# Designing elliptic filters with maximum selectivity

Electronic-filter design, whether analog, digital, or distributed, is an essential part of many electrical engineers' workdays. Frequency-selective networks are useful for suppressing noise, rejecting unwanted signals, or in some way manipulating the input signal's characteristics. Although applications abound, engineers typically use classical filters that are polynomial approximations to the brick-wall filter (see sidebar "A new look at the brick-wall filter"). These classical filters include Butterworth, Chebyshev, and elliptic filters.

Filter requirements often call for highly selective filters, especially in bandpass filters designed to reject out-of-band carriers. If the cutoff-rate specification is stringent, the classical Butterworth and Chebyshev filters result in high orders. A higher order adds complexity to the filter, and the resulting design is more difficult to tune. The sensitivity of the filter to its components also increases. These issues apply to both lumped-element realizations and microwave structures. For microwave structures, the physical features of the implementation directly influence the overall characteristics of the filter.

When selectivity is an issue, you can rely on elliptic filters, which provide the lowest order implementation of the classical filters for the same frequency and rejection requirements. Elliptic filters are equiripple in the passband and the stopband (Figure 1). The finite zeros of transmission, which allow the filter to have a narrower transition band, determine the ripple response in the stopband. The price of a narrower transition band is asymptotic roll-offs of –20 (order *n* odd) or –40 (order *n* even) dB/decade (Reference 1) and the additional complexity of achieving the transmission zeros.

Despite these limitations, the elliptic filter is the filter of choice for stringent magnitude-response requirements. The elliptic filter has the additional advantage of providing several degrees of freedom for controlling its response, including band-edge selectivity. Many designers resort to ad hoc and often wasteful techniques to obtain superior selectivity. However, a new technique allows you to maximize the band-edge selectivity (BES) of elliptic filters without increasing filter order. The technique effectively narrows the transition band by moving the notch frequency closer to the passband. This change increases the lobe levels to the original stopband-rejection requirement and impacts delay performance in the passband. A design example shows the ease with which you can design elliptic filters with maximum selectivity without increasing filter order. By maximizing the selectivity without increasing the filter order, you can reject more noise or unwanted signal components closer to the band edge—a desirable function.

Make better filters with no added cost

You can use a recently derived formulation for the band-edge selectivity of elliptic filters and use a method for maximizing selectivity without increasing the filter order (Reference 2). This useful method, in conjunction with the sensitivity calculations, can result in superior filters at no additional cost. The following design example highlights the power and ease of this method.

The BES of a filter is:

The selectivity is the slope of the magnitude response of the filter at the normalized corner frequency, or band edge. Selectivity is a measure of the cutoff rate, and the "larger-the-better" characteristic applies here. Most designers generally accept selectivity as a property of a filter and not as a goal of filter design. However, you can treat filter selectivity as a design parameter that you can optimize.

The BES of an elliptic filter is (Reference 2)

where *n* is the order,

is the passband ripple parameter, and

W_{S} is the stopband corner frequency, and is the stopband ripple parameter (Figure 1). If you're familiar with filter theory, you'll recognize the first term in the parentheses of **Equation 2** as the BES of the Chebyshev filter. However, for the elliptic filter, the new term (1–*m'*)/(1–*m*) scales this selectivity. As *m'*0, **Equation 2** reduces to

The result of **Equation 4** is that the BES of an elliptic filter is greater than that of the Chebyshev filter for any W_{S}>1, given the same order and passband ripple. Figure 2 shows a plot of the scaling factor. If the passband and stopband ripple are fixed, then W_{S} is the only degree of freedom for maximizing the BES without increasing the filter order *n*.

