Effect of convolution and modulation on the time-varying spectrum of a signal, with application to target recognition

Abstract

In active sonar or radar, the received signal can often be modeled as a convolution of the transmitted signal with the channel impulse response and the target impulse response. Because the received signal may have a time-varying spectrum, due for example to target motion or to changes in the channel impulse response, time-frequency methods have been used to characterize propagation effects and target effects, and to extract features for classification. In this paper, we consider the time-varying spectrum, in particular the Wigner time-frequency representation, of a received signal modeled as the convolution of the transmitted signal with the channel and target responses. We derive a simple but insightful approximation that shows the effects of the magnitude and phase of the frequency response of the target and of the channel on the Wigner representation of the transmitted signal. We also consider time-varying effects on the Wigner representation, such as changes in reflected energy, which we model by amplitude modulation.