1

WAVE AND OPTICS WAVE MODULATION

by:

Gde Parie Perdana (1113021059)

Ni Kadek Fitria (1113021061)

ii

ACKNOWLEDGEMENT

Om Swastiastu,

Worship and praise the author prayed to God Almighty, because over the mercy

paper entitled "Wave Modulation", can be resolved. The author would like to thank those

who helped in the preparation of this paper. So that this paper can be completed on time.

Preparation of this paper aims to fulfill the task for subjects Waves and Optics.

The author realizes that this paper is still a lot of shortcomings. Therefore, criticism

and constructive suggestions are encouraged. Hopefully, this paper can be useful for writers

and readers to increase knowledge and understanding of the oscillation motion.

Om Santih, Santih, Santih Om.

April, 2014

Author

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TABLE OF CONTENT

ACKNOWLEDGEMENT .................................................................................... I

TABLE OF CONTENTS ...................................................................................... II

CHAPTER I INTRODUCTION

1.1 Background ............................................................................................... 1

1.2 Problems Formulation ............................................................................... 2

1.3 Purpose ...................................................................................................... 2

CHAPTER II ANALYSIS OF OSCILLATORY MOTION

2.1 Modulation ................................................................................................ 3

2.2 Modulation DSB ....................................................................................... 4

2.3 Amplitude Modulation .............................................................................. 6

2.4 Frequency Modulation .............................................................................. 10

2.5 Applications of Modulation Concept ........................................................ 15

CHAPTER III ANALYSIS OF OSCILLATORY MOTION

Conclusion ................................................................................................ 17

References

1

CHAPTER I

INTRODUCTION

1.1 Background

As one of the applications of physics that deals with the concept of waves and optics,

radio certainly is not foreign to us. Radio is a medium of communication that is widely used

by humans. At first the radio used is in the maritime field is to send messages from ship to

land and continue to grow until reaching the field of entertainment. In that transmission,

radio broadcasts using modulation techniques, signal is a voice signal riding, while carrying

a radio signal called the carrier signal. Modulation technique that often used in radio signal

is FM and AM.

Understanding modulation itself is the process of change (varying) a periodic wave

that makes a signal capable of carrying an information (Anonymous, 2011). In

communications engineering, wave or baseband signals (base band) transmitted by

modulating a high frequency carrier wave. Signal Baseband signals called modulation wave.

When listening to the radio, if we want to listen to a particular broadcast, then we will

search the broadcast frequency, 92.0 MHz for example. This frequency is the midpoint of

the frequency band occupied by a subsequent communication system operates. Waves at

these frequencies propagate through the atmosphere and captured by radio that we use.

However, voice broadcast / broadcaster is at the 20 ─ 20,000 Hz, so that the sound will not

be heard. Therefore, it is necessary to modulate the sound wave with modulation (carrier)

frequency is higher than the medium used. Carrier signal is usually determined at only one

frequency. In Indonesia, the carrier signal frequency allocation for FM broadcasts on

frequency 87.5 MHz set - 108 MHz and set to broadcast on AM 530 kHz - 1600 kHz. In the

technique of AM radio transmitters, the amplitude of the carrier wave will changed in line

with changes to the information signal (sound) are included. The frequency of the carrier

wave is relatively fixed. Then, the signal passed to the RF (Radio Frequency) amplifier to

strengthen in order to send to long distances.

Modulation method first used almost universally in radio transmissions in the AM

(amplitude modulation) and FM (frequency modulation). Then note that the lines of

communication have greater bandwidth than is needed for talks. Therefore, many phone

conversations can shipped together in a phone line with changing frequency such that some

of the sound channels can wrapped into a wider bandwidth. In this case also used modulation

2

DSB (double side band). Under these conditions, the authors raised the issue entitled

"Modulation DSB Waves, Amplitude, and Frequency".

