Orthogonal functions play an important role in factorial experiments and time series models. In the latter half of the twentieth century orthogonal functions became prominent in industrial experimentation methodologies that employ complete and fractional factorial experiment designs, such as Taguchi orthogonal arrays. Exact estimates of the parameters of linear model representations can be computed effectively and efficiently using “fast algorithms.” The origin of “fast algorithms” can be traced to Yates in 1937. In 1958 Good created the ingenious fast Fourier transform, using Yates’s concept as a basis. This paper is intended to illustrate the fundamental role of orthogonal functions in modeling, and the close relationship between two of the most significant of the fast algorithms. This in turn yields insights into the fundamental aspects of experiment design.