# Filter Shapes

## Overview for Filter Shapes

This section of the Genesys documentation contains information for all filter synthesis programs, including FILTER, A/FILTER, M/FILTER, and S/FILTER.

For each filter type such as lowpass, bandpass, etc. there are a number of response shapes. The design of all filters is based on a lowpass filter with that shape of response. This lowpass filter is called the lowpass prototype and the values of parts in that filter are called the prototype "G" values. The values are normalized to 1 ohm input impedance and a cutoff frequency of 1 radian/sec [1].

To design a filter, FILTER computes these prototype values, makes the transformation to the desired type and normalizes to the specified impedances and frequencies.

FILTER computes the prototype G values for Butterworth, Bessel, Chebyshev, singly-equalized delay and elliptic Cauer-Chebyshev shapes. Also FILTER allows storing of frequently used prototype G values in disk files which are automatically read to design other filter responses. Included with Genesys are lowpass prototype G values for several different filter responses. The included response types are discussed later in this chapter. Other lowpass prototype files may be entered by the user using the editor in Genesys.

Dozens of tables of prototype G values have been published through the years for specific applications, and more may be expected in the future. Many tables of values are given in the references cited in this manual.

## Order Help

FILTER, A/FILTER, and M/FILTER can help you determine the order required for Butterworth, Chebyshev and elliptic Cauer-Chebyshev filters.

Suppose you need a bandpass filter which passes 10.5 to 10.9 MHz with a maximum ripple of .1 dB. The rejection at 10.2 MHz and 11.2 MHz must be at least 50 dB. We will determine the minimum order required for a Chebyshev and a Cauer-Chebyshev filter.

Select "Order Help" from the Settings tab. Determine the Chebyshev order first. Select the Bandpass and Chebyshev radio buttons. Enter ".1" for the ripple prompt. Enter "10.5" and "10.9" for the Fl and Fu prompts respectively.

You may then enter up to 10 separate stopband requirements. =FILTER= will automatically determine which requirement is the most stringent and compute the necessary order based on that requirement. In this case, at the F0 and A0 prompts, enter "10.2" and "50". At the next F1 and A1 prompts enter "11.2" and "50". When you are finished entering stopband requirements, enter "0". The necessary order is computed and displayed as 5.39. The filter is therefore designed with the next higher integer order of 6.

Next, determine the necessary order for a Cauer-Chebyshev elliptic bandpass filter. Select the Cauer-Chebyshev radio button. Enter "50" at the prompt Amin, "10.5" at the Fl prompt and "10.9" at the Fu prompt.

For Cauer-Chebyshev bandpass filters, only two stopband requirements are entered, because the required stopband attenuation is assumed equal to Amin. Enter the lower and upper stopband requirements closest in frequency to the passband. For elliptic lowpass filters, only one stopband requirement is used.

Therefore, enter "10.2" at the Fsl prompt and "11.2" at the Fsu prompt. The required order is 4.03. Because this is very close to 4, a 4th order filter might be chosen. Re-running N-Help with Amin of 49 dB gives a required order of 3.98, so you know that a 4th order filter will have between 49 and 50 dB of attenuation.

Select the close button to return to the main window.

## Bessel

The Bessel filter produces a maximally flat group delay in the frequency domain. It is sometimes referred to as a maximally flat group delay filter and sometimes as a Thompson filter.

Maximally flat group delay is often an advantage in pulse communication system applications. This filter has excellent characteristics in the time domain.

The disadvantage is extremely poor selectivity. However, for applications where good phase or time domain performance is critical, it is often the best choice.

Genesys will design Bessel filters through 10th order.

## Bessel Passband Elliptic Stopband

As was discussed previously, Bessel filters have excellent delay characteristics, but poor selectivity. The selectivity of flat delay filters can be improved by adding zeros of transmission at finite frequencies. Elliptic lowpass prototypes for such a class of filters is given in reference [6]. The amplitude and delay responses for 5th order Bessel (solid curves) and elliptic Bessel Amin =50 dB (dotted curves) filters with 1 MHz cutoffs are compared in the accompanying figure.

