2.1 Sampling and Reconstruction of Signals

Let us repeat that we understand a continuous, one-dimensional signal to be a piecewise continuous real or complex function of one continuous real variable, t. Usually the dimensionality will not be emphasised as, with the exception of Chapter 14, we will deal exclusively with one-dimensional signals. Note that the number of independent variables is not, in principle, limited; we can have two, three or multidimensional signals. Signals that are continuous (according to our definition) are also called analogue signals as they can be represented by time courses of physical ('analogue') variables ( t then specifically means time).

The independent variables usually have a physical meaning of time and therefore such signals are sometimes called continuous-time signals. The name can be restrictive as, in a more general sense, the variable can also, for example, have the meaning of a space variable, however, it also includes functions which are not piecewise continuous. In the interests of simplicity, we will limit ourselves to piecewise continuous signals and to some important signals which, although not piecewise continuous, will be interpreted as limit cases of piecewise continuous signals.

Sampling serves to express the continuous signal f( t) by its discrete samples f _n = f( t _n) for specified values t _n of the independent variable t. Usually, the samples are equidistant so that t _n = nT, where T is a suitable real...