Synthesis of Analog Filters

Wanhammar, Lars

doi:10.1007/978-0-387-92767-1_2

Lars Wanhammar²

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In this chapter, we will discuss different methods to synthesize the transfer function of an analog filter, i.e., determine the required filter order and the coefficients in the transfer function, for different types of requirements. The coefficients, which determine the positions of the poles and zeros, will later be used to determine the component values in the filter realizations. Here, we will focus on frequency selective filters, which can be used to separate signals and noise that lie in different frequency bands. Moreover, in this chapter we will focus on transfer functions that can be realized with lumped circuit elements, i.e., the transfer functions are rational functions ofs.

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2.1 Introduction

In this chapter, we will discuss different methods to synthesize the transfer function of an analog filter, i.e., determine the required filter order and the coefficients in the transfer function, for different types of requirements. The coefficients, which determine the positions of the poles and zeros, will later be used to determine the component values in the filter realizations. Here, we will focus on frequency selective filters, which can be used to separate signals and noise that lie in different frequency bands. Moreover, in this chapter we will focus on transfer functions that can be realized with lumped circuit elements, i.e., the transfer functions are rational functions ofs.

2.2 Filter Specification

Because an ideal lowpass filter cannot be realized, we must use an approximation of the ideal frequency response. Several standard approximations (frequency responses) have been proposed. The approximations, which usually are referred to as standard filters (approximations), have been optimized using different optimization criteria, i.e., to have the best possible performance from a certain point of view. Tables with standard filters, which are suitable for use in simple applications, are widely available [11, 100, 146]. Design of filters meeting more general requirements usually require numerical optimization techniques [29, 112, 124]. We will use programs, written in MATLAB™ [75], both to synthesize standard filters and filters that meet more general requirements.

A filter specification contains all relevant performance requirements in terms of acceptable bounds within measurable quantities may vary. Typically, the specification contains information of acceptable bounds within measurable quantities may vary. Typically, the specification contains bounds on the acceptable passband ripple, stopband attenuation, cutoff frequency, temperature range, etc. Furthermore, the specification usually contains other types of constraints, e.g., date of delivery and an agreement between the buyer and the seller, as it is important that the responsibility for design and manufacturing errors, etc., is well defined.

2.2.1 Magnitude Function Specification

For an ideal LP filter, the magnitude function equals one in the passband and zero in the stopband. Because such filters cannot be realized, we have to use an approximation of the ideal filter. An acceptable approximation of the magnitude response of an ideal filter has a magnitude function with sufficiently small variation in the passband to be negligible and an attenuation that is sufficiently large in the stopband. In some approximations, there may also be requirements on the phase and group delay characteristics as well.

In order to give the designer full freedom to solve theapproximation problem, it is common to specify the frequency response in terms of a specification of acceptable variations with the magnitude function and possibly the phase function and the group delay. We will discuss the specification of the group delay in Section2.2.3.

Typically, the tolerance ranges within the magnitude function may vary in the passband and the stopband, as shown in Fig.2.1. In the transition band, the magnitude function is only required to be decaying.

Figure2.1 shows a standard specification for the magnitude function for an LP filter with cutoff angular frequencyω _c and the stopband edge angular frequencyω _s. Sometimes the term ripple edge forω _c is used. The transition band is the band between the stopband and passband edges, i.e.,ω _s – ω _c.

A specification with a small transition band will require a filter of high order, and a filter that meets higher requirements than necessary will be more expensive to implement. Because the cost of a filter increases with the filter order, it is sensible to minimize the filter order. In many cases, however, it is advantageous to use a filter with slightly higher order than necessary.

2.2.2 Attenuation Specification

A typical specification of the attenuation requirements for a lowpass filter is shown in Fig.2.2 whereA _max is the allowed variation in the passband attenuation andA _min is the minimum required stopband attenuation.

From a filtering point of view, we are only interested in the relative attenuation in the passband and stopband. It is therefore convenient to normalize the maximal passband gain in the passband to 1, which corresponds to an attenuation of 0 dB as was done in Figs.2.1 and2.2.

The real passband gain of the filter is then adjusted in the last step of the filter design.

2.2.3 Group Delay Specification

Sometimes there are also requirements on the phase function of a filter, but instead it is more common to use a specification for the group delay. Figure2.3 shows a typical requirement on the group delay for a telephone channel.

In this case, the variation of group delay in the passband may not be too large and the specification has therefore been normalized with respect toτ _gmin. Hence, we have

$$\tau_g (\omega) =\tau_{gmin} +\Delta \tau_g(\omega).$$

In some applications, the total group delay cannot be too large, e.g., as in a telephone connection via satellite. Typically, the communication channel is active in only one direction at a time, i.e., the direction from the speaker that starts to speak. When the speaker stops speaking, the other speaker may take command of the channel in the other direction. If the delay is too large, it will be difficult to have a conversation because the two speakers will tend to speak or listen at the same time.

The variation of the group delay in the passband cannot be too large either. Typically, the high frequency part of the speech signal is delayed more than the low frequency part. This is perceived as a high frequency time-compressed copy of the word at the end of the word. This is very disturbing to the speakers.

2.3 Composite Requirements

It is difficult to synthesize a transfer function,H(s), which at the same time meets requirements on both the magnitude function and the group delay. Traditionally, this problem has been solved by dividing the problem into two separate design problems, one involving the synthesis of a minimum-phase and the other of a maximum-phase filter, i.e., an allpass filter. This approach is illustrated in Fig.2.4.

Definition 2.1

A minimum-phase filter has all zeros in the left half of thes-plane including the jω-axis.

First, we synthesize a minimum-phase transfer function,H _m(s), which meets the requirement on the magnitude function. Then we synthesize a transfer function of allpass type,H _ap(s), which corrects (adds to) the group delay ofH _m(s). This can be summarized

$$H(s) = H_m(s)H_{ap}(s)$$

((2.1))

$${\begin{vmatrix}H(j\omega)\end{vmatrix}}={\begin{vmatrix}H_m(j\omega)H_ {ap}(j\omega)\end{vmatrix}}= {\begin{vmatrix}H_m(j\omega)\end{vmatrix}}$$

((2.2))

$$\Phi(\omega) = \Phi_m(\omega) + \Phi_{ap}(\omega)$$

((2.3))

$$\tau_f(\omega) = \tau_{fm}(\omega)+\tau_{fap}(\omega)$$

((2.4))

$$\tau_g (\omega) = \tau_{gm}(\omega)+ \tau_{gap}(\omega).$$

((2.5))

Figure2.4 illustrates how the magnitude function and the group delay of the filterH _m(s) andH _ap(s) interact so that the combined filters meet the requirements onH(s). However, this method is generally not optimal, i.e., the sum of the orders of the two filters is not always minimal, even though the minimum-phase filter and allpass filter both have minimal order. Computer-based numerical optimization techniques may thus be used to synthesis optimal filters that simultaneously meet attenuation and group delay requirements.

2.4 Standard LP Approximations

Traditionally, we have derived different analytical solutions to the approximation problem, i.e., the problem to find a transfer function that meets the filter specification. Here we first consider the partial problem of finding a transfer function that meets the requirement on the magnitude function.

The analytical solutions, which will be discussed below, are optimal in a certain aspect. This means that one performance measure has been made as high as possible but often at the expense of other performance measures. In real applications, the filter requirements are often more complex. Often we require that the transfer function at the same time shall meet several different requirements, which even may be contradictory. This makes it difficult to synthesize a good filter. The availability of computer-based methods has changed the situation dramatically, and a transfer function that meets several different requirements can easily be found using an optimizing program. Hence, it is often possible to find more efficient and cheaper solutions than those found in filter tables [11, 100, 146].

In this chapter, the classic filter approximations for an LP filter and the corresponding computer-aided design methods are discussed first. In the following sections, we will discuss methods to transform an LP filter to a highpass, bandpass, or bandstop filter [68]. We also discuss computer-aided methods of synthesizing filters that meet non-standard requirements as well as correction of the group delay with the use of allpass filters.

The magnitude function squared for a filter can be written as

$$|H(j\omega)|^2 = \frac{1}{1+\varepsilon^2|C_N (j\omega)|^2}$$

((2.6))

whereC _N(jω) is the characteristic function.C _N(s) is an odd (even) function ofs for an odd (even) order filter. The zeros ofC _N(s), which are called reflection zeros, typically lie in the passband. The poles ofC _N(s) lie in the stopband and are referred to as transmission zeros. ε is a constant that determines the variation in passband of the filter.

2.4.1 Butterworth Filters

The mathematically simplest and therefore most common approximation is Butterworth filters.^{Footnote 1} Butterworth filters are used mainly because they are easy to synthesize and not because they have particularly good properties [29, 135]. Figure2.5 shows the attenuation for Butterworth filters of different orders^{Footnote 2} and Fig.2.6 shows the corresponding group delays. The allowed tolerance in the passband is in this caseA _max = 1.25 dB, which is obtained at the passband edgeω _c = 1 rad/s. Note that the overall group delay is larger and it has a sharper peak for filters of higher order. In an ideal filter, however, the group delay should be constant in the passband.

The area under the group delay isNπ/2, whereN is the filter order. Hence, the group delay in the passband will increase if the bandwidth is reduced and vice versa.

For a given attenuation requirement, the Butterworth approximation requires a relatively high filter order and hence the group delay becomes large in comparison to the standard approximations, which shall be discussed later. A high-order filter is expensive to implement and has large power consumption if the filter is implemented as an activeRC filter.

Figure2.7 shows the attenuation in the passband for Butterworth filters of different order. Note that a linear scale has been used for theω-axis. For Butterworth filters, the attenuation and magnitude functions are maximally flat forω = 0 andω = ∞, and a maximum number of derivatives (N–1) of the functions are zero at these angular frequencies. The magnitude function is monotonously decaying while the attenuation is monotonically increasing.

The magnitude function squared for a Butterworth filter can be written as

$$|H(j\omega)|^2 = \frac{1}{1+\varepsilon^2( \frac{\omega} { \omega_c })^{2\,N}}.$$

((2.7))

whereε is a constant that determines the variation in the passband, 0 – ωc. The characteristic function squared^{Footnote 3} for a Butterworth filter is

$$|C_N(j\omega)|^2 = \left (\frac{\omega}{\omega_c}\right) ^{2\,N}.$$

((2.8))

The attenuationA _max at the passband edge, i.e.,ω = ω _c is

$$A_{max} = -20\log(|H(j\omega_c)|) = -10 \log \left(\frac{1}{1+\varepsilon^2}\right).$$

((2.9))

The attenuationA _min at the stopband edge, i.e.,ω =ω _s is

$$\begin{aligned}A_{min} &= -20\log(|H(j\omega_s)|) \\&= 10 \log\left( 1+\varepsilon^2\left(\frac {\omega_s}{\omega_c}\right)^{2\,N}\right).\end{aligned}$$

((2.10))

The required filter order, which can be derived from Equations (2.9) and (2.10), is

$$N\geq\frac{{\displaystyle \log\left(\frac {10^{0.1A_{min}} -1}{10^{0.1A _{max}} -1}\right)}}{2\log\left(\displaystyle {\frac{\omega_s}{\omega_c}}\right)}.$$

((2.11))

The filter order must be an integer, and we therefore, but not always, selectN to the nearest highest integer.

If the transition bandω _s – ω _c is small, then the ratioω _s/ω _c is small and the required filter order becomes large. Furthermore, the order is affected, but to a lesser degree, by the required stopband attenuation,A _min, and the allowed ripple in the passband,A _max.

2.4.2 Poles and Zeros of Butterworth Filters

The poles and zeros can also be derived from Equation (2.7) as follows.

The magnitude function squared can be written as

$$|H(j\omega)|^2 = H(s)H(-s)|_{s = j\omega}.$$

((2.12))

Equation (2.7) yields

$$H(s)H(-s) = \frac{1}{1+\varepsilon^2(\frac{s}{j\omega_c })^{2\,N}}.$$

((2.13))

The denominator is

$$D(s)D(-s) = 1 + \varepsilon^2\left(\frac{s}{j\omega_c}\right)^{2\,N}.$$

The roots to the denominator are evenly spaced along a circle with the radius,r _p ₀, with an angle spacing π/N.

$${s_{pk} = j\omega_c \varepsilon^{{-1}/N} e^{j\pi (2\,k - 1)/2\,N}\ \ \ \ \ \textrm{for }k = 1,2, \ldots, 2\,N}$$

((2.14))

which can be written as

$${s_{pk} =\ \ \ r_{p0}\left(-\sin\left(\frac{\pi\left(2\,k-1\right)}{2\,N}\right) +\,j \ \ \, \cos\left(\frac{\pi\left(2\,k-1\right)}{2\,N}\right)\right)}$$

((2.15))

fork = 1, 2,...,2N where the pole radius,r _p ₀, is

$$\displaystyle r_{p0} = \omega_c \varepsilon^{-1/N}$$

((2.16))

and

$$\varepsilon = \sqrt{10^{0.1A_{max}} -1}.$$

((2.17))

The denominator can be factorized into two polynomialsD(s) andD(–s) belonging toH(s) andH(–s), respectively.D(s) is a Hurwitz ^{Footnote 4}polynomial andD(–s) is an anti-Hurwitz polynomial.

Definition 2.2

A Hurwitz polynomial has all zeros in the left half of the s-plane or on the j-axis whereas an anti-Hurwitz polynomial has its root in the right half of the s-plane or on the j-axis. A strictly Hurwitz polynomial has all roots strictly in the left half of the s-plane.

Definition 2.3

An even (odd) polynomial has only non-zero coefficients for the even (odd) power of s.

We allocate the roots in the left half of thes-plane toD(s) and those in the right half-plane toD(–s). The factorization of an even polynomial into a Hurwitz and an anti-Hurwitz polynomials, which is very difficult to do accurately if the roots are closely clustered, can be computed by either of the programs HURWITZ_POLY and HURWITZ_ROOTS.

The roots belonging toD(s), i.e., the poles of the Butterworth filter, are

$$\begin{aligned}s_{pk} &= r_{p0}\left(\cos\left(\frac{\pi\left(N+2\,K-1\right)}{2\,N}\right)\right.\\& \left. \hskip 12pt +j\sin\left(\frac{\pi\left(N+2\,k-1\right)}{2\,N}\right)\right)\end{aligned}$$

((2.18))

fork = 1, 2,...,N.

In tables, Butterworth filters are often denoted byPN, whereP stands for the German word for exponent, i.e., “potenz,” andN is the filter order (2 digits) [100]. Thus,P08, denotes an eight-order Butterworth filter. The poles of Butterworth filters, which are normalized with a passband edge = 1, are denormalized by multiplying withr _p ₀.

For each complex pole pair, we define itsQ factor. In Chapter 3, we will discuss theQ factor in more detail. Here it is sufficient to note that it is favorable from an implementation point of view to have as lowQ factors as possible.

Definition 2.4

The Q factor for a pole (pair) is defined as

$$Q = -\frac{|s_p|}{2Re \{s_p\}}.$$

((2.19))

The minimalQ factor is 0.5 and it occurs for a real pole and becomes infinite for a complex pole pair on thejω-axis.