## Blinchikoff Flat Delay Bandpass

Transformation of lowpass filters to bandpass filters results in the modification of the group delay characteristics of the lowpass prototype. Consequently, bandpass filters designed using Bessel, minimum phase equiripple error and other controlled delay prototypes do not exhibit the desired delay characteristics. This phenomenon worsens with increasing bandpass filter bandwidth. Some transform types, such as the top-C coupled bandpass, are worse than others, such as the shunt-C coupled bandpass. The reasons for this phenomena is discussed in the EQUALIZE documentation.

Blinchikoff and Savetman [3][4] offered a solution to this dilemma for 2nd and 4th order all-pole filters. The poles of the transfer function were optimized by computer for constant delay over the passband directly as a bandpass filter. The lowpass to bandpass transform is therefore avoided. The process is not numerically efficient, but the effort provides a useful set of constant delay bandpass filters for 30 to 70% bandwidth.

The extremely desirable delay characteristics of a 4th order Blinchikoff bandpass filter with 40% bandwidth centered at 70 MHz are demonstrated in the accompanying response. As might be expected, the amplitude selectivity characteristics of the filter are poor.

## Butterworth

The Butterworth lowpass response is useful when match and delay in the passband, particularly low frequencies, are important. The shape is characterized by monotonically increasing attenuation in the pass band to 3 dB at the cutoff frequency. Attenuation monotonically increases with increasing frequency in the transition and stopbands. Filters with infinite attenuation only at DC or infinite frequency, and therefore which have no zeros of transmission at finite frequencies, are called all-pole filters.

The Butterworth is a simple filter, with suitable characteristics for many applications. For straightforward filtering applications, the Butterworth is the filter response of choice. Its disadvantage is only fair selectivity. The frequency response of the Butterworth lowpass filter is given by:

Atten = 10 LOG10 [ 1 + (f/fc) ^{2N} ] dB

where

f = frequency

fc= 3 dB cutoff

N = order

The equations used in Genesys to compute the lowpass prototype G values for the Butterworth filter are given in reference [2].

The output impedance of the Butterworth lowpass prototype filter is equal to the input impedance, therefore G(0), the first prototype G value, is equal to G(N+1), the last prototype G value.

The normal definition for the cutoff of Butterworth filters is the 3 dB attenuation frequency. Genesys allows the user to specify the desired attenuation at the requested cutoff frequency. For example, the user may specify a cutoff frequency at 1 dB attenuation. Any cutoff attenuation greater than zero may be specified for the Butterworth response.

## Cauer-Chebyshev

Elliptic function filters have zeros of transmission at finite as well as infinite frequencies. Zeros of transmission at finite frequencies can reduce the width of the transition region, therefore increasing the selectivity or sharpness of the response. An important class of elliptic function filters is the Cauer-Chebyshev, which exhibits equiripple passbands and equal minimum attenuation, Amin, in the stopband. Genesys designs Cauer-Chebyshev filters with user specified passband ripple and stopband ripple. These routines are based on work by Pierre Amstutz [5]. In the Amstutz paper, distinctly different routines were given for even and odd order Cauer-Chebyshev prototype filters. The routines used different input data for determining values in the odd or even case. The Genesys routines remove this disadvantage.

Genesys computes prototype G values for both type B and type C Cauer-Chebyshev filters:

- Type B Cauer-Chebyshev filters are similar to Chebyshev filters in that the input and output impedances for even order are dissimilar.
- Type C filters approximate the filter response for equal input and output impedances for even order. The selectivity suffers slightly for type C, but this is a minor inconvenience when equal input and output impedance is required.

## Chebyshev

The Chebyshev lowpass response is characterized by equal attenuation ripple in the passband, with attenuation equal to the ripple at the cutoff frequency. The Chebyshev is an all-pole filter. The Chebyshev filter gives better selectivity than the Butterworth filter. The Chebyshev filter is an excellent choice when passband attenuation and return loss ripple can be tolerated. The frequency response of the Chebyshev lowpass filter above cutoff is given by:

Atten = 10 LOG10 {1 + e cosh ^{2} [ N cosh ^{-1} (f/fc) ]} dB

where

f = frequency

fc = 3 dB cutoff

N = order

e = 10 ^{RIPPLE/10} - 1

The equations used in Genesys to compute the lowpass prototype G values for the Chebyshev filter are given in reference [2].

