MCM 特等奖 论文.docx

MCM 特等奖 论文.docx

《MCM 特等奖 论文.docx》由会员分享，可在线阅读，更多相关《MCM 特等奖 论文.docx（20页珍藏版）》请在冰豆网上搜索。

MCM特等奖论文

Contents

Ⅰ.Introduction................................................................................................................................................2

Ⅱ.Model…...........................................................................................................................................................2

2.1.Descriptionofthecellularmodel................................................................................................3

2.2.BasicModel........................………………………………………………………….………….4

2.2.1Bycalculatingthepopulationdensitytocalculatethelengthofeachcellularside,determinethecellsize.…………...……………………………….…….4

2,2,2.Drawthecellular..Getthenumberanddistributionofrepeaters………...…4

2.2.3.Modelverification……………………………..………………………………………….…5

2.3.ImprovedModel……………………………………………………………………………….…..…6

2.3.1Bycalculatingthepopulationdensitytocalculatethelengthofeachcellularside,determinethecellsize.………………………………….….6

2,3.2.Drawthecellular.Getthenumberanddistributionofrepeaters………...6

2.3.3.Modelverification………………………………………………………..8

2.3.4.Thesituationwhenpeoplein

distribution…………………………...8

2.4.Extramodels…………………………………………………………………..13

2.4.1.Makeassumptions……………………………………………………....13

2.4.2FoundationofModel………………………………………………...…..14

2.4.3.Modelverification……………………………..……………………..…17

Ⅲ.ConclusionandExpectation………………………………...…………………..17

Ⅳ.StrengthsandWeaknesses...................................................................................18

Ⅴ.Reference…………………………………………………………………………18

I.Introduction

Inordertohavebetterunderstandingofourmodel,thefollowingbackgroundisworthmentioning.

TheVHFradiospectrumisthekindofradiowavewhosebandis30MHzto300MHzinwirelesscommunication.Duetoitshigherfrequency,thesky-wavewillgenerallypenetratetheionosphereandfiretothespace,ratherthanbereflectedbacktotheground.AsVHFhasalimitedcapacityofdiffraction，ittransmitsbyline-of-sighttransmission.Thepropagationdistanceisrestrictednotonlybysight,butalsobymountainsandtallbuildingslikelight.Tospreadevenfurther,wemustrelyonrepeaters,forawidendissemination.Repeaterspickupweaksignals,amplifythem,andretransmitthemonadifferentfrequency.However,repeaterscaninterferewithoneanotherunlesstheyarefarenoughapartortransmitonsufficientlyseparatedfrequencies.Inadditiontogeographicalseparation,the“continuoustone-codedsquelchsystem”（CTCSS）canbeusedtomitigateinterferenceproblems.Inthispaper,wetrytousetheminimumnumberofrepeaterstocompletelyaccommodate1,000（or10000）simultaneoususersinacircularflatareaofradius40miles（Assumingthatthespectrumavailableis145to148MHz,thetransmitterfrequencyinarepeateriseither600kHzaboveor600kHzbelowthereceiverfrequency,andthereare54differentPLtonesavailable）anddiscussthecasewheretheremightbedefectsinline-of-sightpropagationcausedbymountainousareas.

Reasoningfromthegiveninformationand[1],wecandrawtheconclusionthatonerepeateronlyhaveoneparticularPL.Inordertosimplifythemodel,hereweusewiredrepeater（eachofthe2repeatersareconnectedbycable）.

Wearenowfacingthefollowingtask:

1.Understandingthemechanismofline-of-sighttransmission;

2.Discussingtheinfluenceofrepeaterscausedbythepopulationdistribution;

3.Evaluatetheinfluenceofthecoverageofrepeaterscausedbythedifferentterrains;

4.Makeacellularmodel,findtheoptimalsolution.

Ourapproachis:

Gettheleastnumberofrepeaters,whichcantotallyaccommodate1,000simultaneoususersbyusingcellularmodelindifferentpopulationdistribution（uniformdistributionor

distribution）anddifferentterrains（flatareasormountainsareas）.

Ⅱ.Model

2.1.Descriptionofthecellularmodel:

Whydoweusethecellularmodel?

Cellularcommunicationshasexperiencedexplosivegrowthinthepasttwodecades.CellularnetworkiswidelyappliedbecauseofaMathematicalconjecture:

Regularhexagonisconsideredtocoverthelargestareaofgraphicswhileusingtheminimumnumberofnetworknodes.Regularhexagonisthebestchoicewhentheconstructioncostistakenintoaccount.Itlookslikeacellularwhentheregularhexagonisarrangedonebyone.

Herewedividetheradiospectruminto4parts:

receiverfrequency

transmitterfrequency

A：

145.0MHz----145.6MHz

145.6MHz----146.2MHz

B：

145.6MHz----146.2MHz

146.2MHz----146.8MHz

C:

146.2MHz----146.8MHz

146.2MHz----146.8MHz

D：

146.8MHz----147.4MHz

147.4MHz----148.0MHz

Asaresult,therepeaterhas4types:

A,B,C,D.

Accordingto[2],inordertopreventthesignalsfrominterfering,,thefrequencybandbetween2usersisatleast25KHz.Sothenumberofusersarepeatercansupportis600/25=24.However,2repeaterscaninterferewithoneanotherwhentheyaretooclose.InthiskindofsituationtheCTCSSisrecommended.Itcanassociatetoeachrepeateraseparatesubaudibletonethatistransmittedbyalluserswhowishtocommunicatethroughthatrepeater.

