Analog Filters
==============

An analog LTI filter takes a signal $x(t)$ as input and produces
output signal $y(t)$ being the convolution of $x(t)$ and the impulse
response $h(t)$.

Many analog filters are designed with analog electronic circuits. We
have seen some examples in the chapter on analog electronics. E.g. a
first order low pass filter for audio application is realized with an
inductor of inductance $L$ in series with the load (speaker) of
resistance $R$.

.. tikz::
   :xscale: 40
	    
   \draw (0,0)
   to[voltage source, v=$u_{in}$] (0,4)
   to[short, i=$i$] (2,4)
   to[L=$Z_{ind}$] (2,2)
   to[R=$R$] (2,0)
   to[short] (0,0);
   \draw (5,0) to (5,0);



In the s-domain the transfer function $H(s)$ is given by

.. math::
   H(s) = \frac{1}{1+s\frac{L}{R}}

Remember that its frequency response is:

.. math::
   H(\w) = H(s){\Large\rvert}_{s=j\w} = \frac{1}{1+j\w \frac{L}{R}}

The canonical low pass filter is given as:

.. math::
   H(s) = \frac{1}{1+\tau s}

Analog filters are often designed and specified in the complex
s-domain in the form of the complex transfer function $H(s)$ being the
Laplace transform of the impulse response function.