The output impedance for Chebyshev prototype filters of odd order are equal to the input impedance, therefore G(0), the first prototype G value is equal to G(N+1), the last prototype G value.

For even order Chebyshev prototype filters, the output impedance is less than or greater than the input impedance, depending on whether the subtype is ML or MC. The difference is related to the passband ripple and is greater for larger passband ripple.

Genesys automatically determines and displays the output impedance.

Two normal definitions for the cutoff of Chebyshev filters are often used. Some contributors have defined the cutoff attenuation as 3 dB, and others define the cutoff attenuation as the passband ripple value, with the latter perhaps somewhat more generally accepted. We will define the cutoff attenuation as the ripple value. However, Genesys allows the user to specify any attenuation equal to or greater than the ripple attenuation as the cutoff. For example, the user may specify a cutoff frequency for a .25 dB ripple Chebyshev at .25, 1.07, 3, 6 dB, etc. attenuation. Any cutoff attenuation greater than the ripple may be specified for the Chebyshev response.

## Linear Phase Equripple Error

Just as the Chebyshev prototypes are an optimum amplitude response solution, the linear phase prototypes are an optimum linear phase solution (constant group delay) given an allowable phase ripple. By allowing some phase ripple, improved selectivity is obtained. The two prototype files are named "LP0R05.PRO" and "LP0R5.PRO". LP is an acronym for linear phase. 0R05 and 0R5 represent respectively 0.05 and 0.5 degrees phase ripple.

## Prototype Files

Some frequently used prototype G values have been supplied with =FILTER=. These files have the same format as files which you manually create. The form of these files is:

Remark line describes type and must be included. 3 1 1 2 1 1 4 1 0.7654 1.848 1.848 0.7654 1 5 1 0.618 1.618 2 1.618 0.618 1 7 1 0.445 1.247 1.802 2 1.802 1.247 0.445 1

This particular file is for Butterworth prototype filters of 3rd through 5th and 7th order.

The remark line is used to describe the type of lowpass prototype for reference. The remark line must be present. One line of data is used for each order. One or more orders may be in a file. The first number in a line is the order. The second number is G(0), the third G(1), etc. up to G(N+1), or up to G(N+M+1) for elliptic function filters.

When reading a file, if a data line for the specified filter order is not found, the message "Data for order N not found in file." is displayed. Either edit the file or specify a new one. An incorrect number of data points in a line of data adversely effects program operation so please carefully check prototype files you create.

We recommend using the extension ".PRO" for all-pole lowpass prototype files and the extension ".PRE" for elliptic lowpass prototype files you create.

### Included Prototype Files

To begin your collection of special purpose lowpass prototype filters kept as files, the =FILTER= disk contains a number of lowpass prototype files. Included are:

- Linear phase 0.05 degree equiripple error: orders 2-10.
- Linear phase 0.5 degree equiripple error: orders 2-10.
- Transitional Gaussian to 6 dB: orders 3-8.
- Transitional Gaussian to 12 dB: orders 3-8.
- Singly-terminated Cauer-Chebyshev: orders 3-7. (Ripple = 72 dB which is 24 files.)
- Bessel passband elliptic: orders 3-4 for Amin = 18, 24: orders 3-8 for Amin = 34, 42, 50, 58, 66, 70 dB.

The prototype files except Bessel passband elliptic are provided with permission from reference [1], *Handbook of Filter Synthesis*, by Anatol I Zverev, published by John Wiley and Sons. This reference includes many other singly-terminated Cauer-Chebyshev prototypes and other useful lowpass prototypes.

The Bessel passband elliptic stopband lowpass prototypes are provided with permission from reference [6], *Electronic Filter Design Handbook*, by Arthur B. Williams and Fred J. Taylor, published by McGraw-Hill.

## Singly-Terminated

Sometimes a filter which is singly-terminated is required. =FILTER= will design Butterworth and Chebyshev filters terminated on the input with a specified impedance and terminated on the output with either zero or an infinite impedance. The input and output of these filters may be swapped. Other filter shapes may be designed for single termination by user or file entry of the lowpass prototype G values. The Help screen in the Shape subwindow may be used to determine which subtype for a given filter type will have an infinite or a zero output impedance.

