Multidimensional chirp algorithms for computing Fourier transforms

Continuous versions of the multidimensional chirp algorithms compute the function G(y)=F(My), where F(y) is the Fourier transform of a function f(x) of a vector variable x and M is an invertible matrix. Discrete versions of the algorithms compute values of F over the lattice L₂=ML₁ from values of f over a lattice L₁, where L₂ need not contain the lattice reciprocal to L₁. If M is symmetric, the algorithms are multidimensional versions of the Bluestein chirp algorithm, which employs two pointwise multiplication operations (PMOs) and one convolution operation (CO). The discrete version may be efficiently implemented using fast algorithms to compute the convolutions. If M is not symmetric, three modifications are required. First, the Fourier transform is factored as the product of two Fresnel transforms. Second, the matrix M is factored as M=AB, where A and B are symmetric matrices. Third, the Fresnel transforms are modified by the matrices A and B and each modified transform is factored into a product of two PMOs and one CO