Electronic Communications Presentation Materials Part 3

Angle Modulation(Frequency Modulation and

Phase Modulation)Prepared by: Armando V. Barretto

Angle Modulation

• Angle modulation results whenever the phase angle of a sinusoidal wave is varied with respect to time.

• The two forms of angle modulation are Frequency Modulation and Phase Modulation.

• Direct Frequency Modulation (FM) is varying the frequency of a constant amplitude carrier directly proportional to the amplitude of the modulating signal, at a rate equal to the frequency of the modulating signal.

• Direct Phase Modulation (PM) is varying the phase of a constant amplitude carrier directly proportional to the amplitude of the modulating signal, at a rate equal to the frequency of the modulating signal.

• When frequency of a signal is varied, its phase is also varied. When phase of a signal is varied, its frequency is also varied.

• The difference between frequency and phase modulation lies in which property of the carrier is directly varied by the modulating signal.

• If the modulating signal varies the frequency directly, frequency modulation results.

• If the modulating signal varies the phase directly, phase modulation results.• FM is indirect PM, and PM is indirect FM.

Advantages of Angle Modulation Compared With Amplitude Modulation

• FM has the ability to suppress noise, which is probably the biggest advantage of FM compared to AM.– FM receivers can be fitted with amplitude limiters to remove amplitude

variations caused by noise, making FM more immune to noise. This is called FM Thresholding.

– In AM, once the signal has been contaminated by noise, noise could not be removed.

– Signal to noise ratio can be improved further by increasing the frequency deviation of FM signals.

– Reduced noise improves system fidelity.• Angle modulated signals are more power efficient.

– Amplitude of FM or PM wave remains constant.– Amplitude is independent of modulation depth.– Low level modulation may be used, but all subsequent amplifiers can be

class C and therefore more power efficient. – All transmitted power in FM are useful, making it more power efficient.

• Standard frequency allocations provide guard band between FM broadcast stations, so there is less adjacent channel interference.

Advantages of Angle Modulation Compared With Amplitude Modulation

• FM broadcasts operate in the upper VHF and UHF frequency ranges, at which there happens to be less noise than in the MF and HF ranges occupied by AM broadcasts.

• For FM broadcast, space wave is used, so the radius of operation is limited to slightly more than line of sight. Several independent transmitters separated by space can use the same frequency with less possible interference.

• With FM and PM, a phenomenon known as capture effect allows a receiver to differentiate between two signals with the same frequency.– Providing one signal is at least twice as high in amplitude as the other, the

receiver will capture the stronger signal and eliminate the weaker signal. This is not possible for AM.

Disadvantages of Angle Modulation Compared With Amplitude Modulation

• Higher bandwidth is needed, up to 10 times as large as that needed for AM.• More complex circuits in transmitter and receiver.• Since reception is limited to line of sight, the area of reception (service

area) is much smaller than AM (in broadcasting). This may be an advantage for co-channel allocations, but it is a disadvantage for FM mobile communications over a wide area. This is not due to the intrinsic properties of FM, but rather to the frequencies employed for its transmission (in broadcasting).

Angle Modulation

• The modulating signal can be expressed as:

vm(t)=Vmsin(ωmt)where: ωm = 2πfm = modulating signal radian frequency (angular velocity in radians per

second)Vm = modulating signal peak amplitude (volts)fm = modulating signal frequency (hertz)

• The instantaneous phase deviation of an angle modulated wave (θ(t)) can be expressed as:

θ(t) = F(vm(t))

(the instantaneous phase deviation of an angle modulated wave is a function of the modulating signal)

Angle Modulation

• An angle modulated wave can be expressed as:

m(t)=Vccos(ωct + θ(t))

=Vccos(2πfct + θ(t))

where:

m(t) = angle modulated wave

Vc = peak carrier amplitude (volts)

fc = carrier frequency (hertz)

θ(t) = instantaneous phase deviation (radians)

2πfc = ωc = carrier radian frequency (angular velocity in radians per second)

Frequency Modulated WaveFrequency of modulated signalwhen amplitude of modulating signal is 0 volt (carrier rest frequency)

Lowest frequency ofmodulated signal (fmin)

Highest frequency ofModulated signal (fmax)

Δθ Δθ

Tmin(max. frequency.)

Tmax(min. frequency)

Tmin = minimum period of frequency modulated signalTmax = maximum period of frequency modulated signalΔT = peak to peak change in period of carrierΔf = frequency deviation (relative displacement of the carrier frequency, in hertz) Δf p-p = peak to peak frequency deviation = (1/Tmin) – (1/Tmax)Δθ = phase deviation (relative angular displacement of carrier with respect to a

reference phase, in radians)fc = carrier rest frequency = carrier frequency when there is no modulating signal

ΔT

fc fc + ffc - f

Δf Δf

Δfp-p

Phase Modulation• When the carrier signal is not modulated, its frequency is constant and is

equal to the carrier rest frequency (fc).• When the carrier signal is modulated by a modulating signal, its

instantaneous phase will change depending on the instantaneous value of the modulating signal voltage.

• The higher is the modulating signal voltage, the higher is the phase deviation.

• The higher is the peak amplitude of the modulating signal (Vm), the higher is the maximum or peak phase deviation ( p ).

• The peak phase deviation ( p ) occurs when the voltage of the modulating signal is at its peak (Vm or peak amplitude).

• While the phase of the carrier is changing, its frequency will also be changing. Indirect FM will result.

