Dorf, R.C., Wan, Z. “Transfer Functions of Filters” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 10 Transfer Functions of Filters 10.1 Introduction 10.2 Ideal Filters 10.3 The Ideal Linear-Phase Low-Pass Filter 10.4 Ideal Linear-Phase Bandpass Filters 10.5 Causal Filters 10.6 Butterworth Filters 10.7 Chebyshev Filters 10.1 Introduction Filters are widely used to pass signals at selected frequencies and reject signals at other frequencies. An electrical filter is a circuit that is designed to introduce gain or loss over a prescribed range of frequencies. In this section, we will describe ideal filters and then a selected set of practical filters. 10.2 Ideal Filters An ideal filter is a system that completely rejects sinusoidal inputs of the form x(t) = A cos wt, –￥ < t < ￥, for w in certain frequency ranges and does not attenuate sinusoidal inputs whose frequencies are outside these ranges. There are four basic types of ideal filters: low-pass, high-pass, bandpass, and bandstop. The magnitude functions of these four types of filters are displayed in Fig. 10.1. Mathematical expressions for these magnitude functions are as follows: (10.1) (10.2) (10.3) (10.4) Ideal low-pass: ** ** H BB B () , , w w w = -￡￡ > ì í ? ? ? 1 0 Ideal high-pass: ** ** H BB B () , , w w w = -<< 3 ì í ? ? ? 0 1 Ideal bandpass: all other ** ** H BB () , , w w w = ￡￡ ì í ? ? ? 1 0 12 Ideal bandstop: all other ** ** H BB () , , w w w = ￡￡ ì í ? ? ? 0 1 12 Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis ? 2000 by CRC Press LLC The stopband of an ideal filter is defined to be the set of all frequencies w for which the filter completely stops the sinusoidal input x(t) = A cos wt, –￥ < t < ￥. The passband of the filter is the set of all frequencies w for which the input x(t) is passed without attenuation. More complicated examples of ideal filters can be constructed by cascading ideal low-pass, high- pass, bandpass, and bandstop filters. For instance, by cascading bandstop filters with different values of B 1 and B 2 , we can construct an ideal comb filter, whose magnitude function is illustrated in Fig. 10.2. 10.3 The Ideal Linear-Phase Low-Pass Filter Consider the ideal low-pass filter with the frequency function (10.5) where t d is a positive real number. Equation (10.5) is the polar-form representation of H(w). From Eq. (10.5) we have and FIGURE 10.1 Magnitude functions of ideal filters:(a) low-pass; (b) high-pass; (c) bandpass; (d) bandstop. |H | B 1 0 –B (a) B 1 1 0 –B 1 (c) B 2 –B 2 1 0 (d) 1 0 (b) w |H | |H ||H | B–B ww w B 1 –B 1 B 2 –B 2 FIGURE 10.2Magnitude function of an ideal comb filter. |H | 1 0 –B 4 –B 3 –B 2 –B 1 B 1 B 2 B 3 B 4 H eBB BB jt d () , , , w w ww w = -￡￡ <- > ì í ? ? ? - 0 **H BB BB () , , , w w ww = -￡￡ <- > ì í ? ? ? 1 0 / () , , , H tB B BB d w ww ww = --￡￡ <- > ì í ? ? ?0 ? 2000 by CRC Press LLC The phase function / H(w) of the filter is plotted in Fig. 10.3. Note that over the frequency range 0 to B, the phase function of the system is linear with slope equal to –t d . The impulse response of the low-pass filter defined by Eq. (10.5) can be computed by taking the inverse Fourier transform of the frequency function H(w). The impulse response of the ideal lowpass filter is (10.6) where Sa(x) = (sin x)/x. The impulse response h(t) of the ideal low-pass filter is not zero for t < 0. Thus, the filter has a response before the impulse at t = 0 and is said to be noncausal. As a result, it is not possible to build an ideal low-pass filter. 