2011年数学建模美赛论文

For office use only

T1________________ T2________________ T3________________ T4________________

Team Control Number

10709

Problem Chosen

B

For office use only

F1________________

F2________________

F3________________

F4________________

2011 Mathematical Contest in Modeling (MCM) Summary Sheet

(Attach a copy of this page to each copy of your solution paper.)

Type a summary of your results on this page. Do not include

the name of your school, advisor, or team members on this page.

The Coordination of Repeaters Based on the Frequency Division Multiplexing Theory

Abstract

A repeater is an automated device that extends the range of communications. Interference occurs if repeaters are not far enough apart or transmit on very close frequencies. Our goal is to create an optimal system which has the minimum number of repeaters while achieving high capacity with limited spectrum width.

We develop a theoretical model to give a rough estimation of the necessary repeater numbers considering factors such as the radius of the area, the radius of the repeater’s coverage, the number of the simultaneous users, and the available spectrum width.

Problem I is solved by two steps. One key issue is to cover the whole area seamlessly and to minimize the overlapping area of the adjacent repeaters’ coverage, thus reducing the co-frequency interference. Regular hexagons are chosen to fill the given area seamlessly instead of circles. Another vital issue is to accommodate 1000 users at the same time. We establish frequency division multiplexing model to solve the insufficiency of the spectrum width. By dividing the available spectrum width, we attain channels that only meet 24.6% of the total demand. We improve the user capacity by divide the area into cells and multiplex the spectrum width in cells far apart to meet the demand. We calculate that at least 13 repeaters are needed.

In problem II, as the number of simultaneous users increases from 1,000 to 10,000, while the given spectrum width and the number of PL remain unchanged, we need further expansion of the user capacity. Our model is the combination of the frequency division multiplexing model and the cell division model, and designed to solve the conflict between the rapidly increasing demand and the limited spectrum width. We adopt cell division model to divide the original cells into a proper number of micro cells, and again apply frequency division multiplexing model to the micro cells, thus greatly improving the user capacity. So the result is that 122 repeaters are needed.

In problem III, the changes of environment in mountainous areas are analyzed on its influence in communication. We contract a simulation model utilizing MATLAB software considering factors such as the height and the gradient of the mountain and apply it to 3 kinds of conditions. These situations involving in communication failure caused by mountains are analyzed. We calculate and generate the placement of the repeaters to overcome the difficulties in line-of-sight propagation caused by mountainous areas.

Combining the frequency division multiplexing model and the cell splitting model, our model can be successfully adopted to solve the insufficiency of spectrum resources and meet the rapidly increasing demand for radio communication.