NATIONAL COLLEGE OF SCIENCE AND TECHNOLOGYAmafel Bldg. Aguinaldo Highway Dasmariñas City, Cavite

ASSIGNMENT # 3

“FREQUENCY MODULATION”

Cauan, Sarah Krystelle P. July 11, 2011Communications 1 / BSECE 41A1 Score:

Eng’r. Grace RamonesInstructor

FREQUENCY MODULATION

Frequency Modulation Principles

While changing the amplitude of a radio signal is the most obvious method to modulate

it, it is by no means the only way. It is also possible to change the frequency of a signal to give

frequency modulation or FM. Frequency modulation is widely used on frequencies above 30

MHz, and it is particularly well known for its use for VHF FM broadcasting.

In FM, the carrier amplitude remains constant, while the carrier frequency is changed by

the modulating signal. As the amplitude of the information signal varies, the carrier frequency

will shift in proportion. As the modulating signal amplitude increases, the carrier frequency

increases. If the amplitude of the modulating signal, decreases the carrier frequency decreases.

The reverse relationship can also be implemented. A decreasing modulating signal will increase

the carrier frequency above its center value, whereas an increasing modulating signal amplitude

varies, the carrier frequency varies above and below its normal center frequency with no

modulation. The amount of change in carrier frequency produced by the modulating signal is

known as the frequency deviation. Maximum frequency deviation occurs at the maximum

amplitude of the modulating signal.

The frequency of the modulating signal determines how many times per second the

carrier frequency deviates above and below its nominal center frequency. If the modulating

signal is 100-Hz sine wave, then the carrier frequency will shift above and below the center

frequency 100 times per second. This is called the frequency deviation rate.

An FM signal is illustrated in Figure 1. With no modulating signal applied, the carrier

frequency is a constant-amplitude sine wave at its normal constant center frequency.

The modulating information signal is a low-frequency sine wave. As the sine wave goes

positive, the frequency of the carrier increases proportionately. The highest frequency occurs at

the peak amplitude of the modulating signal. As the modulating signal amplitude decreases, the

carrier frequency decreases. When the modulating signal is zero amplitude, the carrier will be at

its center frequency point.

Now when the modulating signal goes negative, the carrier frequency will decrease. The

carrier frequency will continue to decrease until the peak of the negative half cycle of the

modulating sine wave is reached. Then, as the modulating signal increases toward zero, the

frequency will again increase. Note in Figure 1 how the carrier sine waves seem to be first

“compressed” and then “stretched” by the modulating signal.

a)

b)

c)

Figure 1 The principle of

frequency modulation: (a) carrier signal, (b) modulating signal

(c) Modulated signal (Frequency Modulation)

Phase Modulation

Another way to produce angle modulation is to vary the amount of phase shift of a

constant frequency carrier in accordance with a modulating signal. The resulting output is a PM

signal. Imagine a modulator circuit whose basic function is to produce a phase shift. Remember

that a phase shift refers to a time separation between two sine waves of the same frequency.

Assume that we can build a phase shifter that causes the amount of phase shift to vary with the

amplitude of the modulating signal. The greater the amplitude of the modulating signals, the

greater the phase shifts. Assume further that positive alternations of the modulating signal

produce a lagging phase shift and negative signals produce a leading phase shift.

If a constant-amplitude frequency carrier sine wave is applied to the phase shifter, the

output of the phase shifter will be a PM wave. As the modulating signal goes positive, the

amount of phase lag increase with the amplitude modulating signal. This means that the carrier

output is delayed. That delay increases with the amplitude of the modulating signal. The result at

the output is as if the constant-frequency carrier signal had been stretched out or its frequency

lowered.

When the modulating signal goes negative, the phase shift becomes leading. This causes

the carrier sine wave to be effectively speeded up or compressed. The result is as if the carrier

frequency had been increased.

Phase modulation produces frequency modulation. Since the amount of phase shift is

varying, the effect is as is the carrier frequency is changed. Since FM is produced by PM, the

later is often referred to as indirect FM.

It is important to point out that it is the dynamic nature of the modulating that causes the

frequency variation at the output of the phase shifter. In other words, FM is only reduced as long

as the phase shift is being varied.