10.4 Ideal Linear-Phase Bandpass Filters One can extend the analysis to ideal linear-phase bandpass filters. The frequency function of an ideal linear- phase bandpass filter is given by where t d , B 1 , and B 2 are positive real numbers. The magnitude function is plotted in Fig. 10.1(c) and the phase function is plotted in Fig. 10.4. The passband of the filter is from B 1 to B 2 . The filter will pass the signal within the band with no distortion, although there will be a time delay of t d seconds. FIGURE 10.3 Phase function of ideal low-pass filter defined by Eq. (10.5). FIGURE 10.4 Phase function of ideal linear-phase bandpass filter. H(w) Bt d 0 B–B –Bt d Slope = –t d w H(w) B 2 t d 0 w Slope = –t d B 1 t d –B 2 –B 1 B 2 B 1 ht B SaBt t t d () [( )], =--￥<<￥ p H eB B jt d () , , w w w w = ￡￡ ì í ? ? ? - 12 0 ** all other ? 2000 by CRC Press LLC 10.5 Causal Filters As observed in the preceding section, ideal filters cannot be utilized in real-time filtering applications, since they are noncausal. In such applications, one must use causal filters, which are necessarily nonideal; that is, the transition from the passband to the stopband (and vice versa) is gradual. In particular, the magnitude functions of causal versions of low-pass, high-pass, bandpass, and bandstop filters have gradual transitions from the passband to the stopband. Examples of magnitude functions for the basic filter types are shown in Fig. 10.5. For a causal filter with frequency function H(w), the passband is defined as the set of all frequencies w for which (10.7) where w p is the value of w for which *H(w)* is maximum. Note that Eq. (10.7) is equivalent to the condition that *H(w)* dB is less than 3 dB down from the peak value *H(w p )* dB . For low-pass or bandpass filters, the width of the passband is called the 3-dB bandwidth. A stopband in a causal filter is a set of frequencies w for which *H(w)* dB is down some desired amount (e.g., 40 or 50 dB) from the peak value *H(w p )* dB . The range of frequencies between a passband and a stopband is called a transition region. In causal filter design, a key objective is to have the transition regions be suitably small in extent. 10.6 Butterworth Filters The transfer function of the two-pole Butterworth filter is Factoring the denominator of H(s), we see that the poles are located at FIGURE 10.5Causal filter magnitude functions: (a) low-pass; (b) high-pass; (c) bandpass; (d) bandstop. 0 w w p -w p 1 0.707 (a) 0 w 1 (b) 0 w 1 (c) 0 w 1 (d) ****.**HHH pp () () . ()www3 1 2 0707 Hs ss n nn () = ++ w ww 2 22 2 sj nn =- ± ww 22 ? 2000 by CRC Press LLC Note that the magnitude of each of the poles is equal to w n . Setting s = jw in H(s), we have that the magnitude function of the two-pole Butterworth filter is (10.8) From Eq. (10.8) we see that the 3-dB bandwidth of the Butterworth filter is equal to w n . For the case w n = 2 rad/s, the frequency response curves of the Butterworth filter are plotted in Fig. 10.6. Also displayed are the frequency response curves for the one-pole low-pass filter with transfer function H(s) = 2/(s + 2), and the two- pole low-pass filter with z = 1 and with 3-dB bandwidth equal to 2 rad/s. Note that the Butterworth filter has the sharpest cutoff of all three filters. 10.7 Chebyshev Filters The magnitude function of the n-pole Butterworth filter has a monotone characteristic in both the passband and stopband of the filter. Here monotone means that the magnitude curve is gradually decreasing over the passband and stopband. In contrast to the Butterworth filter, the magnitude function of a type 1 Chebyshev filter has ripple in the passband and is monotone decreasing in the stopband (a type 2 Chebyshev filter has the opposite characteristic). By allowing ripple in the passband or stopband, we are able to achieve a sharper transition between the passband and stopband in comparison with the Butterworth filter. The n-pole type 1 Chebyshev filter is given by the frequency function (10.9) where T n (w/w 1 ) is the nth-order Chebyshev polynomial. Note that e is a numerical parameter related to the level of ripple in the passband. The Chebyshev polynomials can be generated from the recursion T n (x) = 2xT n – 1 (x) – T n – 2 (x) where T 0 (x) = 1 and T 1 (x) = x. The polynomials for n = 2, 3, 4, 5 are T 2 (x) = 2x(x) – 1 = 2x 2 – 1 T 3 (x) = 2x(2x 2 – 1) – x = 4x 3 – 3x T 4 (x) = 2x(4x 3 – 3x) – (2x 2 – 1) = 8x 4 – 8x 2 + 1 T 5 (x) = 2x(8x 4 – 8x 2 + 1) – (4x 3 – 3x) = 16x 5 – 20x 3 + 5x (10.10) FIGURE 10.6Magnitude curves of one- and two-pole low-pass filters. 