Theory and application of the momentary fourier transform

The discrete Fourier transform (DFT) is a widely used tool in signal or image processing and its efficiency is important. There are applications where it is desirable to use relatively small, successive, overlapped DFTs to obtain the spectrum coefficients. The momentary Fourier transform (MFT) computes the DFT of a discrete-time sequence for every new sample in an efficient recursive form. In this thesis we give an alternate derivation of the MFT using the momentary matrix transform (MMT). Recursive and non-recursive forms of the inverse MFT are also given, which can provide efficient frequency domain manipulation (e.g. filtering). Discussion on the properties and examples of the usage of the MFT is given, followed by a survey on its efficiency. In this work we investigate the applicability of the MFT to synthetic aperture radar (SAR) signal processing, and in particular show what advantages the MFT algorithm offers to the SPECtral ANalysis (SPECAN) method and burst-mode data processing. In the SPECAN algorithm, the received signals are multiplied in the time domain by a reference function, and overlapped short length DFT’s are used to compress the data. The azimuth FM rate of the signal varies in each range cell, which leads to the issue of keeping the azimuth resolution and output sampling rate constant. After the introduction to SPECAN, we show what advantages and disadvantages the MFT has compared to the FFT algorithms. When a SAR system is operated in burst-mode, its azimuth received signal has a segmented frequency-time energy in its Doppler history. It requires that IDFTs be located at specific points in the spectral domain to perform the azimuth signal compression. After the introduction of the burst-mode data properties, we show why the short IFFT (SIFFT) algorithm has the requirement of arbitrary-length, highly-overlapped IDFTs to process burst-mode data, in which case the IMFT is shown to have computational advantages.Applied Science, Faculty ofElectrical and Computer Engineering, Department ofGraduat