Symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms

This paper uses the fact that the discrete Fourier transform diagonalizes a circulant matrix to provide an alternate derivation of the symmetric convo lution-multiplication property for discrete trigonometric transforms. Deriv ed in this manner, the symmetric convolution-multiplication property extend s easily to multiple dimensions using the notion of block circulant matrice s and generalizes to multidimensional asymmetric sequences. The symmetric c onvolution of multidimensional asymmetric sequences can then be accomplishe d by taking the product of the trigonometric transforms of the sequences an d then applying an inverse trigonometric transform to the result. An exampl e is given of how this theory can be used for applying a two-dimensional (2 -D) finite impulse response (FIR) filter with nonlinear phase which models atmospheric turbulence,