Fast Algorithms

The focus of this chapter is on fast algorithms: the fast Fourier transform, the fast Haar transforms, and the fast Walsh transform. To build a fast algorithm we use an original approach stemming from introduction of a recurrent sequence of orthogonal bases in the space of discrete periodic signals. On this way we manage to form wavelet bases which altogether constitute a wavelet packet. In particular, Haar bases are wavelet ones. We pay a lot of attention to them in the book. We investigate an important question of ordering of Walsh functions. We analyze in detail Ahmed–Rao bases that fall in between Walsh basis and the exponential basis. The main version of the fast Fourier transform (it is called the Cooley–Tukey algorithm) is targeted to calculate the DFT whose order is a power of two. In the end of the chapter we show how to use the Cooley–Tukey algorithm to calculate a DFT of any order.