Originally, a ladder filter is a passive filter, containing inductors, capacitors, and damping resistors. The damping resistors can be at the input or the output or both. When transformed into active filters, these ladder filters have the attractive property that the state matrices are sparse, which helps to reduce the number of resistors (or transconductors) in the final realization. Another attractive property of these active ladder filters is that they can have very good dynamic-range properties.
FA can transform any filter into a ladder filter. The command
transform: orthogonal;will transform the current filter into a single-ended ladder filter with all damping at the input. The filter is orthogonal because its state-correlation matrix K (see Section 14.1) is equal to the Identity matrix, which makes the state variables form an orthogonal set.
A symmetrical ladder will result from the command:
transform: ladder;The orthogonal transformation is more robust than the ladder transformation. While the orthogonal transformation should always produce a result, the ladder transformation may fail because solutions do not always exist. For instance, generating a symmetric ladder of a Bessel filter of orders 7 and 8 does not work, but it is successful with order 9. In case of no convergence, FA will generate an error and leave the filter unchanged.
Even though ladder is less robust, it can do much more than orthogonal. It can generate ladders with many different distributions of the ladder damping. Whereas in passive ladder filters the damping is only at the input or output, in active ladder filters the damping can be in any integrator. The command
transform: ladder;is equivalent to
transform: ladder(0.5);The factor 0.5 says that half of the damping should be at the input and therefore the other half at the output. Likewise, ladder(1) will result in all damping at the input and no damping at the output. ladder(0) will place all damping at the output. ladder(0.3, 0.2) will result in a ladder with 30% of the damping in the first stage, 20% in the second stage, and the rest (50%) in the last stage.