===========
 Exercises
===========

#. **Sampling.**

   Consider a *real valued* and *even* function $x(t)$ with a Fourier
   spectrum $|X(\w)|$ as sketched below.

   
   .. exec_python:: harmonics  CTP
      :linenumbers:
      :code: shutter
      :Code_label: Show code for figure
      :results: hide

      import numpy as np
      import matplotlib.pyplot as plt

      plt.clf()
      
      w0 = 2
      w = np.linspace(0, 6, 100)
      X = np.maximum(0, 1-w/w0)

      plt.gcf().set_size_inches(5, 4)
      plt.plot(w, X)
      plt.xticks([0, 1, 2, 3, 4], [r'$0$', r'$\omega_0/2$',
	 r'$\omega_0$', r'$3\omega_0/2$', r'$2\omega_0$'])
      plt.xlabel(r'$\omega$')
      plt.ylabel(r'$|X(\omega)|$')
      plt.savefig('source/figures/Xexample_sampling.png')


   .. figure:: /figures/Xexample_sampling.png
      :align: center

   a. Why is it that $X(\w)$ is even and real valued?

   #. Sketch $X(\w)$ for $-4\w_0\leq\w\leq 4\w_0$.

   Let $X_s$ be the spectrum of the 'sampled' signal $x_s$:

   .. math::
      x_s(t) = x(t) p_{\Delta t}(t)

   where $p_{\Delta t}$ is the (infinite length) pulse train with
   period $\Delta t$ corresponding with the chosen sample frequency.

   c. Sketch $X_s$ for sampling frequency $\w_s=2\w_0$.

   #. Sketch $X_s$ for sampling frequency $\w_s=3\w_0/2$.