In FM, maximum deviation occurs at the peak positive and negative amplitudes of the

modulating signal. In PM, the maximum amount of leading ang lagging shift occurs at the peak

amplitude of the modulating signal. The faster the modulating signal voltage varies the greater

the frequency deviation produced. Because of this, the frequency deviation produced in PM

increases with the frequency of the modulating signal. The higher the modulating signal

frequency, naturally the shorter its period and the faster the voltage changes. Higher modulating

voltages produce greater frequency deviation. However, higher modulating frequencies produce

a faster rate of change of modulating voltage and, therefore, also produce greater frequency

deviation.

D eviation

When the audio signal is modulated onto the radio frequency carrier, the new radio

frequency signal moves up and down in frequency. The amount by which the signal moves up

and down is important. It is known as the deviation and is normally quoted as the number of

kilohertz deviation. As an example the signal may have a deviation of ±3 kHz. In this case the

carrier is made to move up and down by 3 kHz.

Assume a carrier frequency of 50 MHz .if the peak amplitude of the modulating signal

causes a maximum frequency shift of 200 kHz, the carrier frequency will deviate up to 50.2 MHz

and down to 59.8 MHz. The total frequency deviation is 50.2 – 49.8 = 0.4 MHz = 400 kHz. In

practice, however, the frequency deviation is expressed as the amount of frequency shift of the

carrier above or below the center frequency. Therefore, the frequency deviation in the example

above is said to be ±200 kHz. This means that the modulating signal varies the carrier above and

below its center frequency to 200 kHz. The frequency of the modulating signal determines the

rate of frequency deviation but has no effect on the amount of deviation which is strictly a

function of the amplitude of the modulating signal.

Advantages of frequency modulation, FM

Although it may not be quite as straightforward as amplitude modulation, nevertheless frequency modulation, FM, offers some distinct advantages. It is able to provide near interference free reception, and it was for this reason that it was adopted for the VHF sound broadcasts. These transmissions could offer high fidelity audio, and for this reason, frequency modulation is far more popular than the older transmissions on the long, medium and short wave bands.

In addition to its widespread use for high quality audio broadcasts, FM is also sued for a variety of two way radio communication systems. Whether for fixed or mobile radio communication systems, or for use in portable applications, FM is widely used at VHF and above.

FM is used for a number of reasons and there are several advantages of frequency modulation. In view of this it is widely used in a number of areas to which it is ideally suited. Some of the advantages of frequency modulation are noted below:

Resilience to noise: One particular advantage of frequency modulation is its resilience to

signal level variations. The modulation is carried only as variations in frequency. This

means that any signal level variations will not affect the audio output, provided that the

signal does not fall to a level where the receiver cannot cope. As a result this makes FM

ideal for mobile radio communication applications including more general two-way radio

communication or portable applications where signal levels are likely to vary

considerably. The other advantage of FM is its resilience to noise and interference. It is

for this reason that FM is used for high quality broadcast transmissions.

Easy to apply modulation at a low power stage of the transmitter: Another advantage of

frequency modulation is associated with the transmitters. It is possible to apply the

modulation to a low power stage of the transmitter, and it is not necessary to use a linear

form of amplification to increase the power level of the signal to its final value.

It is possible to use efficient RF amplifiers with frequency modulated signals: It is

possible to use non-linear RF amplifiers to amplify FM signals in a transmitter and these

are more efficient than the linear ones required for signals with any amplitude variations

(e.g. AM and SSB). This means that for a given power output, less battery power is

required and this makes the use of FM more viable for portable two-way radio

applications.

Sidebands

Any modulation process produces sidebands. As you saw in AM, when a constant-

frequency sine wave modulates a carrier, two side frequencies are produced. The side

frequencies are the sum and difference of the carrier and the modulating frequency. In FM and

PM too, sum and difference sideband frequencies are produced. In addition, a theoretically

infinite number of pairs of upper and lower sidebands are also generated. As a result, the

spectrum of an FM/PM signal usually wider than an equivalent AM signal. A special

narrowband FM signal whose bandwidth is only slightly wider than that of an AM signal can

also be generated.

Figure 2 shows an example of the spectrum of a typical FM signal produced by

modulating a carrier with a single-frequency sine wave. Note that the sidebands are spaced from

the carrier fc and are space from one another by a frequency equal to the modulating frequency

fm. If the modulating frequency is 500 Hz, the first pair of sidebands are above and below the

carrier by 500 Hz. The second pair of sidebands are above and below the carrier by 2 × 500 Hz

1000 Hz or 1 kHz, and so on. Note also that the amplitudes/intensities of the sidebands vary. Is

each sideband is assumed to be sine wave with a frequency and amplitude as indicated in Fig 1

and all these sine waves were added together, then the FM signal producing them would be

created.

