MCM 特等奖 论文.docx

1、MCM 特等奖 论文Contents.Introduction.2.Model.2 2.1. Description of the cellular model.32.2. Basic Model .4 2.2.1 By calculating the population density to calculate the length of each cellular side, determine the cell size.4 2,2,2. Draw the cellular. Get the number and distribution of repeaters.42.2.3. Mo

2、del verification.52.3. Improved Model.62.3.1 By calculating the population density to calculate the length of each cellular side, determine the cell size.62,3.2. Draw the cellular. Get the number and distribution of repeaters.62.3.3. Model verification.82.3.4. The situation when people in distributi

3、on.82.4. Extra models.13 2.4.1.Make assumptions .13 2.4.2 Foundation of Model.14 2.4.3. Model verification.17.Conclusion and Expectation.17. Strengths and Weaknesses.18.Reference18I. Introduction In order to have better understanding of our model, the following background is worth mentioning.The VHF

4、 radio spectrum is the kind of radio wave whose band is 30 MHz to 300MHz in wireless communication. Due to its higher frequency, the sky-wave will generally penetrate the ionosphere and fire to the space, rather than be reflected back to the ground. As VHF has a limited capacity of diffraction ，it t

5、ransmits by line-of-sight transmission. The propagation distance is restricted not only by sight, but also by mountains and tall buildings like light. To spread even further, we must rely on repeaters, for a widen dissemination. Repeaters pick up weak signals, amplify them, and retransmit them on a

6、different frequency. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies. In addition to geographical separation, the “continuous tone-coded squelch system” (CTCSS) can be used to mitigate interference problems. In this

7、 paper, we try to use the minimum number of repeaters to completely accommodate 1,000（or10000） simultaneous users in a circular flat area of radius 40 miles( Assuming that the spectrum available is 145 to 148 MHz, the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the r

8、eceiver frequency, and there are 54 different PL tones available) and discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas. Reasoning from the given information and 1, we can draw the conclusion that one repeater only have one particular PL. In order

9、 to simplify the model, here we use wired repeater (each of the 2 repeaters are connected by cable).We are now facing the following task:1. Understanding the mechanism of line-of-sight transmission;2. Discussing the influence of repeaters caused by the population distribution;3. Evaluate the influen

10、ce of the coverage of repeaters caused by the different terrains;4. Make a cellular model, find the optimal solution.Our approach is :Get the least number of repeaters, which can totally accommodate 1,000 simultaneous users by using cellular model in different population distribution (uniform distri

11、bution or distribution) and different terrains (flat areas or mountains areas). .Model2.1. Description of the cellular model:Why do we use the cellular model?Cellular communications has experienced explosive growth in the past two decades. Cellular network is widely applied because of a Mathematical

12、 conjecture: Regular hexagon is considered to cover the largest area of graphics while using the minimum number of network nodes. Regular hexagon is the best choice when the construction cost is taken into account. It looks like a cellular when the regular hexagon is arranged one by one.Here we divi

13、de the radio spectrum into 4 parts:receiver frequencytransmitter frequencyA： 145.0MHz-145.6MHz145.6MHz-146.2MHzB： 145.6MHz-146.2MHz146.2MHz-146.8MHzC: 146.2MHz-146.8MHz146.2MHz-146.8MHzD： 146.8MHz-147.4MHz 147.4MHz-148.0MHzAs a result, the repeater has 4 types: A,B,C,D.According to 2, in order to pr

14、event the signals from interfering, the frequency band between 2 users is at least 25KHz. So the number of users a repeater can support is 600/ 25=24. However, 2 repeaters can interfere with one another when they are too close. In this kind of situation the CTCSS is recommended. It can associate to

15、each repeater a separate sub audible tone that is transmitted by all users who wish to communicate through that repeater. Now we are calculating the coverage radius of a repeater (reference to MOTOTRBO XiR R8200).According to 3: The transmit power of a repeater: Pt=30 dBmThe free-space loss:Lbf=32.5

