4.10 FILTER DESIGN BY [PLACEMENT OF](#page-10-0) POLES AND ZEROS OF H(s)

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4.10 FILTER DESIGN BY PLACEMENT OF POLES AND ZEROS OF H(s)

In this section we explore the strong dependence of frequency response on the location of poles and zeros of H(s). This dependence points to a simple intuitive procedure to filter design.

4.10-1 Dependence of Frequency Response on Poles and Zeros of H(s)

Frequency response of a system is basically the information about the filtering capability of the system. A system transfer function can be expressed as

$$H(s) = \frac{P(s)}{Q(s)} = b_0 \frac{(s - z_1)(s - z_2) \cdots (s - z_N)}{(s - \lambda_1)(s - \lambda_2) \cdots (s - \lambda_N)}$$

where z1, z2, … , zN are λ1, λ2, … , λ*^N* are the poles of H(s). Now the value of the transfer function H(s) at some frequency s = p is

$$H(s)|_{s=p} = b_0 \frac{(p-z_1)(p-z_2)\cdots(p-z_N)}{(p-\lambda_1)(p-\lambda_2)\cdots(p-\lambda_N)}$$

(4.53)

This equation consists of factors of the form p−zi and p−λi. The factor p−zi is a complex number represented by a vector drawn from point z to the point p in the complex plane, as illustrated in Fig. 4.48a. The length of this line segment is |p − zi|, the magnitude of p − zi. The angle of this directed line segment (with the horizontal axis) is (p − zi). To compute H(s) at s = p, we draw line segments from all poles and zeros of H(s) to the point p, as shown in Fig. 4.48b. The vector connecting a zero zi to the point p is p − zi. Let the length of this vector be ri, and let its angle with the horizontal axis be φi. Then p−zi = riej^φi . Similarly, the vector connecting a pole λ*ⁱ* to the point p is p − λ*ⁱ* = diej^θi , where di and θ*ⁱ* are the length and the angle (with the horizontal axis),

Figure 4.48 Vector representations of (a) complex numbers and (b) factors of H(s).

respectively, of the vector p−λi. Now from Eq. (4.53) it follows that

$$H(s)|_{s=p} = b_0 \frac{(r_1 e^{j\phi_1})(r_2 e^{j\phi_2}) \cdots (r_N e^{j\phi_N})}{(d_1 e^{j\theta_1})(d_2 e^{j\theta_2}) \cdots (d_N e^{j\theta_N})}$$

= $b_0 \frac{r_1 r_2 \cdots r_N}{d_1 d_2 \cdots d_N} e^{j[(\phi_1 + \phi_2 + \cdots + \phi_N) - (\theta_1 + \theta_2 + \cdots + \theta_N)]}$

Therefore

$$|H(s)|_{s=p} = b_0 \frac{r_1 r_2 \cdots r_N}{d_1 d_2 \cdots d_N} = b_0 \frac{\text{product of distances of zeros to } p}{\text{product of distances of poles to } p}$$

(4.54)

and

$$\angle H(s)|_{s=p} = (\phi_1 + \phi_2 + \dots + \phi_N) - (\theta_1 + \theta_2 + \dots + \theta_N)$$

= sum of angles of zeros to $p$ – sum of angles of poles to $p$ (4.55)

Here, we have assumed positive b0. If b⁰ is negative, there is an additional phase π. Using this procedure, we can determine H(s) for any value of s. To compute the frequency response H(jω), we use s = jω (a point on the imaginary axis), connect all poles and zeros to the point jω, and determine |H(jω)| and H(jω) from Eqs. (4.54) and (4.55). We repeat this procedure for all values of ω from 0 to ∞ to obtain the frequency response.