Frequency Modulation• When the carrier signal is not modulated, its frequency is constant and is

equal to the carrier rest frequency (fc).• When the carrier signal is modulated by a modulating signal, its

instantaneous frequency will change depending on the instantaneous voltage of the modulating signal.

• The frequency of the modulated signal will decrease and /or increase from the rest frequency value, depending on the polarity of the modulating signal.

• The higher is the modulating signal voltage, the higher is the frequency deviation.

• The higher is the peak amplitude of the modulating signal (Vm), the higher is the maximum or peak frequency deviation (f ).

• The peak frequency deviation (f) occurs when the voltage of the modulating signal is at its peak (Vm or peak amplitude).

• While the frequency of the carrier is changing, its phase will also be changing. Indirect PM will result.

Angle Modulation

• Instantaneous phase deviation (θ(t)) – the instantaneous change in the phase of the carrier at a given instant of time, and indicates how much the phase of the carrier is changing with respect to its reference phase. It is measured in radians.

• Instantaneous phase (ωct + θ(t)) – is the precise phase of the carrier at a given instant of time. It can be expressed as:

Instantaneous phase = ωct + θ(t) (radians)= 2πfct + θ(t) (radians)

where: ωct = carrier reference phase (radians)fc = carrier frequency (hertz)θ(t) = instantaneous phase deviation (radians)

Angle Modulation

• Instantaneous frequency deviation (θ’(t)) – is the instantaneous change in frequency of the carrier and is defined as the first time derivative of the instantaneous phase deviation. It is measured in radians per second.

Instantaneous frequency deviation = θ’(t) (radians per second)

• Instantaneous frequency (fit) or (ωit) – is the precise frequency of the carrier at a given instant of time and is defined as the first time derivative of the instantaneous phase.

ωit = fi = Instantaneous frequency =dtωc + θ’(t)=

(ωct + θ(t))

2πfc+ θ’(t)=

d (rad / sec)

(rad / sec)

(rad / sec)

Angle Modulation

• To convert instantaneous frequency deviation or instantaneous frequencyin rad/sec to hertz or cycles per sec, divide them by 2π radian per cycle.

Example: To convert 2π rad/sec to hertz(2π rad/sec )(2π radian/cycle)

• Phase modulation can be defined as angle modulation in which the instantaneous phase deviation, θ(t), is proportional to the modulating signal voltage.

• Frequency modulation can be defined as angle modulation in which the instantaneous frequency deviation, θ’(t), is proportional to the modulating signal voltage.

= 1 cycle /sec = 1 hertz

Angle ModulationFor a modulating signal vm(t) ) = Vmsin(ωmt), the phase and frequency

modulation are:

Phase modulation = instantaneous phase deviationPhase modulation = θ(t) = K vm(t) (rad)

Frequency modulation = instantaneous frequency deviationFrequency modulation = θ’(t) = K1 vm(t) (rad/sec)

Where:K = deviation sensitivity of phase modulator (constant in radians per volt)

= phase deviation per volt of modulating signal (It is a measure of how much sensitive is a phase modulator in deviating the phase in proportion to the modulating signal voltage)

K1 = deviation sensitivity of frequency modulator (constant in radians per volt-second or hertz per volt)= frequency deviation per volt of modulating signal (It is a measure of how much sensitive is a frequency modulator in deviating the frequency in proportion to the modulating signal voltage)

vm(t) = modulating signal

Angle Modulation

Since phase modulation is the first integral of frequency modulation,phase modulation can also be written as,

Phase modulation = θ(t) = K vm(t) rad

Where:K = deviation sensitivity of phase modulator (constant in radians per volt)K1 = deviation sensitivity of frequency modulator (constant in radians per

volt-second)vm(t) = modulating signal

dt (t)v K

dt (t)vK

dt (t)'θ

m

m

1

1

=

=

=

Angle Modulation

Substituting modulating signal vm(t)=Vmcos(ωmt) into the preceding equations, the following equations can be written:

The phase modulated wave can be written as:

v(t) = Vccos (ωct + θ(t))= Vccos(ωct + K Vm cos (ωmt)) = Vccos(ωct + mP cos (ωmt))

The frequency modulated wave can be written as :v(t)

t)]sin(ω m t [ω cos Vc t)]sin(ωω

VK t [ω cos Vc

t)dt)cos(ωVK t (ω cos Vc

(t)dt)vK t (ω cos Vc

(t))θ' t (ω cos Vc

mFcmm

m1c

mm1c

m1c

c

Angle ModulationWhere:

Vm = peak amplitude of modulating signal (volts)

ωm = angular velocity of modulating signal (rad / sec)

Vc = peak amplitude of carrier signal

ωc = angular velocity of unmodulated carrier signal (rad / sec)

θ(t) = instantaneous phase deviation = Phase modulation (rad)

θ’(t) = instantaneous frequency deviation = frequency modulation (rad / sec)

K = deviation sensitivity of phase modulator (in radians per volt)

mP = KVm = modulation index for PM = p = peak phase deviation

K1 = deviation sensitivity of frequency modulator (in radians per volt-second)

mF = K1Vm / fm = modulation index for FM

Note: For phase modulation, the instantaneous phase, ωct + θ(t), is directly proportional to the amplitude of the modulating signal.

For frequency modulation, the instantaneous frequency, ωc + θ’(t), is directly proportional to the amplitude of the modulating signal.

