Ideal Filters
=============

.. ipython:: python
   :suppress:

   from matplotlib.pylab import *
      
   w = logspace(1,5);
   Hlp = w<1000
   subplot(2,2,1)
   semilogx(w, Hlp, linewidth=3)
   xlabel(r'\$\omega\$')
   ylim(-0.1, 1.1)
   title("Low Pass")
   #axis('off')
   
   Hlp = w>1000
   subplot(2,2,2)
   semilogx(w, Hlp, linewidth=3)
   xlabel(r'\$\omega\$')
   ylim(-0.1, 1.1)
   title("High Pass")
   #axis('off')
   
   Hbp = logical_and(w>500, w<5000)
   subplot(2,2,3)
   semilogx(w, Hbp, linewidth=3)
   xlabel(r'\$\omega\$')
   ylim(-0.1, 1.1)
   title("Band Pass")
   #axis('off')

   Hbr = logical_or(w<500, w>5000)
   subplot(2,2,4)
   semilogx(w, Hbr, linewidth=3)
   xlabel(r'\$\omega\$')
   ylim(-0.1, 1.1)
   title("Band Reject")
   #axis('off')
   
   @savefig idealfilters.png
   show()
   

.. image:: idealfilters.png
   
The basic ideal filters specified with their frequency responde are
sketched in the figure below:

- **Ideal Low Pass Filter.** All frequencies below $w_c$ are left
  unchanged and all amplitudes of the frequencies larger then $w_c$
  are set to zero:

  .. math::
     H(j\omega) = \begin{cases}
                  1 &: \omega\leq\omega_c\\
		  0 &: \text{elsewhere}
		  \end{cases}

- **Ideal High Pass Filter.** A high pass filter is the opposite of the
  low pass filter:

  .. math::
     H(j\omega) = \begin{cases}
                  1 &: \omega>\omega_c\\
		  0 &: \text{elsewhere}
		  \end{cases}
     

- **Ideal Band Pass Filter.** The band pass filter passes the frequencies
  between $\omega_L$ and $\omega_H$ and suppresses all other frequencies:

  .. math::
     H(j\omega) = \begin{cases}
                  1 &: \omega_L\leq\omega\leq\omega_H\\
		  0 &: \text{elsewhere}
		  \end{cases}
  
- **Ideal Band Reject Filter.** The band reject filter is the opposite
  of the band pass filter:

  .. math::
     H(j\omega) = \begin{cases}
                  0 &: \omega_L\leq\omega\leq\omega_H\\
		  1 &: \text{elsewhere}
		  \end{cases}
  
  In practical applications often a band pass filter is designed to
  suppress one specific frequency. Such a very narrow band bass filter
  is often called a **notch filter**.