Discrete 1D DGT
- In one dimension, we would like to represent an arbitrary signal, f(n),
as a linear combination of basis functions gm(n),
where f(n) is an N-point signal being represented by M basis functions.
Fm are the coefficients.
- In the spatial domain the basis functions are defined as the Gaussian and
it's derivatives with respect to x.
The nth derivative of a Gaussian can be written as the product of a
polynomial (generalized Hermite polynomial) and the original Gaussian.
- Since these functions are local in x, they are translated to various locations
in order to cover the x-axis.
- The Fourier transforms of the functions defined above are given by: