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Abstract:
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Fourier transform methods are employed heavily in digital signal processing. Discrete Fourier
Transform (DFT) is among the most commonly used digital signal transforms. The exponential
kernel of the DFT has the properties of symmetry and periodicity. Fast Fourier Transform (FFT)
methods for fast DFT computation exploit these kernel properties in different ways. In this thesis,
an approach of grouping data on the basis of the corresponding phase of the exponential kernel of
the DFT is exploited to introduce a new digital signal transform, named the M-dimensional Real
Transform (MRT), for l-D and 2-D signals. The new transform is developed using number theoretic
principles as regards its specific features. A few properties of the transform are
explored, and an inverse transform presented. A fundamental assumption is that the size of the
input signal be even. The transform computation involves only real additions. The MRT is an
integer-to-integer transform. There are two kinds of redundancy, complete redundancy & derived
redundancy, in MRT. Redundancy is analyzed and removed to arrive at a more compact version
called the Unique MRT (UMRT). l-D UMRT is a non-expansive transform for all signal sizes,
while the 2-D UMRT is non-expansive for signal sizes that are powers of 2. The 2-D UMRT is
applied in image processing applications like image compression and orientation analysis. The
MRT & UMRT, being general transforms, will find potential applications in various fields of
signal and image processing. |