Introductory ideas on signal processing

Signal processing plays a central role in a multitude of modern technologies. Think about those industries that are dependent on data communication, or on radar, to give just two examples, and the importance of signal processing becomes self-evident. The Hilbert transform plays a central role in a number of signal processing applications. Pioneering work on the application of Hilbert transforms to signal theory was carried out by Gabor (1946).

A notational alert to the reader is appropriate at the start of this chapter. In the following sections the standard Hilbert transform operator H, the Heaviside step function H(x), the Hermite polynomials Hn(x), the Hilbert transfer function H(ω), the fractional Hilbert transform Hα, and the fractional Hilbert transform filter Hp(ω) all appear, sometimes in close proximity, so the reader should pay careful attention to the particular symbols in use.

Broadly defined, a signal provides a means for transmission of information about a system. For the signals of interest in this chapter, it is assumed that a mathematical representation of the signal is known. There are two important types of signals. The first are the continuous or analog signals – sometimes referred to as continuoustime signals. Unless something explicitly to the contrary is indicated, it is assumed throughout this chapter that all signals of this group belong to the class L2(ℝ). In a number of places, this requirement can be generalized.