Review filter sensitivities

Before describing the filter-maximization process, it is useful to review the sensitivities of the BES of the elliptic filter to the various filter parameters. Recall that, when a dependent variable, *y*, is a function of two or more independent variables, *x*_{i}, where *i*=1,2,...N, the sensitivity of *y* with respect to *x*_{i} is as follows (Reference 3):

You therefore need to calculate the partial derivatives of the BES with respect to the various filter parameters as follows:

These **equations** are fairly complicated. However, by calculating the sensitivity using **Equation 5** you get simplified results (Reference 2):

In most applications, the filter order is fixed, and **Equation 10** always holds. On the other hand, you can control the sensitivity of the BES with respect to the passband ripple parameter using either or . By setting =^{2}, the numerator of **Equation 11** becomes a quadratic of the form ^{2}+(4+^{2})–2^{2}=0. Solving for , you obtain

**Equation 14** strictly depends on . Therefore, minimizing the sensitivity is possible by setting as in **Equation 14**.

You can reduce the sensitivity of the BES with respect to the stopband rejection by making >> for any value of . Alternatively, you can reduce this sensitivity by making small. This interaction of parameters is unique to elliptic filters.

Note from **Equation 13** that the sensitivity of the stopband corner frequency W_{S} increases as you decrease W_{S}. However, decreasing W_{S} increases the BES. Thus, although you can increase BES by reducing W_{S}, you must temper your intent by the resulting increase in sensitivity. Consider the effective change in the BES along with the change in the associated sensitivity. Again, using the assumption that *m'*0, you can rewrite **Equation 4** as:

Taking the derivative of **Equation 15** with respect to W_{S} gives the rate of change of the BES with respect to the parameter you are modifying for the maximization:

where D(W_{S})=1/(W_{S}^{2}–1) is the stopband-frequency factor. As for the sensitivity of **Equation 13**, you can easily calculate

Because W_{S}>1 and W_{S}^{3}>W_{S}, you can improve the BES of the filter at a greater rate than you degrade the corresponding sensitivity (Reference 4).

Maximizing the filter involves solving for the incremental order of the elliptic filter. You can obtain the order of an elliptic filter from a filter nomograph (Reference 1) or calculate the order using the following equation (Reference 5).

In **Equation 18**, K is the complete elliptic integral of the first kind (Reference 6) as follows:

You can find tabulated results of the above integral in mathematical handbooks or easily calculate the results using software packages such as MathCAD (Mathsoft Inc, Cambridge, MA).

The result of **Equation 18** is a real number, and you select the next highest integer, that is

where the subscript *i* denotes an integer. You can always select a higher order to satisfy an arbitrary selectivity requirement, but it is useful to maximize the selectivity with no increase in order. As already noted, and are fixed for most practical cases, and *m'*0. Thus, the parameter W_{S} is the degree of freedom for maximizing the selectivity of the filter while assuring that *ni* remains fixed.

You now need to make a distinction between _{S}, which is the variable, and W_{S}, which is the value of the specified stopband corner frequency. Because

_{S} is the variable, you can write **Equation 20** as

where C is a constant and =1/_{S}, such that

where *n* is the real number from the equality in **Equation 18**. Normalizing **Equation 22** using K(*m*)/K(1–*m*) produces the result

Because you must make _{S}&&font face="symbol">W_{S} to increase the selectivity, make

Substituting **Equation 24** into **Equation 23** yields

This **equation** has the same form as the calculation of the filter order in **Equation 18**. Thus, you can use the same formulation and substitute the appropriate values. Figure 3 shows a plot of versus *b* for various values of W_{S}. To use this plot, follow four steps:

1. Calculate *n* from **Equation 18** and *n*_{i} from **Equation 21**.

2. Set _{max}=*ni*/*n* to set the "excess order."

3. For the given W_{S} curve, read *b* from the point where W_{S}=

_{max}=*ni*/*n*.

4. Calculate _{S}=W_{S}/*b*.

This process results in the minimum stopband corner frequency _{S} that maximizes the BES for the given filter order. All other parameters remain fixed.

This technique can be useful with filter-design packages. Filter-design packages typically provide designs that meet the specifications but do not necessarily maximize the selectivity of the filter. A little extra work using the proposed technique results in a superior filter with no additional complexity. To use this technique with the filter software, you simply substitute the value _{S} for the original W_{S} requirement for maximum selectivity.