2 s + 2 w Two-pole Butterworth filter Two-pole filter with z = 1 One-pole filter H(s) = 1 2 3 4 5 6 7 8 9 010 0.707 1 0.8 0.6 0.4 0.2 0 Passband |H(w)| **H n () (/) w ww = + 1 1 4 **H T n () () w ww = + 1 1 22 1 e / ? 2000 by CRC Press LLC Using Eq. (10.10), the two-pole type 1 Chebyshev filter has the following frequency function For the case of a 3-dB ripple (e = 1), the transfer functions of the two-pole and three-pole type 1 Chebyshev filters are where w c = 3-dB bandwidth. The frequency curves for these two filters are plotted in Fig. 10.7 for the case w c = 2.5 rad. The magnitude response functions of the three-pole Butterworth filter and the three-pole type 1 Chebyshev filter are compared in Fig. 10.8 with the 3-dB bandwidth of both filters equal to 2 rad. Note that the transition from passband to stopband is sharper in the Chebyshev filter; however, the Chebyshev filter does have the 3- dB ripple over the passband. FIGURE 10.7Frequency curves of two- and three-pole Chebyshev filters with w c = 2.5 rad/s: (a) magnitude curves; (b) phase curves. w Three-pole filter Two-pole filter 1 2 3 4 5 6 7 8 9 010 0.707 1 0.8 0.6 0.4 0.2 0 Passband |H(w)| (a) w Three-pole filter Two-pole filter 1 2 3 4 5 6 7 8 9 0 10 –50° –100° –150° –200° –250° –300° 0 |H(w)| (b) **H() [(/) ] w ww = +- 1 12 1 2 1 22 e Hs ss Hs sss c cc c ccc () . .. () . . . = ++ = +++ 050 0645 0708 0251 0597 0928 0251 2 22 3 3223 w ww w www ? 2000 by CRC Press LLC Defining Terms Causal filter: A filter of which the transition from the passband to the stopband is gradual, not ideal. This filter is realizable. 3-dB bandwidth: For a causal low-pass or bandpass filter with a frequency function H(jw): the frequency at which *H(w)* dB is less than 3 dB down from the peak value *H(w p )* dB . Ideal filter: An ideal filter is a system that completely rejects sinusoidal inputs of the form x(t) = A cos wt, –￥ < t < ￥, for w within a certain frequency range, and does not attenuate sinusoidal inputs whose frequencies are outside this range. There are four basic types of ideal filters:low-pass, high-pass, bandpass, and bandstop. Passband: Range of frequencies w for which the input is passed without attenuation. Stopband: Range of frequencies w for which the filter completely stops the input signal. Transition region: The range of frequencies of a filter between a passband and a stopband. Related Topics 4.2 Low-Pass Filter Functions ? 4.3 Low Pass Filters ? 11.1 Introduction ? 29.1 Synthesis of Low-Pass Forms References R.C. Dorf, Introduction to Electrical Circuits, 3rd ed., New York: Wiley, 1996. E.W. Kamen, Introduction to Signals and Systems, 2nd ed., New York: Macmillan, 1990. G.R. Cooper and C.D. McGillem, Modern Communications and Spread Spectrum, New York: McGraw-Hill, 1986. Further Information IEEE Transactions on Circuits and Systems, Part I: Fundamental Theory and Applications. IEEE Transactions on Circuits and Systems, Part II: Analog and Digital Signal Processing. Available from IEEE. FIGURE 10.8 Magnitude curves of three-pole Butterworth and three-pole Chebyshev filters with 3-dB bandwidth equal to 2.5 rad/s. w Three-pole Chebyshev 1 2 3 4 5 6 7 8 9 010 0.707 1 0.8 0.6 0.4 0.2 0 Passband |H(w)| Three-pole Butterworth ? 2000 by CRC Press LLC