16、5+20lgF+20lgD (1)F: frequency (MHz) F=145MHzD: distance (km) Signal sensitivity that users receive: Rss = Pt + Gt + Gr LR Lbf (2)Rss = -80dBm;The gain of transmission: Gt =5dB;The gain of transmission: GR=5dB;The loss of radio in different situations: LR.a. Flat areas: LR=17dB, according to (1) and

17、(2), the coverage radius of the repeater :D =15miles.b. Mountains areas: LR=22dB, according to (1) and (2), the transmit radius of the repeater: D=8miles.2.2. Basic ModelIn this section, we use cellular model to design and calculate the distribution and number of the repeaters in flat area when peop

18、le uniformly distribute and the number of entire population is 1,000.2.2.1. By calculating the population density to calculate the length of each cellular side, determine the cell size.The area of the entire region: (3)Population density: (4)According to 2.1, the coverage radius of the repeater in f

19、lat area is 15mile. So the coverage area is : (5)However, according to 2.1, a repeater can accommodate 24 people at most. So to get fully use of a repeater, the area a repeater truly covers is: (6)So the length of each cellular side is: (7)2.2.2. Draw the cellular and get the number and distribution

20、 of repeatersWhen we get the information above, we can draw the cellular. Firstly, draw numbers of connecting hexagons which length of side is 6.8 miles. Then picture a circle with the radius of 40 miles which represents the entire area above the cellular. Get the numbers of the hexagons covered by

21、the circle. For the hexagons in boundary, if the center of it is not covered by that circle, this one is not counted. This will cause a problem that a few hexagons , some part of which are covered by the circle, near the boundary are without repeaters. Fortunately, the unsaturated repeaters in the n

22、earby hexagons can also cover this area due to the large coverage of a repeater. (The coverage radius of a repeater has more than 2 times than that of s hexagon)Now, with the cellular model, we can get the distribution of the repeaters in figure1.In this picture, every repeater is located in the cen

23、ter of every hexagon. And we can count that the number of the repeaters in the circle is 43. A, B, C, D represent the 4 types of repeaters.With different frequency bands and different PL ,the repeater can distribute in different. The repeater in different will not be interfered even at the same plac

24、e. The types of the repeaters is : (8) Figure1 So the type of the repeater can be distribute automatically with 43 different types and they will not be interfered.2.2.3. Model verificationAccording to 2.1, every repeater can accommodate 24 users at most at the same time. So the ideal least number of

25、 the repeater is: (9) So this model is close to the ideal situation.2.3. Improved ModelIn this section, we use cellular model to design and calculate the number and distribution of the repeaters in flat area when people uniformly distribute and distribution when the number of entire population is 10

26、,000.2.3.1 By calculating the population density to calculate the length of each cellular side, determine the cell size.The area of the entire region: (10)Population density: (11)According to 2.1, the coverage radius of the repeater in flat area is 15mile. So the coverage area is : (12)However, acco

27、rding to 2.1, a repeater can accommodate 24 people at most. So to get fully use of a repeater, the area a repeater truly accommodates is: (13)So the length of each cellular side is: (14)2.3.2. Draw the cellular and get the number and distribution of repeatersJust as what 2.2.2. says, we can get the

28、distribution as figure2. we can count that the number of hexagons is 422.However, according to 2.2.2, the types of the repeater are 216, which is less than 422. So maybe some repeater in same type will get interfered when there location is close. So it is very important to distribute the type of the

29、 repeater to ensure the same kind of repeaters dispatched completely. Here is the solution. Create another cellular system with the length of every hexagon equal to the repeaters coverage radius. Then put it on Figure2. Place the 4 types of repeaters into different hexagons. Make sure the repeaters sharing the same high frequency are isolated, just as the figuer3 shows. So communicating in overlapping areas as a or b will not be interfered even if the repe