Discrete-Time Signals and Systems

Discrete-time signals are, in general, infinite-length sequences of numerical values that may either arise from sampling of continuous-time signals, or be generated directly by inherently-discrete-time processes. Deterministic signals have a univocal mathematical description, so that future signal values are exactly predictable. Random signals do not allow for such a description: their treatment requires statistical tools since signal evolution cannot be exactly foreseen. This chapter introduces basic concepts related to discrete-time signals, as well as mathematical operators, called discrete-time systems, that are employed to process them. The main constraints imposed on discrete-time systems, namely linearity, time invariance, stability and causality, are introduced along with the quantities used to describe a system: the impulse response, the transfer function and the frequency response. Finite-impulse-response (FIR) and infinite-impulse-response (IIR) systems are defined. Linear convolution and the linear constant-coefficient-difference equation (LCCDE) are introduced to express the input-output relation of discrete-time systems.