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**.

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 the sample
frequency. The proof of this sampling theorem is the main subject in
this chapter.


.. toctree::
   :hidden:

   samplingtheorem
   interpolation
   sampling_interpolation_practice