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In frequency modulation, the frequency of a rapidly varying

signal is modulated by a slowly varying signal. The

mathematical description of the modulation of the

frequency of a sinusoidal carrier by another sinusoid is

given in the formula below where wc is carrier frequency,

wm is the modulation frequency, and m is the modulationindex. A plot of the resulting signal in the time domain

(amplitude versus time) is shown in the interactive graph

labeled Time Domain.

An expansion of the formula shows that the resulting signal consists of a large

number of frequency components, or sidebands. This is in contrast to the case of

amplitude modulation where only two sidebands are created. The sidebands are

spaced apart by the modulation frequency, wm, and centered about the carrier

frequency, wc. The figure labeled Frequency Domain shows the carrier and

sideband components as a plot of amplitude versus frequency.

Only the most significant components are shown. An infinite number of higher

order components have negligible amplitude. In fact, the amplitude of each

component is given by a Bessel function of the appropriate component order. In

mathematical notation the amplitude of the nth sideband is Jn(m), where m is

the modulation index.

These graphs are interactive. The parameters of the mathematical model can be

adjusted by dragging the sliders with your mouse. You may also click and drag

the graphical objects inside the Time and Frequency Domain graphs. For

example, the figure above shows how the carrier frequency can be adjusted bydragging the center arrow in the Frequency Domain graph. A complete

mathematical definition and analysis of Frequency Modulation is available in the

application note listed below.

Frequency modulation

While AM is the simplest form of modulation to envisage, it is also possible to vary the

frequency of the signal to give frequency modulation (FM). It can be seen from Figure 3-8

that the frequency of the signal varies as the voltage of the modulating signal changes.

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Figure 3-8. A frequency modulated signal.

The amount by which the signal frequency varies is very important. This is known as the

deviation, and is normally quoted in kilohertz. 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.

FM is used for a number of reasons. One particular advantage is its resilience to signal-level

variations and general interference. The modulation is carried only as variations in frequency,

and this means that any signal-level variations will not affect the audio output provided that

the signal is of a sufficient level. As a result, this makes FM ideal for mobile or portable

applications where signal levels vary considerably. The other advantage of FM is its

resilience to noise and interference when deviations much greater than the highest modulating

frequency are used. It is for this reason that FM is used for high-quality broadcast

transmissions where deviations of 75 kHz are typically used to provide a high level of

interference rejection. In view of these advantages, FM was chosen for use in the first-

generation analogue mobile phone systems.

To demodulate an FM signal, it is necessary to convert the frequency variations into voltage

variations. This is slightly more complicated than demodulating AM, but it is still relatively

simple to achieve. Rather than just detecting the amplitude level using a diode, a tuned circuit

has to be incorporated so that a different output voltage level is given as the signal changes its

frequency. There is a variety of methods used to achieve this, but one popular approach is to

use a system known as a quadrature detector. It is widely used in integrated circuits, and

provides a good level of linearity. It has the advantages that it requires a simple tuned circuit

and it is also very easy to implement in a form that is applicable to integrated circuits.

The basic format of the quadrature detector is shown in Figure 3-9. It can be seen that thesignal is split into two components. One of these passes through a network that provides a

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basic 90 phase shift, plus an element of phase shift dependent upon the deviation. The

original signal and the phase-shifted signal are then passed into a multiplier or mixer. The

mixer output is dependent upon the phase difference between the two signals, i.e. it acts as a

phase detector and produces a voltage output that is proportional to the phase difference and

hence to the level of deviation of the signal.

Figure 3-9. Block diagram of an FM quadrature detector.

Modulation index and deviation ratio

In many instances a figure known as the modulation index is of value and is used in other

calculations. The modulation index is the ratio of the frequency deviation to the modulating

frequency, and will therefore vary according to the frequency that is modulating the

transmitted carrier and the amount of deviation:

However, when designing a system it is important to know the maximum permissible values.This is given by the deviation ratio, and is obtained by inserting the maximum values into the

formula for the modulation index:

SidebandsAny signal that is modulated produces sidebands. In the case of an amplitudemodulated signal they are easy to determine, but for frequency modulation thesituation is not quite as straightforward. They are dependent upon not only thedeviation, but also the level of deviation i.e., the modulation index M. The totalspectrum is an infinite series of discrete spectral components, expressed by thecomplex formula:

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In this relationship,Jn(M) are Bessel functions of the first kind, c is the angular frequency of

the carrier and is equal to 2, and m is the angular frequency of the modulating signal. Vc isthe voltage of the carrier.

It can be seen that the total spectrum consists of the carrier plus an infinite number of

sidebands spreading out on either side of the carrier at integral frequencies of the modulatingfrequency. The relative levels of the sidebands can be read from a table of Bessel functions,

or calculated using a suitable computer program. Figure 3-10 shows the relative levels to give

an indication of the way in which the levels of the various sidebands change with different

values of modulation index.

Figure 3-10. The relative amplitudes of the carrier and the first 10 side frequency

components of a frequency modulated signal for different values of modulation index.

