Signals, Systems, and Signal Processing

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.