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Multidimensional chirp algorithms for computing Fourier transforms

机译:用于计算傅立叶变换的多维线性调频算法

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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 L/sub 2/=ML/sub 1/ from values of f over a lattice L/sub 1/, where L/sub 2/ need not contain the lattice reciprocal to L/sub 1/. 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.
机译:多维线性调频算法的连续版本计算函数G(y)= F(My),其中F(y)是向量变量x的函数f(x)的傅立叶变换,M是可逆矩阵。算法的离散版本根据晶格L / sub 1 /上f的值计算晶格L / sub 2 / = ML / sub 1 /上的F值,其中L / sub 2 /不必包含与L倒数的晶格/ sub 1 /。如果M是对称的,则该算法是Bluestein chirp算法的多维版本,该算法使用两个逐点乘法运算(PMO)和一个卷积运算(CO)。离散版本可以使用快速算法来有效地实现以计算卷积。如果M不对称,则需要进行三个修改。首先,将傅里叶变换分解为两个菲涅耳变换的乘积。其次,将矩阵M分解为M = AB,其中A和B是对称矩阵。第三,菲涅耳变换由矩阵A和B修改,每个修改的变换都分解为两个PMO和一个CO的乘积。

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