1. NATIONAL COLLEGE OF SCIENCE AND TECHNOLOGY
Amafel Bldg. Aguinaldo Highway Dasmariñas City, Cavite
ASSIGNMENT # 3
“FREQUENCY MODULATION”
Tagasa, Jerald A. July 11, 2011
Communications 1 / BSECE 41A1 Score:
Eng’r. Grace Ramones
Instructor

2. FREQUENCY MODULATION
Frequency modulation (FM) is a method of impressing data onto an alternating-current (AC)
wave by varying the instantaneous frequency of the wave. This scheme can be used with analog or
digital data
.
In analog FM, the frequency of the AC signal wave, also called the carrier, varies in a continuous
manner. Thus, there are infinitely many possible carrier frequencies. In narrowband FM, commonly used
in two-way wireless communications, the instantaneous carrier frequency varies by up to 5 kilohertz
(kHz, where 1 kHz = 1000 hertz or alternating cycles per second) above and below the frequency of the
carrier with no modulation. In wideband FM, used in wireless broadcasting, the instantaneous frequency
varies by up to several megahertz (MHz, where 1 MHz = 1,000,000 Hz). When the instantaneous input
wave has positive polarity, the carrier frequency shifts in one direction; when the instantaneous input
wave has negative polarity, the carrier frequency shifts in the opposite direction. At every instant in
time, the extent of carrier-frequency shift (the deviation) is directly proportional to the extent to which
the signal amplitude is positive or negative.
In digital FM, the carrier frequency shifts abruptly, rather than varying continuously. The
number of possible carrier frequency states is usually a power of 2. If there are only two possible
frequency states, the mode is called frequency-shift keying (FSK). In more complex modes, there can be
four, eight, or more different frequency states. Each specific carrier frequency represents a specific
digital input data state.
Frequency modulation is similar in practice to phase modulation (PM). When the instantaneous
frequency of a carrier is varied, the instantaneous phase changes as well. The converse also holds: When
the instantaneous phase is varied, the instantaneous frequency changes. But FM and PM are not exactly
equivalent, especially in analog applications. When an FM receiver is used to demodulate a PM signal, or
when an FM signal is intercepted by a receiver designed for PM, the audio is distorted. This is because
the relationship between frequency and phase variations is not linear; that is, frequency and phase do
not vary in direct proportion.

3. TYPES OF FREQUENCY MODULATION
The bandwidth of an FM signal depends on the deviation Kff(t). When the deviation is high, the
bandwidth will be large, and vice-versa. From the equation Δω = Kf f(t) it is clear that deviation is
controlled by Kff (t) (where Δω = frequency deviation). Thus, for a given f (t), the deviation, and hence,
bandwidth will depend on frequency sensitivity Kf. If Kf is too small then the bandwidth will be narrow
and vice-versa. Thus depending on the value of Kf, FM can be divided into two categories.
(1) Narrow band FM: - When Kf is small, the bandwidth of FM is narrow.
(2) Wide band FM: - When Kf has an appreciable value, then the FM signal has a wide
bandwidth. Ideally it is infinite.
Narrow Band Frequency Modulation
The general expression for FM in the phasor form is given by ΦFM(t) = A ej*ωct +Kfg(t)]
For a narrow band FM, Kf g(t) <<1 for all value of t. Hence ejKfg(t) ≈ 1 + j Kfg(t)
And FM phasor expression becomes ΦFM(t) ≈ A *1 + jKfg(t)] ejωct
The FM signal is the real part of its phasor representation, ΦFM(t) ≈ Re* ΦFM(t)+ = A cosωct - A
Kfg(t) sinωct
The above expression of narrow band FM is very much similar to the expression of the AM
(Amplitude modulation) signal, with only a slight modification. In AM all the three components i.e.,
carrier and two sidebands are in phase, but in narrow band FM, the lower sideband is 180 degree out of
the phase with respect to carrier as well as upper sideband. Thus the bandwidth of a narrow band FM is
same as that of the AM.
Generation of Narrowband FM:
Above equation suggest methods for generating the narrow band FM. The sideband terms are
obtained by a balanced modulator, as in Double sideband suppressed carrier amplitude modulation
(DSB-SC) systems and then the carrier term is added t sideband terms. The method for generating
narrow band FM is shown in the figure drawn below. The block diagrams satisfy the corresponding
expression for FM.
Carrier generator generates the carrier signal, and then 90 degree phase-shifter provides the
shift in the carrier signal. After that, this signal is fed to balance modulator in which another signal g (t) is
also fed which is the output of integrator, whereas f(t) is the input to the integrator. After balance
modulator the signal goes to adder where signal is added with unmodified carrier signal. The output of
the adder is required signal i.e., FM signal.
Wide Band Frequency Modulation :
When Kf has larger value, then the signal has a wide bandwidth, ideally infinite and the signal is
called wideband signal. The general equation of wideband FM is given by

