1 Frequency Modulation Background

1.1 Introduction

The comparatively low cost of equipment for an FM broadcasting station, resulted in rapid growth in the years following World War II. Within three years after the close of the war, 600 licensed FM stations were broadcasting in the United States and by the end of the 1980s there were over 4,000. Similar trends have occurred in Britain and other countries. Because of crowding in the AM broadcast band and the inability of standard AM receivers to eliminate noise, the tonal fidelity of standard stations is purposely limited. FM does not have these drawbacks and therefore can be used to transmit music reproducing the original performance with a degree of fidelity that cannot be reached on AM bands. FM stereophonic broadcasting has drawn increasing numbers of listeners to popular as well as classical music, so that commercial FM stations draw higher audience ratings than AM stations.

The integrated chip has also played its part in the wide proliferation of FM receivers, as circuits got smaller it became easier to make a modular electronic device called the "Walkman", which enables the portability of a tape player and an AM/FM radio receiver. This has resulted in the portability of a miniature FM receiver, which is carried by most people when travelling on long trips.

1.2 Technical Background

Frequency	Designation	Abbreviation	Wavelength
3 - 30 kHz	Very Low frequency	VLF	100,000-10,000 m
30 - 300 kHz	Low frequency	LF	10,000 - 1,000 m
300- 3,000 kHz	Medium frequency	MF	1,000 - 100 m
30 - 30MHz	High frequency	HF	100 - 10 m
30 - 300 MHz	Very High frequency	VHF	10 - 1m
300 - 3,000 MHz	Ultra-high frequency	UHF	1m - 10m
3 - 30 GHz	Super-high frequency	SHF	10cm - 1cm
30 - 300 GHz	Extremely-high frequency	EHF	1cm - 1mm

Table 1.1

The main frequencies of interest are from 88MHz to 108MHz with wavelengths between 3.4 and 2.77 meters respectively.

Figure 1.1

With a bandwidth of 200Khz for one station, up to 100 stations can be fitted between 88 & 108Mhz. Station 201 to 300 denote the stations, from 88.1Mhz to 107.9Mhz. Station 201 to 220 (88Mhz to 91.2) are for non-commercial stations (educational) which could be a good area to transmit in, but in recent years the band from 88MHz to 103Mhz has been filled by a lot of commercial channels, making the lower frequencies very congested indeed.

1.2.1 Radio Frequency and Wavelength Ranges

Radio waves have a wide range of applications, including communication during emergency rescues (transistor and short-wave radios), international broadcasts (satellites), and cooking food (microwaves). A radio wave is described by its wavelength (the distance from one crest to the next) or its frequency (the number of crests that move past a point in one second). Wavelengths of radio waves range from 100,000 m (270,000 ft) to 1 mm (.004 in). Frequencies range from 3 kilohertz to 300 Giga-hertz.

1.3 Fm theory

Angle and Amplitude Modulation are techniques used in Communication to transmit Data or Voice over a particular medium, whether it be over wire cable, fibre optic or air (the atmosphere). A wave that is proportional to the original baseband (a real time property, such as amplitude) information is used to vary the angle or amplitude of a higher frequency wave (the carrier).

Where A is the amplitude of the carrier and f(t) is the angle of the carrier, which constitutes the frequency (fC ) and the phase (a) of the carrier. Angle modulation varies the angle of the carrier by an amount proportional to the information signal. Angle modulation can be broken into 2 distinct categories, frequency modulation and phase modulation. Formal definitions are given below :

Phase Modulation (PM) : angle modulation in which the phase of a carrier is caused to depart from its reference value by an amount proportional to the modulating signal amplitude.

Frequency Modulation (FM): angle modulation in which the instantaneous frequency of a sine wave carrier is caused to depart from the carrier frequency by an amount proportional to the instantaneous value of the modulator or intelligence wave.

Phase modulation differs from Frequency modulation in one important way. Take a carrier of the form A Cos(wCt + q) = Re{A.e j(wCt + q)} Pm will have the carrier phasor in between the + and - excursions of the modulating signal. Fm modulation also has the carrier in the middle but the fact that when you integrate the modulating signal and put it through a phase modulator you get fm, and if the modulating wave were put through a differentiator before a frequency modulator you get a phase modulated wave. This may seem confusing at this point, but the above concept will be reinforced further in the sections to follow.

1.3.1 Derivation of the FM voltage equation

Consider a voltage controlled oscillator with a free running frequency of fC , an independent voltage source with voltage VM(t) which causes the VCO to depart from fC by an amount Df, which is equal to the voltage of the independent source multiplied by the sensitivity of the VCO (KO => such as the miller capacitance of a transistor). What is seen at the output of the VCO is a frequency modulated voltage. Now consider the independent voltage source as representing the amplitude of the baseband information .

Figure 1.3.1

Equ. 1.3.1

Equ. 1.3.2

Equ. 1.3.3

Above are the equations which govern the output of the VCO, f is the overall frequency of the frequency modulated output.

Equ. 1.3.4

taking the angle q(t) from equation 1.3.1 and differentiating it will give the angular velocity of the output and equate it to 2(pi) times the effective frequency (f)

Equ. 1.3.5

Equ. 1.3.6

multiply across both sides by the change in time (dt)

Equ. 1.3.7

Equ. 1.3.8

Equ. 1.3.9

Substituting in the equation for the intelligence (baseband) voltage 1.3.8 into equation 1.3.7 and integrating gives equation 1.3.9 which is the angle of the frequency modulated wave of equation 1.3.1.

