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

1-10 Passive, Active, and Digital Filtersthat is, a 0 ¼ b 0 ¼ 1. From the definition of the Laplace transform,H(s) ¼¼¼¼ð10ð10ð10ð10h(t)e st dth(t) 1 st þ s2 t 2h(t)dth(t)dtð1s02! dtð1th(t)dt þ s2t 2 h(t)dt2!0 st D þ s22!t 2 R2p þ t2 D (1:4)Alternatively, from Equation 1.3, direct division givesH(s) ¼ 1 ðb 1 a 1 Þs þ b 2 1a 1 b 1 þ a 2 b 2 s 2 þ (1:5)Now by comparing Equations 1.4 and 1.5, we deduceð10h(t)dt ¼ 1, t D ¼ b 1 a 1andt R ¼ 2p b 2 1 a 2 1 þ 2 a ð 2 b 2 Þ 1=2The previous definitions are based on the assumption that the unit-step response approaches unity ast !1. If this is not the case, i.e., coefficients a 0 and b 0 are not equal to unity, then we can writewhere K ¼ a 0 =b 0 andH(s) ¼ KH 0 (s)H 0 (s) ¼ 1 þ a0 1 s þ a0 2 s2 þþa 0 M sM1 þ b 0 1 s þ b0 2 s2 þþb 0 N sNUsing the coefficients of H 0 (s) in the formulas for t D and t R yields approximate values for the delay timeand rise time, since these parameters are independent of the absolute value of the step response.1.4 Frequency-Domain AnalysisThe frequency response of an analog filter is deduced by finding its steady-state sinusoidal response, aswe shall now demonstrate.