Calculation of Fourier transform using symmetrical properties

A matrix method which computes discrete Fourier transforms using a digital computer program is presented in this paper. The proposed technique takes advantage of symmetry of the complex functions about the real and imaginary axes to reduce the number of calculations necessary in a given Fourier transform. Computationally the method described is not as efficient, especially for a large N, as the well-known Cooley-Tukey method. However, it differs from the Cooley-Tukey formulation in two notable respects: first, the present method is not as restrictive in the selection of values of N as the Cooley-Tukey method; and second, the calculations can proceed with the first value of the time function, thus eliminating the need for storing data before beginning with the transform calculations as is the case with the Cooley-Tukey method. In a number of applications, realizing these two conditions is more important than computational efficiency.The logic of a computer program which calculates the Fourier transform using the symmetry properties is described by a flow chart. This paper also includes numerical examples, using the computer program, of a Fourier transform from the time domain to the frequency domain. In addition, the program calculates the inverse Fourier transform, reconstructing the original time function from its frequency contents.