Figure 2. Frequency Domain Display, fc and sidebands

As the amplitude of the modulating signal varies, of course, the frequency deviation will

change. The number of sidebands produced, their amplitude, and their spacing depend upon the

frequency deviation and modulating frequency. Keep in mind that an FM signal has a constant

amplitude. If that FM signal is a summation of the sideband frequencies, then you can see that

the sideband amplitudes must vary with frequency deviation and modulating frequency if their

sum is to produced a fixed amplitude FM signal.

Although the FM process produces an infinite number of upper and lower sidebands, only

those with the largest amplitudes are significant in carrying the information. Typically any

sideband whose amplitude is less than 1 percent of the unmodulated carrier is considered

insignificant. As a result, this markedly narrows the bandwidth of an FM signal.

MODULATION INDEX

Modulation Index

As indicated earlier, the number of significant sidebands and their amplitudes are dependent

upon the amount of frequency deviation and the modulating frequency. The ratio so the

frequency deviation to the modulating frequency is known as the modulation index, m.

m=f d

f m

where fd is the frequency deviation and fm is the modulating frequency.

For example, assume that the maximum frequency deviation of the carrier is ± 25 kHz while the

maximum modulating frequency is 10 kHz. The modulating index, therefore, is

m=f d

f m

=2510

=2.5

In most communication systems using FM, maximum limits are put on both the frequency

deviation and the modulating frequency. For example, in standard FM broadcasting, the

maximum permitted frequency deviation is 75 kHz, while the maximum permitted modulating

frequency is 15 kHz. This produce a modulating index of

m=f d

f m

=7515

=5

Whenever the maximum allowable frequency deviation and maximum modulating frequency are

used in computing the modulation index, m is known as the deviation ratio.

Knowing the modulation index, you can compute the number and amplitudes of the significant

sidebands. This is done through a complex mathematical process known as the Bessel function.

BESSEL FUNCTION TABLE

Figure 3. the left-hand column gives the modulation index. The remaining columns

indicate the relative amplitudes of the carrier and the various parts of sidebands. Any sideband

with relative carrier amplitude of less than 1 percent has been eliminated. Note that some of the

carrier and sideband amplitudes have negative signs. This means that the signal represented by

the amplitude is simply shifted in phase 180o (phase inversion)

As you can see, the spectrum of an FM signal varies considerably in bandwidth

depending upon the modulation index. The higher the modulation index, the wider the bandwidth

of an FM signals nay be deliberately by putting an upper limit on the modulation index.

Modulation

index

Sideband

Carrier 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.00 1.00

0.25 0.98 0.12

0.5  0.94 0.24 0.03

1.0  0.77 0.44 0.11 0.02

1.5  0.51 0.56 0.23 0.06 0.01

2.0  0.22 0.58 0.35 0.13 0.03

2.41 0    0.52 0.43 0.20 0.06 0.02

2.5  −0.05 0.50 0.45 0.22 0.07 0.02 0.01

3.0  −0.26 0.34 0.49 0.31 0.13 0.04 0.01

4.0  −0.40−0.0

70.36 0.43 0.28 0.13 0.05 0.02

5.0  −0.18−0.3

30.05 0.36 0.39 0.26 0.13 0.05 0.02

5.53 0   −0.3

4−0.13 0.25 0.40 0.32 0.19 0.09 0.03 0.01

6.0  0.15−0.2

8−0.24 0.11 0.36 0.36 0.25 0.13 0.06 0.02

7.0  0.30 0.00 −0.30 −0.17 0.16 0.35 0.34 0.23 0.13 0.06 0.02

8.0  0.17 0.23 −0.11 −0.29−0.1

00.19 0.34 0.32 0.22 0.13 0.06 0.03

8.65 0    0.27 0.06 −0.24−0.2

30.03 0.26 0.34 0.28 0.18 0.10 0.05 0.02

9.0  −0.09 0.25 0.14 −0.18−0.2

7−0.06 0.20 0.33 0.31 0.21 0.12 0.06 0.03 0.01

10.0  −0.25 0.04 0.25 0.06−0.2

2−0.23 −0.01 0.22 0.32 0.29 0.21 0.12 0.06 0.03 0.01

12.0  0.05−0.2

2−0.08 0.20 0.18 −0.07 −0.24 −0.17 0.05 0.23 0.30 0.27 0.20 0.12 0.07 0.03 0.01

Figure 3 A table showing carrier and sideband amplitudes for different modulation indexes of FM signals. Based on the Bessel Function.

