3 - Time-domain representation of discrete-time signals and systems

Abstract

This chapter discusses time-domain representation of discrete-time signals and systems. A discrete-time signal x consists of a sequence of numbers denoted by x(n), or x(nT),where n is an integer index. The latter notation is usually reserved for sampled data sequences with a uniform sampling period of T seconds. Several discrete-time signals are considered basic and important. More complex signals can be constructed from the elementary ones. The unit impulse function is usually denoted as δ (n) and it consists of a single unit-valued sample at the instant n=0, surrounded on both side by zeros. The unit step sequence is used to make an arbitrary sequence zero for all indices less than zero by multiplying the arbitrary sequence with the unit step. In some applications, it is useful to consider a signal as a random signal rather than a deterministic signal. Linear systems possess a unique property, called the superposition principle. This property implies that if one knows the system's responses to some typical inputs such as impulse and step functions, and an arbitrary input can be expressed as a linear combination of these elementary functions, then the system's response to this arbitrary input is known. In real life, all systems are non-linear. However, a non-linear system can usually be approximated as linear within some constraints.