Discrete-Time Systems

Abstract

Let us first consider the context of discrete-time systems. Input and output signals are no longer physical quantities, but sequences of dimensionless numbers. Having in mind a physical meaning behind a discrete-time signal, the physical quantity can always be recovered by multiplying the numbers of the sequence by a reference value of this quantity. This is exactly what digital-to-analog converters do. In the following, the input sequence of the system is denoted by x(n), and the output sequence by y(n) where n points to the nth element of the sequence. If the sequence is related to a real-time signal or process, an association with the actual time axis can be made by writing x(nT) and y(nT) where T is the quantisation period of the time axis. A discrete-time system is the implementation of a mathematical algorithm, which transforms the input sequence x(n) into the output sequence y(n). The algorithm may be realised in dedicated hardware or in software running on Digital Signal Processors (DSPs) or on general purpose processors. The processing of signal sequences may take place in real-time, which requires synchronisation with other processes like A/D and D/A conversion happening at the same time. On the other hand, signal data may be processed off-line without any relationship to the time axis. An example is processing of data that are available on disk and that are stored back to disk after processing.