Design example demonstrates technique

To demonstrate the effectiveness of the technique, consider the following lowpass-filter requirements: passband ripple M_{P}=1.25 dB, stopband rejection M_{S}=40 dB, passband frequency f_{P}=1000 Hz, and stopband frequency f_{S}=2000 Hz. From filter nomographs, you can quickly determine that this set of requirements would result in an eighth-order Butterworth or a fifth-order Chebyshev filter.

From M_{P}=1.25 dB,

=0.5775. From M_{S}=40 dB, =100. From the frequency requirement, W_{S}=f_{S}/f_{P}=2. Using **Equation 18**, you can calculate the order *n*=3.25482. Select the next highest order, so *ni*=4.

Figure 4a shows the fourth-order elliptic filter that meets the requirements. For this filter, the BES calculated from **Equation 2** is 4.62. The sensitivities are S_{e1}^{BES}=1.25, S_{e2}^{BES}=0, and =–0.67.

To maximize the selectivity of the filter, you calculate _{max}=*ni*/*n* 1.23. Reading this value from Figure 3 at the W_{S}=2 curve results in a value reading of *b*=1.35. Thus, the required stopband frequency _{S}=W_{S}/*b*=2/1.35=1.48. You use this value as the stopband corner frequency in the design and recalculate the filter poles and zeros. Figure 4b shows a plot of the fourth-order filter that meets the original requirements with maximum BES. The new BES is 6.36 with new sensitivity =–1.68.

A careful observation of Figure 4a and b highlights the effect of moving in the stopband corner frequency W_{S}. The original specifications resulted in a filter whose transition band just met the –40-dB rejection requirement at 2000 Hz (Figure 4a). The secondary lobe is down around –53 dB with a notch at 2350 Hz. The proposed technique moved the notch closer to the passband to around 1750 Hz (Figure 4b). This notch movement results in an increase of the secondary lobe up to the required –40-dB rejection level. However, rejection in the transition band is superior. For example, the original filter had 25 dB of rejection at 1500 Hz. The modified filter has more than 30 dB of rejection at 1500 Hz.

Compare response to Chebyshev filter

It is interesting to compare the elliptic filter to the Chebyshev filter, which like the elliptic filter provides selectivity that is proportional to *n*^{2}. A seventh-order Chebyshev filter is necessary to meet this new requirement of W_{S}=1.48. Therefore, the elliptic filter is the clear winner due to its reduced parts count in circuit implementations, even when factoring in the transmission zeros.

Increasing filter selectivity has a negative impact on the delay response in the passband. Elliptic filters exhibit less delay variation than Chebyshev filters but more delay peaking. Negative delay impulses of area –W appear at the zero frequencies, and the effect of reducing W_{S} simply moves the zero impulses closer to the transition band. However, to compensate for the zeros, the pole locations shift closer to the j axis. This shift slightly increases delay variation but severely impacts delay peaking near the band edge. In addition, if the zeros are not purely imaginary but lay off the j axis, they would produce negative delay peaking of nonzero bandwidth, thereby distorting the delay near the passband edge.

Reducing W_{S} also impacts the step response of elliptic filters. From the plots in references 7 and 8, the step response depends on the inverse of the stopband corner W_{S} for constant in-band ripple. The low-frequency delay and thus the delay time decreases as W_{S} decreases. In addition, the overshoot decreases as W_{S} decreases. You can explain this fact by observing that the highest-Q complex-pole pair moves closer to the imaginary zeros as W_{S} decreases, which reduces the residue value for that pole and, therefore, the overshoot.

For fixed W_{S}, the overshoot increases with filter order. Therefore, maximizing selectivity not only reduces step-response overshoot but also ensures that there is no increase because the order remains fixed. Also, the rise time remains relatively constant as long as the 3-dB bandwidth is nearly constant, which is a characteristic of high-order filters.

Due to the increase in the filter's sensitivity to the stopband frequency ratio W_{S}, for practical designs you should select a value for

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