# Chebyshev RF Filter Basics

### - basics of the Chebyshev Filter - its performance, key facts and how it can be used in RF filter applications.

The Chebyshev filter is a form of filter used in RF and many other applications.

The Chebyshev filter provides a steeper roll-off than the more commonly use Butterworth filter.

However the additional roll-off of the Chebyshev filter comes at the expense of ripple, and this makes it unsuitable for a number of applications.

Nevertheless, the Chebyshev filter is widely used in RF applications where ripple may not be such an issue. The steep roll-off can be used to advantage to remove out of band spurious emissions such as harmonics or intermodulation.

## Pafnuty Chebyshev

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.

He 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 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. The in-band ripple of one type of Chebyshev filter prevents this format of filter being used in some applications where a flat in-band response is needed.
• 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 way the filter rolls of faster than other types like the Butterworth 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.

## Chebyshev filter types

There are two types of Chebyshev filter that are available:

• 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.

By Ian Poole

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