Sampling
========

So far CT and DT signals are treated more or less independently of
each other. In most practical applications of discrete time signal
processing the signal $x[t]$ finds its origin in some continuous time
signal $x(t)$.

The most often used way to make a DT signal out of a CT signal is
called **sampling**. We assume the sampling to be equidistant in
time. In that case sampling reduces a CT signal to a series of values

.. math::
   x[n] = x(n T_s)

where $T_s$ is the **sampling period**. The frequency $f_s=1/T_s$ is
called the **sampling frequency**.

It should be noted that sampling as defined *mathematically* above is
what is ideally done. In practice every measurement needs a probe of
finite spatial and temporal extend. In these notes we simply assume
that ideal sampling is close enough to reality.

The process to reconstruct the signal $x(t)$ from its samples $x[n]$
is called **interpolation**. Remarkably as it may seem it *is*
possible to reconstruct $x(t)$ exactly in case the original signal
does not contain frequencies greater then or equal to *half* the
sample frequency. The proof of this **sampling theorem** is the main
subject in this chapter.


.. toctree::
   :hidden:

   samplingtheorem
   interpolation
   sampling_interpolation_practice
   sampling_exercises