Noise and Interference Suppression in Sonar Echoes Using the Fractional Fourier Transform

The Fractional Fourier Transform (FrFT) enables separation of signals from noise and interference by utilizing the entire time-frequency space. Signals are filtered by rotating to a new time axis ‘ta’, with rotational parameter ‘a’, selected using some metric such as mean-square error (MSE) between a desired signal-of-interest (SOI) and its estimate. The FrFT has been applied to numerous problems, but it is most suited for applications such as sonar and radar, when the time-frequency distribution of the SOI and the undesired environment are different. It can greatly outperform the conventional fast Fourier Transform (FFT), which is solely a frequency domain method (a=1), as well as conventional time-based MMSE adaptive filtering (a=0). In this paper, we present a simple FrFT-based algorithm that separates sonar echoes of a desired SOI, e.g. a chirp, from the cluttered background, which could be noise or interference (i.e. another signal). We exploit the fact that we can find the best time axis ‘ta’ in which the SOI becomes a tone, or close to it, with the FrFT, enabling easy notching (zeroing) of the clutter. By searching for the tone peak and notching everywhere except the peak, we can successfully and easily remove the clutter. This algorithm is robust because clutter typically does not correlate with the signal in the FrFT domain, and thus does not impair our ability to estimate the peaks and notch the clutter. We compute the MSE between the true transmitted signal and the received echo with and without this algorithm as a function of signal-to-noise ratio (SNR) and show that 5 dB reduction in MSE is possible with the FrFT.