The usual design procedure is of a trial-and-error nature. Starting from scratch, a series of distribution functions would be chosen a priori and, from the conditions of the problem, the sizes of the paired antennas and the fence location are computed for each case. The cost of each system would then be calculated; presumably, if all the criteria were met, the least costly system would be the one of choice. There is no way of knowing for certain whether a distribution function other than those tested could lead to a lower cost system. We propose to reverse this procedure. Instead of choosing a distribution function in advance and then determining the cost of the system, we will attempt to attack the problem by first setting up a cost function and then computing the particular distribution function that will make this cost function a minimum. The mathematical tools that will be utilized are mainly those of the calculus of variations and optimal control theory. It turns out that when certain reasonable simplifying assumptions (to be specified below) are made, very simple one-term formulas for the optimum radii of the two antennas and the minimum system cost result, even though the intermediate mathematical analysis is rather unwieldy. Likewise, the necessary computer programs require only a few minutes running time. These simplified formulas enable the designer to visualize the effects of parameter changes with very little difficulty. 2. COST FUNCTION FOR THE TRANSMITTER AND RECEIVER Quite obviously, the choice of the cost function is going to affect our analysis in a very substantial way. Too complicated a function will result in losing all insight as to what is going on. Too simple a function will result in essential features of the problem being omitted. The following function, however, seems to be a good compromise. If C is the total cost, we write Here, Ro, Rt, and RF represent the unknown radii of the ground receiver, space transmitter, and fence location, respectively. The quantities a0, at, aF are constants corresponding to the cost per unit area of the receiver, transmitter, and land enclosed by the fence around the installation, in that order. It may be mentioned here that although the above function is a quadratic in terms of Rq, Ri, and RF, the method sketched below will work without much modification if, for example, linear terms and/or cubic terms are added to C. Indeed, the essential feature is that C must be a monotonically increasing function of Ro, Ri, and RF For C to be determined in dollars, a0, a,, and aF must all be known explicitly. However, the minimum cost C will occur at the same values of Ro, Rlt and RF if we divide C by any constant. Thus, we write so that only two ratios are necessary, i.e., and aF/a0. The quantity C/a0 represents the system cost per unit cost of the ground antenna. Engineering cost estimates are such that
RkJQdWJsaXNoZXIy MTU5NjU0Mg==