For Butterworth filters, theQ factors are

$$Q = \frac{1}{2\cos\left(\frac{\pi(2\,k-1)}{2\,N}\right)}$$

((2.20))

and the largestQ factor for, e.g., 10th-order Butterworth filter, isQ = 3.19623, which is a relatively low value.

The finite zeros of a transfer function are the roots of the numerator. Hence, Butterworth filters have no finite zeros because the numerator is a constant. According to Theorem 1.1, the number of zeros is equal to the number of poles. Hence, Butterworth filters haveN transmission zeros ats = ∞ andN reflection zeros ats = 0.

Figure2.8 shows the poles for a fifth-order Butterworth filter withA _max = 1.25 dB and ω_c = 1 rad/s. A semicircle with the radius ω_c has been marked in thes-plane.

An odd-order lowpass filter has one real pole andm complex conjugate pole pairs, i.e.,N = 2m + 1 poles. Even-order filters have onlym complex conjugate poles, i.e.,N = 2m. The transfer function for a Butterworth filter has only poles and lacks finite zeros. According to Section 1.8.1, all five zeros lie therefore ats = ∞. Butterworth filters are therefore said to be of all-pole type.

The transfer function for a Butterworth filter can be written as

$${ H(s) = \begin{cases}\displaystyle\frac{G}{(s-\sigma_0)(s^2 - 2\sigma_1 s+r^2 _{p0})...(s^2 -2\sigma_m s+r^2_{p0})}&N = \textrm{odd}\\ \displaystyle\frac{G}{(s^2 - 2\sigma_1 s + r^2_{p0})...(s^2 - 2\sigma_m s+r^2 _{p0})}&N =\textrm{even} \end{cases}\\ } $$

((2.21))

whereσ ₀ = –r _p ₀.

Theorem 2.1

A second-order polynomial with positive real coefficients has its roots in the left half-plane, and if any coefficient is negative or zero, there is at least one root in the right half-plane or on the jω-axis.

Note thatσ _p < 0 for all poles, and all of the coefficients in the denominator are positive in a stable analog filter.

The gain constantG > 0 is, in the last steps of the design, chosen so that the appropriate gain is obtained in the passband. During the synthesis of the transfer function,G is for the sake of simplicity normalized so that the largest gain within the passband, |H(jω)|_max, equals one. For lowpass filters of Butterworth type, we selectG so that |H(0)| = 1, i.e.,A(0) = 0 dB.

2.4.3 Impulse and Step Response of Butterworth Filters

Figure2.9 shows corresponding impulse and step responses for a fifth-order Butterworth filter. The step response has a relatively small overshoot and the ringing decays relatively rapidly. Note the relation in Equation (1.30) between impulse and step responses. Thus, the step response is obtained by integration of the impulse response.

Example 2.1

Write a MATLAB program with the help of the included toolbox or Signal Processing Toolbox™ [75] that computes the required order, poles and zeros, and impulse and step response for a Butterworth filter that meets the specification shown in Fig.2.10 whereA _max = 0.28029 dB,A _min = 40 dB,ω _c = 40 krad/s, andω _s = 56 krad/s.

Validate the result by plotting the magnitude function, group delay, and poles and zeros in thes-plane and compare the result with Equation (2.11) and Equation (2.18).

Equation (2.11) yields

$$\begin{aligned}N\ge\displaystyle\frac{\log\left(\displaystyle\frac{10^{0.1A_{min}}-1}{10^{0.1A_{max}}-1}\right)}{2\log\left(\displaystyle\frac{\omega_s}{\omega_c}\right)} &=\displaystyle\frac{\log\left(\displaystyle\frac{10^4-1}{10^{0.028029}-1}\right)}{2\log\left(\displaystyle\frac{56}{40}\right)} &=\displaystyle\frac{5.176}{0.29226}=17.711.\end{aligned}$$

The computation above givesN = 17.711, but the filter order must be selected to a larger integer. Often, but not always, the closest higher integer is chosen. Here we choseN = 18, which meets a slightly higher requirement. Thus, there is a so-called design margin. How the design margin may be exploited will be discussed later. Note that an analog filter of orderN = 18 is very high.

The MATLAB function, which is a part of the accompanying toolbox,N = BW_ORDER(Wc, Ws, Amax, Amin)

is used to determine required filter order. Because the order must be chosen to an integer, we get a design marginal. The MATLAB function[G, Z, Zref, P, Wsnew] = BW_POLES(Wc, Ws, Amax, Amin, N)

is used to determine poles and zeros and the gain constant,G, so that |H(jω)|_max = 1. Z is an empty vector as Butterworth filters lack finite zeros. The function BW_POLES uses the whole design marginal to reduce the stopband edge, i.e., so that the attenuationA _min is achieved at a lower frequency thanω _s. We get% Synthesis of a lowpass Butter worth filter Wc = 40000; Ws = 56000; % Requirement for the lowpass filter Amax = 0.28029; Amin = 40; N = BW_ORDER(Wc, Ws, Amax, Amin); N = 18; % Re-run the program after selecting N = integer [G, Z, Zref, P, Wsnew] = BW_POLES(Wc, Ws, Amax, Amin, N); Q = -abs(P)./(2*real(P)); omega = linspace(0, 1e5, 1000); % 1000 values between 0 and 1e5 rad/s H = PZ_2_FREQ_S(G, Z, P, omega); % Compute the frequency response Att=MAG_2_ATT(H); % Compute the attenuation Tg = PZ_2_TG_S(G, Z, P, omega); % Compute the group delay PLOT_A_TG_S(Att, Tg, omega, 60, 6*10^-4);

Figure2.11 shows the resulting attenuation and the group delay.

The poles can, thus, be computed either according to Equation (2.18) or by using the above program and plotted with the following addition.xmax = 10000; xmin = -80000; ymax = 80000; PLOT_PZ_S(Z, P, Wc, Ws, xmin, xmax, ymax) P = 1.0 e+04 * Q = -0.3758591 + 4.2960893i 0.5019 -0.3758591 - 4.2960893i 0.5019 -1.1161570 + 4.1655548i 0.5176 -1.1161570 - 4.1655548i 0.5176 -1.8225411 + 3.9084520i 0.5517 -1.8225411 - 3.9084520i 0.5517 -2.4735482 + 3.5325929i 0.6104 -2.4735482 - 3.5325929i 0.6104 -3.0493977 + 3.0493978i 0.7071 -3.0493977 - 3.0493978i 0.7071 -3.5325929 + 2.4735482i 0.8717 -3.5325929 - 2.4735482i 0.8717 -3.9084520 + 1.8225411i 1.1831 -3.9084520 - 1.8225411i 1.1831 -4.1655548 + 1.1161570i 1.9319 -4.1655548 - 1.1161570i 1.9319 -4.2960893 + 0.3758591i 5.7369 -4.2960893 - 0.3758591i 5.7369 G = 2.6614803 e+83

All zeros of a Butterworth filter lie ats = ∞ while the poles lie on a circle with the radiusr _p ₀, which is shown in Fig.2.12 where two semicircles with the radiiω _c andω _s have been marked. The complex conjugate poles can be combined into a second-order equation according to

$$\begin{aligned}(s-s_p)(s-s^*_p) &= s^2 -2Re(s_p)s+|s_p|^2 \\&= s^2 - 2\sigma_p s + r^2 _p \ .\end{aligned}$$

The transfer function is

$$\begin{aligned}H(s) =&\left(\frac{1.8597653\times10^9}{s^2+85921.785s+1.8597653\times10^9}\right)\\&.\left(\frac{1.8597653\times10^9}{s^2+83311.095s+1.8597653\times10^9}\right)\\&\ .\left(\frac{1.8597653\times10^9}{s^2+78169.04s+1.8597653\times10^9}\right)\\&.\left(\frac{1.8597653\times10^9}{s^2+70651.858s+1.8597653\times10^9}\right)\\&\ .\left(\frac{1.8597653\times10^9}{s^2+60987.955s+1.8597653\times10^9}\right)\\&.\left(\frac{1.8597653\times10^9}{s^2+49470.963s+1.8597653\times10^9}\right).\end{aligned}$$

$$\begin{aligned}\quad \quad \quad \ & .\left(\frac{1.8597653\times10^9}{s^2+36450.822s+1.8597653\times10^9}\right)\\&\ \ \ .\left(\frac{1.8597653\times10^9}{s^2+22323.141s+1.8597653\times10^9}\right)\\&\ \ \ \ \ .\left(\frac{1.8597653\times10^9}{s^2+7517.1822s+1.8597653\times10^9}\right).\end{aligned}$$

We use the following program for computing the impulse and step responses:Wc = 40000; Ws = 56000; % Requirement for the lowpass filter Amax = 0.28029; Amin = 40; tmax = 10^-3; N = 18; [G, Z, Zref, P, Wsnew] = BW_POLES(Wc, Ws, Amax, Amin, N); t_axis = [0:0.01*tmax:tmax]; [h, dirac0, t_axis] = PZ_2_IMPULSE_RESPONSE_S(G, Z, P, t_axis); [s_of_t, t_axis] = PZ_2_STEP_RESPONSE_S(G, Z, P, t_axis); h_scale = 5*10^-5; ymin = -0.4; ymax = 1.3; PLOT_h_s_S(h, h_scale, s_of_t, t_axis, tmax, ymin, ymax)

Figure2.13 shows the corresponding impulse and step response, where the impulse response has been multiplied with a scaling factor in order to fit into the same diagram.

The area under the impulse response equals the final value of the step response, i.e.,$\lim\limits_{t\to\infty} s(t)=H(0)=1$.

Hence, if the bandwidth of the filter is increased, the length of the impulse response becomes shorter and its amplitude becomes larger. Note that the impulse response has a relatively long ringing, which becomes larger and longer for higher-order Butterworth filers. The delay of the step response is almost 0.3 ms.

Note also the long ringing and the larger overshoot in the step response and compare this to the impulse response. The step response can be obtained from the impulse response through integration. The step response approaches 1, because we have normalized the Butterworth filter toH(0) = 1, i.e.,A(0) = 0 dB.

2.4.4 Chebyshev I Filters

A Butterworth filter does not use the allowed passband tolerance efficiently. By allowing the magnitude function to vary within the acceptable passband bounds, a smaller transition band than for a Butterworth filter of the same order is obtained. For a Chebyshev I filter, the magnitude function varies between the two tolerance bounds, that is, the filter has equiripple variation, i.e., the error oscillates with equal peaks across the magnitude response in the passband. An equiripple error is optimal in the Chebyshev sense.

Figure2.14 shows the attenuation for Chebyshev I filters of different orders and Fig.2.15 shows the corresponding group delays. All filters haveA _max = 1.25 dB andω _c = 1 rad/s.

Note that a direct comparison between different filter approximations of the same order, which often, unfortunately, is done in the literature, is not correct as the filters meet different magnitude specifications.

The magnitude function squared for a Chebyshev I filter can be written as

$$|H(j \omega)|^2 = \frac{1}{1+\varepsilon^2 T^2_N \left(\frac{\omega}{\omega_c}\right)}$$

((2.22))

whereT _N is anNth-order Chebyshev polynomial^{Footnote 5} of the first kind, which is defined as

$$T_N (x) = \begin{cases}\cos[N{\rm acos}(x)] &|x| \leq 1 \\ \rm cosh [N{\rm acosh} (x)]& |x| \; >\, 1. \end{cases} 2.23$$

((2.23))

Chebyshev polynomials are easily generated by using the recursion

$$T_{N+1}(x) = 2xT_N(x) - T_{N-1}(x)$$

((2.24))

with$T_0(x) = 1$ and$T_1(x) = x$. The Chebyshev polynomials oscillate between –1 and 1 for values ofx between –1 and 1. Hence, the squared magnitude response oscillates in the passband between a maximum value of 1 and a minimum value of$1/(1+\varepsilon^2)$. Hence,

$$\varepsilon = \sqrt{10^{0.1A_{max}}-1}$$

((2.25))

The use of Chebyshev polynomials was proposed by W. Cauer in 1931. Figure2.16 shows the attenuation in the passband for Chebyshev I filters for different orders. A Chebyshev I filter has equiripple in the passband and monotonically decreasing magnitude in the stopband.

The attenuation is obtained from Equations (2.22) and (2.23)

$$A(\omega) = 10\log\left[1\ +\ \left[\varepsilon\,\,\cosh\left(\ \ \ \ \ \ \ \ \ N{\rm a}\cosh\left(\frac{\omega}{\omega_c}\right)\right)\right]^2\right].$$

((2.26))

The attenuation at the passband edge, i.e.,ω = ω _{c, isA} _max. Note that an even order filter has the attenuationA _max atω = 0.

The filter order can be determined from Fig.2.16 as the order is equal to the sum of the number of minimum and maximum in the passband. Actually, the detailed variation in the passband is of little interest because we are only requiring the variation to be within the tolerance bounds. Furthermore, to obtain filters that are less sensitive to errors in the component values, we will require that the allowed passband variation is small, i.e.,A _max is small. This issue will be further discussed in Chapter 3.

Note that the transition band is smaller and the group delay is larger and varies more within the passband for higher order filters. The group delay also varies more than for a Butterworth filter of the same order. However, we should avoid comparing filters of the same order because they do not perform the same amount of filtering.

The required filter order for a Chebyshev I filter, which can be determined with Equation (2.21) [29, 135], is

$$N \geq \frac{{\rm a\\cosh}\left(\sqrt{\displaystyle\frac{10^{0.1 A_{\textit{min}}-1}}{10^{0.1A_{\textit{max}}}-1}}\right)}{{\rm a\\cosh}\left(\displaystyle\frac{\omega_s}{\omega_c}\right)}$$

((2.27))

or alternatively we may use the functionN = CH_ORDER(Wc, Ws, Amax, Amin).