Equ. 1.3.10

Equ. 1.3.11

Equ. 1.3.12

Tiding up equation 1.3.9, and setting the magnitude of the sine wave as MF , the modulation index for frequency modulation.

Equ. 1.3.13

The above equation represents the standard equation for frequency modulation.

Equ. 1.3.14

The equation for the other form of angle modulation, phase modulation is rather similar but has a few subtle differences. The difference is in the modulation Index and the phase of the varying angle inside the main brackets.

1.3.2 Angle modulation Graphs

Graph 1.3.1

Graph 1.3.2

Graph 1.3.3

Graph 1.3.4

Graph 1.3.5

1.3.3 Analysis of the above graphs

There are 5 significant graphs above, The carrier, the Baseband, FM signal, PM signal and the change of frequency over time. The carrier and baseband are there to show the relative scale, so a link between the carrier and Baseband can be seen.

For FM: the carrier’s frequency is proportional to the baseband’s amplitude, the carrier increases frequency proportional to the positive magnitude of the baseband and decreases frequency proportional to the negative magnitude of the baseband.

For PM: the carrier’s frequency is proportional to the baseband’s amplitude, the carrier increases frequency proportional to the positive rate of change of the baseband and decreases frequency proportional to the negative rate of change of the baseband. In other words when the baseband is a maximum or a minimum, there is Zero rate of change in the baseband, and the carrier’s frequency is equal to the its free running value fc.

Note : the FM wave leads the PM wave by 90 degrees, then all that would be needed to convert an FM generator to a PM generator : would be a differentiator between the baseband and the FM generator, incurring a 90 degree phase lag of the angle modulated wave.

Similarily to convert a PM generator to an FM generator : would be an integrator between the baseband and the FM generator, which will incurr a 90 degree lead of the angle modulated wave.

In both systems the rate of modulation is equal to the frequency of modulation (baseband’s frequency). The last graph shows the relationship between the frequency of FM versus Time, this relationship is used (following a limiter which makes sure the amplitude is a constant) by a discriminator at the receiver to extract the Baseband’s Amplitude at the receiver, resulting in an amplitude modulated wave, the information is then demodulated using a simple diode detector. In common AM/FM receivers for an AM station to be demodulated, the limiter and discriminator can be by passed and the intermediate frequency signal can be fed straight to the diode detector.

1.3.4 Differences of Phase over Frequency modulation

The main difference is in the modulation index, PM uses a constant modulation index, whereas FM varies (Max frequency deviation over the instantaneous baseband frequency). Because of this the demodulation S/N ratio of PM is far better than FM. The reason why PM is not used in the commercial frequencies is because of the fact that PM need a coherent local oscillator to demodulate the signal, this demands a phase lock loop, back in the early years the circuitry for a PLL couldn’t be integrated and therefore FM, without the need for coherent demodulation was the first on the market. One of the advantages of FM over PM is that the FM VCO can produce high-index frequency modulation, whereas PM requires multipliers to produce high-index phase modulation. PM circuitry can be used today because of very large scale integration used in electronic chips, as stated before to get an FM signal from a phase modulator the baseband can be integrated, this is the modern approach taken in the development of high quality FM transmitters.

For miniaturisation and transmission in the commercial bandwidth to be aims for the transmitter, PM cannot be even considered, even though Narrow Band PM can be used to produce Wide band FM (Armstrong Method : integrating the baseband before the PM generator).

1.4 Technical terms associated with FM

Now that FM has been established as a scheme of high quality baseband transmission, some of the general properties of FM will be looked at.

1.4.1 Capture Effect

Simply put means that if 2 stations or more are transmitting at near the same frequency FM has the ability t pick up the stronger signal and attenuated the unwanted signal pickup.

1.4.2 Modulation Index

(Was known as the modulation factor)

Modulation Index is used in communications as a measure of the relative amount of information to carrier amplitude in the modulated signal. It is also used to determine the spectral power distribution of the modulated wave. This can be seen in conjunction with the Bessel function. The higher the modulation index the more side-bands are created and therefore the more bandwidth is needed to capture most of the baseband’s information.

1.4.3 Deviation Ratio

The deviation can be quantified as the largest allowable modulation index.

For the commercial bandwidth the maximum carrier deviation is 75KHz. The human ear can pick up on frequencies from 20Hz to 20KHz, but frequencies above 15KHz can be ignored, so for commercial broadcasting (with a maximum baseband frequency of 15KHz) the deviation ratio is 5 radians.

1.4.4 Carrier Swing

The carrier swing is twice the instantaneous deviation from the carrier frequency.

The frequency swing in theory can be anything from 0Hz to 150KHz.

1.4.5 Percentage Modulation

The % modulation is a factor describing the ratio of instantaneous carrier deviation to the maximum carrier deviation.

1.4.6 Carson’s Rule

Carson’s Rule gives an indication to the type of Bandwidth generated by an FM transmitter or the bandwidth needed by a receiver to recover the modulated signal. Carson’s Rule states that the bandwidth in Hz is twice the sum of the maximum carrier frequency deviation and the instantaneous frequency of the baseband.