Frequency Modulation Lecture Notes

DTC5038

ANALOG COMMUNICATION SYSTEM

Trimester 3 2008-2009

LECTURE NOTES 4 CHAPTER 4: FREQUENCY MODULATION 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 4-1

INTRODUCTION TO FREQUENCY MODULATION

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

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DTC5038

ANALOG COMMUNICATION SYSTEM

Trimester 3 2008-2009

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

Figure 4-4: PM wave

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DTC5038

ANALOG COMMUNICATION SYSTEM

Trimester 3 2008-2009

The equations representing the FM waveforms,

s FM = Ac sin (ω c t + m f sin ω m t ) The equations representing the PM waveforms,

s PM = Ac sin (ω c t + φ m sin ω m t ) where, s FM = instantaneous voltage of the FM wave

s PM = 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 ω c t = carrier phase ω m t = modulation phase m f = FM modulation index

φ m = PM modulation index

Frequency Modulator

Figure 4-5: Frequency modulation block diagram 4-2

FREQUENCY ANALYSIS OF THE FM WAVE

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 3/8 NAS

DTC5038

ANALOG COMMUNICATION SYSTEM

Trimester 3 2008-2009

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,

S FM = Ac J 0 (m f )sin ω c t + Ac (J 1 (m f )(sin (ω c + ω m )t − sin (ω c − ω m )t )) + Ac (J 2 (m f )(sin (ω c + 2ω m )t − sin (ω c − 2ω m )t )) + Ac (J 3 (m f )(sin (ω c + 3ω m )t − sin (ω c − 3ω m )t )) + Ac (J 4 (m f )(sin (ω c + 4ω m )t − sin (ω c − 4ω m )t )) + Ac (J 5 (m f )(sin (ω c + 5ω m )t − sin (ω c − 5ω m )t )) + ... + Ac (J n (m f )(sin (ω c + nω m )t − sin (ω c − nω m )t ))

where Ac = the peak amplitude of the carrier

J n = solution to the nth Bessel function for a modulation index of m f m f = FM modulation index Ac J 0 (m f )sin ω c t = carrier frequency component Ac (J n (m f )(sin (ω c + nω m )t − sin (ω c − nω m )t )) = 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.

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DTC5038

ANALOG COMMUNICATION SYSTEM

Trimester 3 2008-2009

Below is a Bessel function tabulated to the 16-th order for modulation indices ranging from 0-15.

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DTC5038

ANALOG COMMUNICATION SYSTEM

Trimester 3 2008-2009

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. 4-3

MODULATION INDEX

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

mf =

δ fm

where m f = modulation index of FM

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

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DTC5038

ANALOG COMMUNICATION SYSTEM

Trimester 3 2008-2009

f m = frequency of the modulating signal 4-4

BANDWIDTH REQUIREMENTS OF FM

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.

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DTC5038

ANALOG COMMUNICATION SYSTEM

Trimester 3 2008-2009

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

BW = 2(nf m ) where n = the highest of significant sideband components

f m = 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:

BW = 2(δ + f m ) 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

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