the ACF of a nonstationary process [8], both its 2-D power spectrum (2DPS) and its Wigner distribution (WD) map information regarding either the Nyquist frequency for sampling purposes and the existence (or not) of local stationarity of the process. In [3], we present a simple method to obtain a LS co- variance matrix (which is a sampled version of the ACF) of a zero-mean second-order process from data. Here, the ACF of the signal has a single nonzero eigenvalue. We show that the corresponding LS ACF has 2 nonzero eigenvalues, and derive the corresponding eigenvectors as functions of the eigenvector of the signal to detect. II. LOCALLY STATIONARY AUTOCORRELATION FUNCTIONS Let , be a zero-mean nonstationary sto- chastic process characterized by the autocorrelation function (ACF) , which can be expressed in terms of its eigen- functions, , and eigenvalues, , by the Mercer-like expan- sion [7] Equivalently, the process can be described either by the 2-D power spectrum (2DPS) or by the Wigner distribution (WD) where denotes the Fourier transform. 1070-9908/04$20.00 © 2004 IEEE