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 linear￾phase 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