Summary of Equations for FM And PM

Type of Modulation Modulating Signal Angle-modulated Wave, m(t)

Phase

Phase

Phase

Frequency

Frequency

Frequency

Frequency

vm(t)

Vmcos(ωmt)

Vmcos(ωmt)

vm(t)

-Vmsin(ωmt)

Vmcos(ωmt)

Vmcos(ωmt)

Vccos [ωct + K vm (t)]

](t)dtvK t [ω cos Vc m1c

Vccos(ωct + K Vm cos (ωmt))

t)]cos(ωωVK t [ω cos Vc mm

m1c

t)]sin(ωωVK t [ω cos Vc mm

m1c

t)]sin(ω m t [ω cos Vc mFc

Vccos(ωct + mP cos (ωmt))

Phase and Frequency Modulation

Modulating signal(fm)

UnmodulatedCarrier

(fc)

0 volt

0 volt

Frequency Modulated

Wave0 volt

PhaseModulated

Wave

0 volt

Maximum frequencyMinimum frequency

Rest frequency

Maximum frequencyMinimum frequency

Phase and Frequency Modulation

• For frequency modulation, the maximum frequency deviation (f ) occurs during the maximum positive and negative peak amplitudes of the modulating signal. Frequency deviation is directly proportional to the amplitude of modulating signal.

• For phase modulation, the maximum frequency deviation occurs during the zero crossings of the modulating signal. Frequency deviation is directly proportional to the slope or first derivative of the modulating signal.

• For both frequency and phase modulation, the rate at which the frequency changes is equal to the modulating signal frequency.

• The frequency of the modulating signal does not affect the peak frequency deviation and it only affects the rate of change of the frequency deviation.

• Under identical conditions, FM and PM are indistinguishable for a single modulating frequency, unless the FM and PM waves are plotted against the modulating signal.

Modulation Index for Phase Modulation • Modulation Index (for phase modulation) – is the peak phase deviation for a

phase modulated wave (radians)– Modulation index is directly proportional to the amplitude of the

modulating signal, independent of its frequency, and it is expressed as:

Where: Vm = peak amplitude of modulating signal (volts)K = deviation sensitivity of phase modulator (radians per volt)

• The relationship of the modulation index for phase modulation to the modulated signal can be expressed as:

Where:m(t) = phase modulated waveVc = peak amplitude of the unmodulated carrier and of the modulated signalmP = modulation index for phase modulation = peak phase deviation (radians)mP cos (ωmt) = instantaneous phase deviation = θ(t) (radians)ωc = angular velocity of unmodulated carrier signal (rad / sec)

m(t) = Vccos[ωct + mP cos (ωmt)] = Vc cos[ωct + p cos (ωmt)]

mP = KVm = modulation index for PM = p = peak phase deviation (radians)

Modulation Index for Frequency Modulation

• Modulation Index (for frequency modulation) – is used to describe the depth of modulation achieved for a given amplitude and frequency of modulating signal.– Modulation index is directly proportional to the amplitude of the

modulating signal, and inversely proportional to its frequency.

Where: mF = modulation index for frequency modulationVm = peak amplitude of modulating signal (volts)K1 = deviation sensitivity of frequency modulator (rad. per volt-sec.)K1Vm = peak frequency deviation (radians per second)ωm = 2πfm = angular velocity of modulating signal (rad / sec)f = peak frequency deviation (Hertz)fm = ωm / 2 = modulating signal frequency (Hertz)

m

m1

ωVK mF = = modulation index for frequency modulation (unitless)

fmf = modulation index for frequency modulation (unitless)

Modulation Index for Frequency Modulation • Frequency deviation (K1Vm in radians per second) is typically given in peak

frequency shift (Δf in hertz).• To convert radians per second to hertz (cycles per second), simply divide

radians per second with 2π radians per cycle.• ωm = 2πfm (radians per second) can also be expressed in hertz by dividing it

with 2π radians per cycle.• The modulation index for frequency modulation can then be expressed as:

• Δf = peak frequency deviation • The peak to peak frequency deviation (2Δf ) is sometimes called carrier

swing (2Δf ) .• In many cases, deviation sensitivity for FM (K1) is given in hertz per volt. To

convert deviation sensitivity in radians per volt-second to hertz per volt, simply divide it with 2π radians per cycle.

• The maximum peak frequency deviation allowed for FM radio transmission is 75 khz and the maximum peak to peak frequency deviation allowed is 150 Khz.

mff mF = =

/2π)(ω2π / )V(K

m

m1= modulation index for FM (Unitless)

Modulation Index for FM and PM

Example:a. Determine the peak frequency deviation (Δf) and modulation index

(m) for an FM modulator with a deviation sensitivity (K1) = 5 khz / volt and a modulating signal vm(t)=2 cos(2π2000t).

b. Determine the peak phase deviation (m) for a PM modulator with adeviation sensitivity (K) = 2.5 radian per volt and a modulatingsignal of vm(t)=2 cos(2π2000t).

Solution:

a. Peak frequency deviation (Δf) = K1 Vm = (5 khz / v)(2 v) = 10 Khz

Modulation index for FM =

b. Peak phase deviation = m = K Vm = (2.5 rad / v)(2 v) = 5 radians

mff m = = 10,000 / 2000 = 5


Example: In an FM system, when the audio frequency (AF) is 500 hz and the AF voltage is 2.4 volts, the deviation is 4.8 khz. If the AF voltage is now increased to 7.2 volts, what is the new deviation? If the AF voltage is raised to 10 volts while the AF is dropped to 200 hz, what is the deviation? Find the modulation index in each case.

