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

Finite Wordlength Effects 20-7In general, for any N," #s 2 o ¼ E{e2 (n)} ¼ Nh 2 (0) þ XN 1(N þ 1 k)h 2 (k) s 2 x s2 e r(20:32)k¼1Notice that if the order of summation of the product terms in the convolution summation is changed,then the order in which the h(k)’s appear in Equation 20.32 changes. If the order is changed so that theh(k) with smallest magnitude is first, followed by the next smallest, etc., then the roundoff noise varianceis minimized. However, performing the convolution summation in nonsequential order greatly complicatesdata indexing and so may not be worth the reduction obtained in roundoff noise.20.4.2 Roundoff Noise in Fixed-Point IIR FiltersTo determine the roundoff noise of a fixed-point infinite impulse response (IIR) filter realization,consider a causal first-order filter with impulse responserealized by the difference equationDue to roundoff error, the output actually obtained ish(n) ¼ a n u(n) (20:33)y(n) ¼ ay(n 1) þ x(n) (20:34)^y(n) ¼ Q{ay(n 1) þ x(n)} ¼ ay(n 1) þ x(n) þ e(n) (20:35)where e(n) is a random roundoff noise sequence. Since e(n) is injected at the same point as the input, itpropagates through a system with impulse response h(n). Therefore, for fixed-point arithmetic withrounding, the output roundoff noise variance from Equations 20.6, 20.12, 20.25, and 20.33 iss 2 o ¼ D212X 1n¼ 1h 2 (n) ¼ D2 X 1a 2n ¼ 2 2B 112 12 1 an¼02 (20:36)With fixed-point arithmetic there is the possibility of overflow following addition. To avoid overflow it isnecessary to restrict the input signal amplitude. This can be accomplished by either placing a scalingmultiplier at the filter input or by simply limiting the maximum input signal amplitude. Consider thecase of the first-order filter of Equation 20.34. The transfer function of this filter isH(e jv ) ¼ Y(e jv )X(e jv ) ¼ 1e jva(20:37)soH(e jv ) 2 1¼1 þ a 2 2a cos (v)(20:38)andH(e jv ) max¼ 11 jaj(20:39)