Calculation of ambiguity functions with Fermat Number Transform

This paper deals with fast calculation of cross-ambiguity functions. The approach that we develop is based on Gauss-Legendre quadrature associated with Fermat Number Transform. For fixed number of quadrature nodes, these nodes are approximated by their closest neighbors on a regular sampling grid. This enables Gauss quadrature good approximation while preserving the convolution structure of the grid quantized quadrature. The interest of preserving the convolution structure of the cross ambiguity terms in the corresponding discretized problem lies in the possibility of using fast transform Fourier-like algorithms. In a digital processing context, Number Theoretic Transforms (NTT) in finite fields of order a Fermat number are known to be particularly well suited to achieve convolution at very low computational cost. The contribution of this paper lies in the association of both powerful concepts of Gauss quadrature and NTT to realize fast convolution, and in particular fast cross-ambiguity calculation. Simulations are carried out to illustrate calculation of a few standard radar waveforms ambiguity functions.