To analyze the structure of a set of high-dimensional perfect sequences over a composition algebra over R, we developed the theory of Fourier transforms of the set of such sequences. We define the discrete cosine transform and the discrete sine transform, and we show that there exists a relationship between these transforms and a convolution of sequences. By applying this property to a set of perfect sequences, we obtain a parameterization theorem. Using this theorem, we show the equivalence between the left perfectness and right perfectness of sequences. For sequences of real numbers, we obtain the parameterization without restrictions on the parameters.