Nowwearecalculatingthecoverageradiusofarepeater（referencetoMOTOTRBOXiRR8200）.

Accordingto[3]:

Thetransmitpowerofarepeater:

Pt=30dBm

Thefree-spaceloss:

Lbf=32.55+20lgF+20lgD

（1）

F:

frequency（MHz）F=145MHz

D:

distance（km）

Signalsensitivitythatusersreceive:

Rss=Pt+Gt+Gr–LR–Lbf

（2）

Rss=-80dBm;

Thegainoftransmission:

Gt=5dB;

Thegainoftransmission:

GR=5dB;

Thelossofradioindifferentsituations:

LR.

a.Flatareas:

LR=17dB,accordingto

（1）and

（2）,thecoverageradiusoftherepeater:

D=15miles.

b.Mountainsareas:

LR=22dB,accordingto

（1）and

（2）,thetransmitradiusoftherepeater:

D=8miles.

2.2.BasicModel

Inthissection,weusecellularmodeltodesignandcalculatethedistributionandnumberoftherepeatersinflatareawhenpeopleuniformlydistributeandthenumberofentirepopulationis1,000.

2.2.1.Bycalculatingthepopulationdensitytocalculatethelengthofeachcellularside,determinethecellsize.

Theareaoftheentireregion:

（3）

Populationdensity:

（4）

Accordingto2.1,thecoverageradiusoftherepeaterinflatareais15mile.Sothecoverageareais:

（5）

However,accordingto2.1,arepeatercanaccommodate24peopleatmost.Sotogetfullyuseofarepeater,theareaarepeatertrulycoversis:

（6）

Sothelengthofeachcellularsideis:

（7）

2.2.2.Drawthecellularandgetthenumberanddistributionofrepeaters

Whenwegettheinformationabove,wecandrawthecellular.Firstly,drawnumbersofconnectinghexagonswhichlengthofsideis6.8miles.Thenpictureacirclewiththeradiusof40mileswhichrepresentstheentireareaabovethecellular.Getthenumbersofthehexagonscoveredbythecircle.Forthehexagonsinboundary,ifthecenterofitisnotcoveredbythatcircle,thisoneisnotcounted.Thiswillcauseaproblemthatafewhexagons,somepartofwhicharecoveredbythecircle,neartheboundaryarewithoutrepeaters.Fortunately,theunsaturatedrepeatersinthenearbyhexagonscanalsocoverthisareaduetothelargecoverageofarepeater.（Thecoverageradiusofarepeaterhasmorethan2timesthanthatofshexagon）

Now,withthecellularmodel,wecangetthedistributionoftherepeatersinfigure1.

Inthispicture,everyrepeaterislocatedinthecenterofeveryhexagon.Andwecancountthatthenumberoftherepeatersinthecircleis43.‘A,B,C,D’representthe4typesofrepeaters.

WithdifferentfrequencybandsanddifferentPL,therepeatercandistributeindifferent.Therepeaterindifferentwillnotbeinterferedevenatthesameplace.

Thetypesoftherepeatersis:

（8）

Figure1

Sothetypeoftherepeatercanbedistributeautomaticallywith43differenttypesandtheywillnotbeinterfered.

2.2.3.Modelverification

Accordingto2.1,everyrepeatercanaccommodate24usersatmostatthesametime.Sotheidealleastnumberoftherepeateris:

（9）

Sothismodelisclosetotheidealsituation.

2.3.ImprovedModel

Inthissection,weusecellularmodeltodesignandcalculatethenumberanddistributionoftherepeatersinflatareawhenpeopleuniformlydistributeand

distributionwhenthenumberofentirepopulationis10,000.

2.3.1Bycalculatingthepopulationdensitytocalculatethelengthofeachcellularside,determinethecellsize.

Theareaoftheentireregion:

（10）

Populationdensity:

（11）

Accordingto2.1,thecoverageradiusoftherepeaterinflatareais15mile.Sothecoverageareais:

（12）

However,accordingto2.1,arepeatercanaccommodate24peopleatmost.Sotogetfullyuseofarepeater,theareaarepeatertrulyaccommodatesis:

（13）

Sothelengthofeachcellularsideis:

（14）

2.3.2.Drawthecellularandgetthenumberanddistributionofrepeaters

Justaswhat2.2.2.says,wecangetthedistributionasfigure2.wecancountthatthenumberofhexagonsis422.

However,accordingto2.2.2,thetypesoftherepeaterare216,whichislessthan422.Somaybesomerepeaterinsametypewillgetinterferedwhentherelocationisclose.Soitisveryimportanttodistributethetypeoftherepeatertoensurethesamekindofrepeatersdispatchedcompletely.

Hereisthesolution.Createanothercellularsystemwiththelengthofeveryhexagonequaltotherepeater’scoverageradius.ThenputitonFigure2.Placethe4typesofrepeatersintodifferenthexagons.Makesuretherepeaterssharingthesamehighfrequencyareisolated,justasthefiguer3shows.Socommunicatinginoverlappingareasas‘a’or‘b’willnotbeinterferedeveniftherepe