Structural Health Monitoring (SHM) aims to monitor in real time the health state of engineering structures. For thin structures, Lamb Waves (LW) are very efficient for SHM purposes. A bonded piezoelectric transducer (PZT) emits LW in the structure in the form of a short tone burst. This initial wave packet (IWP) propagates in the structure and interacts with its boundaries and discontinuities and with eventual damages generating additional wave packets. The main issues with LW based SHM are that at least two LW modes are simultaneously excited and that those modes are dispersive. Matching Pursuit Method (MPM), which consists of approximating a signal as a sum of different delayed and scaled atoms taken from an a priori known learning dictionary, seems very appealing in such a context, however is limited to nondispersive signals and relies on a priori known dictionary. An improved version of MPM called the Single Atom Convolutional Matching Pursuit method (SACMPM), which addresses the dispersion phenomena by decomposing a measured signal as delayed and dispersed atoms and limits the learning dictionary to only one atom, is proposed here. Its performances are illustrated when dealing with numerical and experimental signals as well as its usage for damage detection. Although the signal approximation method proposed in this paper finds an original application in the context of SHM, this method remains completely general and can be easily applied to any signal processing problem.