2.4.5 Poles and Zeros of Chebyshev I Filters

The transfer function is an all-pole function, i.e., all zeros are at infinity. The poles are obtained by solving the denominator of Equation(2.22) withs = jω

$$1 + \left[\varepsilon \cos \left(N{\rm a\\cos} \left(\frac{s}{\;j\omega_c}\right) \right)\right]^2 = 0.$$

((2.28))

We get

$$\cos\left[N{\rm a\\cos} \left(\frac{s}{\;j\omega_c}\right)\right]=\pm \frac{j}{\upvarepsilon}.$$

((2.29))

By letting

$$N {\rm a}\ \cos \left(\frac{s}{\;j\omega_c}\right)=x+jy$$

((2.30))

we get

$$\begin{aligned}\cos (x+{\textit{jy}})&=\cos(x)\cos({\textit{jy}})-\sin(x)\sin(\textit{jy})=\\&=\cos(x)\cosh(y)-j \sin(x)\sinh(y) =\pm\frac{j}{\upvarepsilon}\end{aligned}$$

which yields the two equations

$$\cos(x)\cosh(y) = 0$$

and

$$-\sin(x) {\rm \sinh}(y)= \pm \frac{1}{\upvarepsilon}.$$

Now, because${\rm cosh}(y)>0,$ the first equation yields

$$\begin{aligned}\cos(x)=& \;0\Rightarrow x = (2\,k-1)\frac{\uppi}{2} \\&{\rm for} \ k = 0, \pm1, \pm2,\ldots\end{aligned}$$

((2.31))

The second equation can now be solved.

$$y=\pm{\;\rm a\\sinh} \left(\frac{1}{\varepsilon}\right).$$

((2.32))

Lets _pk denote the poles. Equations (2.29) – (2.32) yield

$$N{\rm a}\ \cos\left(\,\,\frac{s_{pk}}{j\omega_c}\right)= ({2\,k}-1)\frac{\pi}{2}\pm j\,{\rm a\\sinh} \left(\frac{1}{\varepsilon}\right)$$

and

$$\frac{s_{pk}}{j\omega_c}=\frac{\sigma_{pk}+j\omega_{pk}}{j\omega_c}=\cos\left[\frac{\pi}{2\,N}(2\,k-1)\pm \frac{j}{N}{\rm{a}\\sinh}\left(\frac{1}{\varepsilon}\right)\right].$$

The poles for anNth-order Chebyshev I filter are [5, 135]

$$\begin{aligned}s_{pk}=&\rm -\omega_c\left( {\rm{a} sin} \left(\frac{\pi (2\,k-1)}{2\,N}\right)\right.\\ &\left.+j b\cos\left(\frac{\pi (2\,k-1)}{2\,N}\right)\right)\end{aligned}$$

((2.33))

fork = 1, 2,…,N where^{Footnote 6}

$${a = \sinh \left(\frac{1}{N}\rm{a\\sinh}\left(\frac{1}{\varepsilon}\right)\right)\quad b= \cosh\left(\frac{1}{N}\rm{a\\sinh}\left(\frac{1}{\varepsilon}\right)\right)}$$

((2.34))

$$\varepsilon = \sqrt{10^{0.1A_{\textit{max}}}-1}.$$

((2.35))

All zeros lie ats = ∞. The poles to a Chebyshev I filter can also be computed with the function[G, Z, P] = CH_I_POLES(Wc, Ws, Amax, Amin, N).

Z is an empty vector, as Chebyshev I filters lack finite zeros.P is a column vector with poles andG is the gain constant. Here and in corresponding programs,G is chosen for simplicity so that |H(jω)|_max = 1. After the filter has been synthesized,G can be multiplied with a suitable factor to obtain appropriate passband gain.

The poles positions are shown in Fig.2.17 for a fifth-order Chebyshev I filter withA _max = 1.25 dB and ω_c = 1 rad/s. A semicircle with the radius ω_c has been marked in the figure. Note that here we have chosenA _max = 1.25 dB, which is a relatively large value.

We will show in Chapter 3 that anLC filter that has a large passband ripple will have high sensitivity for errors in the element values. Hence, we may often design a filter with lower passband ripple than required by the application in order to be able to use less precise components. Moreover, both theQ factors for the poles and the group delay is reduced for Chebyshev I filters if the design margin is used to reduceA _max.

In tables, Chebyshev I filters are usually represented withTNρ, whereT stands for Chebyshev (Tshebycheff, according to German transcription),N is the filter order, andρ is the reflection coefficient in the passband.

The poles of Chebyshev I filters, which are normalized with a passband edge = 1, are denormalized by multiplying withω _c.

Definition 2.5

The reflection coefficient, ρ, which is given in %, is related to the ripple in the passband according to

$$A_{max} = -10 \log (1 - \rho^2).$$

((2.36))

Thus, the filterT0815 represents an eighth-order Chebyshev I filter withA _max = 0.09883 dB. Filter tables are often made with respect to the reflection coefficient,ρ, instead of with respect to A _max.

The transfer function for a Chebyshev I filter is

$$ H(s) =\begin{cases}\displaystyle\frac{G}{(s - \sigma_0)(s^2 - 2\sigma_1 s + r^2_{p1}) \cdots (s^2 - 2 \sigma_m s + r^2_{pm})} & N = \textrm{odd}\\ \displaystyle\frac{G}{(s^2 - 2\sigma_1 s + r^2_{p1})\cdots (s^2 - 2\sigma_m s + r^2_{pm})} & N= \textrm{even}\end{cases} 2.37$$

((2.37))

where$r_{pk} = \sqrt{\sigma_{pk}^2 + \omega_{pk}^2}$. The transfer function has only poles and lacks (finite) zeros, i.e., the filter is of all-pole type and all zeros are ats = ∞.

2.4.6 Reflection Zeros of Chebyshev I Filters

It is also of interest to determine points in thes-plane where the attenuation is minimum. These points are referred to as reflection zeros. For Butterworth filters, allN reflection zeros are ats = 0, and for Chebyshev I filters, we get the zeros by finding the minima of Equation (2.26). Hence,

$$\cos \left(N\textrm{acos}\left(\frac{s}{\;j\omega_c}\right)\right) = 0$$

which yields theN reflection zeros for Chebyshev I filters

$$s_{rz} = i \omega_c \cos \left(\frac{\pi(2\,k - 1)}{2\,N}\right)$$

((2.38))

which all lie in the passband and on thejω-axis. As can be seen in Fig.2.16, there is a reflection zero ats = 0 for filters of odd order.

2.4.7 Impulse and Step Response of Chebyshev I Filters

Figure2.18 shows the impulse and step responses. Both the impulse response and step response have a larger and longer ringing than in the Butterworth filter of the same order. However, we shall refrain from comparing different filter types with the same order, as they do not perform the same amount of filtering.

Note, however, that the Chebyshev I filter meets a stricter requirement. Thus, a comparison of different filter approximations of the same order is incorrect. Instead, different filter approximations should be compared when they meet the same specification of the attenuation.

The rise time of the step response is proportional to the width of the first half-period of the impulse response. The rise time and delay of the step response are also larger than for the Butterworth filter of the same order.

Example 2.2

Write a MATLAB program that determines necessary order, poles and zeros, and impulse and step responses for a Chebyshev I filter that meets the specification shown in Fig.2.10 whereA _max = 0.28029 dB,A _min = 40 dB,ω _c = 40 krad/s, andω _s = 56 krad/s, i.e., the same specification as in Example 2.1. Verify the result by plotting the attenuation, the group delay, and poles and zeros in thes-plane and verify the result with Equations (2.21) and (2.22).

Equation (2.21) yields

$$\begin{aligned}N \geq \frac{\textrm{acosh}\left(\sqrt{\displaystyle\frac{10^4 - 1}{10^{0.028029}-1}}\right)}{\textrm{acosh}\left(\displaystyle\frac{56}{40}\right)} &= \frac{\textrm{acosh} (387.277)}{\textrm{acosh}(1.4)} \\&= 7.6726.\end{aligned}$$

We modify the program in Example 2.1 according to the following:% Synthesis of lowpass Chebyshev I filter Wc = 40000; Ws = 56000; Amax = 0.28029; Amin = 40; N = CH_ORDER(Wc, Ws, Amax, Amin) N = 8; % Re-run the program after selecting an integer order [G, Z, Zref, P, Wsnew] = CH_I_POLES(Wc, Ws, Amax, Amin, N); Q = -abs(P)./(2*real(P)) omega = linspace(0, 1e5, 1000); H = PZ_2_FREQ_S(G, Z, P, omega); Att = MAG_2_ATT(H); Tg = PZ_2_TG_S(G, Z, P, omega); lw = 2; fs = 16; fn = 'times '; PLOT_A_TG_S(Att, Tg, omega, 80, 8*10^-4); set(gca, 'FontName ', fn, 'FontSize ', fs); xtick([0:10000:100000])which yieldsN = 8 P = 1.0 e+04 * Q = - 0.2035170 + 4.0543643i 0.6366 - 0.2035170 - 4.0543643i 0.6366 - 0.5795674 + 3.4371241i 1.4151 - 0.5795674 - 3.4371241i 1.4151 - 0.8673839 + 2.2966129i 3.0071 - 0.8673839 - 2.2966129i 3.0071 - 1.0231491 + 0.8064632i 9.9733 - 1.0231491 - 0.8064632i 9.9733 G = 1.9829574 e+35.

The poles and zeros for the eight-order Chebyshev I filter is shown in Fig.2.19. Two semicircles with the radiiω _c andω _s have been marked in the figure. Note that the poles lie closer to thejω-axis and that theQ factors are higher compared with the corresponding Butterworth filter.

The transfer function is

$$\begin{aligned}H(s) = \frac{1.6479289 \cdot 10^9}{(s^2 + 4070.3401s + 1.6479289 \times 10^9)}\\ \frac{1.2149721 \cdot 10^9}{(s^2 + 11591.348s + 1.2149721 \times 10^9)}\\ \frac{0.60267857 \cdot 10^9}{(s^2 + 17347.678s + 0.60267857 \times 10^9)}\\ \frac{0.16972169 \cdot 10^9}{(s^2 + 20462.981s + 0.16972169 \times 10^9)}\end{aligned}$$

The constantG has been chosen so |H(jω)|_max = 1 in the passband and has been divided into four 0factors. These factors should be selected so that the internal signal levels in the corresponding realization are optimized. We will discuss this issue later. This optimization is referred to as scaling the signal levels in the filter realization.The attenuation and the group delay are shown in Fig.2.20.

Note that the difference in the group delay of the filter shown in Fig.2.20 and the filters in Fig.2.14 is mainly caused by the difference inA _max. The group delay is largest just above the passband edge and it varies somewhat within the passband, whereas for Butterworth filters the group delay increases monotonically in the passband.

By modifying the program for computing the impulse and step responses for Butterworth filters, we can compute the impulse and step responses for Chebyshev I filters. Figure2.21 shows the impulse and step responses. The step response can also be computed by integration of the impulse response. Note the long ringing in the impulse and step responses.

The step response approaches |H(0)| = 10^–0.05Amax for a normalized Chebyshev I filter of even order. Here we have |H(0)| = 0.968246. However, for filters of odd order, the step response approaches |H(0)| = 1.

2.4.8 Chebyshev II Filters

The attenuation for Butterworth and Chebyshev I filters of LP type approaches infinity for high frequencies. Thus, the filters have a much larger attenuation in the stopband than necessary. In most practical applications, we only require that the attenuation in the stopband is sufficiently large; for example, at least 60 dB attenuation in the whole stopband. This extra attenuation comes at a higher cost.

A filter approximation that is similar to a Butterworth filter in the passband, but has equiripple stopband attenuation, is theChebyshev II filter, which is also called inverse Chebyshev filter. Chebyshev II filters do not provide more attenuation than necessary in the stopband.

Figures2.22 and2.23 show the attenuation and the group delay for Chebyshev II filters of different orders.

The attenuation for Chebyshev II filters is monotonically increasing in the passband and has equiripple stopband attenuation. The Chebyshev II filter has finite zeros, which are situated on thejω-axis. In the passband, the filter resembles a Butterworth filter. Due to the finite zeros, the filter gets a smaller transition band than a Butterworth filter of the same order, but the same transition band as for a Chebyshev I filter of the same order. Figure2.24 shows the corresponding attenuations in the passband for Chebyshev II filters of different order. All filters haveA _max = 1.25 dB.

The magnitude function squared for a Chebyshev II filter is

$$|H(\;j\omega)|^2 = \frac{1}{1 + \varepsilon^2 \left(\displaystyle\frac{T^2_N\left(\frac{\omega_s}{\omega_c}\right)}{T^2_N\left(\displaystyle\frac{\omega_s}{\omega}\right)}\right)}$$

((2.39))

whereT _N is anNth-order Chebyshev polynomial, which is either an even or odd function ofω.

The required order for Chebyshev II filters can be determined either by using the function CH_ORDER, which is the same for Chebyshev I filters, or from Equation (2.21).

The pole-zero configuration for a fifth-order Chebyshev II filter withA _max = 1.25 dB,A _min = 40 dB, ω_c = 1 rad/s, andω _s = 1.4 rad/s is shown in Fig.2.25. Circles with radiiω _c andω _s are marked in the figure. The filter has two finite zero pairs; one pair is close to the stopband edge, the other pair is further inside the stopband, and finally, a zero ats = ∞. Note the large difference between the pole positions of Chebyshev I and Chebyshev II filters. Transfer functions with poles close to thejω-axis are more difficult to realize than if the poles lay far from thejω-axis. A simple measure of how difficult it is to realize a transfer function isQ factors for the poles.

The transfer function has finite zeros. For odd-order filters, there is a zero ats = ∞, but for even-order filters, the magnitude function approachesA _min. For Chebyshev II lowpass filters, we can chooseG so that |H(0)| = 1.

The transfer function for a Chebyshev II filter can be written as

$$ H(s)=\begin{cases}\displaystyle\frac{G(s^2 + r^2_{z1}) \cdots (s^2 + r^2_{zm})}{(s - \sigma_0)(s^2 - 2\sigma_1s + r^2_{p1})\cdots (s^2 - 2\sigma_m s + r^2_{pm})} & N = \textrm{odd}\\ \displaystyle\frac{G(s^2 + r^2_{z1}) \cdots (s^2 + r^2_{zm})}{(s^2 - 2\sigma_1 s + r^2_{p1}) \cdots (s^2 - 2\sigma_m s + r^2_{pm})} & N = \textrm{even}.\end{cases} 2.40$$

((2.40))

2.4.9 Poles and Zeros of Chebyshev II Filters

The squared magnitude response as given by Equation (2.39) can be rewritten as

$$\begin{aligned}|H(j\omega)|^2 &= H(s)H(-s) \\&= \frac{T^2_N \left(j\displaystyle\frac{\omega_s}{s}\right)}{T^2_N\left(j\displaystyle\frac{\omega_s}{s}\right) + \varepsilon^2 T^2_N\left(\displaystyle\frac{\omega_s}{\omega_c}\right)} \quad \textrm{for } s = j\omega.\end{aligned}\\$$

((2.41))

The transmission zeros are obtained when the numerator is zero, i.e.,

$$T^2_N\left(j\frac{\omega_s}{s_z}\right) = \cos^2 \left[N\textrm{acos}\left(j\frac{\omega_s}{s_z}\right)\right] = 0.$$

All of the transmission zeros lie on the imaginary axis, and at these frequencies the attenuation is infinite. Associating half of them toH(s) and the other half toH(–s) yields

$$s_{zk} = j\frac{\omega_s}{\cos\left[\frac{\pi}{2\,N}(2\,k - 1)\right]} \quad \textrm{for}\ k = 1,2, \cdots, N.$$

((2.42))

ForN = odd, one zero is obtained at infinity.

