Chebyshev filters are one of the common active filter topologies used to build higher-order filters, and they are sometimes integrated into analog front-ends/RF front-ends in ICs on an I/O buffer. Chebyshev filters are just one of the filter types available for these purposes, but their construction from active devices and some simple passives makes them easy to build and analyze.
Uses of Chebyshev filters include usage in image processing, specifically in filtration in images to extract useful data, as well as in brick-wall filters for precision waveform synthesis. The goal in using a Chebyshev filter is to design a circuit that provides a steep roll-off in its transfer function curve. While Chebyshev filters give an optimal approximation of an ideal filter's magnitude response at the stop-band edge, there is a tradeoff to this behavior as seen in the pass-band.
Types of Chebyshev Filters
Chevbyshev filters are used to provide active filtering that can be extended to higher-order filtering through cascading of active filter circuit blocks. These filters are built using an amplifier and some additional passives to create low-pass, high-pass, or bandpass functionality. Because they use an amplifier it is also possible to provide gain in the passband.
An example of a 4th order Chebyshev filter is shown below to illustrate the topology involved in building these filter stages. Each amplifier circuit is a 2nd order filter stage; a first order filter stage would be created by omitting C2/C4 in one of these circuit blocks. The filter could also be changed to high-pass functionality by swapping R1-R4 for capacitors to implement AC coupling and C1-C4 with resistors to prevent shunting high frequencies to ground.
Example of 4th order low-pass Chebyshev filter.
Chebyshev filters come in two varieties, called Type 1 and Type 2. Both offer steep roll-off and attenuation, and they are easy to cascade to create higher-order filters. They differ in terms of where the ripple appears in the filter transfer function. In the Type 1 filter, the ripple appears in the passband, while in the Type 2 filter, the ripple appears in the stop-band. A comparison is shown in the graph below.
The main difference between these is seen in the passband and stop-bands, as was mentioned above. This leads to different transient responses for a given filter order. In both cases, the higher order filter produces steeper roll-off in the transfer function shown above, as one would expect.
Ripple and Overshoot
Another way to classify a Chebyshev filter is based on its passband ripple value, expressed in dB. Common filter designs aim for 1 dB or 0.5 dB ripple as seen in the transfer function. This ripple level, as well as the filter order number (equal to number of poles in the transfer function) will determine the transient response when driven with a pulse or a perfect square wave.
As is the case with any filter that contains passband ripple, this will affect the transient response when driven with a fast edge rate signal. The appropriate usage in certain applications may demand examining the transient response of the circuit for various filter orders to determine if the overshoot and turn-on edge rate are acceptable.
The example below compares a 4th-order Chebyshev filter with a critically-damped RLC filter and a Butterworth filter to examine the transient responses. The curves are ordered as follows:
Blue: step input
Red: critically damped RLC circuit
Purple: 4th order Butterworth
Black: 4th order Chebyshev (type 1)
In general, higher order Chebyshev filters with more ripple will produce greater overshoot. There will also be a delay in the rise time of the transient response, and that delay in the upswing will be larger when the filter order/ripple are larger.
If the transient response observed in a simulation includes too much overshoot, then an alternative filter type could be useful. An example would be a higher-order Bessel filter, which can provide much lower transient response with comparable roll-off to a Butterworth filter. While this will be less ideal brick-wall behavior at a certain filter order, some additional cascading in a Bessel filter could help increase the rolloff so that it is comparable to a Chebyshev filter of lower order.
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