1.2. Signals¶

\(\newcommand{\setR}{\mathbb R}\) \(\newcommand{\setC}{\mathbb C}\) \(\newcommand{\setZ}{\mathbb Z}\)

In these notes we will mostly confine ourselves to a simple type of signal: a mapping of the time \(t\) to a scalar value \(x(t)\). We will call \(x\) a signal. We will also denote the signal as \(x(t)\).

We will distinguish real and complex signals. A real valued signal \(x\) is a mapping from time \(t\in\setR\) to the real value \(x(t)\in\setR\).

A complex valued signal \(x\) is a mapping from time \(t\in\setR\) to the complex value \(x(t)\in\setC\).

Mathematically there is little (if none) difference between a signal and a function that is dependent on time. Later in these lecture notes we will look at functions depending on other parameters than time (e.g. frequency). Such functions can be complex valued as well and a lot of what is to be said in this section applies to those functions as well.

The second way to characterize signals is by looking at the time domain. Traditionally time was (and is) represented as a real value running from \(t=-\infty\) (the infinite past) to \(t=\infty\) the infinite future. Note that most often the zero time is chosen arbitrarily. These signals are called continuous time signals.

With the use of computers we started to represent signals with a discrete and countable stream of numbers: the signal samples. At every time instance \(t=n \Delta t\) the signal value \(x(n \Delta t)\) is stored in computer memmory. All signal values inbetween these sample times are simply ignored. A signal that is known only at a countable set of time instances is called a discrete time signal.

To distinguish between CT signals and DT time signals we use the notation \(x(t)\) for CT signals and \(x[n]\) for DT signals.

It should be noted that these lecture notes in a way are a bit old fashioned in the sense that we stress the relation between CT and DT signals. It is possible to develop almost all notions in DT signals and systems processing without refering once to CT signals. In that view a DT signal is just a stream of (real or complex) numbers.