Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Two-Dimensional IIR Filters 23-25–1H – N× + H N–1+FIGURE 23.152D high-pass filter configuration.As can be seen in Figure 23.15, the complexity of the high-pass configuration tends to increase rapidlywith the number of rotations, and consequently, the number of rotations should be kept to a minimum.It should also be mentioned that the coefficients of the rotated filters must be properly adjusted, by usingEquation 23.65, to ensure that the zero-phase is approximated. However, the use of this transformationleads to another problem: the 2-D digital transfer function obtained has spurious zeros at the Nyquistpoints. These zeros are due to the fact that the transformation in Equation 23.65 does not have typepreservation in the neighborhoods of the Nyquist points but their presence does not appear to be ofserious concern.With the availability of circularly symmetric low-pass and high-pass filters, bandpass and bandstopfilters with circularly symmetric amplitude responses can be readily obtained. A bandpass filter can beobtained by connecting a low-pass filter and a high-pass filter with overlapping passbands in cascade,whereas a bandstop filter can be realized by connecting a low-pass filter and a high-pass filter withoverlapping passbands in parallel.23.4.3 Design of Filters Satisfying Prescribed SpecificationsA similar approach to that described in Section 23.3.3 can be used for the design of circularly symmetricfilters satisfying prescribed specifications. Assume that the maximum=minimum passband and themaximum stopband gain of the 2-D filter are (1 Dp) and Da, respectively, if K rotated filter sectionsare cascaded where half of the rotations are in the range of 1808–2708 and the other half are in the range2708–3608. Then, we can assign the passband ripple d p and the stopband loss d a to be [2]andd p ¼ DpK(23:70)d a ¼ Da 2=K (23:71)The lower (or upper) bound of the passband gain would be achieved if all the rotated sections were tohave minimum (or maximum) passband gains at the same frequency point. Although it is possible for allthe rotated sections to have minimum (or maximum) gains at the origin of the (v 1 , v 2 ) plane, the gainsare unlikely to be maximum (or minimum) together at some other frequency point and, in effect, thepreceding estimate for d p is low. A more realistic value for d p isd p ¼ 2DpK(23:72)If Dp and Da are prescribed, then the passband ripple and minimum stopband attenuation of the analogfilter can be obtained from Equations 23.72 and 23.71 as KA p ¼ 20 log(23:73)K 2Dp