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##

Ladder Filters

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.

** Next:** Analysis in the Frequency
** Up:** Filter Networks
** Previous:** Transposition
** Contents**
2009-06-03