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

FIR Filters 18-451N ove ¼ N optL þ 2L(v sv p )=p(18:77)At these values of L, N F decreases and N 1 N 2 increases inversely proportionally to L with the minimumof N ove given byrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(v s v p )N ove ¼ 2N optp(18:78)taking place atrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(v s v p )L opt ¼ 1p(18:79)If for L ¼ L opt , u þ f is not approximately equal to p, then L minimizing the filter complexity can befound in the near vicinity of L opt . The following example illustrates the use of the above estimationformulas.Example 18.1Consider the specifications: v p ¼ 0.4p, v s ¼ 0.402p, d p ¼ 0.01, and d s ¼ 0.001. For the optimum conventionaldirect-form design, N opt ¼ 2541, requiring 1271 multipliers when the coefficient symmetry isexploited. Equation 18.79 gives L opt ¼ 16. Table 18.12 shows, for the admissible values of L in the vicinityof this value, l, u, f, the estimated orders for the subfilters, and the sum of the subfilter orders as well aswhether the overall filter is a Case A or Case B design. For N F , the minimum even order larger than orequal to the estimated order is used, whereas N 2 is forced to be even (odd) if N 1 is even (odd).Also with the estimated filter orders of Table 18.12, L ¼ 16 gives the best result. The actual filter ordersare N F ¼ 162, N 1 ¼ 70, N 2 ¼ 98. The responses of the subfilters as well as that of the overall design aredepicted in Figure 18.25. The overall number of multipliers and adders for this design are 168 and 330,respectively, which are 13 percent of those required by an equivalent conventional direct-form design(1271 and 2541). The overall filter order is 2690, which is only 6 percent higher than that of the directformdesign (2541).TABLE 18.12 Estimated Filter Orders for the Admissible Values of L in the Vicinity of L opt ¼ 16L Case l u f N F N 1 N 2 N F þ N 1 þ N 28 B 2 0.784p 0.8p 318 98 26 4429 B 2 0.382p 0.4p 282 38 58 37811 A 2 0.4p 0.422p 232 47 69 34812 A 2 0.8p 0.824p 212 162 38 41213 B 3 0.774p 0.8p 196 155 43 39414 B 3 0.372p 0.4p 182 58 92 33216 A 3 0.4p 0.432p 160 70 98 32817 A 3 0.8p 0.834p 150 236 54 44018 B 4 0.764p 0.8p 142 210 58 41019 B 4 0.362p 0.4p 134 78 128 34021 A 4 0.4p 0.442p 122 92 128 34222 A 4 0.8p 0.844p 116 314 68 498