Fourier extension and hough transform for multiple component FM signal analysis

Conventional representations in the time or frequency domain are inadequate for non-stationary signals which have statistical properties varying with time. In particular the advent of pulse compression techniques and the use of time varying chirp-type signals widespread in radar, sonar and seismic technologies means there is a need for time-frequency representation. We present a technique which extends the Fourier transform to non-stationary signals. We call the technique Fourier Extension analysis. We show that the analysis extends naturally to a time-frequency representation using the Hough transform projection, and investigate the resolutions obtainable with regard to separation of chirp signals compared with the usual matched filter approach common in radar processing. A visual interpretation of the magnitude and phase of the analytic results is introduced allowing a range of transform orders to be viewed simultaneously. Using frequency modulated signals, we demonstrate significantly higher resolution both in rate and time separation. Examples are given using synthetic and real world chirp signals illustrating improvements in time-frequency resolution using the new approach compared to the commonly used quadratic transforms.