Symmetry relations in multidimensional Fourier transform pairs

A relation between the types of symmetries that exist in signal and Fourier transform domain representations is derived for continuous as well as discrete domain signals. The symmetry is expressed by a set of parameters, and the relations derived in this paper will help to find the parameters of a symmetry in the signal or transform domain resulting from a given symmetry in the transform or signal domain respectively. A duality among the relations governing the conversion of the parameters of symmetry in the two domains is also brought to light. The application of the relations is illustrated by a number of two-dimensional examples.Notation R the set of real numbers - R^m R × R × ... × R m-dimensional real vector space - continuous domain real vector - L {¦ – _i , i = 1,2,..., m} - m-dimensional frequency vector - W { – _i ,i=1,2,..., m} - m-dimensional normalized frequency vector - _P {¦ – _i , i=1,2,...,m} - g(ol) g (₁,₂,..., _m) continuous domain signal - () (₁₂,...,_m)=G (j₁,j₂,..., j_m) Fourier transform ofg (ol) - (A,b,,,) parameters ofT- symmetry - N the set of integers - N^m N × N × ... × N m-dimensional integer vector spacem-dimensional lattice - h(n) h (n₁,.,n_m) discrete domain signal - H() Fourier transform ofh (n) - v₁,v₂,..., v_m m sample-direction and interval vectors - V (v₁v₂ ...v_m) sampling basis matrix - [x]^* complex conjugate ofx - detA determinant ofA - X {x¦ – x_i , i=1,2,..., m} - A^–t [A^–1]^t,t stands for transposeThis work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-7739 to M. N. S. Swamy and in part by Tennessee Technological University under its Faculty Research support program to P. K. Rajan.