Filter Synthesis

Filters can be implemented (1) as a cascade if biquadratic filters, or biquads, that implements one or two poles at a time, or (2) as ladder filters.

Biquadratic Filters

To implement a single real pole, we can use a simple RC cicuit, as shown in Fig. 1, whose transfer function is given by:

$H\left(s\right)={\frac {1}{1+sRC}}$

(1)

To implement two complex conjugate poles, we can use a lossy LC circuit, as shown in Fig. 2. We can then express the transfer function of this circuit as:

$H\left(s\right)={\frac {1}{s^{2}LC+s{\frac {L}{R}}+1}}$

(2)

A more complex filter can then be implemented by cascading a 1- or 2-pole filter, as shown in Fig. 3 for a fifth order filter. Note that filters built as a cascade of biquads are easy to implement, but they are very sensitive to mismatch since the filter transfer function is dependent not only on the pole locations, but on the pole placement relative to other poles.

Ladder Networks

A better alternative to cascading biquads is to expand the filter's driving point or input impedance using continued fraction expansion. Consider a general ladder network shown in Fig. 4. The input impedance, $Z_{11}$, can then be expressed as the continued fraction:

$Z_{11}={\frac {1}{Y_{1}+{\frac {1}{Z_{2}+{\frac {1}{Y_{3}+\ldots }}}}}}$

(3)

Consider the filter shown in Fig. 5, where it is driven by a source with input resistance, $R_{S}$, and loaded by the resistance, $R_{L}$.