1.2 Problems Formulation

Formulation of the problem that the authors take is as follows:

a. How does the concept of modulation in general?

b. How does the concept of the DSB modulation, Amplitude and Frequency?

c. How does the application of the concept of modulation wave?

1.3 Purpose

From the above formulation of the problem, as for several objectives, namely:

a. Knowing and understanding the modulation wave in general.

b. Explain and describe the DSB wave modulation, Amplitude and Frequency.

c. Identify and explain the application of the concept of wave modulation.

3

CHAPTER II

DISCUSSION

2.1 Modulation

Modulation wave the process of changing certain characteristics of a wave by

oscillation pattern of the other. Through that process, the characteristics of the second wave

can be "superimposed" on the first wave, and then separated again if necessary. In

communications engineering, a wave or signal baseband (to form an audio signal, video

signal, or other signal) generally sent to a distant target to modulate a carrier wave, which

has a relatively high frequency and power. In this case, the baseband signal known as

modulation wave.

The major advantages derived from modulation techniques in communications

systems, among others:

a. Enables sending a weak signal by "piggybacking" high-power carrier wave

(adjustable).

b. Reduction of the size of the antenna because the signal transmission carried out via

the carrier wave has a high frequency (short λ).

c. Allows setting and allocation of separate frequency regions for the distribution of a

number of signals simultaneously through the same medium.

d. Allows the frequency shift signal to the frequency range more easily processed by

the equipment available.

Modulation techniques can be divide into groups of analogue techniques and digital

techniques. In analog modulation group, still can distinguished several different ways such

as the following:

a. Carrier wave in the form of a continuous wave or a pure harmonic wave, namely:

Linear modulation (DSB, AM, SSB, VSB)

Angle modulation (FM, AM)

b. A carrier wave pulse:

Pulse amplitude modulation (PAM)

Pulse width modulation (PDM)

Pulse position modulation (PPM)

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2.2 Modulation DSB (Double Side Band)

By reviewing sections that vary with time (local oscillation), the carrier wave is

generally sinusoidal waveform.

tt ccc cos ............................................................. (1)

Suppose modulation wave (signal wave) is also sinusoidal

tt mmm cos ............................................................ (2)

Then the results obtained from the modulation multiplication of equation (1) and (2) is,

tt

tt

ttt

mcmcmc

mmcc

mcDSB

)cos(cos2

1

)cos(cos

............ (3)

Which resulted in the expansion of the frequency of mc the two components of the

side band. Each frequency region representation for each wave at equation (1), (2), and (3)

can be obtained by Fourier transformation, namely,

ccccg ................................... (4a)

mmmmg ................................ (4b)

mcmc

mcmc

mcDSBg

......... (4c)

From the picture above shows that the spectrum due to modulation as the modulation

frequency translation so far mcmc menjadidari . DSB modulation effects in

the region of time and frequency region as described above further clarified in Figure 1

below.

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Figure 1 Illustration DSB modulation results in the area of t and ω. (a) wave modulation; (b) carrier wave; (c) Wave

DSB

In addition, as a result of DSB wave modulation bandwidth to 2 times the signal wave,

but the amplitude becomes half time, as shown in Figure 2. The band broadening connected

with the appearance of the components on both sides c called the tape side (upper side

band, abbreviated USB) and lower side bands (lower side band, abbreviated as LSB).

Shown in Figure 2 B transmission bandwidth for the same DSB wave with twice the signal

bandwidth wave, namely:

mB 2 .............................................................................. (5)

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Figure 2 Illustration of the DSB in the region due to the modulation frequency (a) Spectrum of the signal, (b) Spectrum of

DSB wave

Average power N are forwarded directly proportional to

dttT

NT

T

T

2lim2

2

1

.......................................................... (6)

Substituting equation (3) into equation (6) obtained average power signal DSB,

mcDSB NNN ....................................................................... (7a)

With

22lim

2

11 2

2

ccT

c dttT

N

T

T

............................................ (7b)

