A common theme in many SP applications is to decompose an original signal into its primitive or fundamental constituents and to perform simple operations separately on each component, thereby accomplishing extremely sophisticated operations by a combination of individually simple operations. The classic and still pervasive example is Fourier analysis, the theory and practice that breaks signals into sinusoidal (smooth oscillating) components just as complex sounds can be decomposed into rich combinations of simple tones. Fourier analysis lies at the heart of the development during World War II of modern analog communications and radar technology, and discrete versions play a corresponding role in DSP. Such decompositions have been a key theoretical tool for the modern analog and digital communications techniques that underly transmitting signals through the physical pathways of the NII. They have also proved equally important to the understanding and advancement of error-control coding for reliable communications, SP for removing unwanted noise and signals, and SP for improving the quality and utility of signals. During recent years Fourier methods have been supplemented by other approaches, most notably the many methods now subsumed under the general heading of wavelets. These alternatives hold promise for providing more useful ways of analyzing and processing signals for different applications, but the classic methods still provide a useful theoretical tool for evaluating the new tools, and continue to dominate in applications.