1

The Beauty of Mathematics in

Communications

R. C. T. LeeDept. of Information Management

&Dept. of Computer ScienceNational Chi Nan University

2

Operating systems and compilersCan be built without mathematics.

Most drugs were invented without Mathematics.

3

Can communications systems bebuilt without mathematics?

Ans: Absolutely no.

Modern communication systemsare totally based upon mathematics.

4

For computer scientists, data are stored in memory as bits, either 1 or 0.

  How are the data transmitted?  They are transmitted as pulses: A

pulse represents a 1 and no pulse represents a 0.

5

Fig. 1

6

Is this possible when the transmission is done in a wireless environment?

Impossible.

Fact: Wireless communication is done every day.

How is this possible?

7

Can we mix together two bits and send out?

Impossible if the two bits are represented as pulses.

Fact: We often mix 256 bits together and send them at the same time.

How is this done?

8

Is an antenna open-circuited?

Yes, it must be. You can easily prove this by looking at your mobile phone antenna.

9

If an antenna is open-circuited, then there must be no current on it.

How can it induce electromagnetic fields without any current?

10

Can we broadcast our voice signals directly through some antenna?

Impossible. Some kind of modulation must be done.

Why?

11

All of these questions can be answered by mathematics and only by mathematics.

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-4

-3

-2

-1

0

1

2

3

4

Time

t

f(t)

Fig. 2

13

Fig. 3.The Discrete Fourier Transform Spectrum of the Signal in Fig. 2 after Sampling.

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

Frequency

f

14Fig. 4 A Signal with Some Noise.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-150

-100

-50

0

50

100

150

Time

t

f(t)

15

Fig. 5 The Discrete Fourier Transform of the Signal in Fig. 4 after Sampling.

0 50 100 150 200 2500

10

20

30

40

50

60

70

Frequency

f

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-150

-100

-50

0

50

100

150

Time

t

f(t)

Fig. 6 The Signal Obtained by Filtering Out the Noise.

17Fig. 7 A Music Signal Lasting 1 Second.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time

t

18

0 2000 4000 6000 8000 10000 12000 14000 160000

50

100

150

Frequency

f

Fig. 8 A Discrete Fourier Transform Spectrum of the Signal in Fig. 7.

19

By using Fourier transform, we can see that the frequency components in our human voice are roughly contained in 3k Hertz.

20

For a signal with frequency f, its wavelength can be found as follows:

where v is the velocity of light.

f

v

21

If , .3103f km100m10103

103 53

8

22

It can also be proved that the length of an antenna is around

.

For human voice, this means that the wavelength is 50km.

No antenna can be that long.

2

23

What can we do?

Answer: By amplitude modulation.

24

Let be a signal. The amplitude modulation is

defined as follows:

where fc is the carrier frequency?

)(tx

)2cos()( tftx c

25

What is the Fourier transform of ? )2cos()( tftx c

26Fig. 9

27

The effect of amplitude modulation is to lift the baseband frequency to the carrier frequency level, a much higher one.

Once the frequency becomes higher, its corresponding wavelength becomes smaller.

An antenna is now possible.

28

After we receive , how can we take out of it?

Answer: Multiply by .

)2cos()()( tftxts c)(tx

)(ts )2cos( tfc

29

Thus is recovered.

)2(cos)()2cos()( 2 tftxtfts cc

)))2(2cos()()((2

1tftxtx c

)(tx

30Fig. 10

31

Our next question: How is a bit transmitted?

Answer: A bit is usually represented by a cosine function.

32

Let us assume that bit 1 is represented by and bit 0 is represented by .

When the receiver receives a bit, how can it detect whether 1, or 0, is sent?

)2cos( 0tmf)2cos( 0tnf

33

The basic scheme behind the detection is the inner product property of cosine functions:

where .

T

dttnftmftnftmf0 0000 )2cos()2cos()2cos(),2cos(

0

1fT

34

It can be proved that

nmiftnftmf 0)2cos(),2cos( 00

nmifT

tnftmf 2

)2cos(),2cos( 00

35

This inner product property gives us the fundamental mechanism of detecting 1 or 0.

Let the sent signal be denoted as . We perform two inner products:

and

Decision rule: If , say that 1 is sent. If , say that 0 is sent.

)(ts

)2cos(),(2

)( 01 tmftsT

ty

)2cos(),(2

)( 02 tnftsT

ty

1)(1 ty1)(2 ty

36

Suppose that we have two bits to send.

Can we bundle them together and

send the bundled result at the same time?

Answer: Of course, we can.

37

Let the two bits be demoted as and . or 0. or 0.

Let if Let if

1m 2m

11 m

12 m

)0(11 s

)0(12 s

)0(11 m

)0(12 m

38

The sent signal is

Our job is to determine the values of

and .

)2cos()2cos()( 0201 tnfstmfsts

1s 2s

39

We perform inner product again.

and

101 )2cos(),(2

)( stmftsT

ty

202 )2cos(),(2

)( stnftsT

ty

40

Can we bundle 256 bits together and send them at the same time?

Answer: Yes, as along as the signals are orthogonal to one another.

This is the basic principle of ADSL: OFDM (Orthogonal Frequency Division Method).

41

Can we extend the above idea to two users case?

