  In this paper, a new variant to fractional signal processing is proposed
known as the Reduced Order Fractional Fourier Transform. Various properties
satisfied by its transformation kernel is derived. The properties associated
with the proposed Reduced Order Fractional Fourier Transform like shift,
modulation, time-frequency shift property are also derived and it is shown
mathematically that when the rotation angle of Reduced Order Fractional Fourier
Transform approaches 90 degrees, the proposed Reduced Order Fractional Fourier
Transform reduces to the conventional Fourier transform. Also, the Reduced
Order Fractional Fourier Transform of various kinds of signals is also derived
and it is shown that the obtained analytical expressions of different Reduced
Order Fractional Fourier Transform are a reduced form of the conventional
fractional Fourier transform. It is also shown that proposed definition of
Fractional Fourier Transform is easier to be handled analytically. Finally, the
convolution theorem associated with the proposed Reduced Order Fractional
Fourier Transform is derived with its various properties like shift
convolution, modulation convolution, and time-frequency shift modulation
properties. It has been shown that with this proposed new definition of
fractional Fourier transform, the convolution theorem has been reduced to
multiplication in the fractional frequency domain.
