The simple filters you learn to design in an electronics class are almost always first order or second order (e.g., RLC circuit). Sometimes, these filters simply contain passive components and they do not provide steep attenuation in the filter transfer function. An alternative type of filter is a brick-wall filter, which is a generalization of a higher order low-pass filter with very steep roll-off in the transfer function.
Brick-wall filters are very simple to design for some topologies; these topologies are standardized and can be found in many textbooks and design guides. Those familiar with brick-wall filters will know that these filters provide very steep attenuation outside their passband. No matter which topology you use, the goal in designing a brick-wall filter is to have very high attenuation outside the pass band such that high frequency noise beyond a certain limit is very quickly eliminated.
Brick-Wall Filter Topologies
All brick-wall filters are implemented as low-pass filters, with the goal being to severely attenuate all frequencies above a specific cutoff. The filter response is intended to approximate a low-pass filter with an infinitely steep roll-off in its transfer function. In order to reach this behavior, brick-wall filters are always higher-order low-pass filters. These can be constructed as active filters, as cascaded filters, or cascaded passive/active elements.
There are several options for designing brick-wall filters depending on the desired level of attenuation. These filters can be designed as cascaded elements using standard topologies with passive components. They can also be designed with active components, such as amp based filter circuits. There are five common topologies that are often used to build higher order filters:
Chebyshev (type 1 and type 2)
The level of attenuation, passband/stop-band ripple, roll-off curvature, and gain at pole frequencies can be determined from a transfer function simulation in SPICE.
When selecting one of these filter topologies for a brick-wall filter, the main reason we want to choose a specific topology is to ensure a desired transient response. This requires taking an input waveform (in the Laplace domain) and calculating the inverse Laplace transform of the following products:
The other approach (or rather the inverse approach) is to select the filter based on its attenuation in a specific bandwidth. This is the standard method used to select filters, including a brick-wall filter. This approach is normally used when designing an RF filter that will be placed on a module or MMIC.
Transfer Functions and Transients
The selected filter topology should be based on what type of signal is expected to be filtered as well as the transfer function. When looking at the transfer function, the higher-order nature of the above topologies creates both strong roll-off and ripple in the transfer function. Only portions of the power spectrum within the passband will be affected by the ripple and attenuation in the transfer function.
The ripple in the passband essentially illustrates the behavior due to the multiple poles present in the transfer function. This will have different effects whether the circuit is driven in with broadband or narrow-band signals:
If broadband, ripple in the passband can create typical transient behavior with overshoot/ringing
If narrow-band, ripple in the passband does not matter, what matters is the gain or attenuation at the frequency of interest
Consider the set of transfer functions shown below, which compares Chebyshev filters with a Butterworth filter.
The transfer function of the Chebyshev filter shows ripple in the passband, while the Butterworth filter of the same order does not show any ripple. This is actually one reason a Butterworth filter may be preferable: among the brick-wall filter topologies with flat ripple in the passband, the Butterworth filter provides the steepest roll-off slope for a given filter order.
The ripple in the passband produces some odd-looking transient responses when driven with a square wave, again depending on the filter topology. The example below compares four curves:
Blue: step input
Red: critically damped RLC circuit
Purple: 4th order Butterworth
Black: 4th order Chebyshev (type 1)
The response to the square wave is somewhat non-intuitive because the ripple in the pass band behaves like a filter with multiple poles. In addition, the ripple outside the pass band also affects the transient response because the transfer function never falls precisely to zero. If this type of response is unwanted, then an alternative filter topology should be used for a brick-wall filter.
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