A Comrade-Matrix-Based Derivation of the Eight Versions of Fast Cosine and Sine Transforms

The paper provides a full self-contained derivation of fast algorithms to compute discrete Cosine and Sine transforms I IV. For the Sine I/II and Cosine I/II transforms a unified derivation based on the concept of the comrade matrix is presented. The comrade matrices associated with different versions of the transforms differ in only a few boundary elements; hence, in each case algorithms can be derived in a unified manner. The algorithm is then modified to compute Sine III/IV and Cosine III/IV transforms as well. The resulting algorithms for the versions III/IV are direct and recursive, such algorithms were missing in the existing literature. Finally, formulas reducing Cosine and Sine transforms of the types III and IV to each other are presented. Part I: Versions I and II Of The Transforms