Solution: K1 = Δf / Vm = 4,800 / 2.4 = 2,000 hz / volt = FM deviation sensitivity in v/sec

When Vm = 7.2 volts 500 hz, Δf = K1Vm = (2,000)(7.2) = 14,400 hz

When Vm = 10 volts and AF is dropped to 200 hz,Δf = K1Vm = (2,000)(10) = 20,000 hz (Δf is not affected by modulating

signal frequency)

When AF voltage is 2.4 volts 500 hz, mF = Δf / fm = 4800 / 500 = 9.6When AF voltage is 7.2 volts 500 hz, mF = Δf / fm = 14,400 / 500 = 28.8When AF voltage is 10 volts 200 hz , mF = Δf / fm = 20,000 /200 = 100

(FM Modulation index is affected by modulating signal frequency.)


Example: Find the carrier and modulating frequencies, the modulation index, and the maximum deviation of the FM wave represented by the voltage equation v = 12 sin (6x108t + 5 sin 1250t). What power will this FM dissipate in a 10 ohm resistor?

Solution: fc = 6x108 / (2Mhz = carrier frequency

fm = 1250 / (2) = 199 hz = modulating signal frequency

mF = 5 = modulation index

Δf = (mF)(fm) = (5)(199) = 995 hz

Ec = peak amplitude of carrier = 12 volts (modulated or not)

P = Ec2 / (2R) = 122 / [(2)(10)] = 7.2 watts = power which will be dissipated by the FM wave in a 10 watt resistor.


Example: A 25 Mhz carrier is modulated by a 400 hz sine wave. If the carrier voltage is 4 volts and the maximum deviation is 10 khz, write the equation of this modulated wave for (a) FM and (b) PM. If the modulating signal is now changed to 2 khz, all else remaining constant, write a new equation for (c) FM and (d) PM.

Solution: Calculating the frequencies in radians, we have:c = (2x 108 rad/sm = (2)(400) = 2,513 rad/s

mF = Δf / fm = 10,000 / 400 = 25 = modulation index for FMSince the peak frequency deviation for PM and FM are the same, modulation

index for FM = modulation index for PM. (mF = mP = 5)

The equations for FM and PM are:

a) For FM: v=4 sin [1.57 x 108 t + 25 sin(2513 t)]b) For PM: v=4 sin [1.57 x 108 t + 25 sin(2513 t)]


If modulating frequency (fm) is changed to 2 khz, and deviation sensitivities for FM (K1) and PM (K) remain the same, the new modulation indices are:

For FM: mF = Δf / fm = 10,000 / 2,000 = 5 = modulation index for FMFor PM: mP = 25 = peak phase deviation = modulation index for PM

The modulation index for FM is affected by the modulating signal frequency whereas the modulation index for PM is unaffected by the modulating signal frequency.

The frequency of the modulating signal in radians will be:

m = (2)(2000) = 12,566 rad/s

The equations for FM and PM are:

c) For FM: v=4 sin [1.57 x 108 t + 5 sin(12,566 t)]d) For PM: v=4 sin [1.57 x 108 t + 25 sin(12,566 t)] (unchanged)

Percent Modulation for Angle Modulated Wave

• Percent modulation for an angle modulated wave is expressed as:

% modulation =

where: Δf (actual) = Actual peak frequency deviation of angle modulated signalΔf (allowed) = Maximum peak frequency deviation allowed by law in the area where the angle modulated wave is present.

For example: If Δf (actual) is 50 Khz, and Δf (allowed) is + or -100 khz, then

(allowed) f(actual) f

X 100 (in percent)

% modulation = (50,000 / 100,000) 100 = 50 %

Actual peak to peak frequency deviation (actual carrier swing) = 100 khzAllowed peak to peak frequency deviation (allowed carrier swing) = 200 khz

The maximum peak frequency deviation allowed in the US is + or - 75 Khz.

Phase and Frequency Modulators

• Phase modulators – is a circuit in which the carrier is varied in such a way that its instantaneous phase is proportional to the modulating signal.– Unmodulated carrier is a single frequency which is commonly

called rest frequency.– Phase modulation is not used in practical analog transmission

systems.• Frequency modulator (also called frequency deviator)– is a circuit in

which the carrier is varied in such a way that the instantaneous phase is proportional to the integral of the modulating signal.– Unmodulated carrier is a single frequency which is commonly

called rest frequency.– If modulating signal, v(t), is differentiated prior to being applied to

a frequency modulator, the resulting wave is a PM wave.

Phase and Frequency Modulators

• The following equivalences can be applied for phase and frequency modulators and demodulators:– PM modulator = differentiator followed by FM modulator– PM demodulator = FM demodulator followed by an integrator– FM modulator = integrator followed by a PM modulator– FM demodulator = PM demodulator followed by a differentiator

• A low pass filter (1/f filter or predistorter or frequency correction network) could be an integrator.

• If an FM signal is received by a PM receiver, bass (low) frequencies would considerably have more phase deviations and would be boosted more than if a PM was was received. (mf = f / fm)

• If a PM signal is received by an FM receiver, high frequencies would considerably have more frequency deviations and would be boosted more than if a FM was received.

Frequency Analysis of Angle Modulated Wave

• With angle modulation, the frequency components of the modulated wave are much more complexly related to the frequency components of the modulating signal, than those in amplitude modulation.

• The modulated signal produced from a single modulating signal is composed of:– the original carrier frequency– an infinite number of pairs of side frequencies displaced on either side

of the carrier by an integral multiple of the modulating signal frequency– A sideband set includes an upper and a lower side frequency (fc+fm’

fc+2fm, fc+3fm, …..)– The bandwidth for all the side frequencies is infinite.

