This chapter examines discrete-time LTI systems in detail. It shows that the input–output behavior of an LTI system is characterized by the so-called impulse response. The output is shown to be the so-called convolution of the input with the impulse response. It is then shown that exponentials are eigenfunctions of LTI systems. This property leads to the ideas of transfer functions and frequency responses for LTI systems. It is argued that the frequency response gives a systematic meaning to the term “filtering.” Image filtering is demonstrated with examples. The discrete-time Fourier transform (DTFT) is introduced to describe the frequency domain behavior of LTI systems, and allows one to represent a signal as a superposition of single-frequency signals (the Fourier representation). DTFT is discussed in detail, with many examples. The z-transform, which is of great importance in the study of LTI systems, is also introduced and its connection to the Fourier transform explained. Attention is also given to real signals and real filters, because of their additional properties in the frequency domain. Homogeneous time-invariant (HTI) systems are also introduced. Continuous-time counterparts of these topics are explained. B-splines, which arise as examples in continuous-time convolution, are presented.

When most students first approachfeedback control, they are still coming to grips with its foundation inlinear system theory. This unsteadiness with thefoundationmakes understanding feedback dramatically moredifficult. Experienced professionals canhavesimilar problems, painfully compounded by the fog of imperfect memory of the linear system theorystudied years before. It is thusvery common for all types of students to sit down to study feedback, glimpse the breadth and depth of knowledge required as a prerequisite, and simply give up. The purpose of this chapter is to strengthen the reader in linear systemfundamentals. It will fill in the gaps for those who need gaps filled,deepen the understanding of thosefluent in the mechanics of solving problems but who were nevershown the overarching conceptual logic, and serve as a handy reference for those who are truly comfortable.

Useful discrete-time signals and systems properties are introduced. This is followed by a brief review of the z-transform.Spectral analysis of seismic data and useful transforms are discussed. Signal analysis in the spectral or other domains is very important and assists in obtaining a better understanding of signals. Particularly when dealing with seismic data, it becomes almost standard to analyze seismic data sets in the 2-D frequency-wavenumber domain. Also, other discrete transforms such as the Radon transform are very useful for processing seismic data sets, which can be used, for example, for seismic wavefield decomposition as well as seismic multiple removal.