Frequency Modulation Lecture Notes

Subtopic:

4-1 Introduction to Frequency Modulation 4-2 Frequency Analysis of the FM wave 4-3 Modulation Index

4-4 Bandwidth Requirements for FM

A major problem in AM is its susceptibility to noise superimposed on the modulated carrier signal. To improve on this, the first frequency modulation (FM) radio communication system was developed in 1936, which is much more immune to noise than its AM counterpart.

Unlike the AM, FM is difficult to treat mathematically due to the complexity of the sideband behavior resulting from the modulation process.

4-1-1 Angle Modulation

In AM, the amplitude of the carrier signal varies as a function of the amplitude of the modulating signal. But when the modulating signal can be conveyed by varying the frequency or phase of the carrier signal, we have angle modulation. Angle modulation can be subdivided by frequency modulation (FM) and phase modulation (PM).

In Frequency Modulation, the carrier’s instantaneous frequency deviation from its unmodulated value varies in proportion to the instantaneous amplitude of the modulating signal In Phase Modulation, the carrier’s instantaneous phase deviation from its unmodulated value varies as a function of the instantaneous amplitude of the modulating signal

4-1 INTRODUCTION TO FREQUENCY MODULATION

LECTURE NOTES 4

Below is the figures illustrates the FM and PM waveforms for sine wave modulation

Figure 4-1: Carrier wave

Figure 4-2: Modulation wave

Figure 4-3: FM wave

The equations representing the FM waveforms,

(

t m t

)

A

sFM = csin

ω

c + f sin

ω

m

The equations representing the PM waveforms,

(

t t

)

A

sPM = csin

ω

c +

φ

msin

ω

m

where,

s

_FM

=

instantaneous voltage of the FM wave

=

PM

s

instantaneous voltage of the PM wave Ac = peak amplitude of the carrier

ω

_c = angular velocity of the carrier

ω

_m = angular velocity of modulating signal = t c

ω

carrier phase = t m

ω

modulation phase = f m FM modulation index = m

φ

PM modulation index

Figure 4-5: Frequency modulation block diagram

Recall that in AM, the frequency component consists of a fixed carrier frequency with upper and lower sidebands equally displayed above and below the carrier frequency. The frequency spectrum of the FM wave is much more complex, that it will produce an infinite number of sidebands

Frequency Modulator

Analysis of the frequency components and their respective amplitudes in FM wave requires use of a complex mathematical integral known as Bessel function of the first kind of the nth order. Evaluating this integral for sine wave modulation yields,

( )

+ = A J m t SFM c 0 f sin

ω

c _A

(

_J

( )

_m

(

(

+

)

_t−

(

−

)

_t

)

)

+ m c m c f c 1 sin

ω

ω

sin

ω

ω

A_c

(

J₂

( )

m_f

(

sin

(

ω

_c+2

ω

_m

)

t−sin

(

ω

_c −2

ω

_m

)

t

)

)

+ _A

(

_J

( )

_m

(

(

+

)

_t−

(

−

)

_t

)

)

+ m c m c f c 3 sin

ω

3

ω

sin

ω

3

ω

_A

(

_J

( )

_m

(

(

+

)

_t−

(

−

)

_t

)

)

+ m c m c f c 4 sin

ω

4

ω

sin

ω

4

ω

_A

(

_J5

( )

_m

(

sin

(

+5

)

_t−sin

(

−5

)

_t

)

)

+...+ m c m c f c

ω

ω

ω

ω

_A

(

_J

( )

_m

(

(

_n

)

_t

(

_n

)

_t

)

)

m c m c f n c sin

ω

+

ω

−sin

ω

−

ω

where A_c =the peak amplitude of the carrier =

n

J solution to the nth Bessel function for a modulation index of mf

=

f

m FM modulation index

( )

m t =

J

A_{c 0 f} sin

ω

_{c carrier frequency component}

( )

(

(

)

(

)

)

(

J m +n t− −n t

)

=

A_{c n f} sin

ω

_c

ω

_m sin

ω

_c

ω

_{m the nth-order sideband}

From above equation, shows that FM wave contains an infinite number of sideband component whose individual amplitudes are preceded by J_n

( )

m_f coefficients.

Below is the plot of the Bessel functions illustrates the relationship between the carrier and sideband amplitudes for sine wave modulation as a function of modulation index,

m

.

Figure 4-6: Spectral components of a carrier of frequency, f_c frequency modulated by a sine wave with modulating frequency, f_m

From either one of both figures above, we can obtain the amplitudes of the carrier and sideband components in relation to the unmodulated carrier.

The modulation index for an FM signal is defined as the ratio of the maximum frequency deviation to the modulating signal frequency,

m f

f m =

δ

where mf = modulation index of FM

δ

= maximum frequency deviation of the carrier caused by the amplitude of the modulating signal

fm = frequency of the modulating signal

In theory, the FM wave contains an infinite number of sidebands, thus suggesting an infinite bandwidth requirement for transmission or reception. In practice, the bandwidth of the FM is depending on the modulation index. The higher the modulation index, the greater the required system bandwidth as shown in the Bessel functions. Figure below shows a graphical illustration of how the FM system’s bandwidth requirements grow with an increasing modulation index.

So, the modulation frequency, f_m is held constant, whereas the carrier frequency deviation,

δ

is increased (and, consequently, m_f as well) in proportion to the amplitude of the modulation signal.

The bandwidth requirements for an FM signal can be computed by

(

nfm

)

BW ₌2

where

n

=

the highest of significant sideband components fm = the highest modulation frequency

In establishing the quality of transmission and reception desired, a limitation must be placed on the number of significant sidebands that the FM system must pass. This can be implemented by using Carson’s Rule:

(

fm

)

BW =2

δ

+

The Carson’s rule will give results that agree with the bandwidths used in telecommunications industry. But it is only an approximation used to limit the number of significant sidebands for minimal distortion.

Reference:

1. James Martin, Telecommunication and the Computer, 2nd _{Edition, Prentice-Hall, 1976} 2. E. Chambi, Bessel Functions, Dover Publication, 1948