It can be gathered that for small levels of deviation (that is, what is termed narrowband FM)

the signal consists of the carrier and the two sidebands spaced at the modulation frequency

either side of the carrier. The spectrum appears the same as that of an AM signal. The major

difference is that the lower sideband is out of phase by 180.

As the modulation index increases, other sidebands at twice the modulation frequency start to

appear (Figure 3-11). As the index is increased, further sidebands can also be seen. It is also

found that the relative levels of these sidebands change, some rising in level and others

falling as the modulation index varies.

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Figure 3-11. Spectra of frequency-modulated signals with various values of modulation index

for a constant modulation frequency. It can be seen that for small values of the modulationindex M (e.g. M = 0.5), the signal appears to consist of the carrier and two sidebands. As the

modulation index increases, the number of sidebands increases and the level of the carriercan be seen to decrease for these values.

Bandwidth

It is clearly not acceptable to have a signal that occupies an infinite bandwidth. Fortunately,

for low levels of modulation index all but the first two sidebands may be ignored. However,

as the modulation index increases the sidebands further out increase in level, and it is often

necessary to apply filtering to the signal. This should not introduce any undue distortion. To

achieve this it is normally necessary to allow a bandwidth equal to twice the maximum

frequency of deviation plus the maximum modulation frequency. In other words, for a VHFFM broadcast station with a deviation of 75 kHz and a maximum modulation frequency of

15 kHz, this must be (2 75) + 15 kHz, i.e. 175 kHz. In view of this a total of 200 kHz is

usually allowed, enabling stations to have a small guard band and their centre frequencies on

integral numbers of 100 kHz.

Improvement in signal-to-noise ratio

It has already been mentioned that FM can give a better signal-to-noise ratio than AM when

wide bandwidths are used. The amplitude noise can be removed by limiting the signal. In

fact, the greater the deviation, the better the noise performance. When comparing an AM

signal with an FM signal, an improvement equal to 3D2 is obtained where D is the deviation

ratio. This is true for high values of D i.e. wideband FM.

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An additional perceived improvement in signal-to-noise ratio can be achieved if the audio

signal is pre-emphasized. To achieve this, the lower-level high-frequency sounds are

amplified to a greater degree than the lower-frequency sounds before they are transmitted.

Once at the receiver, the signals are passed through a network with the opposite effect to

restore a flat frequency response.

To achieve the pre-emphasis, the signal may be passed through a capacitorresistor (CR)

network. At frequencies above the cut-off frequency, the signal increases in level by 6 dB per

octave. Similarly, at the receiver the response falls by the same amount.

Frequency shift keying

Many signals employ a system called frequency shift keying (FSK) to carry digital data

(Figure 3-12). Here, the frequency of the signal is changed from one frequency to another,

one frequency counting as the digital 1 (mark) and the other as a digital 0 (space). By

changing the frequency of the signal between these two it is possible to send data over the

radio.

Figure 3-12. Frequency shift keying.

There are two methods that can be employed to generate the two different frequencies needed

for carrying the information. The first and most obvious is to change the frequency of the

carrier. Another method is to frequency-modulate the carrier with audio tones that change in

frequency, in a scheme known as Audio Frequency Shift Keying (AFSK). This second

method can be of advantage when tuning accuracy is an issue.

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Frequency Modulation

The instantaneous frequency deviation of the modulated carrier with respect to

the frequency of the unmodulated carrier is directly proportional to the

instantaneous amplitude of the modulating signal.

Bandwidth of FM signals

In practice, the spectrum of an FM signal is not infinite. The sideband amplitudes

become negligibly small beyond a certain frequency offset from the carrier,

depending on the magnitude of . We can determine the bandwidth required for

low distortion transmission by counting the number of significant sidebands. For

high fidelity, significant sidebands are those sidebands that have a voltage at

least 1 percent (40 dB) of the voltage of the unmodulated carrier for any

between 0 and maximum. We shall now investigate the spectral behavior of an

FM signal for different values of . In figure 22, we see the spectra of a signal for

= 0.2, 1, 5, and 10.

The sinusoidal modulating signal has the constant frequency fm, so isproportional to its amplitude. In figure 23, the amplitude of the modulating

signal is held constant and, therefore, is varied by changing the modulating

frequency. Note: in figure 23a, b, and c, individual spectral components are

shown; in figure 23d, the components are not resolved, but the envelope is

correct.

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Figure 22. Amplitude-frequency spectrum of an FM signal (sinusoidal

modulating signal; f fixed; amplitude varying). In (a), = 0.2; in (b),

= 1; in (c), = 5; in (d), = 10

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Figure 23. Amplitude-frequency spectrum of an FM signal (amplitude of

delta f fixed; fm decreasing.) In (a), = 5; in (b), = 10; in (c), = 15;

in (d), >

Two important facts emerge from the preceding figures:

(1) For very low modulation indices ( less than 0.2), we get only one

significant pair of sidebands. The required transmission

bandwidth in this case is twice fm, as for AM. (2) For very highmodulation indices ( more than 100), the transmission

bandwidth is twice fp.

For values of between these extremes we have to count the significant sidebands.