This paper is concerned with the approach to teaching an introductory course on Fourier theory for engineers. Commonly called Signals and Systems (or some variation), this course generally introduces four transforms: the Fourier Transform, the Fourier Series, the Discrete-Time Fourier Transform (DTFT), and the Discrete Fourier Transform (DFT). Our concern is that the method and order of topic p...

--The fractional fourier transform(FRFT) which is a generalisation of classical fourier transform, was introduced number of years ago in the mathematics literature but appears to have remained largely unknown to a signal processing community to which it may, however, be a potentially useful. The fractional fourier transform depends on a parameter ‘α’ and can be interpreted as a rotation by an a...

A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) of an input vector. Efficient means that the FFT computes the DFT of an n-element vector in O(n logn) operations in contrast to the O(n2) operations required for computing the DFT by definition. FFTs exist for any vector length n and for real and higher-dimensional data. Parallel FFTs have b...

The terms Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) are used to denote efficient and fast algorithms to compute the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT) respectively. The FFT/IFFT is widely used in many digital signal processing applications and the efficient implementation of the FFT/IFFT is a topic of continuous research.

Based on the coherent optical theory, we extend the fractional Fourier transform of first-order correlation to a fractional Fourier transform of second-order correlation. An optical system for implementing a second-order fractional Fourier transform was designed. As a numerical example, we investigate the second-order fractional Fourier transform for a single slit.

This paper will explore the heuristic principle that a function on the line and its Fourier transform cannot both be concentrated on small sets. We begin with the basic properties of the Fourier transform and show that a function and its Fourier transform cannot both have compact support. From there we prove the Fourier inversion theorem and use this to prove the classical uncertainty principle...