Since singly-terminated filters are not really matching networks, the insertion loss of these filters is not 0 dB in the passband. Consider a lowpass filter consisting of series inductors and shunt capacitors inserted between a source of 50 ohms and a zero impedance load. As the frequency approaches zero, the capacitors effectively become open circuits and the inductors effectively disappear. How can these components match the source to the load if they are effectively absent? The answer is they can not. Singly-terminated filters preserve the shape of the amplitude response and the shape of the voltage response, but when the power transferred between the source and load is calculated or measured, the insertion loss is proportional to the difference in input and output impedances.

When writing Genesys schematics for singly-terminated filters in order to avoid infinite attenuation in the calculated loss of the passband, FILTER sets the output impedance at 398 or 1/398 times the input impedance, instead of infinity or zero. The response of a filter changes little when the ratio is greater than about 20. The ratio of 398 results in an attenuation offset of approximately 20 dB, so singly-terminated filters analyzed using Genesys require adding a 20 dB offset to the scale of S21 and S12.

### Singly Equalized Delay

All ladder, passive, filter structures are minimum-phase; the amplitude and phase responses are inseparably related via the Hilbert transform. Selectivity and flat delay cannot be simultaneously achieved.

To resolve this difficulty, selective filters are sometimes cascaded with non-ladder all-pass delay equalization networks. Unfortunately several all-pass sections may be required and each requires several components.

Developed by Eagleware, the singly-equalized prototype offers selectivity far better than Bessel and other controlled phase prototypes but it is perfectly delay equalized with a single all-pass network. This prototype may be selected from the Shape dialog box. The theory behind the development of these prototypes is described as an example in HF Filter Design and Computer Simulation.

### Singly Terminated Cauer-Chebyshev

Genesys synthesizes doubly-terminated Cauer-Chebyshev elliptic functions directly. However, it does not synthesize singly-terminated Cauer-Chebyshev functions. This class is useful for designing contiguous elliptic diplex filters, so a number of prototype files for this class are included on disk.

The convention for naming these files is CCnNPPAA, where CC represents Cauer-Chebyshev, nN are the lowest and highest order included, PP is the reflection coefficient in percent, and AA is the approximate Amin in decibels. For the most part, the orders included are 3rd through 7th. The reflection coefficients are 1, 2, 4, 8, 10, 15, 20, and 25%, which represents a passband ripple of .00043, .0017, .007, .028, .044, .098, .18 and .28 dB respectively. Reference [1], which was so popular it has gone back into print, contains many other prototype files.

## Transitional Gaussian

Also included are Gaussian to 6 dB (G06.PRO) and Gaussian to 12 dB (G12.PRO) transitional lowpass prototype filters for 3rd through 8th order. This class of filter approximates a Gaussian response to 6 or 12 dB and is a compromise of delay and selectivity characteristics.

Both the linear phase equiripple error and transitional Gaussian prototypes are doubly-terminated unsymmetrical filters. The component values on the input and output are not mirror images of each other.

## User Filters

The powerful optimizing routines in Genesys can be used to develop prototype G values with a limitless variety of amplitude or delay characteristics. An example of using Genesys to develop a new lowpass prototype class is given in the Examples manual.

G(0) is the normalized input impedance. It is normally 1. For all-pole filters, G(1) through G(N) are the prototype values, and G(N+1) is the normalized output impedance.

For elliptic filters designed by Genesys, the number of finite frequency zeros of the lowpass prototype, M is given by:

For N odd,

M = .5(N-1)

and for N even,

M = .5(N-2)

There are N+M prototype G values for elliptic filters. The normalized output impedance is G(N+M+1). The form of the elliptic lowpass prototype filter is given on the previous page. For even order filters, the final series branch is non-elliptic. The Cauer-Chebyshev and many other elliptic function filters utilize this topology. The general case for filters with zeros of transmission at finite frequencies may not utilize this topology. For example, the older class of m-derived filters have designer selected topologies.