Figure 4 Flowchart of the CGA illustrating its two phases

Figure 5 Action of the community crossover operator. Note that individual chromosomes are untouched

Finally, the community mutation operator is applied. Each bit in the community chromosomes of the population created by community crossover is negated with probability pmc . At this point, all chromosomes are re-evaluated; if the design goal is not met, the individual loop is entered after Ngenc

iterations of community evaluation, selection, crossover, and mutation.

In the individual loop, each community acts like its own population, and hence the individual operators are the same as their SGA counterparts, except that they only operate on the individual chromosomes. Stochastic binary tournament selection is used, along with one-point crossover applied with probability pci . and standard mutation applied with probability pmi .. After Ngeni individual generations, the algorithm is returned to the community loop, and the process begins anew.

3. NUMERICAL RESULTS

This selection relates the results of applying the methods described in the preceding two sections to the optimization of a fourand a six-post E-plane filter. In both examples, considering only symmetric filters accelerated the optimization. For instance, for the four-post structure, it was assumed that s1 s s4 , s2 s s3, t1 s t4 , t2 s t3, and l1 s l3. This al-

lowed the GA to operate on fewer parameters, and was also used to accelerate the analysis by eliminating redundant computations. The width of the waveguide was taken as

a s 7.112 mm as in w4x.. The CGA parameters were Ngenc s 1,

Ngeni s 4, Npopc s 70, Npopi s 70, Nselc s Nseli s 4, and pcc s pci

s 0.8, and the mutation rates pmc and pmi varied between 0.005 and 0.015. The limit parameters used to decode s, t,

and l were in micrometers. smin s 70, smax s 4000, tmin s 70, tmax s 1000, lmin s 500, and lmax s 7000, with Ns s 7, Nt s 5, and Nl s 7.

The four-post filter was designed to have a passband between 34.8 and 35.3 GHz with a passband rejection of less than y25 dB. The CGA was run for ten iterations consisting of a single community generation followed by four individual iterations. The frequency response of the resulting design is shown in Figure 6, and design parameters are listed in Table 1.

The six-post filter was designed to have the same passband as the four-post filter; however, a faster rolloff was desired. To encourage this, the stopband test frequencies were placed very close to edges of the passband, and the filter was allowed

TABLE 1 Optimized Dimensions of the Four-Post Filter

a passband rejection of y15 dB. the CGA parameters remained unaltered. The characteristics of the resulting filter are shown in Figure 7, and its design parameters are listed in Table 2.

While the results achieved by the CGA are comparable to those in the literature see, e.g., w4x., the algorithm is interesting for two reasons. First, no design knowledge is used in the CGA whatsoever; it is a fully automated algorithm that begins with randomly generated designs. Second, it demonstrates how GAs can be used to efficiently design devices that are expensive to analyze numerically.