  The affine Weyl groups with their corresponding four types of orbit functions
are considered. Two independent admissible shifts, which preserve the
symmetries of the weight and the dual weight lattices, are classified. Finite
subsets of the shifted weight and the shifted dual weight lattices, which serve
as a sampling grid and a set of labels of the orbit functions, respectively,
are introduced. The complete sets of discretely orthogonal orbit functions over
the sampling grids are found and the corresponding discrete Fourier transforms
are formulated. The eight standard one-dimensional discrete cosine and sine
transforms form special cases of the presented transforms.
