Exercises
=========


#. **Continous Time Convolution**

   On http://pages.jh.edu/~signals/convolve/index.html you can find a
   Java applet that might help you understand the convolution of
   continous time signals.


#. **Continous Time Convolution**
   
   Consider the function:

   .. math::
      p_1(t) = \begin{cases}
      1 &:-1 \leq  t \leq 1\\
      0 &: \mbox{elsewhere}
      \end{cases}

   #. Calculate the convolution $p_1 \ast p_1$. (Hint: this can be done 'graphically').
      
   #. Calculate $p_1 \ast p_1 \ast p_1$.

   #. Let $x(t)=\sin(t)$. Calculate $x \ast p_1$.


#. **Discrete Convolution**
   
   Show that any *discrete time* linear time invariant system is
   completely characterized by its impulse response $h[n]$ and that
   the system response $y[n]$ on input $x[n]$ is given by:

   .. math::
      y[n] &= x[n] \ast h[n] \\
           &= \sum_{k=-\infty}^{\infty} x[k]\, h[n - k]


#. **Discrete Convolution**

   Consider a part from a discrete signal $x[n]$:

   .. math::
      \matrix{\cdots& 0& 0& 0& 1& 2& 1& 3& 2& 3& 1& 2& \underline{3}& 8& 7& 8& 9& 9& 7& 8& 8& 8& \cdots}

   The underlined value indicates the origin (i.e. $x[0]$). Calculate
   the convolution $x[n]\ast h[n]$ with:

   #. $h[n] = \matrix{1& \underline{1}& 1}$

   #. $h[n] = \matrix{\underline{1}& 1& 1}$

   #. $h[n] = \matrix{\underline{0}& 0& 1}$

   #. $h[n] = \matrix{1& \underline{0}& -1}$

   In all above functions $h$ the origin is indicated with the
   underlining and we have used the convention that all values in the
   (in principle) infinite function (from $-\infty$ to $+\infty$) that
   are not given are equal to zero.


#. **Eulers Formula**

   #. Prove that:

      .. math::
	 \sin(\phi) = \frac{e^{j\phi}-e^{-j\phi}}{2j}

   #. Prove that:

      .. math::
	 \cos(\phi) = \frac{e^{j\phi}+e^{-j\phi}}{2}


#. **Eigenfunctions**

   #. What is the response of an LTI system given an input
      $x(t)=\sin(\omega t)$?

   #. What is the response of an LTI system given an input
      $x(t)=\cos(\omega t)$?