What is a Chebyshev RF Filter - the basics

The Chebychev filter topology is used in many RF applications because of its fast transition from pass-band to stop-band using LC combinations.

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The Chebychev filter is popular in RF application - using inductor and capacitor, LC combinations it provides the fastest transition from passband to stopband.

The fast transition between pass-band and stop-band comes at the price of in-band ripple, and this may not make it acceptable for all applications.

The Chebyshev RF filter is still widely used in many RF applications where ripple may not be such an issue. The steep roll-off is used to advantage to provide significant levels of attenuation of unwanted out of band spurious emissions such as harmonics or intermodulation.

The fast transition between pass-band and stop-band enables the best attenuation of unwanted signals to be achieved.

Chebyshev filter development

The Chebyshev filter is named after Pafnuty Chebyshev, who developed the polynomials on which the filter design was based.

He was a Russian mathematician who lived between 16 May 1821 to 8 December 1894 (dates using current calendar - using the original Julian calendar used in Russia at the time he was born on 4 May and died on 26 November).

Chebyshev was brought up in Moscow from the age of 11 and had an interest in mathematics, ultimately studying at Moscow University where he took his Master's degree.

After gaining his degree, Chebychev moved to St Petersburg where he was elected an extraordinary professor at the University in 1850, then ordinary professor in 1860 and, ultimately he became merited professor in 1872. Then in 1882 he retired from the university.

Chebyshev is known for work in a variety of areas of mathematics, but chiefly for his contributions to the areas of probability, statistics and number theory. Chebyshev is also credited with his work on orthogonal polynomials, and was probably the first person to comprehend the concept.

Chebyshev filter basics

Some of the key features of the Chebyshev RF filter can be summarised as below:

Roll-off: One of the main aspects of the Chebyshev filter is that it has a steep roll-off. It reaches its ultimate roll-off faster than other forms of filter. Accordingly is widely used in RF applications where a steep transition between pass-band and stop-band is required to remove unwanted products such as intermodulation of harmonics.
Ripple: Although the Chebyshev filter provides a steep roll-off, this is at the cost of ripple. Although there are different types of Chebychev filter, this aspect of its performance may prevent it from being used.
Cut-off frequency: The common definition of the cut-off frequency of the point at which the response falls to -3 dB does not hold for Chebyshev filters in view of the in-band ripple. Instead, the cut-off is taken as the point at which the gain falls to the value of the ripple for the final time. This can be seen from the diagram of the typical response of a Chebychev filter.
Chebyshev filter name: The name of the Chebyshev filter comes from the fact that the format and calculations for the filter are based on Chebyshev polynomials.

The gain (or amplitude) response, G_n as a function of the angular frequency, ω for an n-th order Chebychev filter can be expressed in the form of the function below:

G_{n} (ω) = | H_{n} (ω) | = \frac{1}{\sqrt{1 + ε^{2} T_{n}^{2} (\frac{ω}{ω_{c}})}}

Where
ε = ripple factor
ω_c is the cut-off frequency
T_n is the Chebychev polynominal of the nth order

The pass-band of the Chebychev filter provides an equi-ripple behaviour. The in-band ripple is determined by the ripple factor ε. In the passband, the Chebyshev polynomial alternates between -1 and 1. This means that the actual response / gain alternates between unity as the maximum and a minimum level determined by the formula below:

G = \frac{1}{\sqrt{1 + ε^{2}}}

Response of a 4th order Chebychev low pass filter showing the ripple

Chebyshev filter types

There are two types of Chebyshev filter that are available, each with its own characteristics - it is important to define the type of filter to be used when defining it:

Chebyshev type I filter: These are the most common Chebyshev filters. It has the steepest roll-off but exhibits in-band ripple.
Chebyshev type II filter: The type 2 Chebyshev filter may also be known as the inverse Chebyshev. It is less commonly used than the Type 1 filter because it does not roll off as fast, and also requires more components. However, its big advantage is that it has no ripple in the pass-band, but does have what is termed equi-ripple in the stopband.

The Chebychev filter is popular in many areas and in particular with RF designers where it provides the high levels of out of band reduction required to remove unwanted signals.