Interesting eigenvectors of the Fourier transform

ABSTRACT

It is well known that a function can be decomposed uniquely into the sum of an odd and an even function. This notion can be extended to the unique decomposition into the sum of four functions – two of which are even and two odd. These four functions are eigenvectors of the Fourier Transform with four different eigenvalues. That is, the Fourier transformof each of the four components is simply that component multiplied by the corresponding eigenvalue. Some eigenvectors of the discrete Fourier transform of particular interest find application in coding, communication and imaging. Some of the underlying mathematics goes back to the times of Carl Friedrich Gauss.

Keywords:

• function decomposition
• Fourier transform
• discrete Fourier transform
• coded apertures
• coded aperture imaging
• Gaussian integers
• Eisenstein integers
• Legendre symbol
• Legendre sequence
• Legendre symbol sequence
• bi-level auto-correlation
• ideal auto-correlaton
• flat power spectrum
• periodic phase sequences