22lim

2

11 2

2

mmT

m dttT

N

T

T

........................................... (7c)

2.3 Amplitude Modulation

DSB modulation and AM are both amplitude modulation, which is a characteristic

wave that changed. Modulation Amplitude modulation formulated as a deficiency in DSB

has demodulation wherein the regenerated carrier signal must be coherent with the existing

carrier signal at the transmitter (modulation process). Regenerated carrier signal will result

in no coherent modulated signal that has passed through a low pass filter generates the wave

equation cos2

0 tma

ty . Therefore, it interpreted:

a. Errors in phase

If 02

ty , No signal is received the receiver.

7

If 2

received signal damped with a particular factor, because

1cos .

If tf , A function of a random time, then the signal y (t) is the product of

the (modulated) signal m (t) with f (t). Signals of information we want to get

distracted by the signal f (t) is.

b. Errors in frequency

Appropriate output signal of pure information, it multiplied by the harmonic

signal with unknown frequency. This multiplication leads to beating which is a

distortion to the signal that we have.

Amplitude modulation obtained by adding the identity of the carrier to the DSB

signal able to overcome the first problem on the DSB modulation, i.e. the phase error.

In the region t, the expression form of AM signals:

)cos()(1)( ttt cmcAM .......................................... (8)

or

)cos()()( ttAt cAM ........................................................ (9)

with

)(1)( ttA mc ............................................................. (10)

Which is the modulation factor which expresses the change in amplitude envelope

happens AM.

In the area , equation (10) becomes:

ccmcccAM gg 222

1)( ..... (11)

Figure 3 (a) sinusoidal signal (b) the signal (carrier); and (c) results of modulation AM

8

Results for the case of amplitude modulation sinusoidal signal and the more general

case further clarified in Figure 3. From Figure 3 (c) shows that the amplitude function A (t)

for the AM band never cut the axis t. This happens because the signal wave generally made

to comply 1)( tm (so that the demodulation process is easy to do). While the results of

the modulation bandwidth satisfies the equation:

B=2ωm .................................................................................. (12)

Figure 4 (a.) spectrum signal ψm(t) and (b.) AM wave spectrum

For sinusoidal signal that meets

)(cos)( tmt mm ............................................................. (13)

With m known as the index is a dimensionless factor that is useful for determining

the ratio of sideband to the carrier, formulated as:

amplitudepeak carrier

AmplitudePeak DSB m

E

EEmor

E

EEm minmaks

9

minmax

min

maks

EEEE

amplitudeCarrier E

BAM amplitude MinimumE

AAM amplitude MaxsimumE

1)()(

)()(

minmax

minmax

tAtA

tAtAm ................................................. (14)

The magnitude of the modulation index has a range between 0 and 1. Modulation

index of zero, meaning no modulation, while the modulation index of the maximum

modulation is possible. When the wave amplitude modulation becomes larger than the

carrier amplitude, resulting in the modulation index is greater than one, the resultant wave

will have more peaks than the wave modulation, and the original signal will not restored.

The shape of the graph is as follows:

Average power AM signal, obtained as follows.

MCCAm NNNN ............................................................. (15)

Because the AM signal, the carrier component contains no information and therefore

is part of "useless", can be defined transmission power efficiency.

m

m

MCC

MC

N

N

NNN

NN

1 ................................................. (16)

Considering that 1)( tm clear 1MN the maximum efficiency of power

transmission can be searched by substituting Nm = 1 in equation (16):

Figure 5 Shape in the case: a. 0<m<1, b. m = 1, c. m > 1

10

%50%1002

1%100

11

1%100

1

m

m

MCC

MC

N

N

NNN

NN

So, %50 for the AM band in general. Special for AM wave if the wave signal

satisfies the equation (13) then then %3,33 .

Modulation wave is also an energy transmission, because:

In modulation, which is transmitted frequency or amplitude. These quantities

searched from the phase velocity and the speed of its group. Energy and

momentum of the wave can be determined from the velocity.