GAIN ENHANCEMENT BY A POLE

To understand the effect of poles and zeros on the frequency response, consider a hypothetical case of a single pole −α + jω0, as depicted in Fig. 4.49a. To find the amplitude response |H(jω)| for a certain value of ω, we connect the pole to the point jω (Fig. 4.49a). If the length of this line is d, then |H(jω)| is proportional to 1/d,

$$|H(j\omega)| = \frac{K}{d} \tag{4.56}$$

where the exact value of constant K is not important at this point. As ω increases from zero, d decreases progressively until ω reaches the value ω0. As ω increases beyond ω0, d increases

Figure 4.49 The role of poles and zeros in determining the frequency response of an LTIC system.

progressively. Therefore, according to Eq. (4.56), the amplitude response |H(jω)| increases from ω = 0 until ω = ω0, and it decreases continuously as ω increases beyond ω0, as illustrated in Fig. 4.49b. Therefore, a pole at −α + jω⁰ results in a frequency-selective behavior that enhances the gain at the frequency ω⁰ (resonance). Moreover, as the pole moves closer to the imaginary axis (as α is reduced), this enhancement (resonance) becomes more pronounced. This is because α, the distance between the pole and jω⁰ (d corresponding to jω0), becomes smaller, which increases the gain K/d. In the extreme case, when α = 0 (pole on the imaginary axis), the gain at ω⁰ goes to infinity. Repeated poles further enhance the frequency-selective effect. To summarize, we can enhance a gain at a frequency ω⁰ by placing a pole opposite the point jω0. The closer the pole is to jω0, the higher is the gain at ω0, and the gain variation is more rapid (more frequency selective) in the vicinity of frequency ω0. Note that a pole must be placed in the LHP for stability.

Here we have considered the effect of a single complex pole on the system gain. For a real system, a complex pole −α + jω⁰ must accompany its conjugate −α − jω0. We can readily show that the presence of the conjugate pole does not appreciably change the frequency-selective behavior in the vicinity of ω0. This is because the gain in this case is K/dd , where d is the distance of a point jω from the conjugate pole −α − jω0. Because the conjugate pole is far from jω0, there is no dramatic change in the length d as ω varies in the vicinity of ω0. There is a gradual increase in the value of d as ω increases, which leaves the frequency-selective behavior as it was originally, with only minor changes.

GAIN SUPPRESSION BY A ZERO

Using the same argument, we observe that zeros at −α ± jω⁰ (Fig. 4.49d) will have exactly the opposite effect of suppressing the gain in the vicinity of ω0, as shown in Fig. 4.49e). A zero on the imaginary axis at jω⁰ will totally suppress the gain (zero gain) at frequency ω0. Repeated zeros will further enhance the effect. Also, a closely placed pair of a pole and a zero (dipole) tend to cancel out each other’s influence on the frequency response. Clearly, a proper placement of poles and zeros can yield a variety of frequency-selective behavior. We can use these observations to design lowpass, highpass, bandpass, and bandstop (or notch) filters.

Phase response can also be computed graphically. In Fig. 4.49a, angles formed by the complex conjugate poles −α±jω⁰ at ω =0 (the origin) are equal and opposite. As ω increases from 0 up, the angle θ¹ (due to the pole −α +jω0), which has a negative value at ω = 0, is reduced in magnitude; the angle θ² because of the pole −α − jω0, which has a positive value at ω = 0, increases in magnitude. As a result, θ¹ + θ2, the sum of the two angles, increases continuously, approaching a value π as ω → ∞. The resulting phase response H(jω) = −(θ¹ +θ2) is illustrated in Fig. 4.49c. Similar arguments apply to zeros at −α ± jω0. The resulting phase response H(jω) = (φ¹ + φ2) is depicted in Fig. 4.49f.

We now focus on simple filters, using the intuitive insights gained in this discussion. The discussion is essentially qualitative.

4.10-2 Lowpass Filters

A typical lowpass filter has a maximum gain at ω = 0. Because a pole enhances the gain at frequencies in its vicinity, we need to place a pole (or poles) on the real axis opposite the origin (jω = 0), as shown in Fig. 4.50a. The transfer function of this system is

$$H(s) = \frac{\omega_c}{s + \omega_c}$$

We have chosen the numerator of H(s) to be ω*^c* to normalize the dc gain H(0) to unity. If d is the distance from the pole −ω*^c* to a point jω (Fig. 4.50a), then

$$|H(j\omega)| = \frac{\omega_c}{d}$$

with H(0) = 1. As ω increases, d increases and |H(jω)| decreases monotonically with ω, as illustrated in Fig. 4.50d with label N = 1. This is clearly a lowpass filter with gain enhanced in the vicinity of ω = 0.

