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

Finite Wordlength Effects 20-9To obtain an output noise-to-signal ratio we note thatand, using the approach of [6],whereH(e jv ) ¼11 2r cos (u)e jv þ r 2 e j2v (20:48)H(e jv ) 2 max ¼ 1n 4r 2 sat 1 þ r22rcos (u) 1 þ r 2þ 21 r2rcos (u)2 2o (20:49)2rsin (u)sat(m) ¼(1 m > 1m 1 m 11 m < 1(20:50)Following the same approach as for the first-order case then givess 2 os 2 y¼ v 2 2B121 þ r 21 r 2 3(1 þ r 2 ) 2 4r 2 cos 2 (u)1 n 4r 2 sat 1 þ r22rcos (u) 1 þ r 2þ 21 r2rcos (u)2 2o (20:51)2rsin (u)Figure 20.1 is a contour plot showing the noise-to-signal ratio of Equation 20.51 for v ¼ 1 in units ofthe noise variance of a single quantization 2 2B =12. The plot is symmetrical about u ¼ 908, so only therange from 08 to 908 is shown. Notice that as r ! 1, the roundoff noise increases without bound. Alsonotice that the noise increases as u ! 08.901.01 1.2 2 5 20 100 1000Pole angle (degrees)80706050403020101E61E800.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99Pole radiusFIGURE 20.1Normalized fixed-point roundoff noise variance.