High Pass Filter
================

.. figure:: /figures/highpass.*
   :align: right
   :figwidth: 40%

   1st order highpass filter

As for the low pass filter we design a high pass filter using just one
passive element, in this case a capacitor in series with the driver.

The transfer function in this case is:

.. math::
   H(\omega) = \frac{R}{R+\frac{1}{j\omega C}} = \frac{j\omega RC}{1+j\omega RC}

The bode plot for this system with $R=8\Omega$ and $C=10\mu F$ is
given below. The cutoff frequency appears to be near 2 KHz.

Let's do some quick and dirty analysis to see where the cutoff
frequency is in terms of R and C. For low frequencies we have

.. math::
   H(\omega) \approx j\omega R C

and thus

.. math::
   \log |H(\omega)| = \log(RC) + \log(\omega)

and for large frequencies:

.. math::
   H(\omega) \approx 1

and thus

.. math::
   \log |H(\omega)| = 0

The cutoff frequency is thus at $\omega_x = 1/(RC)$ and for this
choice of resistor and capacitor $f_x = \omega_x/(2\pi) = 1989 Hz$