Dorf, R.C., Wan, Z. "Transfer Functions of Filters The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
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 1 Introduction 10.2 Ideal filters Richard C. Dorf 10.3 The Ideal Linear -Phase Low -Pass Filter 10.4 Ideal Linear-Phase Bandpass Filters University of California, davis 10.5 Causal Filters Zhen wa 10.6 Butterworth 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 n ideal filter is a system that completely rejects sinusoidal inputs of the form x(t)=A cos t, -oo< t< oo, for o 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: Ideal low-pass:H(O) B≤0≤B (10.1) Ideal high-pass: H(O)/=Jo,-B<O<B (102 ≥B j,B1sol≤B2 Ideal bandpass: (H(o)o,all other o (103) Ideal bandstop:H(O) ≤o≤B (10.4) all other o c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 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 B B B ( ) , , w w w = - £ £ > Ï Ì Ô Ó Ô 1 0 Ideal high-pass: * * * * H B B B ( ) , , w w w = - < < ³ Ï Ì Ô Ó Ô 0 1 Ideal bandpass: all other * * * * H B B ( ) , , w w w = Ï £ £ Ì Ô Ó Ô 1 0 1 2 Ideal bandstop: all other * * * * H B B ( ) , , w w w = Ï £ £ Ì Ô Ó Ô 0 1 1 2 Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis
FIGURE 10.1 Magnitude functions of ideal filters: (a) low-pass;(b)high-pass;(c) bandpass;(d)bandstop. The stopband of an ideal filter is defined to be the set of all frequencies o for which the filter ompletely stops the sinusoidal input x(0)=A cos ot-∞<t<∞. The passband of the filter is the set of all frequencies o for which the input x(r) 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 B1 B2 B3 B4 of B, and B,, we can construct an ideal comb filter, whose magnitude function is illustrated in Fig. 10.2. FIGURE 10.2 Magnitude function of an ideal comb filter 0.3 The ideal linear - Phase Low-Pass Filter Consider the ideal low-pass filter with the frequency function (10.5) 0<-B,o>B t, is a positive real number. Equation(10.5)is the polar-form representation of H(o). From Eq (10.5) 0<-B,0>B Ho)=」-ota,-B≤0≤B e 2000 by CRC Press LLC
© 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, highpass, bandpass, and bandstop filters. For instance, by cascading bandstop filters with different values of B1 and B2, 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 td 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) B1 1 0 –B1 (c) B2 –B2 1 0 (d) 1 0 (b) w |H| H| |H|| –B B w w w B1 –B1 B2 –B2 FIGURE 10.2 Magnitude function of an ideal comb filter. |H| 1 0 –B4 –B3 –B2 –B1 B1 B2 B3 B4 H e B B B B j td ( ) , , , w w w w w = - £ £ < - > Ï Ì Ô Ó Ô - 0 *H * B B B B ( ) , , , w w w w = - £ £ < - > Ï Ì Ô Ó Ô 1 0 / ( ) , , , H t B B B B d w w w w w = - - £ £ < - > Ï Ì Ô Ó Ô0
FIGURE 10.3 Phase function of ideal low-pass filter defined by Eq (10.5) Bald Bl'd Slope a-t FIGURE 10.4 Phase function of ideal linear-phase bandpass filter. The phase function /H(o) of the filter is plotted in Fig. 10.3. Note that over the frequency range 0 to B, the hase function of the system is linear with slope equal to -tr 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(O). The impulse response of the ideal lowpass filter is h(t)=-Sa[B(t-ta)] ∞<t<o (10.6) x)/x. The impulse response h(n) of the ideal low-pass filter is not zero for t< 0. Thus, the before the impulse at t=0 and is said to be al. 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 H(0)= ro,B1≤ol≤B all other o where te B,, and B, are positive real numbers. The magnitude function is plotted in Fig. 10.(c)and the function is plotted in Fig. 10.4. The passband of the filter is from B, to B. The filter will pass the signal the band with no distortion, although there will be a time delay of ta seconds e 2000 by CRC Press LLC
© 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 –td. 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 linearphase bandpass filter is given by where td, B1, and B2 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 B1 to B2. The filter will pass the signal within the band with no distortion, although there will be a time delay of td 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) Btd –B 0 B –Btd Slope = –t d w H(w) B2t d 0 w Slope = –t d B1t d –B2 –B1 B2 B1 h t B Sa B t t t d ( ) = [ ( - )], - • < < • p H e B B j td ( ) , , w w w w = Ï £ £ Ì Ô Ó Ô - 1 2 0 * * all other
FIGURE 10.5 Causal filter magnitude functions: (a)low-pass;(b)high-pass;(c)bandpass;(d)bandstop 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 transition 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(o), the passband is defined as the set of all frequencies o for Ho)2|H(o)=0707H(o, (10.7) where is the value of o for which H(o) is maximum. Note that Eq (10.7) is equivalent to the condition that H(o)laB is less than 3 dB down from the peak value H(op)laB. 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 o for which H(ollas is down some desired amount(e.g,40 or 50 dB)from the peak value H(O laB. 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 H(s) Factoring the denominator of H(s), we see that the poles are located at ±j e 2000 by CRC Press LLC
© 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 wp 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(wp)* 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(wp)* 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 10.5 Causal filter magnitude functions: (a) low-pass; (b) high-pass; (c) bandpass; (d) bandstop. 0 w wp -wp 1 0.707 (a) 0 w 1 (b) 0 w 1 (c) 0 w 1 (d) *H * *H *. *H * p p ( ) w ³ (w ) . (w ) 1 2 0 707 H s s s n n n ( ) = + + w w w 2 2 2 2 s j n n = - ± w w 2 2