The poles are obtained by solving for the roots of the denominator of Equation (2.41)

$$T^2_N\left(j\frac{\omega_s}{s_{pk}}\right) + \varepsilon^2T^2_N\left(\frac{\omega_s}{\omega_c}\right) = 0$$

or

$$\cos^2\left[N\textrm{acos}\left(j\frac{\omega_s}{s_{pk}}\right)\right] + \varepsilon^2 \textrm{cosh}^2\left[N\textrm{acosh}\left(\frac{\omega_s}{\omega_c}\right)\right] = 0.\\$$

((2.43))

Letting$K = \varepsilon \ \textrm{cosh}\left[N\textrm{acosh}\left(\displaystyle\frac{\omega_s}{\omega_c}\right)\right]$ yields

$$\quad\quad\quad\cos^2\left[N\textrm{acos}\left(j\frac{\omega_s}{s_{pk}}\right)\right] = -K^2.$$

In the same way as was done for Chebyshev I filters, we get

$$\frac{j\omega_s}{s_{pk}} = \cos\left[\frac{\pi}{2\,N}(2\,k - 1) \pm \frac{j}{N} \textrm{asinh} (K)\right]$$

and by using a polar representation, i.e.,$s_{pk} = r_{pk}e^{\;j\phi_{k}}$, we get

$$\begin{cases}\displaystyle\frac{\omega_S}{r_{pk}}\sin(\phi_k) = \cos \left[\frac{\pi}{2\,N}(2\,k -1)\right] \textrm{cosh}[N\textrm{asinh}(K)]\\ \displaystyle\frac{\omega_s}{r_{pk}}\cos (\phi_k) = \pm \sin\left[\frac{\pi}{2\,N}(2\,k -1)\right]\textrm{sinh}[N\textrm{asinh}(K)].\end{cases}$$

Selecting the poles in the left-hands-plane yields

$$\begin{cases}r_{pk} = \displaystyle\frac{\omega_s}{\sqrt{\textrm{cos}^2\left[\displaystyle\frac{\pi}{2\,N}(2\,k -1)\right] + \textrm{sin}h^2\left[\displaystyle\frac{1}{N}\textrm{asinh} (k)\right]}}\\ \phi_k = \textrm{atan} \left\{-\cot\left[\displaystyle\frac{\pi}{2\,N}(2\,k -1)\right] \coth \left[\displaystyle\frac{1}{N}\textrm{asinh}(K)\right]\right\}+ \pi\\ \textrm{for} \ k = 1 \cdots N\end{cases}$$

((2.44))

The transfer function can now be written in the form

$$H(s) = \frac{G}{D_0(s)}\prod\limits_{k =1}^{n} \frac{s^2 +\omega^2_{zk}}{(s^2 - 2\sigma_k s + r^2_{pk})}$$

((2.45))

whereω _zk = |s _zk|, σ _k = r _pk cos(φ _k),n = 2 floor (N/2), and

$$D_0(s) = \begin{cases}s - \sigma_{n +1} \quad &\textrm{for } N \textrm{ odd}\\ 1 \quad &\textrm{for } N \textrm{ even}.\end{cases} 2.46$$

((2.46))

Poles and zeros can be determined with the function CH_II_POLES. The reflection zeros for Chebyshev II filters are ats = 0.

2.4.10 Impulse and Step Response of Chebyshev II Filters

Figure2.26 shows the impulse and step responses. Note that the impulse and step responses for Butterworth and Chebyshev II filters are similar and that the size of the ringing and its duration in the step response is small.

For Chebyshev II filters of even order, the impulse response has an impulse,δ(t) att = 0, and the corresponding step response has a small initial step. The step response approaches 1 for high frequencies for Chebyshev II filters becauseH(0) = 1 for both odd and even filter orders.

Example 2.3

Determine poles and zeros for a Chebyshev II filter that meets the same attenuation requirement as the filter in Example 2.1.

We know that an eighth-order Chebyshev I filter meets the requirement and, hence, an eighth-order Chebyshev II filter meets the same requirement, i.e.,A _max = 0.28029 dB,A _min = 40 dB,ω _c = 40 krad/s, andω _s = 56 krad/s.

The poles and zeros for the Chebyshev II filters can be computed by modifying the program for computing the poles and zeros for Chebyshev I filters. We therefore modify the program in Example 2.2 and instead call the function CH_II_POLES.

The above, modified program yieldsN = 8 Z = 1.0 e+05 * 0 + 0.5709710i 0 - 0.5709710i 0 + 0.6735063i 0 - 0.6735063i 0 + 1.0079734i 0 - 1.0079734i 0 + 2.8704653i 0 - 2.8704653i P = 1.0 e+04 * Q = -0.5295096 + 4.5906227i 0.5265 -0.5295096 - 4.5906227i 0.5265 -1.8485257 + 4.7708137i 0.7463 -1.8485257 - 4.7708137i 0.7463 -4.0650817 + 4.6840490i 1.3380 -4.0650817 - 4.6840490i 1.3380 -7.1772824 + 2.4619622i 4.1989 -7.1772824 - 2.4619622i 4.1989 G = 0.0100000.

The poles and zeros of the Chebyshev II filter are shown in Fig.2.27. Note that theQ factors are lower than for the corresponding Chebyshev I filter. ForN = even, all zeros are finite, and forN = odd, there are (N–1) finite zeros and one zero ats = ∞. Two semicircles with the radiiω _c andω _s have been marked in the figure.

The poles of Chebyshev II filters, which in tables often are normalized with a stopband edge = 1, are denormalized by multiplying withω _c3, i.e., the 3-dB passband frequency.

The attenuation and the group delay are shown in Fig.2.28. The constantG has been chosen so |H(0)| = 1.

The transfer function for the Chebyshev II filter is

$$\begin{aligned}H(s)&=\frac{0.06987403 (s^2 +82.395710 \times 10^9)} {(s^2 +143545.65s+5.7574641\times 10^9)} \frac{0.37858268(s^2 + 10.160103\times 10^9)}{(s^2 + 81301.635s + 3.8465204\times10^9)}\\&\cdot\frac{0.57708826(s^2+4.5361070\times10^9)}{(s^2+36970.513s+2.6177710\times10^9)}\frac{0.65506019(s^2+3.2600794\times 10^9)}{(s^2+10590.192s+2.1354197\times 10^9)}.\end{aligned}$$

The impulse response and step response for the Chebyshev II filter can be computed by modifying the previously discussed program for computing the impulse and step responses for Chebyshev I filters. Figure2.29 shows the corresponding impulse and step responses.

The impulse response, in this case, has an impulse,δ(t) att = 0, because the filter order is even. The impulse is thus so small that it does not show up in the figure. Note that the delay of the step response is smaller than for the two earlier filter approximations, which agrees with the fact that the group delay is smaller.

In order to realize a Chebyshev II filter of even order with a passiveLC filter, the poles and zeros positions must be modified. A necessary requirement for anLC realizable lowpass filter is that there is at least one zero ats = ∞. This modification, which also is necessary for Cauer filters of even order, will be discussed later.

2.4.11 Cauer Filters

Cauer filters, also known as elliptic filters or Zolotarev filters, have equiripple in both the passband and stopband and therefore exploit the acceptable tolerances in the filter specification optimally. This means that a Cauer filter meets a standard magnitude specification with lower filter order than any other filter approximation. The main contribution to the development of Cauer, Chebyshev I, and Chebyshev II filters was made by the German scientist Wilhelm Cauer (1933).

The characteristic function for anNth-order Cauer filter involves the Chebyshev rational function$R_N(x, L)$.R _N is a complicated function and its details are beyond the scope of this book, and the interested reader is referred to [5, 29, 96]. The Chebyshev rational functionR _N has the following properties

R _N is odd (even) forN odd (even).
TheN zeros lie in the interval –1 <x < 1 and theN poles lie outside this interval.
R _N oscillates between ±1 in the interval –1 <x < 1 andR _N(1,L) = 1.
1/R _N oscillates between ±1/L in the interval |x| >x _L = ω _s/ω _c where

$$ {L = \sqrt{(10^{0.1A_\textit{min}}-1) /(10^{0.1A_\textit{max}}-1)}}.$$

((2.47))

The characteristic function for anNth-order lowpass Cauer filter is

$$ {\left|C_N(j\omega)\right|^2 = R_N^2 \left(\frac{\omega}{\omega_c}, L\right)}$$

((2.48))

where

$$ {R_N(s, L) = \begin{cases} c_1 s \prod \limits_{i=1}^{N/2} \frac{s^2 + (x_L / x_i)^2}{s^2 +x^2_i} & N \, \textrm{odd} \\ c_2 \prod\limits_{i=1}^{(N - 1)/2} \frac {s^2 + (x_L / x_i)^2}{s^2 + x_i^2} & N \, \textrm{even}.\end{cases} 2.49}$$

((2.49))

The normalizing constantsc ₁ andc ₂ are determined fromR _N(1,L) = 1. Obviously, forx = x _L/x _i we have a reflection zero,ω _ri, becauseR _N(x _L/x _i,L) = 0, i.e., |H(jω)| = 1, and a transmission zero forx = x _i = ω _zi. Note that the reflection and transmission zeros are related according to

$$\omega_{ri}=\frac{\omega_s}{\omega_c} \frac{1}{\omega_{zi}}.$$

((2.50))

The expression for the transmission zeros,ω _si, involves the elliptic sine function, which is the reason why the name elliptic filter sometimes is used.

Figures2.30 and2.31 show the attenuation and the group delay for Cauer filters of different order. Figure2.32 shows the passband for corresponding attenuations for Cauer filters of different order.

The order of a Cauer filter can be determined from the passband response as the sum of the number of maxima and minima in the passband. This is also true for Chebyshev I filters. Filters of even order have the attenuationA _max atω = 0 while odd-order filters haveA _max = 0. Note, however, we are not really interested in the detailed behavior in the passband. In fact, we are only interested in that the requirement is met.

The required filter order for a Cauer filter can be determined with the functionCA_ORDER(Wc, Ws, Amax, Amin).

In tables, Cauer filters are usually represented byCNρθ, whereC stands for Cauer-Chebyshev (the prefixCC is also used),N is the filter order,ρ is the reflection coefficient (%), see Equation (2.36), andθ is the modular angle (degrees). The three quantities are given with two digits. Cauer filters, which in tables are normalized with a passband edge of 1, are denormalized by multiplying the poles and zeros withω _c.

Definition 2.6

The modular angle is defined as

$$\theta=\arcsin \left(\frac{\omega_c}{\omega_s}\right).$$

((2.51))

2.4.12 Poles and Zeros of Cauer Filters

The poles and zeros are complicated to derive. An algorithm for computing the poles and zeros is given in [5]. The transfer function has finite zeros. Filters of odd order have a zero ats = ∞, but for filters of even order, the magnitude function approaches the stopband attenuation,A _min. The gain constantG is chosen in the programs so that |H(jω)|_max = 1.

The poles and zeros of Cauer filters, which are normalized with respect toω _c, can be determined with the function[G, Z, R_ZEROS, P, Wsnew] = CA_POLES(Wc, Ws, Amax, Amin, N).

The transfer function for a Cauer filter can be written

$$H(s)= \begin{cases}\displaystyle\frac{G(s^2 + r^2_{z1})\ldots(s^2 + r^2_{zm})} {(s-\sigma_0) (s^2 - 2\sigma_1 s+r^2_{p1})\ldots (s^2 - 2\sigma_m s +r^2_{pm})} \quad N =\textrm{odd} \\ \displaystyle\frac{G(s^2 + r^2_{z1})\ldots(s^2 + r^2_{zm})} {(s^2 -2\sigma_1 s + r^2_{p1})\ldots(s^2 - 2\sigma_m s +r^2_{pm})} \quad \qquad N =\textrm{even}. \end{cases}$$

((2.52))

Thus,C042056, denotes a Cauer filter withN = 4,A _max = 0.1772877 dB, andω _s/ω _c = 1.2062 = sin(56°).

The poles and zeros for a fifth-order Cauer filter withA _max = 1.25 dB,A _min = 40 dB,ω _c = 1 rad/s, andω _s = 1.205 rad/s is shown in Fig.2.33. Note that one of the pole pairs lies close to thejω-axis and that the lower finite zero pair lies close to the stopband edge.

2.4.13 Impulse and Step Response of Cauer Filters

Figure2.34 shows the impulse and step response for the Cauer filterC055056. The impulse response contains a small impulse fort = 0 for Cauer filters of even order. The step response approaches asymptotically 1 and |H(0)| = 10^–0.05A _max for normalized odd-order and even-order Cauer filters, respectively. The impulse response has larger ringing than any of the previous filters, but note that they do not meet the same requirements on the magnitude function. Hence, we should not compare these filters. A proper comparison will be done in Section 2.4.14.

Example 2.4

Determine the poles and zeros for a Cauer filter that meets the same attenuation requirement as the filter in Example 2.1.

We modify the program by instead callingCA_ORDER andCA_POLES for computing of poles and zeros. We getN = 5 Z = 1.0 e+04* 0 - 7.9217042i 0 + 7.9217042i 0 - 5.4610294i 0 + 5.4610294i P = 1.0 e+04* Q = -1.2952788 - 3.0512045i 1.2795 -1.2952788 + 3.0512045i 1.2795 -0.3208531 - 4.1105968i 6.4249 -0.3208531 + 4.1105968i 6.4249 -2.1649281 0.5000 G = 2.1607653 e+03.

The locations of the poles and zeros for the Cauer filter are shown in Fig.2.35.

ForN = even, all zeros are finite, and forN = odd, there is a zero ats = ∞. Two semicircles with radiiω _c andω _s are also shown in the figure.

Note that is a zero pair,s _z _3,4 = ± j 54610.294, in the transition band, i.e., inside the utter semicircle because 54,610.294 < 56,000. This is because the design margin has been used to reduce the stopband edge.

The transfer function can be written as

$$\begin{aligned}H(s)&=\frac{21607653}{(s + 2164.9281)} \\ &\cdot \frac{(s^2 + 62753397)}{(s^2 + 2590.5577s+1.0987596 \times 10^9)} \\ &\cdot\frac{(s^2 + 29822842)}{(s^2 + 6417.0618s + 1.6999953 \times 10^9)}.\end{aligned}$$

((2.53))

The two finite zero pairs can be combined with either of the two second-order sections to optimize the signal levels inside the filter realization, see Section 7.3.

The attenuation and the group delay are shown in Fig.2.36. The constantG has been chosen so |H(0)| = 1. Note that the magnitude function approaches zero whenω → ∞ as there is a transmission zero ats = ∞.

By modifying the above program in the same way as done earlier, we get the impulse and step responses. Figure2.37 shows the impulse and step responses. Note that the ringing has a somewhat longer duration than for corresponding Butterworth and Chebyshev I, and Chebyshev II filters that meet the same specification. The impulse and step responses for filters with large group delays are delayed proportionately to the group delay.

2.4.13.1 Cauer Filters with MinimumQ Factors

A less expensive circuit, with smaller element spread, is required to implement a pole pair with a lowQ factor. The importance ofQ factors will be further discussed in Chapter 6. Cauer filters with the following relationship betweenA _max andA _min have minimalQ factors [71]

$$A_{\textit{max}} = 10 \log \left(\frac{10^{0.1A_{\textit{min}}}}{10^{0.1A_{\textit{min}}}-1}\right).$$

((2.54))

Hence, for an arbitrary specification it may be favorable to modify the specification so that Equation (2.54) holds. For example,A _min = 40 dB yieldsA _max = 0.0004343 dB, which is a very small passband ripple. It may appear that this is an unreasonable small ripple, but in fact it is advantageous to design the filter for a smaller ripple than required, as it results in a less sensitiveLC filter. This issue will be further discussed in Sections 3.3.8 and 3.3.9.