In modulating the efficiency of power transmission indicated by equation (15)

and equation (16). Energy is the power per unit of time. So from both equations,

has indirectly shown that the energy involved.

2.4 Frequency Modulation

Frequency modulation (FM) is the angle modulation is the process of changing the

phase angle of the carrier wave signal according to the pattern of the wave changes

(modulation) that are non-linear (can not be described by the superposition principle). In

terms of technical, angle modulation is more difficult than linear modulation and requires a

wide transmission bandwidth anyway. The advantage mainly lies in improving the quality

of the signal by increasing the ratio of S / N.

Figure 6 (a) Carrier wave; (b) Modulation wave; (c) FM wave

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Carrier signal can be a sine wave, whereas the modulation signal (information) can

be any waveform (sinusoidal, square, triangle, or other signals e.g. audio signals). Figure 1

illustrates the modulation frequency of the carrier signal using a sinusoidal modulation signal

is also shaped sinusoidal signal. Carrier wave shown by function:

tt ccc cos ...................................................... (18)

Then the angle modulation means changing the characteristics of the constants be a

function (t) accordance with the relevant characteristics of the modulation wave. To

formulate the relationship between (t) and we write the signal modulation results in the

form by:

t

ttt

c

ccFM

cos

cos

.................................................. (19)

Thus,

tdt

tdt c '

..................................................... (20)

If the wave signal is to be reviewed m(t), Then the frequency deviation angle ’(t) and

phase deviation (t) clearly satisfy the relationship:

tkdt

tdt mF

' or, ................................................ (21)

n

mF dttkt0

.............................................................. (22)

Where kF is the frequency deviation constant. FM modulation so that the results can be

write as follows more.

n

mFccFM dttktt0

cos .................................. (23)

If the signal wave is a sinusoidal function, namely: tt mmm cos , Then equation

(22) will be:

tk

dttkt m

m

mF

n

mmF

sincos

0

......................... (24)

According to equation (21), it obtained:

tdt

tdt m

cos'' .................................................. (25)

With, mFk '

12

Based on that defined a parameter called the FM modulation index, namely:

mm

mFk

' .................................................................. (26)

Therefore, the expression results for the FM modulation signal in a single FM modulation

index, namely:

ttt mccFM sincos ....................................... (27)

Spectral characteristics of the function in equation (28) can been studied by Fourier series

description. So that the equation can be written in the form.

t)(ωsin βtω i expRe ψtψ mccFM ............................. (28)

Given the complex exponential function is periodic,

m

mω

2πtsin(ω β i exp . With

period 𝑇𝑚 =2𝜋

𝜔𝑚, then the function can be describe in Fourier series as follows:

exp i [β sin(ωmt)] = ∑ Cneinωmt∞

𝑛=−∞ .............................. (29)

Thus,

𝐶𝑛 =1

2𝜋∫ 𝑒𝑖𝛽(sin 𝜃+𝑛𝜃) 𝜋

−𝜋𝑑𝜃 .............................................. (30)

With, 𝜃 = 𝜔𝑚𝑡 = (2𝜋

𝑇𝑚) 𝑡

Form of the integral equation (30) is a Bessel function of the first kind of order to-n, which

is real, so

𝐶𝑛 =1

2𝜋∫ 𝑒𝑖𝛽(sin 𝜃+𝑛𝜃) 𝜋

−𝜋𝑑𝜃 = 𝐽𝑛(𝛽) ................................. (31)

In Bessel functions, equation (29) becomes:

exp 𝑖[𝛽 sin(𝜔𝑚𝑡)] = ∑ 𝐽𝑛(𝛽)𝑒𝑖𝑛𝜔𝑚𝑡

∞

𝑛=−∞

Which satisfy the relation (for n round)

𝐽−𝑛(𝛽) = (−1)𝑛 𝐽𝑛 (𝛽)