• Generally, most of the side frequencies are negligible and can be ignored.

• The angle modulated wave can be expressed as:m(t) = Vccos[ωct + m cos (ωmt)]

• The above equation can be expanded using Bessel function identities shown below:


)2

nπnβ(m)cos(αJ-n

n

m(t) = Vc

cos(mcos

• The angle modulated wave can then be expressed as:

....)tω2-(ω(m)cosJ)tω2(ω(m)cosJ2π-)tω(ω(m)cosJ

2π)tω(ω(m)cosJt(m)cosω[J

mc2mc2

m-c1mc1c0

Where:Vc = peak amplitude of unmodulated carrier (volts)Jn = relative amplitude of frequencies relative to amplitude of unmodulated carrier (Vc)(can be found on Bessel function table, its value eventually decreases as n increases)m = modulation index; used as argument for the Bessel functionωc = 2πfc =angular velocity of unmodulated carrier (rad / sec)ωm = 2πfm = angular velocity of modulating signal (rad / sec)

Frequency Analysis of Angle Modulated Wave• The modulation index (m) determines the number of side frequencies.• If modulation index = 0, there are no side frequencies.• The higher the modulation index (m), the more are the side frequencies

that have significant amplitudes.• A side frequency is not considered significant unless it has an amplitude

equal to or greater than 1 % of unmodulated carrier amplitude (Jn >= 0.01)• The bandwidth of an angle modulated wave is a function of the modulation

index.– The higher the modulation index, the higher is the bandwidth.

• The theoretical bandwidth required to pass all frequencies in an angle modulated wave is infinite. However, this is impractical and the bandwidth used in practice are approximates of the theoretical (infinite) bandwidth.

• The value of J eventually decreases, but not in a simple manner. It fluctuates as the value of n increases.

• When m (modulation index) = 2.4 or 5.4 (or some other values in Bessel table), J0 = 0, and there is no component of the modulated wave whose frequency = fc.

– This is called carrier null.– The values of m where the carrier component disappears is called

eigenvalues.


Example:Given: An FM modulator with a modulation index m= 1 has a modulating signal

of vm(t)=Vm cos(2π1000t), and an unmodulated carrier of vc(t)=10 sin(2π500000t).

Determine: a. Number of sets and amplitude of significant side frequenciesb. Draw the frequency spectrum showing their relative amplitudes

Solution:From Bessel table, m = 1 yields a reduced carrier component and three sets of side frequencies with the following amplitudes:J0 = .77(10) = 7.7 voltsJ1 = .44(10) = 4.4 voltsJ2 = .11(10) = 1.1 voltsJ3 = .02(10) = 0.2 volt

Note: PM modulator with m = 1 radian will yield the same results.

v

vv

vvvv

Bandwidth of Angle Modulated WavesAngle modulated waves can be classified into:• Low-Index (narrowband)

– Peak phase deviation (modulation index) is less than 1 radian.• Medium-Index

– Peak phase deviation (modulation index is 1 radian to 10 radians.• High-Index

– Peak phase deviation (modulation index) is greater than 10 radians.

The bandwidth of angle modulated waves can be approximated using the following:

• For low-index angle modulation, most of the signal information is on the first set of sidebands, and the bandwidth required is approximated by:

• For high-index signal, quasi-stationary approach may be used. Modulating signal is assumed to be very low, thus bandwidth is approximated by:

B = 2fmmax = (2)(highest modulating frequency)

B = (2Δf) = peak to peak frequency deviation

Bandwidth of Angle Modulated Waves• The actual bandwidth required to pass all the significant sidebands for an angle

modulated wave can be computed by:

• Using Carson’s rule, the bandwidth required can be approximated by:

Where:Δf = peak frequency deviationfm(max) = highest modulating frequency

This formula gives a narrower bandwidth compared to those obtained using the Bessel table.

Note: For FM broadcast, standard frequency range occupied is 200 Khz, 180 khz for the signal and 20 khz for the guard band.

B = 2(nfm(max)) = (2)(number of significant set of sidebands)(modulating frequency)

B = 2(Δf + fm(max)) (hertz)

Bandwidth of Angle Modulated WavesExample: What is the bandwidth required for an FM signal in which the modulating

frequency is 2 khz and the maximum deviation is 10 khz ? Use all the formulas for FM bandwidth.

Solution: mF = f / fm = 10,000 / 2,000 = 5 = modulation index

From the Bessel table, the number of significant sets of sidebands (J) = 8B = 2 (number significant set of sidebands) (fm) = (2)(8)(2,000) = 32,000 hz

Using Carson’s rule: B = 2(Δf + fm(max)) = 2(10,000 + 2,000) = 24,000 hz

Using quasistationary approach: B = (2Δf) = 2(10,000) = 20,000 hz

The higher is the allotted bandwidth, the better is the quality of the FM system.

Bandwidth of Angle Modulated Waves

• Bandwidth used for FM is dependent on its application.• Wideband FM has been defined as that in which the modulation index

normally exceeds unity (1), which produces more significant side frequencies.

• Wideband FM is usually used for entertainment such as FM radio broadcast.• The modulation index for narrowband FM is near unity, since the maximum

modulating frequency is usually 3 khz, and the maximum frequency deviation is 5 khz.

• Narrowband FM is usually used for communications by police, military, and other similar services.

Deviation ratio (DR)

• Deviation ratio – is the worst case modulation index (m), and is equal to the maximum peak frequency deviation divided by the maximum modulating signal frequency.– The worst-case modulation index produces the widest output

frequency spectrum.

