Signals
=======

.. sidebar:: Complex Numbers

   Signal processing makes heavy use of complex numbers $z=a+jb$ where
   the imaginary unit is written as $j$ (instead of the more common
   $i$). For a short introduction to complex numbers we refer to :doc:`this
   chapter</Math/complex>` in the math part of these lecture notes.
   

In these notes we will mostly confine ourselves to a simple type of
signal $x$ being a mapping of the time $t$ to a scalar value
$x(t)$. As a somewhat sloppy way of notation 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 at all any) 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.

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 the time $t=0$ is chosen arbitrarily and often
denotes the present (which of course is relative). 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.


.. toctree::
   :hidden:

   plottingsignals
   signalproperties
   signaltransformations
   basicsignals