An elliptic filter (also known as a Cauer filter, named after
Wilhelm Cauer
Wilhelm Cauer (24 June 1900 – 22 April 1945) was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis. Prio ...
, or as a Zolotarev filter, after
Yegor Zolotarev
Yegor (Egor) Ivanovich Zolotarev (russian: Его́р Ива́нович Золотарёв) (31 March 1847, Saint Petersburg – 19 July 1878, Saint Petersburg) was a Russian mathematician.
Biography
Yegor was born as a son of Agafya Izoto ...
) is a
signal processing filter
In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspe ...
with equalized
ripple
Ripple may refer to:
Science and technology
* Capillary wave, commonly known as ripple, a wave traveling along the phase boundary of a fluid
** Ripple, more generally a disturbance, for example of spacetime in gravitational waves
* Ripple (electri ...
(equiripple) behavior in both the
passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
and the
stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
between the
passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
and the
stopband, for the given values of ripple (whether the ripple is equalized or not). Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.
As the ripple in the stopband approaches zero, the filter becomes a type I
Chebyshev filter
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error betwee ...
. As the ripple in the passband approaches zero, the filter becomes a type II
Chebyshev filter
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error betwee ...
and finally, as both ripple values approach zero, the filter becomes a
Butterworth filter.
The gain of a
lowpass
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter ...
elliptic filter as a function of angular frequency ω is given by:
:
where R
n is the ''n''th-order
elliptic rational function (sometimes known as a Chebyshev rational function) and
:
is the cutoff frequency
:
is the ripple factor
:
is the selectivity factor
The value of the ripple factor specifies the passband ripple, while the combination of the ripple factor and the selectivity factor specify the stopband ripple.
Properties
![Elliptic Filter Response (4th Order)](https://upload.wikimedia.org/wikipedia/commons/d/de/Elliptic_Filter_Response_%284th_Order%29.svg)
* In the passband, the elliptic rational function varies between zero and unity. The gain of the passband therefore will vary between 1 and
.
* In the stopband, the elliptic rational function varies between infinity and the discrimination factor
which is defined as:
:
:The gain of the stopband therefore will vary between 0 and
.
* In the limit of
the elliptic rational function becomes a
Chebyshev polynomial
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshe ...
, and therefore the filter becomes a
Chebyshev type I filter, with ripple factor ε
* Since the Butterworth filter is a limiting form of the Chebyshev filter, it follows that in the limit of
,
and
such that
the filter becomes a
Butterworth filter
* In the limit of
,
and
such that
and
, the filter becomes a
Chebyshev type II filter with gain
::
Poles and zeroes
![Elliptic Filter s-Plane Response (8th Order)](https://upload.wikimedia.org/wikipedia/commons/e/e8/Elliptic_Filter_s-Plane_Response_%288th_Order%29.svg)
The zeroes of the gain of an elliptic filter will coincide with the poles of the elliptic rational function, which are derived in the article on
elliptic rational functions
In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic filter, elliptic electronic filters. (These functions are someti ...
.
The poles of the gain of an elliptic filter may be derived in a manner very similar to the derivation of the poles of the gain of a type I
Chebyshev filter
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error betwee ...
. For simplicity, assume that the cutoff frequency is equal to unity. The poles
of the gain of the elliptic filter will be the zeroes of the denominator of the gain. Using the complex frequency
this means that:
:
Defining
where cd() is the
Jacobi elliptic cosine function and using the definition of the elliptic rational functions yields:
:
where
and
. Solving for ''w''
:
where the multiple values of the inverse cd() function are made explicit using the integer index ''m''.
The poles of the elliptic gain function are then:
:
As is the case for the Chebyshev polynomials, this may be expressed in explicitly complex form
:
:
:
:
where
is a function of
and
and
are the zeroes of the elliptic rational function.
is expressible for all ''n'' in terms of Jacobi elliptic functions, or algebraically for some orders, especially orders 1,2, and 3. For orders 1 and 2 we have
:
:
where
:
The algebraic expression for
is rather involved (See ).
The nesting property of the
elliptic rational functions
In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic filter, elliptic electronic filters. (These functions are someti ...
can be used to build up higher order expressions for
:
:
where
.
Minimum Q-factor elliptic filters
See .
Elliptic filters are generally specified by requiring a particular value for the passband ripple, stopband ripple and the sharpness of the cutoff. This will generally specify a minimum value of the filter order which must be used. Another design consideration is the sensitivity of the gain function to the values of the electronic components used to build the filter. This sensitivity is inversely proportional to the quality factor (
Q-factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
) of the poles of the transfer function of the filter. The Q-factor of a pole is defined as:
:
and is a measure of the influence of the pole on the gain function. For an elliptic filter, it happens that, for a given order, there exists a relationship between the ripple factor and selectivity factor which simultaneously minimizes the Q-factor of all poles in the transfer function:
:
This results in a filter which is maximally insensitive to component variations, but the ability to independently specify the passband and stopband ripples will be lost. For such filters, as the order increases, the ripple in both bands will decrease and the rate of cutoff will increase. If one decides to use a minimum-Q elliptic filter in order to achieve a particular minimum ripple in the filter bands along with a particular rate of cutoff, the order needed will generally be greater than the order one would otherwise need without the minimum-Q restriction. An image of the absolute value of the gain will look very much like the image in the previous section, except that the poles are arranged in a circle rather than an ellipse. They will not be evenly spaced and there will be zeroes on the ω axis, unlike the
Butterworth filter, whose poles are arranged in an evenly spaced circle with no zeroes.
Comparison with other linear filters
Here is an image showing the elliptic filter next to other common kind of filters obtained with the same number of coefficients:
![Filters order5](https://upload.wikimedia.org/wikipedia/commons/b/bd/Filters_order5.svg)
As is clear from the image, elliptic filters are sharper than all the others, but they show ripples on the whole bandwidth.
References
*
*
{{DEFAULTSORT:Elliptic Filter
Linear filters
Network synthesis filters
Electronic design