This special case is related to digital half-band filters where the poles lie on the imaginary axis in thez-plane.

2.4.13.2 Cauer and Chebyshev II Filters of Type B

In order for an LP filter to be realized with anLC ladder network without the couples coils (see Chapter 3), the transfer function must have at least one zero ats = ∞. Cauer and Chebyshev II filters of even order lack this zero. However, the transfer functions can be modified to circumvent these problems. The modified filters are denoted typeb and typec [29].[G, Z, R_ZEROS, P, Wsnew] = CH_II_B_POLES(G, Z, P, Wc, Ws, Amax, Amin) [G, Z, R_ZEROS, P, Wsnew] = CA_B_POLES(G, Z, P, Wc, Ws, Amax, Amin)

Figure2.38 shows the attenuation for a fourth-order Cauer filter and corresponding modified filter.

The highest zero pair has been moved tos = ∞ so the modified filter obtains a double zero ats = ∞. The remaining finite zeros are also moved slightly, which has the effect that the stopband edge is increased. Such filters are provided with a suffixb, e.g.,C045042b. The passband edge,ω _c, andA _max andA _min do not change, but the transition band increases with this modification. Sometimes we use the suffixa for the unmodified Cauer filter to avoid misunderstandings.

2.4.13.3 Cauer and Chebyshev I Filters of Type C

Cauer and Chebyshev I filters of even order must be modified so that |H(0)| = 1 in order to be able to use the same source and load resistances in a ladder network (see Section 3.3.3). In typeb filters, the highest finite zero pair is moved to infinity as was done in typeb filters. In addition, the lowest reflection zero is moved to the origin. The passband edgeω _c and attenuationsA _max andA _min are not affected by this modification, but the transition band becomes larger compared withC045042b. This modification is indicated with the suffixc, e.g.,C045042c. Figure2.39 shows how the attenuation is changed.

The programs[G, Z, P, Wsnew] = CH_I_C_POLES(G, Z, P, Wc, Ws) [G, Z, R_ZEROS, P, Wsnew] = CA_C_POLES(G, Z, P, Wc, Ws, Amax, Amin)

generate the modified pole and zeros for a typec filter. Note that for these two filter approximations, it is not valid that the sum of the number of maxima and minima in the passband is equal to the filter order.

2.4.14 Comparison of Standard Filters

In comparing the standard approximations Butterworth, Chebyshev I, Chebyshev II, and Cauer filters, we find that the two latter have less variation in the group delay. In the literature it is often stated that Cauer filters have larger variation in the group delay than, i.e., Butterworth filters and that this is a valid reason for using the Butterworth approximation. The mistake is that the two filter approximations are compared using the same filter order. This is obviously not correct, which is evident of the following example, as Cauer filters can handle a considerably stricter requirement on the magnitude function. Even the step response for a Cauer filter is better. The difference between Chebyshev II and Cauer filters is however relatively small, the latter has a somewhat smaller group delay however the order is on the other hand larger.

Example 2.5

Compare Butterworth, Chebyshev I, Chebyshev II, and Cauer filters that meet the same standard LP specification:A _max = 0.01 dB,A _min = 40 dB,ω _c = 2 rad/s, andω _s = 3 rad/s [135]. Note thatA _max has been chosen very small because this will make the element sensitivity in a correspondingLC realization small. This will be discussed in detail in Section 3.3.

We get the following filter orders with the four standard approximations.

Butterworth:	N _B = 18.846	⇒	N _B = 19
Chebyshev I and Chebyshev II:	N _C = 8.660	⇒	N _C = 9
Cauer:	N _Ca = 5.4618	⇒	N _Ca = 6

Note the large difference between the required orders for standard approximations that meet the same requirement.

Figure2.40 shows the attenuation for the different approximations. The allowed passband ripple is very small, and we are only interested in that the requirement is met, and not in detailed variation inside the passband. Figure2.41 shows the corresponding group delays.

The attenuation in the transition band and the stopband varies between the different filters. The Chebyshev II filter has a more gradual transition between the passband and stopband compared with the Chebyshev I filter in spite of the fact that they have the same order. Note that the Cauer filter has a smaller transition band than required. The order has been rounded up from 5.46 toN = 6.

The attenuation approaches infinity for all of the filters, except for the Cauer filter, as its order is even.

The differences in the group delays are large between different approximations. The peaks in the group delays lie above the passband edgeω _c = 2 rad/s. The Butterworth and Chebyshev I filters have larger group delay in the passband whereas the Chebyshev II and Cauer filters have considerably smaller group delay and the difference between the latter two is small.

In the literature, it is commonly stated that the Butterworth filter has the best group delay properties. This is obviously not correct and is based on an unfair comparison between the standard approximations of the same order. According to Fig.2.41, Chebyshev II and Cauer filters have considerably better properties. TheQ factors for the four filters are shown in Table2.1.

Table 2.1 Comparison ofQ factors

Full size table

The element sensitivity for anLC filter is proportional to the group delay, see Section 3.3.9. For example, by using a Cauer filter instead of a Butterworth filter, the group delay can be reduced with about a factor 3 and the component tolerance is increased with the same factor. Components with large tolerances are considerably cheaper than those with small tolerances. Besides, the number of components is fewer. It is therefore important to use an approximation with small group delay. Cauer is often the preferred approximation because the require order is significantly lower than for Chebyshev II and the group delay andQ factors are similar.

The conclusion is that Cauer is the best approximation in most cases, i.e., when we have requirements on both the magnitude function, the group delay, and step response. We will later see that the Cauer filter has almost as low sensitivity to errors in the element values as Chebyshev II filters when it is realized with anLC filter.

2.4.15 Design Margin

The filter order must be an integer. Often, but not always, the order is chosen to the closest larger integer. The difference between theoretical and chosen filter order allows that the slightly more stringent requirement can be met. The design margin can then be used to reduce the passband and the stopband ripples and reduce the width of the transition band in excess of what is required in the specification.

For example, we first compute the case where the whole design margin is used to reduce the stopband edge,ω _smin, and afterwards choose a suitable reduction of this as shown in Figure2.42.

In the same manner, we can then use the remaining part of the design margin to determine the largest possible attenuation in the stopband and afterwards choose a suitable increase of the attenuation requirement withd ₁. Finally, in the same manner the remaining part of the design margin is used to increase the passband edge and reduce the passband ripple.

The intention is to obtain a safety margin for the errors, which always are present in the element values in the circuit that realizes filter. By exploiting the design margin appropriately, we can minimize the number of filters that violates the specification with a given statistical error distribution of the component values.

It can be shown that theQ factors of the poles become smaller if we use the design margin to reduce the passband ripple, and, hence, the filter will be simpler to implement.

2.4.16 Lowpass Filters with Piecewise-Constant Stopband Specification

Cauer filters have equiripple in both the passband and stopband, i.e., they meet a filter specification with a maximum attenuation ofA _max in the passband and has at least the attenuationA _min in the stopband. However, we often have different requirements in different parts of the stopband as illustrated in Fig.2.43. In such cases, the Cauer filter is not the best filter approximation. The zeros must in these cases be computed using numerical optimizing methods.

A program that is based on a well-known method [29] is called the PolePlacer where the ancient term pole refers to attenuation poles, i.e., the zeros of the transfer function.

The transfer function can be optimized by making the distanced _i between the attenuation function and attenuation requirement equal. In this way, we get an equally large attenuation margin in the whole stopband. It is however not certain that there exists an optimum where alld _i become equal.

The passband requirement is constant in the passband, and we can let the filter have equiripple or maximally flat passband.

In the same way can highpass and bandpass filters, which meet a piecewise-constant stopband requirement, be synthesized using the functionsPOLE_PLACER_HP_MF_S POLE_PLACER_HP_EQ_S POLE_PLACER_BP_MF_S POLE_PLACER_BP_EQ_Swhere MF and EQ indicate maximally flat and equiripple passband, respectively. Note that we must provide reasonable initial values for the number of finite and non-finite zeros and their placement in order for the programs to find an optimal solution.

Example 2.6

Below is shown how the POLE_PLACER_LP_EQ_S program may be used. Amax = 0.5; % POLE_PLACER_LP_EQ_S Wc = 1; Amin = [75 55]; % Amin = 75 dB from 2 and 55 dB above 2.5 rad/s Wstep = [2 2.5]; % Break frequencies Wfi = [2.5 4]; % Initial guess of finite zeros ±j2.5 and ±j4 rad/s NIN = 1; % Number of zeros at infinity, N = 2Nfi + NIN % Hence, a 5th-order LP filter with equiripple passband [G, Z, Zref, P, dopt] = POLE_PLACER_LP_EQ_S(Amax, Wc, Amin, Wstep, Wfi, NIN) W = (0:1000)*4*pi/1000; H = PZ_2_FREQ_S(G, Z, P, W); axis([0 8 0 100]), subplot(‘position’, [0.1 0.4 0.88 0.5]); PLOT_ATTENUATION_S(W, H)

If the program does not converge, a new initial placement of the finite zeros or an increase of the number of zeros may be tried. Ifdopt < 0, the number of zeros might be too small. If instead the function POLE_PLACER_LP_MF_S is used, a filter with maximally flat passband is obtained. However, this requires an increase of the filter order. Figure2.44 shows the resulting attenuation function.

All design margins in the stopband are equal withdopt. The program yields the following poles and zeros. The design margin in the stopband is at least 2.27 dB. Zref is a vector with the reflection zeros. For the maximally flat case, all reflection zeros are at the origin.N = 5 Z = P = 0 - 2.0472076i -0.0921637 - 1.0123270i 0 - 2.3868661i -0.0921637 + 1.0123270i 0 + 2.0472076i -0.2886563 + 0.6753131i 0 + 2.3868661i -0.2886563 - 0.6753131i -0.4023776 G = 0.009392200 dopt = 2.2734442

2.5 Miscellaneous Filters

2.5.1 Filters with Diminishing Ripple

The sensitivity to errors in the element values in the double terminated ladder that implements an LP filter is low forω = 0 and increases for increasing frequency and becomes largest where the group delay is maximal, i.e., just above the passband edge. The sensitivity can be reduced by reducing the ripple, i.e., use an approximation where the ripple decays toward the passband edge. We will discuss this case in more detail in Chapter 3.

2.5.2 Multiple Critical Poles

Another way of reducing the sensitivity emanates from the fact that the most sensitive pole pair is the one that has the highestQ factor, see Section 6.4. By increasing the filter order, i.e., higher than necessary, we can make the pole pair with the highestQ factor a multiple pole pair and reduce theirQ factors. This technique is called multiple critical root and can be used for both maximum flat passband ( MUCRMAF) and equiripple passband ( MUCROER) [94].

2.5.3 Papoulis MonotonicL Filters

In the literature, there exist a number of different types of all-pole approximations, e.g., Papoulis, parabolic, and Halpern approximations [68, 88]. These filters can have, in some cases, either better group delay, step response, or lowerQ factors.

However, the interest for different standard approximation has diminished with the event of effective optimizing programs that are able to optimize several different parameters at the same time.

Papoulis developed this class of filters whose attenuation increases monotonically in the passband and has maximal attenuation rate at the cutoff edge. The approximation maximizes the rate of change of the magnitude function atω = 1 under the constraint of monotonic attenuation response.

The filters combine the desirable features of the Butterworth and Chebyshev responses. The step response of these filters is good because the magnitude response monotonically decreases. The characteristic function for these filters is an Legendre polynomial of the first kind, and they are therefore referred to as MonotonicL filters.

2.5.4 Halpern Filters

Halpern filters are closely related to Papoulis filters but optimize the shaping factor under the conditions of a monotonically decreasing response. Halpern filters have a monotonic step response and maximum asymptotic cutoff rate. The characteristic function is a polynomial related to the Jacobi polynomials.

From the point of view of the stopband attenuation, little can be gained from maximizing the asymptotic attenuation cutoff rate instead of maximizing the attenuation rate at the edge as done for the Papoulis filters. In addition, the Papoulis filters yield much smaller passband magnitude distortion than the Halpern filters, and the latter are therefore only of academic interest.

2.5.5 Parabolic Filters

This class of filters has all the poles located in a parabolic contour in the left-hand side of thes-plane. Parabolic filters have a monotonic passband magnitude response, similar to the Halpern filters, and the variation between the maximum and minimum values of their group delay is better, and the step response has the fastest response without overshoot [68].

2.5.6 Linkwitz-Riley Crossover Filters

Loudspeakers are not capable of covering the entire audio spectrum with acceptable loudness and distortion. However, high-quality loadspeakers can be manufactured for smaller frequency ranges. Audio crossover filters are therefore used to split the audio signal into separate frequency bands, which are sent to individual loudspeakers that have been optimized for those bands.

Linkwitz-Riley crossover filters, named after S. Linkwitz and R. Riley, consist of a parallel combination of a lowpass and a highpass filter. Each filter consist of two cascaded Butterworth filters. Because each Butterworth filter has 3 dB attenuation at the cutoff frequency, the resulting Linkwitz-Riley filter will have 6 dB attenuation at the cutoff frequency. The overall attenuation at the cutoff frequency of the lowpass and highpass filter will be 0 dB, and the crossover filter behaves like an allpass filter with a smoothly changing phase response.

2.5.7 Hilbert Filters

Hilbert filters are used in single-sideband modulation schemes. The Hilbert filter is an allpass filter that approximates the phase function −90° forω > 0 and 90° forω < 0. Hilbert filters have the ideal frequency response

$$H(j\upomega)=\begin{cases}-j \ \quad\omega\,> 0\\0\qquad\omega=0\\j\qquad\omega\,<0.\end{cases}$$

((2.55))

Thus, a Hilbert filter generates an output signal that is ±90° different compared to the input signal. It is, however, often simpler to realize the Hilbert filter as a filter with two outputs with 90° difference in their phase responses.

2.6 Delay Approximations

In previous sections, transfer functions that meet a given magnitude specification have been discussed. In this section, we will discuss filter approximations, which approximate linear phase or constant group delay. A group delay requirement is more stringent than a linear-phase requirement. There exist standard approximations with maximally flat or equiripple group delay.

2.6.1 Gauss Filters

Characteristic for a Gauss filter [11, 29, 68, 88, 146] is that the step response does not have any overshoot or ringing and that the impulse response is approximately symmetric around the timet ₀, i.e., the phase function is almost linear.

Both Gauss and Bessel filters are nowadays of limited interest because filters that meet demands on both the magnitude function and the phase function can be determined with the help of optimization programs.

2.6.2 Lerner Filters

Lerner filters have approximately linear phase in most of the passband and a relatively small transition band [88].