WALL OF POLES

An ideal lowpass filter characteristic (shaded in Fig. 4.50d) has a constant gain of unity up to frequency ωc. Then the gain drops suddenly to 0 for ω>ωc. To achieve the ideal lowpass

Figure 4.50 Pole-zero configuration and the amplitude response of a lowpass (Butterworth) filter.

characteristic, we need enhanced gain over the entire frequency band from 0 to ωc. We know that to enhance a gain at any frequency ω, we need to place a pole opposite ω. To achieve an enhanced gain for all frequencies over the band (0 to ωc), we need to place a pole opposite every frequency in this band. In other words, we need a continuous wall of poles facing the imaginary axis opposite the frequency band 0 to ω*^c* (and from 0 to −ω*^c* for conjugate poles), as depicted in Fig. 4.50b. At this point, the optimum shape of this wall is not obvious because our arguments are qualitative and intuitive. Yet, it is certain that to have enhanced gain (constant gain) at every frequency over this range, we need an infinite number of poles on this wall. We can show that for a maximally flat† response over the frequency range (0 to ωc), the wall is a semicircle with an infinite number of poles uniformly distributed along the wall [11]. In practice, we compromise by using a finite number (N) of poles with less-than-ideal characteristics. Figure 4.50c shows the pole configuration for a fifth-order (N = 5) filter. The amplitude response for various values of N is illustrated in Fig. 4.50d. As N → ∞, the filter response approaches the ideal. This family of filters is known as the Butterworth filters. There are also other families. In Chebyshev filters, the wall shape is a semiellipse rather than a semicircle. The characteristics of a Chebyshev filter are inferior to those of Butterworth over the passband (0,ωc), where the characteristics show a rippling effect

^† Maximally flat amplitude response means the first 2*^N* ⁻ 1 derivatives of ^|H(jω)^| with respect to ^ω are zero at ω = 0.

instead of the maximally flat response of Butterworth. But in the stopband (ω>ωc), Chebyshev behavior is superior in the sense that Chebyshev filter gain drops faster than that of the Butterworth.

4.10-3 Bandpass Filters

The shaded characteristic in Fig. 4.51b shows the ideal bandpass filter gain. In the bandpass filter, the gain is enhanced over the entire passband. Our earlier discussion indicates that this can be realized by a wall of poles opposite the imaginary axis in front of the passband centered at ω0. (There is also a wall of conjugate poles opposite −ω0.) Ideally, an infinite number of poles is required. In practice, we compromise by using a finite number of poles and accepting less-than-ideal characteristics (Fig. 4.51).

4.10-4 Notch (Bandstop) Filters

An ideal notch filter amplitude response (shaded in Fig. 4.52b) is a complement of the amplitude response of an ideal bandpass filter. Its gain is zero over a small band centered at some frequency ω⁰ and is unity over the remaining frequencies. Realization of such a characteristic requires an infinite number of poles and zeros. Let us consider a practical second-order notch filter to obtain zero gain at a frequency ω = ω0. For this purpose, we must have zeros at ±jω0. The requirement of unity gain at ω = ∞ requires the number of poles to be equal to the number of zeros (M = N). This ensures that for very large values of ω, the product of the distances of poles from ω will be equal to the product of the distances of zeros from ω. Moreover, unity gain at ω = 0 requires a pole and the corresponding zero to be equidistant from the origin. For example, if we use two (complex-conjugate) zeros, we must have two poles; the distance from the origin of the poles and of the zeros should be the same. This requirement can be met by placing the two conjugate poles on the semicircle of radius ω0, as depicted in Fig. 4.52a. The poles can be anywhere on the semicircle to satisfy the equidistance condition. Let the two conjugate poles be at angles ±θ with respect to the negative real axis. Recall that a pole and a zero in the same vicinity tend to cancel out

Figure 4.51 (a) Pole-zero configuration and (b) the amplitude response of a bandpass filter.

Figure 4.52 (a) Pole-zero configuration and (b) the amplitude response of a bandstop (notch) filter.

each other’s influences. Therefore, placing poles closer to zeros (selecting θ closer to π/2) results in a rapid recovery of the gain from value 0 to 1 as we move away from ω⁰ in either direction. Figure 4.52b shows the gain |H(jω)| for three different values of θ.

EXAMPLE 4.31 Notch Filter Design

Design a second-order notch filter to suppress 60 Hz hum in a radio receiver.

