The discrete trigonometric transforms and their fast algorithms: an algebraic symmetry perspective

It is well-known that the discrete Fourier transform (DFT) can be characterized as decomposition matrix for the polynomial algebra C[x]/(xⁿ - 1). This property gives deep insight into the DFT and can be used to explain and derive its fast algorithms. In this paper we present the polynomial algebras associated to the 16 discrete cosine and sine transforms. Then we derive important algorithms by manipulating algebras rather than matrix entries. This makes the derivation more transparent and explains their structure. Our results show that the relationship between signal processing and algebra is stronger than previously understood.