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

Frequency Transformations 3-5r 2 = 1+–V in+V i–r 1 = 1g m V ig m = 10r L = 1 c L = 1+V out–FIGURE 3.4Normalized transconductance amplifier.capacitance, and transconductance gain values are in the range of kV, pF, and mA=V, which areappropriate for the implementation of the circuit as an integrated circuit in CMOS technology.SolutionThe required location of the pole and range of values for the circuit elements can be achieved usingfrequency and impedance scaling factors v o ¼ 2p310 7 and a ¼ 10 4 , respectively. These result inR 1 ¼ ar 1 ¼ 10 kV, R 2 ¼ ar 2 ¼ 10 kV, g ¼ g m =a ¼ 1000 mA=V, R L ¼ ar L ¼ 10 kV, and C L ¼ c L =av o ¼ 1.59 pF.3.3 Low-Pass to High-Pass TransformationThe LPP to high-pass transformation is defined bys ¼ v2 op(3:8)Using this substitution in the LPP transfer function (Equation 3.4), it becomesT HP (P) ¼ K pn m ðp p z1 Þðp p z2 Þðp p zm Þ (3:9)p p p1 p pp2 p ppnwhere the poles and zeros of Equation 3.8 are given byp zi ¼ v2 os zifor i 2 f1, 2, ..., mgp pj ¼ v2 os pjfor j 2 f1, 2, ..., ng(3:10)It can be seen that zeros and poles of T HP (p) are reciprocal to those of T LPP (s) and scaled by the factorv 2 o . T HP (p) has n m zeros at s ¼ 0, which can be considered to originate from n m zeros at 1 inT LPP (s).Let us consider now the transformation of the imaginary axis in s to the imaginary axis in p. For s ¼ jV,p takes the form p ¼ jv wherev ¼v2 oV(3:11)From Equation 3.11, it can be seen that positive frequencies in the LPP transform to reciprocal and scaledfrequencies of the HP filter. Specifically, the frequency range from 0 to 1 in V maps to the frequencyrange 1 to 0 in v, while the range 1 to 0 in V maps to 0 to 1 in v. The passband edge frequency