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