Up to now, we have described signals using a discrete-time paradigm, that is, a mathematical model in which events can be successfully described as countable sequences of complex numbers. In physics and engineering, however, the "traditional" model of the world is one in which physical quantities such as time possess an infinite granularity and are therefore best described in terms of real numbers. This so-called continuous-time paradigm speaks the language of calculus and, in it, signals are described by functions of a real-valued variable.

This model, which appeared long before the "invention" of discrete time, has been extremely successful in both abstract and applied domains and it is still the framework of choice in applications such as electronics. And, indeed, we have already used the continuous-time model, however informally, in previous modules when we have been called to label our plots with units that took into account a real-world experimental setup (such as the real frequency values in a DFT plot).

In this module we will finally build a solid mathematical bridge between these two worlds, the discrete- and continuous-time paradigms. We will soon see that for most signals of interest, these two models are equivalent, in the sense that the description of a signal can be freely translated from one language to the other and back again with no loss of information.

Historically, most of mankind's attempts to record and understand the world have taken place in discrete time: time series , as we have seen in the Introduction, represented the only available way to commit a phenomenon to paper (or clay tablet); it was only during the 17th century that a great intellectual drive to idealized abstraction put calculus at the forefront of scientific research, where it solidly remains even today. Still, because of the current ubiquity of digital processing devices, the granular nature of our data records has become "native" again; we will therefore start our journey in the world of continuous time by considering interpolation first, that is, the way to turn a sequence of samples into a function of a real variable.

Afterwards, we will show that under very mild conditions, a continuous-time function can indeed be sampled, i.e., reduced to a countable set of values, with no loss of information. The result, known as the Sampling Theorem, is perhaps the cornerstone of digital processing, since it allows us to move from the analog world to the digital world with a precise knowledge of how information is preserved.