DR = Deviation ratio = Δ f(max) / fm(max) (unitless)

where:Δ f(max) = maximum peak frequency deviation (hertz)fm(max) = maximum modulating signal frequency (hertz)

Example: If Δ f(max) = 50 khz, and fm(max) = 15 khz, then

DR = 50,000 / 15,000 = 3.33

Commercial FM broadcast• A 20 Mhz band of frequencies has been assigned by FCC for FM radio

broadcast.• The frequency range is from 88 Mhz to 108 Mhz.• The frequency range is divided into 100 channels, each is 200 Khz wide.• The first channel has a center frequency is 88.1 Mhz, and its range is from

88 Mhz to 88.2 Mhz.• The maximum frequency deviation allowed is + or – 75 Khz with a

maximum modulating signal frequency of 15 Khz.• The deviation ratio for commercial FM is 5 (75 Khz / 15 Khz) which

produces 8 significant side frequencies based on the Bessel table.

Average Power of Angle Modulated wave

• The total power in an angle modulated wave is equal to the power of unmodulated carrier.

• The power in an unmodulated carrier is redistributed among the carrier and the sidebands in the modulated wave.

• The average power in an angle modulated wave is independent of the modulating signal, the modulation index, and the frequency deviation.

• The average power in an angle modulated wave is equal to the average power of the unmodulated carrier, and it is expressed as:

Where: Pc = unmodulated carrier power (watts) = average power in modulated waveVc = peak voltage of unmodulated carrier (volts)R = load resistance (ohms)

• The instantaneous power in an angle modulated wave is:

Pc = Vc2 / (2R) = (.707Vc)2/ R

Pt = [Vc2 cos2 (ωct + θ(t))] / R

Vc

2

Rcos2 (ωct + θ(t))

Average Power of Angle Modulated wave

• The power of the modulated wave is the sum of the power of the modulated carrier and the sidebands and can be expressed as:

Where: Pc = power of modulated wave = power of unmodulated carrier (watts)Vc = peak voltage of unmodulated carrier (volts)R = load resistance (ohms)Jn = relative amplitude of frequencies relative to amplitude of unmodulated

carrier (Vc) from Bessel table

Example: An FM modulator with a modulation index m= 1 has a modulating signal of vm(t)=Vm cos(2π1000t), and an unmodulated carrier of vc(t)=10 sin(2π500000t). Determine the unmodulated carrier power and total power in modulated wave assuming a load resistance of 50 ohms.

Pc = Vc2 / (2R) = (10)2 / (2)(50) = 1 watt = power of unmodulated carrier

Pc = [(10)(.77)]2 / (2)(50) + 2[(10)(.44)]2 / (2)(50) + 2[(10)(.11)]2 / (2)(50) + 2[(10)(.02)]2 / (2)(50) = 1.0051 w = total power in modulated wave

Pc = (J0Vc)2 / (2R) + 2 (J1Vc)2 / (2R) + 2 (J2Vc)2 / (2R)+…

Noise in Phase Modulation

PMDemodulator

Carrier with noise

Carrier = vc(t) = Vc sin ( ct)Noise = vn (t) = Vn sin (nt)

Noise at demodulator output:vnd (t) = Vnd sin (nd t)

Where : Vc = peak voltage of carrier at input of demodulator (volts)Vn = peak voltage of noise at input of demodulator (volts)

= maximum deviation in amplitude of signal affected by noise atinput of demodulator

Vnd = peak amplitude of noise at output of demodulatorc = 2fc = angular velocity of carrier at demodulator input n = 2 fn = angular velocity of noise at demodulator inputnd = 2 fnd = n - c

= relative angular velocity of noise at input of demodulator relative to the carrier

= angular velocity of noise at demodulator outputfnd = fn – fc = frequency of noise at output of demodulatorfn = frequency of noise at input of demodulatorfc = frequency of carrier at input of demodulator

Noise in Phase Modulation• If there is noise in a phase modulated wave, the noise will add vectorially with the

phase modulated signal.• As the noise alters the amplitude of the phase modulated wave, it also causes phase

changes on the PM wave, and the phase changes at the input of the demodulatorwill generate noise voltage at the output of the PM demodulator.

• The vector relationship of noise and the carrier signal voltage in phase modulation is shown below:

Where : Vc = peak voltage of unmodulated carrier (volts)Vn = peak voltage of noise at input of demodulator (volts)

= maximum deviation in amplitude of signal affected by noiseθ = instantaneous phase deviation due to noise

Vc

VnLocus of resultant

θ

Amplitude change

Phase change

c n

nd

Noise in Phase Modulation• Since the noise vector rotates about the carrier voltage with a relative angular

velocity of nd =n - c , the noise angular velocity at the output of the demodulator can be computed as:

nd = n - c = 2fnd = relative angular velocity of noise at input of demodulatorrelative to the carrier

= angular velocity of noise at output of demodulator fnd = fn – fc = frequency of noise at output of demodulator (Hz) fn = frequency of noise or interfering signal at input of demodulator (Hz) fc = frequency of carrier at input of demodulator (Hz)

• The maximum phase change (peak phase deviation) at the input of the demodulator due to noise occurs when the noise is perpendicular to the resultant voltage and it can be computed as:

Peak phase deviation = peak =Sin-1 (Vn / Vc) (radians)

Vc

VnLocus of resultant

peakAmplitude change

Maximum Phase change


• Assuming that Vc >> Vn, the peak phase deviation (Δθ (peak) ) due to the noise at the input of the demodulator can be approximated as:

