where [k.sub.1,2] is the

normalized coupling coefficient between [C.sub.y1] and [C.sub.y3].

where [k.sub.cs] is the coupling spring constant, [k.sub.cc] is the

normalized coupling coefficient between the resonator tanks for a given filter type, and [k.sub.cl] is the resonator stiffness at the coupling location.

The relationship between the de-normalized coupling coefficient, k, and the normalized coupling coefficient, M, is given by

In this case, the relationship between the de-normalized coupling coefficient, k, and the normalized coupling coefficient, M, is given by

In Figure 8, the normalized threshold gain al was plotted as a function of Bragg frequency deviation [xi], for various values of the normalized coupling coefficient [k.sub.3]L (which takes values from 0.01 to 50).

In Figure 12, the normalized threshold gain aL was plotted as a function of Bragg frequency deviation S, for various values of the normalized coupling coefficient [k.sub.3]L (which takes values from 0.01 to 50).

In Figure 16, the normalized threshold gain [[alpha].sub.L] was plotted as a function of Bragg frequency deviation [xi], for various values of the normalized coupling coefficient [k.sub.3]L (which takes values from 0.01 to 50).

In Figure 20, the normalized threshold gain [[alpha].sub.L] was plotted as a function of Bragg frequency deviation [xi], for various values of the normalized coupling coefficient [k.sub.3]L (which takes values from 0.01 to 50).

They were made for the normalized coupling coefficients [absolute value of [k.sub.2]L] = 8, [absolute value of [k.sub.3]L] = 4 and filling factor f = 0.16.

They were made for the normalized coupling coefficients [absolute value of [k.sub.1]L] = 10.96, [absolute value of [k.sub.3]L] = 4 and filling factor f = 0.16.

The proposed the topology can be implemented by the structure shown in Figure 3, and its

normalized coupling coefficients using the gradient-based optimization method [36] are synthesized to be: [M.sub.S1] = 0.7856, [M.sub.S2] = 0.5703, [M.sub.1L] = -0.7856, [M.sub.2L] = 0.5703, [M.sub.23] = -1.156, [M.sub.SL] = 0.2116.

In the general case of a filter with N resonators, to calculate the coupling bandwidth for each resonator, the desired 3 dB bandwidth of the filter is multiplied by the

normalized coupling coefficients. Then using Equation 7, for the ferrite whose 4[pi][M.sub.s] value and diameter are known, the coupling loop diameter for the input, output and interstages can be calculated.