𝐽𝑛(𝛽) = (−1)𝑛 𝐽𝑛 (−𝛽) .................................................... (32)

1)(Jn

2

n

...................................................................... (33)

Overall expression of equation (28) becomes

tnJt mc

n

ncFM )ωω()()(

................................... (34)

While the FM frequency spectral region with TF product

13

𝑔(𝜔) = 𝜋(𝜓𝐶)[𝛿(𝜔 − 𝜔𝐶 − 𝑛𝜔𝑚) + 𝛿(𝜔 − 𝜔𝐶 − 𝑛𝜔𝑚)] .... (35)

Figure 7 Illustration of the basic pattern variations (oscillations with smaller amplitude)

Based on the expression (35) we can conclude:

1. FM signal with a single tone signal contain component carrier and component

frequency side bands are not finite numbers:

𝜔 = 𝜔𝐶 ± 𝑛𝜔𝑚, n = 1, 2, 3

2. Amplitude of each component frequency depend on β, which depends also with

characteristic ψ m.

3. For case of narrow band

J0 (β) ≈ 1 ............................................................................... (36a)

J1 (β) ≈ β/2 ............................................................................ (36b)

J0 (β) ≈ 0, n>1 ....................................................................... (36c)

So in the case of β << 1 (narrow band), the spectrum contains only a component frequency

ω c and ω c ± ω m as well as the AM band. Whereas in the case of wide band (wide band) β

>> 1, FM side band clearly contains components which are quite large, and therefore have

a large bandwidth but still limited.

14

The following will be reviewed power and transition bandwidth, the ratio S / N on the FM

wave. As in the AM band, to determine the transmission on FM tape will be reviewed

comparison of the average price of the total transmission power N T and N n to the power

transmission tape containing the low frequency components.

𝐴𝑛 =𝑁𝑛

𝑁𝑇 ............................................................................... (37)

With Nn satisfies the equation (7) and tnJt mc

n

ncn )ωω()()(

or

l

lc JNn )(2

1 22

.......................................................... (38)

Because the tribes cross the sum 𝜓𝑛2(𝑡) yields the average price of zero. And along with it

can be formulated for NT.

222

2

1)(

2

1c

l

lcT JN

............................................. (39)

Having used the identity (36b). These results indicate that the amplitude of the FM signal

is constant. With the above results, the equation (37) becomes

l

ln Ja )(2

.................................................................... (40)

In numeric can be shown that the FM signal with a single tone modulation, 𝑎𝑛 ≥ 98%, for

𝑛 ≥ 𝛽 + 1............................................................................. (41)

Therefore, the transmission bandwidth can be express by the concerned.

B ≈ 2 (β+1) ωm = 2 (ω’+ ωm)

It is clear that for narrow band B ≈ 2 ω m. For more general FM signal (non-sinusoidal), the

modulation index β can not be defined, and the rules applicable to B Carson:

B = 2 (D+1) ωm = 2 (ω’+ ωm) .............................................. (42)

With D stated ratio, D = ω’max / ωm

As a small note on the angle modulation advantages may be mentioned that the ratio S/ N of

the results of the ideal modulation is proportional to (kF)2 for the FM signal. This means an

increase in the price of the S / N can be achieve by increasing the sensitivity of the modulator

in question (k). Yet another result that will happen is widening B because D is directly

proportional to k F without increasing the transmission power, because the amplitude of the

carrier wave remains the same.

15

Furthermore, these parameters, B, S/N will determine the capacity of the FM signal channels

according to the Shannon-Hartley formula for the ideal system:

C = B log 2 (1 + S / N) bits / s ............................................ (43)

2.5 Application of Modulation Concept

Application of the modulation wave is on the radio. Radio broadcasts in operation

using modulation techniques, where the signal is riding sound signals, while carrying a radio

signal called the carrier signal. Of the many modulation techniques, AM and FM modulation

which is widely applied in radio broadcasts. Both of these techniques is use because it is

easy to compare in other techniques. That way, a series of radio transmitters and receivers it

is more simple and easy to make.