• When the carrier component is much larger than the interfering noise voltage, the instantaneous phase deviation due to the noise is approximately:

Vn) Vc(When (radians) VcVn / (radians) Vc)(Vn / Sin peak change phase Maximum -1

θ (t) = (Vn / Vc)[sin(ωndt + θn)] radians

Where : Vc = peak voltage of carrier at input of demodulator (volts)Vn = peak voltage of noise at input of demodulator (volts) c = 2fc = angular velocity of carrier at demodulator input n = 2 fn = angular velocity of noise at demodulator inputnd = 2 fnd = n – c = angular velocity of noise at demodulator outputfnd = fn – fc = frequency of noise at output of demodulatorfn = frequency of noise at input of demodulatorfc = frequency of carrier at input of demodulator

Noise in Phase Modulation• Note that in the book by Tomasi, when Vc > Vn, the peak phase deviation

due to an interfering single-frequency sinusoid occurs when signal and noise voltages are in quadrature.

– If such is the case, the peak phase deviation can be computed as:

– For small peak phase deviation (Vc>> Vn), the peak phase deviation can be approximated by:

– For small peak phase deviation (Vc>>Vn), the difference of the resultant voltage and the signal voltage is small, and

(radians) Vc)(Vn / Tan peak deviation phasePeak -1

Vn) Vc(When (radians) VcVn / (radians) Vc)(Vn / Tan peak deviation PhasePeak -1

(radians) Vc)(Vn / Tan Vc)(Vn / Sin -1-1


• For FM receivers, a process called amplitude limiting, which is limiting the amplitude of the received signal, is usually done to reduce noise.

• For PM, the effect of amplitude limiting is shown by the diagram below:

• Due to limiting, the noise signal has been transposed into noise sideband pairwith amplitude Vn / 2.

• The peak phase deviation is still Vn / Vc radians.• The interference in the demodulated signal is not reduced.• For a given range of noise frequencies with the same amplitudes, the output

of a PM demodulator is the same for all the noise frequencies.

Vc

Vn /2

Locus of resultant

θVn /2

Noise in Frequency Modulation• FM is much more immune to noise than AM.• FM is significantly more immune to noise than PM.• Unlike noise at the output of a phase modulation (PM) demodulator, noise

voltage at the output of an FM demodulator increases linearly with frequency of noise at the output of the demodulator.– This is called the FM noise triangle, which is shown below.– The demodulated noise voltage is inherently higher for higher noise

frequencies at the output of demodulator.

• Noise frequencies that produce components at the high end of the modulating signal frequency spectrum produces more frequency deviation for the same phase deviation than frequencies that fall at the low end of the modulating signal.

fc fc + fmfc - fm

FM noise triangleAssuming that noise is equally distributed throughout the modulating signal frequency range, noise will create side frequencies just like the side frequencies created by the modulating signals.

Noise distribution at output of demodulator of AM and PM

Noise in Frequency Modulation• For frequency modulation (FM), the instantaneous frequency deviation,

Δf (t), is the first time derivative of the instantaneous phase deviation Δθ (t).• When the carrier component is much larger than the interfering noise voltage,

the instantaneous phase deviation due to the noise is approximately:

• The instantaneous frequency deviation at the input of the demodulator is equal to the first time derivative of the instantaneous phase deviation and it can be computed as:

θ (t) = (Vn / Vc)[sin(ωnd t + θn)] radians

Δ ω(t) = (Vn / Vc) ωnd [cos(ωnd t + θn)] radians / sec

Where : Vc = peak voltage of carrier at input of demodulator (volts)Vn = peak voltage of noise at input of demodulator (volts) c = 2fc = angular velocity of carrier at demodulator input n = 2 fn = angular velocity of noise at demodulator inputnd = 2 fnd = n – c = angular velocity of noise at demodulator outputfnd = fn – fc = frequency of noise at output of demodulatorfn = frequency of noise at input of demodulatorfc = frequency of carrier at input of demodulatormn = FM modulation index due to noise (mn << 1)

Noise in Frequency Modulation• The peak frequency deviation (Δ ωpeak ) due to noise is:

Δ ωpeak = (Vn / Vc) ωnd = peak frequency deviation (radians / sec)Δ fpeak = (Vn / Vc) fnd = mnfnd = peak frequency deviation (Hz)

And the modulation index is

mn = Δ fpeak / fnd = modulation index

• From the preceding equations, it can be seen that the peak frequency deviation due to noise is directly proportional to the frequency of the noise at the output of the demodulator. (Δ fpeak = (Vn / Vc) fnd fnd = fn - fc

• The output voltage of an FM demodulator is directly proportional to the frequency deviation (f) at its input.

• Therefore, for a given range of noise frequencies with equal amplitudes, noise with higher frequencies at the output of the demodulator will produce higher voltage at the output of the FM demodulator. (This is not the case for PM, because all noise frequencies with the same amplitudes will produce the same voltages at the output of a PM demodulator.)