TYPES OF FREQUENCY MODULATION

Wide Band Frequency Modulation – Broadcast stations in the VHF portion of the

frequency spectrum between 88.5 and 108 MHz use large values of deviation, typically ±75 kHz.

This is known as wide-band FM (WBFM). These signals are capable of supporting high quality

transmissions, but occupy a large amount of bandwidth. Usually 200 kHz is allowed for each

wide-band FM transmission.

For b > 0.3 there are more than 2 significant sidebands. As b increases the number of

sidebands increases. This is referred to as wideband FM (WBFM).

Narrow Band Frequency Modulation – For communications purposes less bandwidth is

used. Narrow band FM (NBFM) often uses deviation figures of around ±3 kHz. It is narrow band

FM that is typically used for two-way radio communication applications. Having a narrower

band it is not able to provide the high quality of the wideband transmissions, but this is not

needed for applications such as mobile radio communication.

From the graph/table of Bessel functions it may be seen that for small b, (b £ 0.3) there

is only the carrier and 2 significant sidebands, i.e. BW = 2fm. FM with b £ 0.3 is referred to as

narrowband FM (NBFM) (Note, the bandwidth is the same as DSBAM).

The block diagrams satisfy the corresponding expression for FM.

POWER IN FREQUENCY MOULATION

From the equation for FM

we see that the peak value of the components is VcJn(b) for the nth component.

Single normalised average power = then the nth component is

Hence, the total power in the infinite spectrum is

Total power

By this method we would need to carry out an infinite number of calculations to find PT. But,

considering the waveform, the peak value is Vc, which is constant.

Since we know that the RMS value of a sine wave is

and power = (VRMS)2 then we may deduce that

Hence, if we know Vc for the FM signal, we can find the total power PT for the infinite spectrum

with a simple calculation.

Now consider – if we generate an FM signal, it will contain an infinite number of sidebands.

However, if we wish to transfer this signal, e.g. over a radio or cable, this implies that we require

an infinite bandwidth channel. Even if there was an infinite channel bandwidth it would not all

be allocated to one user. Only a limited bandwidth is available for any particular signal. Thus we

have to make the signal spectrum fit into the available channel bandwidth. We can think of the

signal spectrum as a ‘train’ and the channel bandwidth as a tunnel – obviously we make the train

slightly less wider than the tunnel if we can.

vs( t )=V c ∑n=−∞

∞

J n( β )cos (ωc+nωm)t

(V pk

√2 )2

=(V RMS)2

(V c J n ( β )

√2 )2

=(V c J n( β ))2

2

PT= ∑n=−∞

∞ (V c J n( β ))2

2

(V pk

√2 )2

=V c

√2

PT=(V c

√2 )2

=V c

2

2= ∑

n=−∞

∞ (V c J n( β ))2

2

However, many signals (e.g. FM, square waves, digital signals) contain an infinite number of

components. If we transfer such a signal via a limited channel bandwidth, we will lose some of

the components and the output signal will be distorted. If we put an infinitely wide train through

a tunnel, the train would come out distorted, the question is how much distortion can be

tolerated? Generally speaking, spectral components decrease in amplitude as we move away

from the spectrum ‘centre’.

In general distortion may be defined as

With reference to FM the minimum channel bandwidth required would be just wide enough to

pass the spectrum of significant components. For a bandlimited FM spectrum, let a = the number

of sideband pairs, e.g. for b = 5, a = 8 pairs (16 components). Hence, power in the bandlimited

spectrum PBL is

= carrier power + sideband powers.

D=Power in total spectrum - Power in Bandlimited spectrumPower in total spectrum

D=PT−PBL

PT

PBL= ∑n=−a

a (V c J n ( β ))2

2

Since

Distortion

Also, it is easily seen that the ratio

= 1 – Distortion

i.e. proportion pf power in band limited spectrum to total power =

PT=V c

2

2

D=

V c2

2−

V c2

2 ∑n=−a

a

( J n( β ))2

V c2

2

=1− ∑n=−a

a

( Jn ( β ))2

D=Power in Bandlimited spectrumPower in total spectrum

=PBL

PT

= ∑n=−a

a

( J n( β ))2

∑n=−a

a

(J n ( β ))2