At the radio station, the radio waves generated by electric charges accelerated

through a wire conductor. The electrical charge generated by an insulator. Prior transmitted

through a transmitting antenna, radio waves first modulated (bundled) with the audio signal.

The radio waves that carry the audio signal to transmit through a transmitting antenna. In

this case serves as a radio wave (carrier wave) which carries the audio signal.

On the radio transmitter, amplitude modulation techniques (AM), the amplitude of

the carrier wave will be change in line with changes in the information signal (sound)

entered. Relatively fixed frequency carrier wave then passed to the RF (Radio Frequency)

amplifier to strengthen in order to send to long distances. After that, transmitted through the

antenna.

In the journey reaches the receiver, the waves will experience attenuation (fading)

by air, gets interference from other frequencies, noise, or other forms of interference. The

disturbances are generally in the form of amplitude variations that will inevitably affect the

amplitude of the waves sent. This resulted in the information that is sent will be changed and

the quality of information will decrease.

How to reduce losses caused by attenuation, noise, and interference is quite difficult.

Disturbance amplitude reduction (which has a smaller amplitude), will have an impact on

the reduction of the original signal. Meanwhile, the increase in the amplitude of the original

signal also causes an increase in the amplitude of the disturbance. The dilemma can

overcome by using techniques that are more complicated. However, this led receiver circuit

will be expensive, while the results obtained have not been quality Hi Fi and not necessarily

equivalent to the price paid. Consequently, they also have to move the carrier frequency due

to the rules of the carrier frequency locations for different AM broadcast with FM broadcasts.

16

Carrier frequency to broadcast AM located at the Medium Frequency (300 kHz - 3 MHz /

MF), while the carrier frequency of the FM broadcast is in the Very High Frequency (30

MHz - 300 MHz / VHF).

In a radio transmitter with FM modulation technique, the frequency of the carrier

wave will change as you change the sound signals or other information. The amplitude of

the carrier wave is relatively fixed. After the signal power gain (to be sent away), which has

been mixed earlier waves emitted by an antenna. As well as AM waves, these waves will

experience attenuation by air and gets interference from other frequency, noise, or other

forms of interference. However, since it generally shaped disturbance amplitude variations

are less likely to affect the information that riding in the carrier wave frequency.

This result, the quality of information received remains good. The audio quality is

also higher than the modulated audio quality AM. Therefore, the music we listen to music

quality will be similar to that sent by the radio station. Stereo sound transmission technique

is also not too complicated so that the circuit is easy to make.

17

CHAPTER III

CONCLUSION

Based on the description of the material that has had been described, there are several

conclusions that drawn regarding the wave modulation, as follows:

1. Modulation wave are basically the process of changing certain characteristics of a wave

by wave oscillation pattern of the others. Advantages modulation i.e.: allowing weak

signals to bring high waves, the antenna size reduction of high-frequency carrier, the

setting and the allocation of separate frequency regions through the same medium, and

the shift frequency signal into the area easily processed.

2. Carrier wave generally sinusoidal waveform. DSB modulation wave band width to 2

times the signal wave, but the amplitude becomes half time.

3. Modulation Amplitude modulation formulated as a deficiency in DSB has demodulation

wherein the regenerated carrier signal must be coherent with the existing carrier signal

at the transmitter (modulation process).

4. Frequency modulation (FM) is the angle modulation is the process of changing the phase

angle of the carrier wave signal according to the pattern of the wave changes

(modulation) that are non-linear.

5. Application of the modulation wave is on the radio. Radio broadcasts in operation using

modulation techniques, where the signal is riding sound signals, while carrying a radio

signal called the carrier signal. Of the many modulation techniques, AM and FM

modulation which is widely applied in radio broadcasts.

18

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