OVERVIEW

We begin our study of digital signal processing (DSP) with an introduction to Fourier analysis. Although a somewhat complex place to start, it is necessary to the discussion of physical signals that are measurable, such as temperature, pressure, voice, and sound. These signals, though arbitrary in nature, must be defined as mathematical functions if further processing is required; for example, removing the unwanted noise from the music or designing a control system for maintaining a constant action. The central idea of the Fourier analysis is to look through these events as if they are functions made up of superimposed sinusoidal frequencies. How to identify these component frequencies is the topic of discussion in this chapter.

The seventeenth century mathematician Jean Baptiste Joseph Fourier (1768 1830) discovered that a continuous time periodic function (one that repeats a pattern periodically after a certain interval) could be approximated as a series of simple sin and cosine functions. In his honor, such an approximation of the function is called the Fourier series of the function. Although it is hard to find functions in nature that are truly periodic for all time, we could still define a proper domain of interest and approximate a function using its Fourier series. When we deal with an arbitrary signal and we don't know its period (see Chapter 3, "The Fourier Transform"), we can remove this periodic dependency: we obtain what we call the relative magnitudes of the component frequencies, and that becomes the Fourier Transform of the...