Signal Orthogonal Transforms

Abstract

In this chapter we learn about orthogonal signal decomposition into a sum of weighted orthogonal functions (decomposition basis) like in Fourier series analysis. The decomposition weights are found by orthogonal transformation of signal samples, i.e. by multiplying them by rectangular orthogonal matrix having complex-conjugated decomposition functions in its rows. There are many sets of decomposition functions. We learn about: discrete cosine transforms (DCTs), discrete sine transform (DST), discrete Fourier transform (DFT), Hartley, Haar, and Walsh–Hadamard transform, as well as optimal Karhunen-Loève transform. Transform weights are called the signal spectrum in respect to the chosen set of functions. When only a few weight are significant, we a telling that the transformation has compact support. It is the case when basis functions very well fit to signal components. Orthogonal transformations of signal samples are perfect reversible—doing the inverse transformation of the weights one obtains the original signal samples. When we modify the weights and do the signal synthesis—a signal filtering operation is performed. In this chapter we become familiar with all these aspects of orthogonal signal transformations and their applications.

Everything can be taken to pieces and assembled back. A signal also.