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

Two-Dimensional IIR Filters 23-5a 1 s 1 þ a 2 s 2g 4 (s 1 , s 2 ) ¼1 þ b 1 s 2 1 þ (23:13)ð s2 2Þþb 2 s 1 s 2and they proved that the stability of g 4 (s 1 , s 2 ) is ensured ifb 1 > 0 (23:14)andb 1 > b2 24b 2 1 > 0 (23:15)However, it is necessary as earlier to include a guard filter, which may have the simple form ofG(z 1 , z 2 ) ¼ (1 þ z 1)(1 þ z 2 )(d 1 þ z 1 )(d 2 þ z 2 )(23:16)in order to remove the high-pass regions along all radii except the coordinate axes.Then, through an optimization procedure, the coefficients of g 4 (s 1 , s 2 ) and G(z 1 , z 2 ) are calculatedsubject to the constraints of Equations 23.14 and 23.15, so that the cutoff frequency of the 1-D filter ismapped into a desired cutoff boundary in the (V 1 , V 2 ) plane.23.2.2 Spectral TransformationsSpectral transformation is another kind of important transformation in the design of both 1-D and 2-DIIR filters. In this section, three groups of spectral transformations are discussed. Among them, thelinear transformations map frequency axes onto frequency axes in the (V 1 , V 2 ) plane, the complextransformation is of wide applications to the design of fan filters, and the Constantinides transformationstransform a discrete function into another discrete function and through which any transformation of alow-pass filter to another low-pass, high-pass, bandpass, or bandstop filter becomes possible.23.2.2.1 Linear TransformationsConsider a group of linear transformations that map frequency axes onto themselves in the (V 1 , V 2 )plane. There are eight possible such transformations [7,8] and they have the algebraic structure of a finitegroup under the operation of multiplication [2]; each transformation can be expressed as v 1v: ¼ D(T) 1(23:17)v 2v 2where D(T) isa23 2 unitary matrix representing transformation Td. The eight transformations andtheir effect on the frequency response of the digital filter are as follows with a multiplication table beingillustrated in Table 23.1 [2].1. Identity (I):2. Reflection about the v 1 axis (r v1 ):D(I) ¼ 1 00 1D(r v1 ) ¼ 1 00 1