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

High-Order Filters 15-9where we introduced the abbreviationsY if i ¼ F i A j q j andY ik i ¼ aK i A j q j (15:26)To realize the prescribed third-order functionj¼1j¼1H LP (p) ¼ V oV i¼ a 3p 3 þ a 2 p 2 þ a 1 p þ a 0p 3 þ b 2 p 2 þ b 1 p þ b 0(15:27)we compare coefficients between Equations 15.25 and 15.27. For the denominator terms we obtainb 2 ¼ q 1 þ q 2 þ q 3 þ f 1b 1 ¼ q 1 q 2 þ q 1 q 3 þ q 2 q 3 þ f 1 (q 2 þ q 3 ) þ f 2b 0 ¼ q 1 q 2 q 3 þ f 1 q 2 q 3 þ f 2 q 3 þ f 3These are three equations in six unknowns, f i and q i , i ¼ 1, . . . , 3, which can be written more convenientlyin matrix form:0101 01@1 0 0q 2 þ q 3 1 0A f 1@ f 2A ¼ @q 2 q 3 q 3 1 f 3b 2 (q 1 þ q 2 þ q 3 )b 1 (q 1 q 2 þ q 1 q 3 þ q 2 q 3 )b 0 q 1 q 2 q 3A (15:28)The transmission zeros are found via an identical process: the unknown parameters k i are computedfrom an equation of the form (Equation 15.28) with f i replaced by k i =(aK 0 ) and b i replaced by a i =a 3 ,i ¼ 1, . . . , 3. Also, K 0 ¼ a 3 =a.The unknown parameters f i can be solved from the matrix expression (Equation 15.28). It is a set oflinear equations whose coefficients are functions of the prescribed coefficients b i and of the numbers q iwhich for given Q are determined by the quality factors Q i of the second-order sections T i (s). Thus, the Q iare free parameters that may be selected to satisfy any criteria that may lead to a better-working circuit.The free design parameters may be chosen, for example, to reduce a circuit’s sensitivity to elementvariations. This leads to a multiparameter (i.e., the nQ i -values) optimization problem whose solutionrequires the availability of the appropriate computer algorithms. If such software is not available, specificvalues of Q i can be chosen. The design becomes particularly simple if all the Q i -factors are equal, a choicethat has the additional practical advantage of resulting in all identical second-order building blocks,T i (s) ¼ T(s). For this reason, this approach has been referred to as the ‘‘Primary Resonator Block’’ (PRB)technique. The passband sensitivity performance of PRB circuits is almost as good as that of fullyoptimized FLF structures. The relevant equations are derived in the following:With q i ¼ q for all i we find from Equation 15.280 101 0 1It shows that1 0 0@ 2q 1 0A f 1@ f 2A ¼ @q 2 q 1 f 3b 2b 1 3q 2b 0 q 33qA (15:29)F 1 A 1 q ¼ f 1 ¼ b 2 3qF 2 A 1 A 2 q 2 ¼ f 2 ¼ b 1 3q 2 2qf 1(15:30)F 3 A 1 A 2 A 3 q 3 ¼ f 3 ¼ b 0 q 3 q 2 f 1 qf 2