数学建模 中继器的分布 2011美赛B题

Abstract

Repeater is a network equipment for expanding the distance of the signal

transmission. PL can eliminate the impact of repeaters.

With the theory of comb, the distribution of repeater we build spend from

the circle in the middle to all directs. From the wireless communication

theory, repeater service area is a circle of 18km radius. We determine the

minimum number of repeater with gradually approximation method. We

obtain the result, and we need 19 repeaters to meet the requirements of

1000

users

and

circular

flat

area

of

radius

40

miles

radius,

radius

of

repeater

is

11.09

miles.

But

for

the

requirement

of

10000

users,

19

repeaters

can

’

t

meet.

We

solve

this

question

through

increasing

the

number of repeater. We need to set 37 repeaters after calculating.

As

for

impact

of

mountain,

we

set

another

repeater

on

the

peak,

when

peak

when

this

peak

hider

the

transmission

of

signal.

Because

the

influence

of

mountain

area

is

the

obstacle

of

peak

when

signal

transmission.

Introduction

Repeater Coordination

The

VHF

radio

spectrum

involves

line-of-sight

transmission

and

reception. This limitation can be overcome by “repeaters,” which pick up

weak signals, amplify them, and retransmit them on a different frequency.

Thus,

using

a

repeater,

low-power

users

(such

as

mobile

stations)

can

communicate

with

one

another

in

situations

where

direct

user-to-user

contact would not be possible. 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), sometimes nicknamed “private line” (PL),

technology can be used to mitigate interference problems. This system

associates to each repeater a separate subaudible tone that is transmitted

by all users who wish to communicate through that repeater. The repeater

responds only to received signals with its specific PL tone. With this

system, two nearby repeaters can share the same frequency pair (for

receive and transmit); so more repeaters (and hence more users) can be

accommodated in a particular area.

For a circular flat area of radius 40 miles radius, determine the minimum

number of repeaters necessary to accommodate 1,000 simultaneous users.

Assume 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

receiver frequency, and there are 54 different PL tones available.

How does your solution change if there are 10,000 users?

Discuss the case where there might be defects in line-of-sight propagation

caused by mountainous areas.

Repeater service area

Repeater is a network equipment for expanding the network distance of

the

signal

transmission,

through

retransmitting

signal.

Thus,

using

a

repeater, low-power users can communicate in a long distance. Repeater

can

received

and

transmission

signal

Within

a

certain

distance.

This

distance

is

associated

with

the

frequency

of

signal

and

position

of

repeater. In our model, we let this distance

d

?

18

km

?

1

?

area is a circle of 18km radius, due to the transmission is in all directions.

Analysis

For a circular flat area of radius 40 miles radius, determine the minimum

number of repeaters necessary to accommodate 1,000 simultaneous users.

Repeater service area is a circle, too. This question is equivalent to that

fill a big circle with some number of small circle.

However,

repeaters

can

interfere

with

one

another

unless

they

are

far

enough apart or transmit on

sufficiently separated frequencies. This question is equivalent to coloring

question that can’t dye the same color in the adjacent area, and frequency

is equivalent to color.

Assumptions

?

1 Users is uniform distribution

?

2 All of repeater is the same.

Model

With the theory of comb, we use hexagon insteading of repeater service

area approximately, and build the distribution of repeater as Fig.1

Fig.1

Left in Fig.1 is distribution of repeaters, right is repeater instead of

hexagon.

Where

R

3

is radius of area.

The distribution of repeater spend from the circle in the middle to all

directs. Circle in the middle is in the first layer,

L

?

1

. The representative

of the

L

is layer.

The number of repeaters

n

n

?

3

L

< br>2

?

3

L

?

1

,

L

?

1

,

2

,

?

We can obtain the relationship between

R

3

and radius of repeater

r

,

r

2

R

3

< br>?

7

r

?

12

2

Now, we determine the minimum number of repeater with gradually

approximation method. The basic idea is as Fig.2