4. ΦFm (t) = AJ0(mf)cosωct + AJ1(mf)*cos(ωc+ωm)t - cos(ωc-ωm)t] +AJ2(mf)*cos(ωc + 2ωm)t + cos(ωc - 2ωmt)] +
AJ3(mf) *cos(ωc + 3ωm)t - cos(ωc - 3ωm)t] + ...
Wide Band Frequency Modulation : Frequency Components
The FM signal has the following frequency components:
Carrier term cosωct with magnitude AJ0 (mf), i.e. the magnitude of the carrier term is reduced by
a factor J0 (mf) here, J0 is the Bessel function coefficient and mf is known as modulation index of the FM
wave. It is defined as the “ratio of frequency deviation to the modulating frequency. The modulation
index mf decides whether an FM wave is a narrowband or a wideband because it is directly proportional
to the frequency deviation. Normally mf = 0.5 is the transition point between a narrowband and a
wideband FM. If mf<0.5, then FM is a narrowband, otherwise it is a wideband. As discussed, when m f is
large, the FM produces a large number of sidebands and the bandwidth of FM is quite large. Such
systems are called wideband FM.
Theoretically infinite numbers of sidebands are produced, and the amplitude of each sideband is
decided by the corresponding Bessel function Jn (mf). The presence of infinite number of sidebands
makes the ideal bandwidth of the FM signal is infinite. However, the sidebands with small amplitudes
are ignored. The sidebands having considerable amplitudes are known as significant sidebands. They are
finite in numbers.
Power content in FM signal: Since the amplitude of FM remains unchanged, the power of the
FM signal is same as that of unmodulated carrier. (i.e.` A^2/2` ) where A is amplitude of signal.

5. MODULATION INDEX
As with other modulation indices, this quantity indicates by how much the modulated variable varies
around its unmodulated level. It relates to the variations in the frequency of the carrier signal:
where is the highest frequency component present in the modulating signal xm(t), and is
the Peak frequency-deviation, i.e. the maximum deviation of the instantaneous frequency from the
carrier frequency. If , the modulation is called narrowband FM, and its bandwidth is
approximately .
If , the modulation is called wideband FM and its bandwidth is approximately . While
wideband FM uses more bandwidth, it can improve signal-to-noise ratio significantly. For example,
doubling the value of while keeping fm constant, results in an eight-fold improvement in the signal
to noise ratio.[1] Compare with Chirp spread spectrum, which uses extremely wide frequency deviations
to achieve processing gains comparable to more traditional, better-known spread spectrum modes.
With a tone-modulated FM wave, if the modulation frequency is held constant and the modulation
index is increased, the (non-negligible) bandwidth of the FM signal increases, but the spacing between
spectra stays the same; some spectral components decrease in strength as others increase. If the
frequency deviation is held constant and the modulation frequency increased, the spacing between
spectra increases.
Frequency modulation can be classified as narrow band if the change in the carrier frequency is about
the same as the signal frequency, or as wide-band if the change in the carrier frequency is much higher
(modulation index >1) than the signal frequency. [2] For example, narrowband FM is used for two way
radio systems such as Family Radio Service where the carrier is allowed to deviate only 2.5 kHz above
and below the center frequency, carrying speech signals of no more than 3.5 kHz bandwidth. Wide-band
FM is used for FM broadcasting where music and speech is transmitted with up to 75 kHz deviation from
the center frequency, carrying audio with up to 20 kHz bandwidth.

6. BESSEL FUNCTION TABLE
The carrier and sideband amplitudes are illustrated for different modulation indices of FM signals
Modulation Sideband
index 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 −0.07 0.36 0.43 0.28 0.13 0.05 0.02
5.0 −0.18 −0.33 0.05 0.36 0.39 0.26 0.13 0.05 0.02
5.53 0 −0.34 −0.13 0.25 0.40 0.32 0.19 0.09 0.03 0.01
6.0 0.15 −0.28 −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 −0.10 0.19 0.34 0.32 0.22 0.13 0.06 0.03
8.65 0 0.27 0.06 −0.24 −0.23 0.03 0.26 0.34 0.28 0.18 0.10 0.05 0.02
9.0 −0.09 0.25 0.14 −0.18 −0.27 −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 −0.22 −0.23 −0.01 0.22 0.32 0.29 0.21 0.12 0.06 0.03 0.01
12.0 0.05 −0.22 −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

7. POWER IN FREQUENCY MOULATION
From the equation for FM vs (t ) Vc J n ( ) cos( c n m )t
n th
we see that the peak value of the components is VcJn( ) for the n component.
2 2 2
Single normalised average power = V pk then the nth component is Vc J n ( )
2
Vc J n ( )
(VRMS ) 2
2 2
Hence, the total power in the infinite spectrum is
(Vc J n ( ))2
Total power PT
n 2
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.
2
V pk Vc
Since we know that the RMS value of a sine wave is
2 2
2 2
2 Vc Vc2 Vc J n ( )
and power = (VRMS) then we may deduce that PT
2 2 n 2
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.
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’.

8. In general distortion may be defined as
Power in total
spectrum- Power in Bandlimite spectrum
d
D
Power in total
spectrum
PT PBL
D
PT
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 = 5, a = 8 pairs (16 components). Hence, power in the bandlimited spectrum PBL is
a
(Vc J n ( ))2 = carrier power + sideband powers.
PBL
n a 2
2
V c
Since PT
2
Vc2 Vc2 a
( J n ( ))2 a
Distortion D 2 2 n a
1 ( J n ( ))2
Vc2 n a
2
Also, it is easily seen that the ratio
a
Power in Bandlimite spectrum PBL
d
D ( J n ( ))2 = 1 – Distortion
Power in total
spectrum PT n a
a
i.e. proportion pf power in band limited spectrum to total power = ( J n ( ))2
n a