2.6.3 Bessel Filters

A Bessel filter, which also is called a Thomson filter, approximates a lowpass filter with linear phase [88]. Bessel filters were first described by Z. Kiyasu 1943 and later by W.E. Thomson 1952. Bessel filters have a maximally flat group delay forω = 0. Figure2.45 shows the magnitude function for Bessel filters with orders 1–5. The magnitude function decays monotonically and the filter has relatively poor attenuation in the stopband and a wide transition band. The passband edge depends on the filter order. There exists no simple expression for computing the required filter order of the poles given an attenuation and group delay requirement.

Figure2.45 shows the attenuation and Fig.2.46 shows the corresponding group delays for Bessel filters with orders 1–5. The filters have been normalized to haveτ _g(0) = 1 s. Note that the frequency range over which the group delay is approximately constant increases with increasing filter order. In fact, the bandwidth-group delay product^{Footnote 7} is fixed and increases with the filter order. Hence, we may obtain a larger group delay over a small bandwidth or vice versa.

The transfer function for a Bessel filter is

$$H(s)=\frac{B_N(0)}{B_N(s)}=\frac{b_0}{\sum\limits_{k=0}^N b_ks^k}$$

((2.56))

where$b_k=\displaystyle\frac{(2N-k)!}{2^{N-k}k!(N-k)!}$ andB _N(s) is a Bessel polynomial [140].

The Bessel polynomial can easily be computed by the recursion

$$B_n=(2n-1)B_{n-1}+s^2B_{n-2}$$

((2.57))

withB ₀ =1 andB ₁ = s +1. Bessel filters do not have finite zeros.

The poles for a fifth-order Bessel filter are shown in Fig.2.47 withτ _g(0) = 3.93628 s, and Fig.2.48 shows the impulse and step responses of the corresponding filter.

Note that a filter with linear phase has an impulse response that is symmetric around the timet ₀ ≈τ _g. The impulse response for the Bessel filter is almost symmetric, which results in an almost linear phase response. Note that because the impulse response only has a small undershoot, the ripple in the step response will also be small. The overshoot and the ringing in the step response are much smaller compared with the previously discussed filters.

The required order for a Bessel filter with a given group delay atω = 0, a maximal deviation in the group delay fromτ _g(0), and a maximum ripple in the passband atω ₀, can be computed with the function BESSEL_ORDER.

Filters that are similar to Bessel filters are used in the read channel for hard drives to equalize both magnitude function and the group delay and to reduce the noise. The resulting read channel should have linear phase in order to reliably detect the ones and zeros. In this application, the filter’s bandwidth varies depending on which track is being read. A typical filter has order seven and a bandwidth that can be varied between 10 and 100 MHz. The allowed variation in phase function is less than ±0.05°.

Because hard drives are manufactured in several hundred million units annually, these filters are economically important, and large development efforts have been made to integrate the filters on the same chip as other digital logic that is a part of the read channel.

It is common to use different compromising solutions between, e.g., Chebyshev II filter and Bessel filter. Such a filter can therefore obtain a relatively good group delay and at the same time a good attenuation and narrow transition band because the finite zeros give large stopband attenuation and do not affect the group delay [11, 29]. Note that zeros on thejω-axis do not affect the group delay. Moreover, the area under the group delay for anNth-order all pole filter depends only on the filter order. We have

$$\begin{aligned}\int_0^\infty\tau_g(\omega)d\omega=\int_0^\infty-\frac{\partial\varPhi}{\partial\omega}d\omega=-[\varPhi(\infty)-\varPhi(0)]=-\left[-N\frac{\uppi}{2}-0\right]=N\frac{\uppi}{2}.\end{aligned}$$

((2.58))

Hence, because the area under the group delay is constant, an increase in the bandwidth will result in a decrease in the group delay and vice versa.

2.6.4 Lowpass Filters with Equiripple Group Delay

Lowpass filters with equiripple group delay response can be designed by using numerical optimization routines. The equiripple group delay filters have a wider equiripple–group delay band and similar stopband attenuation as the corresponding Bessel filter of the same order and are therefore often preferred.

2.6.5 Equiripple Group Delay Allpass Filters

An allpass filter can be derived from any of the previously discussed allpole, lowpass filters by forming the transfer function using their denominator, i.e.,

$$H(s)=\frac{D(-s)}{D(s)}$$

((2.59))

The group delay of this filter is twice that of the corresponding lowpass filter, e.g.,

$$\uptau_{gAP}=2\uptau_{g\textit{Bessel}}.$$

The ripple will also be twice as large as in the lowpass filter.

2.7 Frequency Transformations

In the following sections, we will discuss a method to compute the poles and zeros of highpass, bandpass, or bandstop filters from the poles and zeros of a lowpass filter [68]. This technique is based on a frequency transformation. These transformations, which are also called reactance transformations, result in a filter that meets a magnitude specification. However, the resulting filters are in general not optimal, except for highpass filters, which are optimal if the corresponding lowpass filter is optimal. Hence, bandpass and bandstop filters designed using reactance transformations result in suboptimal solutions, but the technique is often used due to its simplicity. The group delay with the lowpass filter is not retained by the transformation. Thus, it is not meaningful to frequency transform Bessel filters because the group delay is distorted.

Optimal frequency selective filters can, however, be designed by using numerical optimization techniques. We discuss how bandpass filters that are optimal with respect to the magnitude specification can be synthesized with a PolePlacer program. It is also possible to synthesize corresponding bandstop filters.

2.8 LP-to-HP Transformation

To design a highpass filter, we first design a corresponding lowpass filter according to the methods that have been described earlier. The requirement on the magnitude function of the LP filter depends on the requirement on the highpass filter.

Figure2.49 shows the specification for the highpass filter, which shall be synthesized, and the corresponding specification for the lowpass filter, which is used in the synthesis. Note thatA _max andA _min are the same in the highpass filter and the lowpass filter.

To separate the frequency variable for the lowpass filter from the corresponding frequency variable for the desired frequency transformed filter, capital letters and lowercase letters are used, respectively. The complex frequency for the lowpass filter and highpass filter is$S=\varSigma+j\,\Omega$ and$s=\upsigma+j\omega$, respectively.

The relation betweens andS is

$$S=\frac{\omega_I^2}{s}$$

((2.60))

whereω _I is the transformation angular frequency. It is often favorable, as it simplifies the computations, to chooseω _I = ω _c.

Figure2.50 shows how the poles and zeros are transformed whenω _I ²= 3.5. Poles on the real axis are mapped onto the real axis and zeros on thej Ω-axis are mapped onto thejω-axis. A complex pole pair is mapped to a complex pole pair. Zeros in the LP filter atS = ∞ are mapped tos = 0.

The design process for highpass filter is as follows:

1.
Determine the LP specification from the HP specification using Equation (2.60). That is,Ω _c = −ω _I ²/ω _c,Ω _s = −ω _I ²/ω _s and with the sameA _max andA _min.
2.
Determine required filter order, poles and zeros for the lowpass filter.
3.
Transform poles and zeros of the lowpass filter as well as any zeros atS = ∞ using Equation (2.60).

Example 2.7

Write a MATLAB program that determines necessary order and poles and zeros for a Butterworth filter that meets the specification shown in Fig.2.51. Verify the result by plotting the attenuation and poles and zeros in thes-plane.

We getWc = 30000; Ws = 8000; Amax = 0.1; Amin = 40; % Requirements for the highpass filter WI = Wc; % We select WI = Wc Omegac = Wc; % Omegac = WI^2/Wc = Wc Omegas = WI^2/Ws; NLP = 5; % Synthesis of 5th-order lowpass Butterworth filter [GLP, ZLP, ZrefLP, PLP, Wsnew] = BW_POLES(Omeagc, Omegas, Amax, Amin, NLP); QLP = -abs(PLP)./(2*real(PLP)) % Transform the LP to a HP filter [GHP, ZHP, PHP] = PZ_2_HP_S(GLP, ZLP, PLP, WI^2) N = NLP; % LP and HP filter has the same order QHP = -abs(PHP)./(2*real(PHP)) Here we use the function PZ_2_HP_S to transform the LP poles and zeros. The program yieldsNLP = 5 PLP = 1.0e+04 * QLP = -4.369168 0.50000 -3.534731 - 2.568132i 0.61804 -3.534731 + 2.568132i 0.61804 -1.350147 - 4.155326i 1.618034 -1.350147 + 4.155326i 1.618034 GLP = 1 GHP = 1 ZHP = 0 0 0 0 0 PHP = 1.0e+04 * QHP = -2.05989 0.50000 -1.666485 - 1.210772i 0.61804 -1.666485 + 1.210772i 0.61804 -0.6365407 - 1.959071i 1.618034 -0.6365407 + 1.959071i 1.618034

The poles and zeros for the highpass filter are shown in Fig.2.52. The attenuation and the group delay for the highpass filter are shown in Fig.2.53. The transfer function of the highpass filter is

$$\begin{aligned}H(s)=&\frac{s^5}{(s+20598.888)(s^2+12730.813s+424314200)}\\ \cdot&\frac{1}{s^2+33329.701s+424314200}.\end{aligned}$$

((2.61))

Note that the orders of the highpass and lowpass filters are the same and that a Butterworth filter of highpass type hasN zeros ats = 0 while the corresponding lowpass filter hasN zeros atS = ∞. TheQ factors are the same in the lowpass and highpass filters.

Note that all poles and zeros are mirrored in a circle with radiusω _I ². This means that if the poles with the LP filter lie on a circle with radiusR _p ₀, as is the case for Butterworth filters, the poles with corresponding highpass filters will also lie on a circle, but with the radiusr _p ₀

$$r_{p0}=\frac{\omega^2_I}{R_{p0}}.$$

An HP filter with piecewise-constant stopband requirements can be synthesized with the programs POLE_PLACER_HP_MF_S and POLE_PLACER_HP_EQ_S where MF and EQ denote maximally flat and equiripple passband, respectively.

2.8.1 LP-to-HP Transformation of the Group Delay

The group delay of the highpass filter can be expressed in terms of the group delay for the corresponding lowpass filter. We have for the highpass and lowpass filters

$$\begin{aligned}\tau_{gHP}(\upomega)&=-\frac{d}{d\upomega}\varPhi_{HP}(\upomega)=-\frac{d}{d\upomega}\varPhi_{LP}(\varOmega)\\&=-\frac{d\varOmega d}{d\omega d\varOmega}\varPhi_{LP}(\varOmega)=\frac{d\varOmega}{d\omega}\tau_{gLP}(\varOmega)\end{aligned}$$

and the relation between the lowpass and highpass angular frequencies is, according to Equation (2.60),$\varOmega=-\omega^2_I/\omega$. We get

$$\frac{d\varOmega}{d\omega}=-\frac{d}{d\omega}\frac{\omega^2_I}{\omega}=\frac{\omega^2_I}{\omega^2}.$$

Finally, we get

$$\tau_{gHP}(\omega)=\frac{\omega^2_I}{\omega^2}\tau_{gLP}(\varOmega).$$

((2.62))

The transfer function can be expanded into a partial faction expansion. Each term in this expression consists of a first- or second-order transfer function. The group delay for the lowpass filter is the sum of contributions for each individual pole pair

$$\uptau_{gLP}(\varOmega)=\sum\limits_{n=1}^{N}\uptau_{gn}(\varOmega)$$

where$\uptau_{g1}(\varOmega)=\frac{-\upsigma_p}{\varOmega^2+\upsigma^2_p}$ and$\uptau_{g2}(\varOmega)=\displaystyle\frac{-2\upsigma_p(\Omega^2+r^2_p)}{\varOmega^4+2(2\upsigma^2_p-r^2_p)\Omega^2+r^4_p}$ for a first-order pole and second-order pole pair, respectively. ReplacingΩ withω according with$\varOmega=-\omega^2_I/\omega$ yields for a real pole

$$\tau_{gHP}(\omega)=\left(-\frac{\omega^2_I}{\upsigma_p}\right)\frac{1}{\upomega^2+\displaystyle\frac{\omega^4_I}{\upsigma^2_p}}$$

and for a second-order pole pair

$$\tau_{gHP}(\omega)=-2\frac{\upsigma_p\omega^2_I}{r^2_p}\cdot\frac{\omega^2+\displaystyle\frac{\omega^4_I}{r^2_p}}{\upomega^4+\displaystyle\frac{2\omega^4_I}{r^4_p}(2\upsigma^2_p-r^2_p)\upomega^2+\frac{\omega^8_I}{r^4_p}}.$$

The group delay for the highpass filter has a similar shape as the group delay for the lowpass filter, but distorted according to Equation (2.62). The group delay for the highpass filter approaches zero forω → ∞ and toward a constant forω → 0. Figure2.54 shows the group delays for a fifth-order lowpass Butterworth filter withA _max = 0.1 dB andΩ _c = 1 rad/s and the corresponding highpass filter, which is obtained withω _I = Ω _c. Hence, they have the same passband edges. Note that the group delay for the highpass filter is small in the passband.

2.9 LP-to-BP Transformation

In the same manner as for highpass filters, a lowpass filter can be frequency transformed to a bandpass filter [68]. The frequency relation between the bandpass filter and the lowpass filter for the LP-BP transformation is

$$S=\frac{s^2 + \omega_I^2}{s}.$$

((2.63))

However, this frequency transformation requires that the product of the passband and the stopband edges meet a condition for geometric symmetry, i.e.,

$$\omega_I^2 = \omega_{s1}\omega_{s2} = \omega_{c1}\omega_{c2}$$

((2.64))

whereω _I is the transformation angular frequency.

Figure2.55 shows the specifications for a bandpass filter and the specification for a corresponding lowpass filter.

The relations between the edges in the LP and BP filters are

$$\begin{aligned}\varOmega_c &= \omega_{c2}-\omega_{c1}\\ \varOmega_s &= \omega_{s2}-\omega_{s1}.\end{aligned}$$

((2.65))

The bandwidth of the lowpass filter is equal to the bandwidth of the bandpass filter, and the stopband edge corresponds to the difference between stopband edges for the bandpass filter. Normally, the specification does not meet the condition in Equation (2.64). The specification must therefore be sharpened by changing at least one of the band edges.

If the transition bands or the attenuation requirements in the upper and lower stopbands differ, the bandpass filter must be designed to meet the most stringent requirement. This means that bandpass filters often get higher order than necessary because the frequency response meets a higher requirement than necessary. The bandpass filter that is obtained is a geometric symmetric bandpass filter.

Figure2.56 illustrates how the poles and zeros are mapped by the LP-BP transformation. A complex conjugate pole pair is mapped to two complex conjugate pole pairs, both with the sameQ factors.

From Fig.2.56 it is obvious that this way of designing bandpass filters results in a filter with the same number of zeros in the upper and lower stopbands. An optimizing program must be used to design a more complicated bandpass filter with, for example, different number of zeros in the two stopbands.

The design process for geometric symmetrical bandpass filters is

1.
Compute the LP specification from the BP specification. If needed, change the band edges so that the symmetry constraint is satisfied.
2.
Determine necessary order and the poles and zeros of the LP filter.
3.
Transform the poles and zeros of the LP filter. Transform also the zeros atS = ∞.

Example 2.8

Write a MATLAB program that determines necessary order and poles and zeros for a Cauer filter that meets the specification shown in Fig.2.57. Validate the result by plotting the attenuation and poles and zeros in thes-plane.

