Exercises
=========

#. **FIR Bandpass Filter**

   Show that an ideal bandpass filter can be seen as the combination
   of two low pass filters. If the pass band starts at
   $\Omega_\text{low}$ and ends at $\Omega_\text{high}$ what are the
   two low pass filters needed and how to combine them. The impulse
   response of a low pass filter is given in the table of the section
   on DTDT's. Show that you get the same result as stated in this
   section:

   .. math::
      h[n] = \frac{\sin(\Omega_\text{high}n)}{\pi n} - \frac{\sin(\Omega_\text{low}n)}{\pi n}


#. **Idempotency.**

   In the introduction of this chapter there was a remark in a
   footnote explaining that the name filter in math is often reserved
   for idempotent operators. Let $\op F$ be such a filter operating on
   signal $x$ like $\op F x$. Then the filter is idempotent in case
   $\op F \op F = \op F x$ or more generally $\op F \op F = \op F^2 =
   \text{id}$, where $\text{id}$ is the identity operator.

   a. Show that an ideal low-pass filter *is* an idempotent operator
      (and thus a real filter in the mathematical sense).

   #. Proof of the first part is easy in the frequency domain. In the time domain
      it is harder (and rather counter intuitive). But analysis in the
      time domain clearly shows that idempotence is impossible in
      practice. Why?