Answer: We can.

42

Let User 1 use to represent 1 and to represent 0. Let User 2 use to represent 1 and to represent 0.

if i=j and if i≠j. and are orthogonal.

)(1 t

)(2 t)(1 t

)(2 t

1)(),( tt ji 0)(),( tt ji

)(1 t )(2 t

43

The sent signal is denoted as where and

.

To determine, we perform inner products:

)()()( 2211 tststs 11 s12 s

11 )(),( stts

22 )(),( stts

44

Fig. 11 Signature Signals Generated from Hadamand matrix H8.

All of the signals are orthogonal to each other.

45

We can also view the problem as a vector analysis problem.

Assume that User 6 sends 1 and User 8 sends 0.

)1,1,1,1,1,1,1,1(6 s)1,1,1,1,1,1,1,1(8 s

46

V6=(1,-1,1,-1,-1,,1,-1,1)V8=(1,1,-1,-1,-1,-1,1,1)

The inner product of v6 and v8 is1-1-1+1+1-1-1+1=0

47

The sent signal is

1 is sent.

0 is sent.

)0,2,2,0,0,2,2,0()( 86 sss

1)2222(8

1

8

16 ss

1)2222(8

1)(

8

18 ss

48

This is the principle of CDMA (code division multiple access).

It can be extended to more than two users.

It was used by the military as an encryption method before.

49

Suppose we send a signal entirely in digital form, can we say that this signal is an analog signal?

Yes, we can because according to Fourier series analysis, a pulse also contains a set of cosine functions.

50

)(tx

t1/100 1

)( fX

f

51

)(tx

t

1/200 1

)( fX

f

52

Obviously, the smaller the pulse-width is, the morefrequency components it contains. One may even say that the smaller the pulse-width is, the moreinformation it may contain.

Note that if a pulse has a small pulse-width, it meansthat within a second, a large number of pulses canbe sent. This corresponds to “high bit rate”.

Now, we know why a wire which has a high bit ratemay be called broadband.

53

It is important to observe the following:

Bits are represented by analog signals.

There are no digital signals in the world.

54

It i

Maxwell’s Equations

Equations Concerning with Electromagnetic Waves

55

Electric Field Induced by Charges

Fig. 12 Coulomb’s Law .

56

Magnetic Field Induced by a Current Segment

Fig. 13 Magnetic Flux Density Induced by a Current.

57

Do the electric field and magnetic field affect each other?

No, not in the static field.

Yes, if the fields are time-varying.

58

The curl of a vector.

▽ A zyx ˆˆˆ

y

A

x

A

x

A

z

A

z

A

y

A xyzxyz

59

Faraday’s law

t

B

E

60

Fig. 14 The Voltage Caused by the Movement of a Magnet Inside a Coil

The changing of magnetic field with time causes an electric field.

N S

Voltmeter

61

Ampere’s law

Fig. 15 Magnetic Flux Density Induced by a Current.

▽ JH

62

The Ampere’s law modified by Maxwell

The changing of electric field with time will induce a magnetic field.

Maxwell modified Ampere’s law without performing any experiments.

▽ t

DJH

63

Differential form Integral form ▽

▽

▽

▽

Maxwell’s Equations

t

B

E sB

lE dt

dsc

t

D

JH sD

sJlH dt

ddssc

vD dvdv vs sD

0B 0s dsB

64

Plane Electromagnetic Waves

With specical boundary conditions, Maxwell’s equations reduce to

2

2

2

2

t

E

z

E xx

)(cos0 vtzkEEx

65

The speed of the wave:

Implication: The electromagnetic waves travel with the speed of light.

1v

sec/103)10(91

36/10

)10(4

816

9

7

mv

66

Maxwell was not able to prove his theory.

Hertz proved the correctness of Maxwell’s equations.

67

Fig. 16 The Traveling of a Wave.

z

E

z=vt

t = 0 t = t

68Fig. 17 The Electric and Magnetic Fields in a

Plane Electromagnetic Wave.

69

Fig. 18 The Wavelength.

We can prove that . f

v

z

E

70

Transmission Line:

Any electric wire which carries currents

with high frequency can be considered

as a transmission line.

71

Fi

b

a

iwd

y z

x8

i

Fig. 20 Twin-Strip Parallel Plate Transmission Line

Fig. 19 A Co-Axial Cable Transmission Line

72

v

dv

dx

Fig. 21 An Equivalent Circuit of a Lossless Transmission Line.

73

The above equations show that there are waves on the transmission line.

2

2

2

2

t

VLC

x

V

2

2

2

2

t

ILC

x

I

74

It can be proved that the velocity of the waves is roughly the speed of light.

vva

tXLCVf tXLCVb

tXLCIf tXLCIb

RS

X=0 X=A

RL

Fig. 22 The Waves on a Transmission Line.

75

Standing Waves

y

)y(V

)y(I

43

2

4

Fig. 23 The Case of Open-Circuited Load

76

2

I

I

Fig. 24 A Half Wave Dipole Antenna

77

In ancient times, human beings built spectacularbuildings.

But, modern communications systems werepossible only recently.

Why?

Answer: Modern communication systems cannot exist without sophisticated mathematics.

78

Thank you.