The requirement of geometric symmetry, Equation (2.64), is not met becauseω _c ₁ ω _c ₂ = 800 [krad/s]² andω _s ₁ ω _s ₂ = 720 [krad/s]². The symmetry requirement can be met, i.e., if the lower transition band is made smaller, i.e.,ω _s ₁ is changed soω _s ₁ = 13.333333 krad/s.Wc1 = 25000; Wc2 = 32000; % Requirement for the bandpass filter Ws1 = 12000; Ws2 = 60000; Amax = 0.28; Amin = 60; if Wc1*Wc2<= Ws1*Ws2; % Modify band edges if needed Ws2 = Wc1*Wc2/Ws1; else Ws1 = Wc1*Wc2/Ws2; end WI2 = Wc1*Wc2; Omegac = Wc2 - Wc1; % Requirements for lowpass filter Omegas = Ws2 - Ws1; NLP = CA_ORDER(Omegac, Omegas, Amax, Amin)% Synthesis of LP filter NLP = 3; % Select the next higher integer [GLP, ZLP, R_ZEROSLP, PLP, Wsnew] = CA_POLES(Omegac, Omegas, Amax, Amin, N); QLP = -abs(PLP)./(2*real(PLP)) % Transform the lowpass to a bandpass filter [GBP, ZBP, PBP] = PZ_2_G_SYM_BP_S(GLP, ZLP, PLP, WI2); figure(1) PLOT_PZ_S(ZBP, PBP, 0,0, -40000, 10000, 80000) ZBP PBP GBP QBP = -abs(PBP)./(2*real(PBP)) W = [0:100:100000]; H = PZ_2_FREQ_S(GBP, ZBP, PBP, W); figure(2) axis_Amax = 80; axis_Tg_max = 1.6*10^-3; Att = MAG_2_ATT(H); % Compute the attenuation Tg = PZ_2_TG_S(GBP, ZBP, PBP, W); % Compute the group delay PLOT_A_TG_S(Att, Tg, W, axis_Amax, axis_Tg_max); set(gca,'FontName', 'times', 'FontSize', 16);

We use the function PZ_2_G_SYM_BP_S to synthesize a geometric symmetrical bandpass filter. The program yieldsWs1 = 13333.333 NLP = 3 ZLP = 1.0 e+04 * 0 - 5.064487i 0 + 5.064487i PLP = 1.0 e+03 * QLP = -2.563182 - 7.566248i 1.558365 -2.563182 + 7.566248i 1.558365 -5.257168 0.500000 GLP = 1.3080498 e+02

For the BP filter we getN = 6 ZBP =1.0 e+04 * 0 - 1.264104i 0 + 1.264104i 0 - 6.328590i 0 + 6.328591i 0 PBP = 1.0 e+04 * QBP = -0.111152 - 2.472474i 11.13352 -0.111152 + 2.472474i 11.13352 -0.145166 + 3.22910i 11.13352 -0.145166 - 3.22910i 11.13352 -0.262858 - 2.81619i 5.380143 -0.262858 + 2.81619i 5.380143 GBP = 1.3080498 e+02

Poles and zeros and two semicircles, which indicated the passband and the stopband edges for the highpass filter, are shown in Fig.2.58. Note that every pole (or zero) in the LP filter is mapped to a complex conjugate pole (or zero) pair in the BP filter while the zero atS = ∞ in the LP filter is mapped tos = 0 ands = ∞. There are equally many zeros in the upper and lower stopband. The order for the bandpass filter is twice as high as for the corresponding lowpass filter, and theQ factors are much larger. The transfer function is

$$\begin{aligned}H(s)&=\frac{130.80050s(s^2+15979600)}{(s^2+2223.036s+61254838)(s^2+2903.327s+1044815429)}\\ &\cdot\frac{(s^2+4005106465)}{(s^2+5257.168s+8000000000)}.\end{aligned}$$

((2.66))

Note that the design margin for Cauer filters is used so the stopband edge is reduced and the lower stopband edge has been increased so an infinite zero pair lies in the specification’s lower transition band. The attenuation and the group delay for bandpass filter are shown in Fig.2.59.

The group delay of the LP filter is mapped to two peaks for the BP filter, see Problem 2.34. Note that the group delay is largest at the lower end of the passband. The impulse response is shown in Fig.2.60. The frequency of the ringing is related to the center frequency of the bandpass filter, and the envelope is related to the impulse response of the corresponding lowpass filter.

2.10 LP-to-BS Transformation

Bandstop filters can be designed in a similar way as the bandpass filter [68]. Figure2.61 shows the specifications for a bandstop filter and corresponding lowpass filter.

The frequency relation between the bandstop filter and lowpass filter using the LP-BS Ltransformation is

$$S=\frac{\omega_I^2s}{s^2+\omega_I^2}.$$

((2.67))

The frequency transformation of BS filters yields a geometric symmetrical BS filter, i.e., the frequency transformation results in that the product of passbands and respective stopband edges will meet the geometric symmetry constraint

$$\omega_I^2 = \omega_{s1}\omega_{s2} = \omega_{c1}\omega_{c2}$$

((2.68))

whereω _I is the transformation angular frequency. The relations between the LP and BS filters band edges are

$$\begin{aligned}\varOmega_c = \frac{\omega_I^2}{\omega_{c2}-\omega_{c1}}\\ \varOmega_s = \frac{\omega_I^2}{\omega_{s2}-\omega_{s1}}\end{aligned}$$

((2.69))

Because Equation (2.68) must be met, the bandstop filters often need to meet a stricter requirement than necessary [29, 112].

Figure2.62 illustrates how the poles and zeros are mapped by the LP-BS transformation.

The design process for geometric symmetrical bandstop filters is

1.
Compute the LP specification from the BS specification. If needed, adjust the band edges to meet the symmetry constraint.
2.
Determine necessary order and poles and zeros for the LP filter.
3.
Transform the poles and zeros of the LP filter. Transform also any zeros atS = ∞.

Example 2.9

Write a MATLAB program that determines necessary order and poles and zeros for a Cauer filter that meets the specification shown in Fig.2.63. Verify the result by plotting the attenuation and poles and zeros in thes-plane.

The requirement on geometric symmetry, Equation (2.64), yieldsω _c ₁ ω _c ₂ = 540 [krad/s]² andω _s ₁ ω _s ₂ = 650 [krad/s]². The symmetry requirement can be met if the transition band is made smaller, i.e.,ω _s ₁ is changed toω _s ₁ = 20.769 krad/s, as it is the lower transition band that determines the filter order. We getWc1 = 12000; Wc2 = 45000; % Requirement for the bandstop filter Ws1 = 25000; Ws2 = 26000; Amax = 0.28; Amin = 60; % Modify band edges if needed if Wc1*Wc2 >= Ws1*Ws2; Ws2 = Wc1*Wc2/Ws1; else Ws1 = Wc1*Wc2/Ws2; end WI2 = Wc1*Wc2; % Requirements for lowpass filter Omegac = WI2/(Wc2 - Wc1) Omegas = WI2/(Ws2 - Ws1) % Synthesis of lowpass filter (Cauer) NLP = CA_ORDER(Omegac, Omegas, Amax, Amin) NLP = 3; % Select next higher integer [GLP, ZLP, R_ZEROSLP, PLP, Wsnew] = CA_POLES(Omegac, Omegas, Amax, Amin, N); QLP = -abs(PLP)./(2*real(PLP)) % Transform the lowpass to a bandstop filter [GBS, ZBS, PBS] = PZ_2_G_SYM_BS_S(GLP, ZLP, PLP, WI2); QBS = -abs(PBS)./(2*real(PBS)) figure(1) PLOT_PZ_S(ZBS, PBS, 0,0, -50000, 10000, 50000); alfa = linspace(pi/2, 3*pi/2, 200); plot(Wc1*cos(alfa), Wc1*sin(alfa), '_', 'linewidth', 1); plot(Wc2*cos(alfa), Wc2*sin(alfa), '_', 'linewidth', 1); plot(Ws1*cos(alfa), Ws1*sin(alfa), '_', 'linewidth', 1); plot(Ws2*cos(alfa), Ws2*sin(alfa), '_', 'linewidth', 1); N = 2*NLP % The BS filter has the order 2*NLP ZBS PBS GBS W = [0:10:100000]; H = PZ_2_FREQ_S(GBS, ZBS, PBS, W); figure(2) axis_Amax = 80; axis_Tg_max = 0.8*10^-3; Att = MAG_2_ATT(H); % Compute the attenuation Tg = PZ_2_TG_S(GBS, ZBS, PBS, W); % Compute the group delay PLOT_A_TG_S(Att, Tg, W, axis_Amax, axis_Tg_max); set(gca, 'FontName', 'times', 'FontSize', 16); xtick([0:10000:100000])

We getN = 6 ZBS = 1.0e+04 * 0 - 2.563104i 0 + 2.563104i 0 - 2.106821i 0 + 2.106821i 0 - 2.323790i 0 + 2.323790i PBS = 1.0e+04 * -0.2257678 - 1.298334i -0.2257678 + 1.298334i -0.7020074 - 4.037086i -0.7020074 + 4.037086i -2.197004 - 0.7570833i -2.197004 + 0.7570833i GBS = -1.000000e+00 QLP = QBS = 1.558365 2.918536 2.918536 1.558365 2.918536 2.918536 0.500000 0.5288544 0.5288544

The poles, zeros, and semicircles, which indicate the passband and stopband edges for the bandstop filter, are shown in Fig.2.64. TheQ factors are not affected by the transformation. Because the MATLAB routine for the Cauer filter uses the design margin to lower the stopband edge for the lowpass filter, the width for the corresponding stopband will increase, which is evident as two finite zeros lie in the transition band. The stopband thus becomes larger than necessary.

The attenuation and the group delay for the bandstop filter are shown in Fig.2.65. Note that the group delay becomes largest in the lower transition band.

The transfer function is

$$\begin{aligned}H(s) &= \frac{-(s^2+6.569500\cdot10^8)(s^2+5.400000\cdot10^8)}{(s^2+4.394007\cdot10^4 s+5.400000\cdot10^8)(s^2+1.404015\cdot10^4 s+16.79088\cdot10^8)}\\ &\cdot\frac{(s^2+4.438669\cdot10^8)}{s^2+0.4515357\cdot10^4 s+1.736657\cdot10^8}.\end{aligned}$$

2.11 Piecewise-Constant Stopband Requirement

Frequency transformations of lowpass filter to bandpass filter result in both stopbands meeting the same attenuation requirementA _min. Furthermore, one of the transition bands normally becomes smaller than necessary due to the geometric symmetry constraint. The filter, thus, gets unnecessary high order. In many applications, the requirements are different in the two stopbands and the requirements usually vary, which is shown in Fig.2.66.

With the help of an optimizing program of the type PolePlacer, the poles and zeros can be determined to a filter that meets a partially constant stopband requirement. The passband can be chosen with equiripple or as a maximum flat. By choosing a suitable number of zeros in both stopbands, the filter order can be minimized.

A way of optimizing the filter, so the design margin is used well, is by making the distancesd _i between the attenuation function and the specification equal. It is however not certain that the program always will find such a solution. In such cases, we have to try with another set of the start values for the zeros or increasing the number of zeros if the filter order is too small. There are, however, in both cases certain limits on how the zeros can be placed in order for the transfer function to be realized with anLC network.

Example 2.10

Below is shown the use of POLE_PLACER_BP_MF_S for a maximum flat passband. Using the program POLE_PLACER_BP_EQ_S, a BP filter with equiripple passband can be synthesized. Note that the program is sensitive to the initial positions of the zeros.% POLE_PLACER_BP_MF_S with maximally flat passband Amax = 0.5; Wc1 = 3; Wc2 = 4; % Passband edges 3 and 4 rad/s Amin_low = [40]; % 40 dB between 0 and 2 rad/s Wstep_low = [2]; Amin_high = [35 60]; % 35 dB between 5 and 6 rad/s % 60 dB from 6 rad/s and higher Wstep_high = [5 6]; Wi_low = [0.4]; % Initial finite zeros in the lower stopband Wi_high = [7 8]; % and in the upper stopband NIN = 1; % One zero at infinity NZ = 1; % One zero at the origin [G, Z, P, dopt] = POLE_PLACER_BP_MF_S(Amax, Wc1, Wc2, ... Amin_low, Amin_high, Wstep_low, Wstep_high, Wi_low, Wi_high, NZ, NIN) W = (0:1000)*15/1000; H = PZ_2_FREQ_S(G, Z, P, W); Att = MAG_2_ATT(H); subplot('position', [0.08 0.4 0.90 0.5]); PLOT_ATTENUATION_S(W, Att) axis([0 15 0 100]); color = [0.7 0.7 0.7]; % Gray patch([0 Wstep_low(1) Wstep_low(1) 0], [0 0 Amin_low(1) ... Amin_low(1)], color); V = axis; patch([Wstep_high(1)V(2)V(2)Wstep_high(2)Wstep_high(2)... Wstep_high(1)], [0 0 Amin_high(2) Amin_high(2) Amin_high(1)... Amin_high(1)], color); Z = P = 0 -2.328055e-01 - 4.103456e+00i 0 - 1.777425i -2.328055e-01 + 4.103456e+00i 0 + 1.777425i -6.190255e-01 + 3.779765e+00i 0 - 5.678486i -6.190255e-01 - 3.779765e+00i 0 + 5.678486i -6.267145e-01 + 3.224695e+00i 0 - 7.574665i -6.267145e-01 - 3.224695e+00i 0 + 7.574665i -2.323394e-01 + 2.894913e+00i ∞ -2.323394e-01 - 2.894913e+00i G = 1.631020e-02 dopt = 1.008892e+00

With the specification given in the program we obtain the attenuation shown in Fig.2.67.

2.12 Equalizing the Group Delay

By connecting a minimum-phase filter that meets a specification for the magnitude function with an allpass filter, we can, according to Fig.2.4, equalize the group delay so that the two combined filters meet both a magnitude and group delay specification.

