Exercises
=========

#. Write Python functions to plot signals/functions. You should be
   able to plot real as well as complex signals and CT as well as DT
   signals. You will be using this functionality quite a lot in this
   course.

   
#. Plot the following functions:
   
   #. Step functions $u(t)$ and $u[n]$.

   #. Translated stepfunctions $u(t-2)$ and $u[n-2]$

   #. Scaled and translated step function $3\, u(\frac{t-2}{3})$

#. Plot the $x_a$ function for several values of $a$. See for yourself
   that it starts to look as the pulse function.

   
#. Defining the CT pulse function as the limit of the $x_a$ function
   is not a unique definition. If we take the function $g_s(t)$ being
   the Gaussian function (you may well remember...) and look at the
   limit for $s\rightarrow0$ you will see that in the limit we arrive
   at the pulse function too. Plot the $g_s$ for several values of $s$
   to see for yourself.

#. Consider the CT complex exponential functions

   .. math::
      x(t) &= e^{j 3\pi t / 4}\\
      x[n] &= e^{j 3\pi n / 4}


   .. exec_python:: exercises systemsignalsexercies
      :linenumbers:
      :code: shutter
      :Code_label: Show code for figure
      :results: hide

      import numpy as np
      import matplotlib.pyplot as plt
		
      plt.clf()
      t = np.linspace(0, 15, 1000)
      n = np.arange(16)
      xct = np.exp(1j*3*np.pi*t/4)
      xdt = np.exp(1j*3*np.pi*n/4)
      plt.subplot(2,1,1)
      plt.plot(t, xct.imag)
      plt.subplot(2,1,2)
      plt.stem(n, xdt.imag, use_line_collection=True)
      plt.savefig('source/figures/complexexponentialexample.png')


   .. figure:: /figures/complexexponentialexample.png
      :align: center
      :figwidth: 80%

      **Complex exponential** $e^{j 3\pi t / 4}$ (top) and
      $e^{j 3\pi n / 4}$ (bottom).
       

  #. what is the period of $x(t)$?

  #. what is the period of $x[n]$?