1 Optimal convolutive filters for real-time detection and arrival time estimation of transient signals Michal Natora, Felix Franke, and Klaus Obermayer Abstract—Linear convolutive filters are fast in calculation and in application, and thus, often used for real-time processing of continuous data streams. In the case of transient signals, a filter has not only to detect the presence of a specific waveform, but to estimate its arrival time as well. In this study, a measure is presented which indicates the performance of detectors in achieving both of these tasks simultaneously. Furthermore, a new sub-class of linear filters within the class of filters which minimize the quadratic response is proposed. The proposed filters are more flexible than the existing ones, like the adaptive matched filter or the minimum power distortionless response beamformer, and prove to be superior with respect to that measure in certain settings. Simulations of a real-time scenario confirm the advantage of these filters as well as the usefulness of the performance measure. Index Terms—Adaptive matched filter, minimum variance distortionless response, beamforming, Capon beamformer, linear filters, performance measure I. I NTRODUCTION F OR detection of signals in single data samples corrupted by Gaussian noise, linear filters, in particular the adaptive matched filter (AMF), have been proven to be powerful. Their performance is measured with respect to the probability of detection and of false alarm; see [1] for a performance analysis of the AMF and other filters. The AMF has been applied amongst others in radar and antenna systems [2]. In other applications, however, the incoming data stream does not consist of a few data samples, but of a continuous data stream, whereas the signal is present only in a few of the samples (transient signals). In this case, the signal must not only be detected, but also its arrival time must be estimated. The research field of optimal simultaneous detection and estimation has been mainly initiated by the work presented in [3]. Based on this theory some detectors were developed [4]–[6], and most of these approaches rely on order statistics. In the work of [5], however, the authors mention, that in the case of long signals, linear convolutive filters prove to be superior to order statistics. Moreover, linear convolutive filters are computationally much more efficient, and thus, more suitable for real-time applications than order statistics. This raises the question of which detectors should be used for the mentioned task, and how their performance should be compared. This study focuses in particular on the performance of linear filters, since they are easy to implement and are M. Natora and K. Obermayer are with the Institute for Software Engineering and Theoretical Computer Science, Berlin Institute of Technology, 10623 Berlin, Germany. Email: natora@cs.tu-berlin.de F. Franke is with the Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany. optimal in the class of linear transformations [7]. Although the performance of various detectors for transient signals was compared (see [8]–[10]), these studies compared only the detection performance and linear convolutive filters were rarely used for comparison. Linear convolutive filters, in the following abbreviated sim- ply by the term linear filters, are a convenient approach for the task of simultaneous detection and arrival time estimation of transient signals, and, thanks to their computational efficiency, suitable for real-time applications. For example, they are used for extracting information from bio-medical data [11]–[13] or in speech processing (see [14] for a survey). However, to the knowledge of the authors, no work exists to date which would propose a measure assigning a performance to detectors with respect to their ability of simultaneously detecting the presence as well as estimating the arrival time of transient signals. This work is organized as follows: In Sec. II-B the general optimization problem is presented to which linear filters are the solution. By modifying the optimization criteria, a new class of linear filters is derived. In Sec. II-C a measure of performance of detectors with respect to simultaneous detection and arrival time estimation is presented. In Sec. III-A different linear filters are compared with respect to this measure. The results from simulations in Sec. III-B agree with the theoretical findings and demonstrate the usefulness of these new filters and of the performance measure. The work is summarized and discussed in Sec. IV and a brief outlook on further research directions is given. II. METHOD A. Notation A notation is used in which symbols for scalar quantities are represented by lower case letters, vectorial quantities are represented by bold lower case letters, and matrices are represented by bold upper case letters. A vector ξ usually represents a set of sampled data points, the index set being centered around zero, i.e. ξ =(ξ −b ,...,ξ b ) ⊤ . T ξ denotes the dimension of the vector ξ, i.e. T ξ =2b +1. The symbol δ y (x) denotes the usual Kronecker delta func- tion, i.e. δ y (x)=1, if x = y, and δ y (x)=0 otherwise. The noncyclic cross-correlation between two vectors x and y is denoted as x ⋆ y = z, where z t = ∑ τ x τ y τ +t . It is T z = T x + T y − 1. The notion of variance is slightly abused by attributing the variance to a probability density function (pdf) f (x) rather World Academy of Science, Engineering and Technology 55 2009 235