2011年 美国大学生数学建模竞赛 B题 论文

A non-linear programming method for the distribution of

VHF repeaters

1.Introduction

With the development of technology, the limitation of VHF radio line-of-sight transmission and reception has been solved by the use of repeaters. What’s more, the technology of “continuous tone-coded squelch system” (CTCSS) help to mitigate the interference problems between repeaters which are closed in geographical distance.

In order to use this technology better, the number of repeaters and how to distribute them are in view. In this paper, seven models are used to simulate the repeaters’ distribution in the circular flat area of 40 miles radius and find out the minimum of repeaters for 1,000 and 10,000 simu ltaneous users. To simplify and to easily understand the model, evenly distributed principle is applied to the models. To find out the optimal solution, we calculate the minimum distance and PL number as constraints by using the regularity of trans mission between repeaters. We firstly come to one-dimension model and follows in second- dimension flat model, non-flat model and the three-dimension model to solve the similar problem such as the distribution of satellite, and then we have a lot of results and analysis.

2.Assumptions

●The users meet the evenly distributed principle in the circle. To meet the minimum number of repeaters, the repeaters meet the evenly distributed principle too.

● One repeater for one user service.

●The system must be a network, in which a user can transmit information to another user wherever in the area.

● Each repeater has only one receiv e frequency.

● Only the same repeaters which share the same frequency and PL number cause interference.

● The transmission cycle: According to the regularity of transmission—each repeater can only transmit frequency to the either 600 kHz above or 600 kHz below its own receive frequency, the same repeaters can only transmit frequency in this way: transmit from repeater A to B, from B to C, from C to B, finally from B to A (we can add more repeaters in this circle) to avoid the interference problems.

●All of the transmission cycle for each pair of same repeaters can be included in a circle—called including circle, and the whole area can be covered by many including circles.

● The distance of interfere and the distance of communication is Proportional, we as sume it’s 2.When two repeaters want to communicate , their distance must be less than the communication distance, and if two repeaters have intersections in their communication distance, The user’s transition in the intersection may be in trouble.