Example 2.11

The program below computes the poles and zeros of an allpass filter for equalizing the group delay of a fifth-order Cauer filter in Example 2.4. The frequency range has been normalized toω _c = 1.% Determine poles and zeros for the Cauer lowpass filter Wc = 1; Ws = 56/40; Amax = 0.28029; Amin = 40; N = CA_ORDER(Wc, Ws, Amax, Amin); N = 5; [GLP, ZLP, R_ZEROSLP, PLP, Wsnew] = CA_POLES(Wc, Ws, Amax, Amin, N); W1 = 0; W2 = 1; % Equalization range = passband Nap = 7; % 7th-order allpass filter [PAP, Wpas] = EQ_TG_LP_S(W1, W2, GLP, ZLP, PLP, Nap); figure(1) PLOT_PZ_S(-PAP, PAP, Wc, Ws,-1.5, 0.5, 2.5) PLOT_PZ_S(ZLP, PLP, Wc, Ws, -1.5, 0.5, 2.5) TgH = PZ_2_TG_S(GLP, ZLP, PLP, Wpas); TgAP = PZ_2_TG_S(1, -PAP, PAP, Wpas); figure(2); subplot('position', [0.08 0.4 0.90 0.5]); plot(Wpas, TgAP + TgH, 'linewidth', 2) hold on; grid on; axis([0, 1, 0, 25]); plot(Wpas, TgH, 'linewidth', 2) plot(Wpas, TgAP, 'linewidth', 2)

The poles and zeros for the allpass filter areZAP = 0.1834844 - 0.560061i 0.1834844 + 0.560061i 0.1873279 - 0.2830502i 0.1873279 + 0.2830502i 0.1649636 - 0.8441111i 0.1649636 + 0.8441111 0.1921410 PAP = -0.1834844 - 0.560061i -0.1834844 + 0.560061i -0.1873279 - 0.2830502i -0.1873279 + 0.2830502i -0.1649636 - 0.8441111i -0.1649636 + 0.8441111 -0.1921410

Note that zeros of an allpass filters lie in the right half-plane and that they are mirror images in thejω-axis of corresponding poles. That is, poles and zeros of the allpass filter have the same imaginary part and the same real part except for the different signs. Typically, most of the poles of the allpass filter lie closer to thejω-axis than the poles in the lowpass filter. That is, most of theQ factors are higher than in the lowpass filter.

Figure2.69 shows the resulting group delays for the lowpass filter in Example 2.4, allpass filter, and resulting overall group delay using a seventh-order allpass filter. Figure2.68 shows the pole-zero configurations for the lowpass and allpass filters. Synthesis of the allpass filter can be made with the help of the above optimizing program, which yields an approximately equiripple solution. In this case, the minimax error is about 1.7372.

2.13 Problems

2.1
1. a)
  Derive an expression for the required filter order for a Butterworth filter.
2. b)
  Derive an expression for the poles for a Butterworth filter.
2.2
Derive the relation betweenA _max,ε, andρ.
2.3
The poles in a normalized Butterworth filter are denormalized by multiplying with the factor,r _p ₀, and not with the passband edge,ω _c, which is done with the other approximations. Determine the attenuation atω = r _p ₀.
2.4
Show that the first 2N−1 derivatives of the magnitude function squared of a Butterworth filter are zero atω = 0.
2.5
Determine how the impulse and step responses are affected by a scaling of the frequency with a factork.
2.6
Determine the extreme values ofT _n(x) = cos(n acos(x)).
2.7
Determine the poles and zeros and transfer function for a third-order Butterworth filter withA _max = 0.1 dB andω _c = 5 Mrad/s and determine the attenuation atω = 10 Mrad/s andω = 20 Mrad/s.
2.8
Repeat Problem 2.7 for a Chebyshev I filter.
2.9
Repeat Problem 2.7 for a Chebyshev II filter withA _min = 40 dB.
2.10
Repeat Problem 2.7 for a Cauer filter withA _min = 40 dB.
2.11
1. a)
  Determine the required filter order and the poles and zeros for a filter with maximally flat magnitude function that meets the following attenuation requirement:A _max = 0.40959 dB (ρ = 30%),A _min = 50 dB,ω _c = 3 Mrad/s,ω _s = 12 Mrad/s.
2. b)
  Repeat a) but only a fourth-order filter can be afforded. How much of the stopband attenuation must be sacrificed?
2.12
Design a fifth-order Butterworth lowpass filter for use in a high data rate Bluetooth system, which is required to meet the requirements: 3 dB cutoff edge at 1 MHz and passband gain of 5.
2.13
1. a)
  Write a MATLAB program that computes required filter order, poles and zeros for a Butterworth filter that meets the requirement:ρ = 30%,A _min = 35 dB,ω _c = 10 krad/s,ω _s = 30 krad/s.
2. b)
  Validate the program by plotting the attenuation.
3. c)
  Plot the group delay.
4. d)
  Plot the poles and zeros in thes-plane.
5. e)
  Plot the impulse and step responses in the same diagram.
2.14
Repeat Problem 2.13 for a Chebyshev I filter.
2.15
Repeat Problem 2.13 for a Chebyshev II filter withA _min = 40 dB.
2.16
Repeat Problem 2.13 for a Cauer filter withA _min = 40 dB.
2.17
Compare the filters in Problems 2.13, 2.14, 2.15, and 2.16 and plot the magnitudes, group delays, and poles and zeros.
2.18
Determine the constant G in Example 2.2 so that the gain atω = 0 is 32. Determine also the rate of attenuation increase in dB per octave at high frequencies.
2.19
Determine and compare the required filter order and the poles and zeros for a
1. a)
  Butterworth filter
2. b)
  Chebyshev I filter
3. c)
  Chebyshev II filter
4. d)
  Cauer filter
that meets the following requirement:ρ = 15%,A _min = 60 dB,ω _c = 5 Mrad/s, andω _s = 2.5 Mrad/s. Mark the poles and zeros in thes-plane and determine the transfer function.
2.20
Compare using MATLAB the Butterworth, Chebyshev I, Chebyshev II, and Cauer approximations with respect to
1. a)
  Required order for a given standard lowpass specification.
2. b)
  The width of the transition band for filter of orderN = 4 andN = 5. Select typical values forA _max andA _min.
2.21
Determine the required filter order and the poles and zeros for a
1. a)
  Butterworth filter
2. b)
  Chebyshev I filter
3. c)
  Chebyshev II filter
4. d)
  Cauer filter
that meets the following requirements:ρ = 15%,A _min = 60 dB,ω _c = 10 Mrad/s,ω _s = 25 Mrad/s. Mark the poles and zeros in thes-plane and determine the transfer function.
2.22
An anti-aliasing filter in front of an analog-to-digital converter (ADC) in a video system is used to bandlimit the input signal before sampling. The passband is up to 8 MHz and the stopband edge is at 12 MHz. The acceptable passband ripple corresponding toρ = 5% and the stopband attenuation is at least 40 dB. Select a suitable filter approximation and find the transfer function find and the poles and zeros.
2.23
LP (long-playing) records are engraved with reduced bass levels and increased treble levels because of two main reasons. Low signal frequencies require a larger groove, which has the drawback of shorter recording time and difficulties for the stylus to follow, and, thus, causes distortion. In addition, at high frequencies the stylus has difficulty to accurately follow the groove, which causes high frequency noise. RIAA (Recording Industry Association of America) has standardized a scheme where the high frequencies are amplified during recording to obtain a higher signal-to-noise ratio and attenuated at playback through a filter described by the RIAA frequency curve
$$N=-10\log(1+\omega^2\tau^2_1)+10\log(1+\omega^2\tau^2_2)-10\log(1+\omega^2\tau^2_3)$$
whereN = level in dB; treble time constant,τ ₁ = 75 µs; medium time constant,τ ₂ = 318 µs; and bass time constant,τ ₃ = 3.180 ms. Find the corresponding transfer function and its poles and zeros. The attenuation should be normalized to 0 dB at 1 kHz.
2.24
Find a second-order allpass transfer function with gain = –2 andΦ _AP(ω) = –π/4 rad atω = 2π Mrad/s.
2.25
Determine how the group delay of a lowpass filter is transformed by the LP-HP transformation and mark the positions of the poles and zeros in thes-plane.
2.26
Determine how the group delay of a lowpass filter is transformed by the LP-BS transformation.
2.27
Compute by hand the poles and zeros for a highpass filter of Butterworth type that meets the requirement:A _max = 1 dB,ω _c = 70 Mrad/s,A _min = 25 dB, andω _s = 20 Mrad/s.
2.28
Design an allpass filter that minimizes the overall ripple in the group delay of the cascade of the filter in Problem 2.22 and the allpass filter to less than 5%.
2.29
A filter design program found on the Internet, a so-called shareware program, produces the following transfer function for given specification
$$H(s) = \frac{s^6-31s^4+175s^2-625}{s^3-5s^2+(3+4j)s-15-20j}.$$
Give several reasons why you should not buy the program, at least not in this version.
2.30
Design an LP filter with rise time of 3 ms and a delay of 1.5 ms.
2.31
Determine the poles and zeros for a Cauer filter that meets the attenuation requirement:ρ = 15%A _min = 45 dB,ω _c = 5.5 Mrad/s, andω _s = 3.5 Mrad/s.
2.32
Write a MATLAB program that computes the required filter order, poles and zeros, and transfer function, for a
1. a)
  Chebyshev I filter
2. b)
  Cauer filter that meets the following requirement:ρ = 30%,ω _c = 10 krad/s,A _min = 35 dB, andω _s = 6 krad/s.
3. c)
  Plot the impulse and step responses.
2.33
Show that lowpass and highpass Butterworth filters can be designed so that: |H _LP(jω)|² + |H _HP(jω)|² = 1.
2.34
Determine how the group delay of a lowpass filter is transformed by the LP-BP transformation.
2.35
Determine the poles and zeros for a Chebyshev I filter that meets the specification:ρ = 30%,A _min = 60 dB,ω _s ₁ = 2 krad/s,ω _c ₁ = 6 krad/s,ω _c ₂ = 8.5 krad/s, andω _s ₂ = 25.5 krad/s.
2.36
Compute by hand the poles and zeros for an analog filter that meets the requirement:ρ = 30%,ω _c ₁ = 10 krad/s,ω _c ₂ = 12 krad/s,A _min = 35 dB,ω _s ₁ = 5 krad/s, andω _s ₂ = 32 krad/s. The filter approximation should be of Chebyshev I type.
2.37
Write a MATLAB program that computes the required filter order and the poles and zeros as well as the transfer function for a
1. a)
  Butterworth filter
2. b)
  Chebyshev I filter
3. c)
  Chebyshev II filter
4. d)
  Cauer filter
  
  that meets the requirement:ρ = 30%,ω _c ₁ = 10 krad/s,ω _c ₂ = 12 krad/s,A _min = 35 dB,ω _s ₁ = 5 krad/s, andω _s ₂ = 17 krad/s.
2.38
Add plotting of the impulse and step responses to the MATLAB program developed in Problem 2.37.
2.39
Write a MATLAB program that computes the required filter order and the poles and zeros as well as the transfer function for a Chebyshev I filter that meets the specification:

A _max ₁ = 0.409586 dB ω _c ₁ = 2π 49.5 krad/s

A _max ₂ = 0.409586 dB ω _c ₂ = 2π 50.5 krad/s

A _min = 29 dB ω _s ₁ = 2π 48.5 krad/s ω _s ₂ = 2π 51.5 krad/s
2.40
Determine the required filter order and the poles and zeros for a
1. a)
  Butterworth filter
2. b)
  Chebyshev I filter
3. c)
  Chebyshev II filter
4. d)
  Cauer filter
  
  that meets the requirement:ω _s ₁ = 3 krad/s,ω _c ₁ = 5 krad/s,ω _c ₂ = 8 krad/s, andω _s ₂ = 15 krad/s.
Mark the poles and zeros in thes-plane and determine the transfer function.
2.41
Design an equiripple passband filter withρ = 30% andω _c = 15 Mrad/s that meets the following piecewise-constant stopband requirement:A _min ₁ = 65 dB atω _s ₁ = 23 Mrad/s, andA _min ₂ = 40 dB atω _s ₂ = 30 Mrad/s.
2.42
Use the POLE_PLACER_BP_EQ_S program to design a BP filter that meets the specification in Problem 2.40 and discuss the pros and cons of using the geometric symmetric LP-BP transformation.

Notes

1.
S. Butterworth, UK, 1930.
2.
The order of a rational function is the greater of the degree of the numerator and denominator polynomials. For a causal filter, the degree of the numerator is equal to or less than the degree of the denominator.
3.
In some literature, the factorε ² is included in the characteristic function.
4.
Adolf Hurwitz, (1859–1919), Germany.
5.
Pafnuty L. Chebyshev, (1821–1894), St. Petersburg, Russia.
6.
$\rm{a\\cosh(x)} = \rm{In} (x+\sqrt{x^2-1})and \ a\\sinh(x)=In (x+\sqrt{x^2+1})$.
7.
This product is an alternative formulation of the Heisenberg uncertainty principle.

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Dept. Electrical Engineering Division of Electronics Systems, Linköping University, SE-581 83, Linköping, Sweden
Dr. Lars Wanhammar

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Wanhammar, L. (2009). Synthesis of Analog Filters. In: Analog Filters Using MATLAB. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-92767-1_2

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DOI: https://doi.org/10.1007/978-0-387-92767-1_2
Published: 30 April 2009
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Synthesis of Analog Filters

Keywords

2.1 Introduction

2.2 Filter Specification

2.2.1 Magnitude Function Specification

2.2.2 Attenuation Specification

2.2.3 Group Delay Specification

2.3 Composite Requirements

Definition 2.1

2.4 Standard LP Approximations

2.4.1 Butterworth Filters

2.4.2 Poles and Zeros of Butterworth Filters

Definition 2.2

Definition 2.3

Definition 2.4

Theorem 2.1

2.4.3 Impulse and Step Response of Butterworth Filters

Example 2.1

2.4.4 Chebyshev I Filters

2.4.5 Poles and Zeros of Chebyshev I Filters

Definition 2.5

2.4.6 Reflection Zeros of Chebyshev I Filters

2.4.7 Impulse and Step Response of Chebyshev I Filters

Example 2.2

2.4.8 Chebyshev II Filters

2.4.9 Poles and Zeros of Chebyshev II Filters

2.4.10 Impulse and Step Response of Chebyshev II Filters

Example 2.3

2.4.11 Cauer Filters

Definition 2.6

2.4.12 Poles and Zeros of Cauer Filters

2.4.13 Impulse and Step Response of Cauer Filters

Example 2.4

2.4.13.1 Cauer Filters with MinimumQ Factors

2.4.13.2 Cauer and Chebyshev II Filters of Type B

2.4.13.3 Cauer and Chebyshev I Filters of Type C

2.4.14 Comparison of Standard Filters

Example 2.5

2.4.15 Design Margin

2.4.16 Lowpass Filters with Piecewise-Constant Stopband Specification

Example 2.6

2.5 Miscellaneous Filters

2.5.1 Filters with Diminishing Ripple

2.5.2 Multiple Critical Poles

2.5.3 Papoulis MonotonicL Filters

2.5.4 Halpern Filters

2.5.5 Parabolic Filters

2.5.6 Linkwitz-Riley Crossover Filters

2.5.7 Hilbert Filters

2.6 Delay Approximations

2.6.1 Gauss Filters

2.6.2 Lerner Filters

2.6.3 Bessel Filters

2.6.4 Lowpass Filters with Equiripple Group Delay

2.6.5 Equiripple Group Delay Allpass Filters

2.7 Frequency Transformations

2.8 LP-to-HP Transformation

Example 2.7

2.8.1 LP-to-HP Transformation of the Group Delay

2.9 LP-to-BP Transformation

Example 2.8

2.10 LP-to-BS Transformation

Example 2.9

2.11 Piecewise-Constant Stopband Requirement

Example 2.10

2.12 Equalizing the Group Delay

Example 2.11

2.13 Problems

Notes

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