Direct construction of a ξ-pulse from natural frequencies and evaluation of the late-time residuals

An important type of target identification utilizes the natural frequencies of the target, these being poles in the complex-frequency s-plane as expressed in the singularity expansion method. Since the location of these natural frequencies s_α in the s-plane is aspect independent, this simplifies the identification problem to the comparison of the target pole patterns stored in some target library. In order to implement the pole pattern as a target discrimination scheme one needs some scheme to find the s_α in experimental data (scattered fields), either explicitly or implicitly. Some of the early schemes used what is referred to as the Prony method in which one fits a sum of dumped sinusoids to a waveform. This had various limitations which have been overcome to some degree by a matrix pencil method. These and related techniques are explicit in that they determine a set of s_α from one or more waveforms which are then compared to the target library. In contrast, implicit methods do not generate a set of s_α from the data, but utilize predetermined sets of s_α for various targets which are then manipulated with the data in some way which transforms it to a form in which the discrimination can be more readily performed. Those used to date have used temporal functions with two-sided Laplace transforms with zeros corresponding to the s_α (poles) of preselected targets. Convolving these with target impulse response removes (annihilates) the late-time response for the properly chosen target. These go by various names such as K-pulse and E-pulse. In the present paper we go further into this latter type of target identification