We use the poles and zeros in Fig. 4.52a with ω⁰ = 120π. The zeros are at s = ±jω0. The two poles are at −ω⁰ cos θ ±jω⁰ sin θ. The filter transfer function is (with ω⁰ = 120π)

$$H(s) = \frac{(s - j\omega_0)(s + j\omega_0)}{(s + \omega_0 \cos \theta + j\omega_0 \sin \theta)(s + \omega_0 \cos \theta - j\omega_0 \sin \theta)}$$

=

$$\frac{s^2 + \omega_0^2}{s^2 + (2\omega_0 \cos \theta)s + \omega_0^2} = \frac{s^2 + 142122.3}{s^2 + (753.98 \cos \theta)s + 142122.3}$$

and

$$|H(j\omega)| = \frac{-\omega^2 + 142122.3}{\sqrt{(-\omega^2 + 142122.3)^2 + (753.98\omega\cos\theta)^2}}$$

The closer the poles are to the zeros (the closer θ is to π/2), the faster the gain recovery from 0 to 1 on either side of ω⁰ = 120π. Figure 4.52b shows the amplitude response for three different values of θ. This example is a case of very simple design. To achieve zero gain over a band, we need an infinite number of poles as well as an infinite number of zeros.

MATLAB easily computes and plots the magnitude response curves of Fig. 4.52b. To illustrate, let us plot the magnitude response using θ = 60◦ over a frequency range of 0 ≤ f ≤ 150 Hz. The result, shown in Fig. 4.53, matches the θ = 60◦ case of Fig. 4.52b.

f = (0:.01:150); omega0 = 2*pi*60; theta = 60*pi/180;

• H = @(s) (s.^2+omega0^2)./(s.^2+2*omega0*cos(theta)*s+omega0^2);

• plot(f,abs(H(1j*2*pi*f)),‘k-’);

• xlabel(‘f [Hz]’); ylabel(‘|H(j2\pi f)|’);

DR ILL 4.16 Magnitude Response from Pole-Zero Plots

Use the qualitative method of sketching the frequency response to show that the system with the pole-zero configuration in Fig. 4.54a is a highpass filter and the configuration in Fig. 4.54b is a bandpass filter.

4.10-5 Practical Filters and Their Specifications

For ideal filters, everything is black and white; the gains are either zero or unity over certain bands. As we saw earlier, real life does not permit such a worldview. Things have to be gray or shades of gray. In practice, we can realize a variety of filter characteristics that can only approach ideal characteristics.

An ideal filter has a passband (unity gain) and a stopband (zero gain) with a sudden transition from the passband to the stopband. There is no transition band. For practical (or realizable) filters, on the other hand, the transition from the passband to the stopband (or vice versa) is gradual and takes place over a finite band of frequencies. Moreover, for realizable filters, the gain cannot be zero over a finite band (Paley–Wiener condition). As a result, there can be no true stopband for practical filters. We therefore define a stopband to be a band over which the gain is below some small number Gs, as illustrated in Fig. 4.55. Similarly, we define a passband to be a band over which the gain is between 1 and some number Gp (Gp < 1), as shown in Fig. 4.55. We have selected the passband gain of unity for convenience. It could be any constant. Usually the gains are specified in terms of decibels. This is simply 20 times the log (to base 10) of the gain. Thus,

$$G(\text{dB}) = 20\log_{10} G$$

A gain of unity is 0 dB and a gain of ^√ 2 is 3.01 dB, usually approximated by 3 dB. Sometimes the specification may be in terms of attenuation, which is the negative of the gain in dB. Thus, a gain of 1/ ^√2, that is, 0.707, is ⁻3 dB, but is an attenuation of 3 dB.

Figure 4.55 Passband, stopband, and transition band in filters of various types.

In a typical design procedure, Gp (minimum passband gain) and Gs (maximum stopband gain) are specified. Figure 4.55 shows the passband, the stopband, and the transition band for typical lowpass, bandpass, highpass, and bandstop filters. Fortunately, the highpass, bandpass, and bandstop filters can be obtained from a basic lowpass filter by simple frequency transformations. For example, replacing s with ωc/s in the lowpass filter transfer function results in a highpass filter. Similarly, other frequency transformations yield the bandpass and bandstop filters. Hence, it is necessary to develop a design procedure only for a basic lowpass filter. Then, by using appropriate transformations, we can design filters of other types. The design procedures are beyond our scope here and will not be discussed. The interested reader is referred to [1].