Filtering of chirped ultrasound echo signals with the fractional Fourier transform

The fractional Fourier transform represents a generalisation of the conventional Fourier transform. Previous work (M. Bennett et al, Proc. IEEE EMBS vol. 1, pp. 882-885, 2003) has shown that the application of the fractional Fourier transform to conventional, un-coded ultrasound signals has little advantage over conventional filtering techniques such as band-pass filtering. However, the fractional Fourier transform can be ´tuned´ to be sensitive to signals of a particular chirp rate (C. Capus et al, IEE Seminar on Time-Scale and Time-Freq. Analysis and Appl., 2000) and can achieve levels of pulse compression similar to those obtained using a matched filter. To this end a system was developed which could generate and transmit linear chirp coded ultrasound signals. The fractional Fourier transform was then used to process the signals received from a simple phantom arrangement. When the transform was used with the optimum transform order corresponding to the chirp rate of the signals, the transform domain signals demonstrated a degree of pulse compression similar to that given by a matched filter. Results are also presented which demonstrate that a chirp signal identified in the fractional Fourier domain may be completely recovered in the time domain through the use of the inverse transform. Matched filtering was found to give a greater degree of pulse compression, but the fraction Fourier method can be applied without a-priori knowledge of the transmitted signal. Further work will be carried out to determine the best way of extracting useful information from the fractional domain signals.