• Higher modulating frequencies are also more affected by noise because their signal to noise ratio will be lower.

c n

nd

c + mc - m

Noise in Frequency Modulation• The signal to noise ratio at the output of an FM demodulator due to unwanted

frequency deviation from an interfering sinusoid is:

Example: For an angle modulated carrier vc=6 cos (2π110 Mhz t) with 75 khzfrequency deviation due to the information signal and a single frequency interferingsignal vn= 0.3 cos (2π109.985 Mhz t), determine:

a. Frequency of the demodulated interference signalb. Peak phase and frequency deviations due to the interfering signalc. Voltage signal to noise ratio at the output of the demodulatora. Frequency of demodulated noise interference = fnd = fc – fn

= 110 Mhz – 109.985 Mhz = 15 khzb. Δθ (peak) = Vn / Vc = 0.3 / 6 = 0.05 radian

Δ fpeak = (Vn / Vc) fnd = (0.3 / 6)(15 khz) = 750 hzc. The voltage signal to noise ratio before demodulation is:

S / N = 6 / 0.3 = 20 The voltage signal to noise ratio after demodulation is:

S / N = 75 khz / 750 hz = 100 The signal to noise ratio improvement = 100 / 20 = 5 = 20 log 5 = 14 db

S / N = Δ f (due to signal) / Δ f (due to noise)

Comparison of Effects of Noise in AM, FM, and PMFor AM:• Effects of noise with different frequencies on an AM wave are the same.

Only the amplitude of the noise affects the quality of the demodulated AM wave.

• Changes in noise and modulating signal frequency do not affect the signal to noise ratio.

• Under conditions of very low signal to noise ratio, AM is superior than PM and FM.

For PM• PM has all the noise immunity properties of FM except the noise triangle.• Effects of noise with different frequencies on a PM wave are the same. Only

the amplitude of the noise affects the quality of the demodulated PM wave.

For FM:• Under identical conditions, FM will be 4.75 db better than PM for noise.• FM can be made more noise resistant through higher frequency deviation,

use of limiters , and preemphasis / deemphasis.• Having a maximum frequency deviation of 75 khz and 75 s preemphasis,

FM gives a noise rejection of at least 24 db better than AM.

Preemphasis and Deemphasis• With FM, noise at higher modulating signal frequencies (including thermal noise)

is inherently greater in amplitude than noise at the lower frequencies at the output of an FM demodulator.– Higher frequency noise have greater effects in an FM system.– Higher modulating frequencies have lower Signal to Noise ratio than lower

ones.• To compensate for the non-uniform distribution of noise in FM, preemphasis,

which is emphasizing or boosting of amplitude of high frequency modulating signals, is done at the FM transmitter. This is done to improve S/N ratio.

• To compensate for the preemphasis done at the transmitter, deemphasis, which is attenuating or deemphasizing high frequency signals after demodulation, is done at the FM receiver.

• A preemphasis network is a high pass filter while a deemphasis network is a low pass filter.

• The figure below shows the effects of preemphasis and deemphasis:

dbdb

db

17 db

17 db

Preemphasis

DeemphasisNet effect = 0 db

Preemphasis and Deemphasis Circuits+ Vcc

Q1Input

L = 750 mh

R= 10 k

C1

Output

Time constant =L / R = 75 micro sec.

fb = 2.12 khz(used in FM broadcast band)

= 1nfTime constant =RC = 75 micro sec.

fb = 2.12 khz(used in FM broadcast band)

Preemphasis Deemphasis

The break frequency (frequency where preemphasis and deemphasis begins) is determined by the L/R and RC time constant of the network. In the US, standard preemphasis/deemphasis is 75 s corresponding to 2.12 Khz.

The break frequency occurs at the frequency when Xc or XL = R (3 db pt.) or:

πRC21fb

πL/R21fb (hertz) (hertz)

Input Output

Generation of Frequency ModulationThe prime requirements of an FM system are:• Variable output frequency which is proportional to the instantaneous

amplitude of modulating signal• Modulated signal must have constant peak amplitude.• Frequency deviation must be independent of frequency of modulating

signal.The methods used in generating FM are:• Direct Method – The frequency of oscillation is directly varied, either

through: – varying the capacitance or inductance of a tank circuit– varying the voltage across a varactor diode in an oscillator, to change

the capacitance of the diode. (Varactor diode capacitance changes as the voltage across it changes.)

– Varying the reactance of a semiconductor device such as FET which is used as a load of a tank circuit.

• Indirect Method – FM is generated without varying the frequency of an oscillator, such as varying the phase of a signal. This is sometimes called Armstrong system. Advantage of this is more stable frequency.

• Integrated circuits designed for FM and possibly other types of modulation.

Varactor Diode FM Modulator (Direct Method)

R3

Q1

RFCCc

Cc

Vcc

Cc

FM Output

Modulating SignalInput

R1

R2 R4

RFC

crystal

RFCCc

VD1

•The crystal is used to generate the carrier signal.•The varactor diode is used to deviate the frequency of the crystal oscillator.•R1 and R2 develop a DC voltage that reverse biases the varactor diode (VD1).•The modulating signal voltage adds to or subtracts from the DC bias, which changes the capacitance of VD1, and thus the frequency of oscillation.•Positive alterations of the modulating signal increase the reverse bias on VD1 which decreases the capacitance of VD1, and increases the frequency of oscillation.

•A varactor diode is a semiconductor diode whose junction capacitance varies linearly with the reverse biased voltage.

Because a crystal is used, the peak frequency deviation is limited to relatively small values. Thus, crystals cannot be used for medium and high index FM systems.

Voltage Controlled Oscillator for FM (Direct Method)Using Varactor Diode

R1

Cbp

T1

ModulatingSignalInput

Q1

Cc

L

Vcc

C

FM Output

• The varactor diode is used to transform changes in the modulating signal amplitude to changes in oscillator frequency.

• Circuit is also direct FM modulator

LCπ21fc (Hz)

Where: fc = Oscillator center frequencyf = frequency when modulating signal

is appliedL = inductance of primary winding (hz)C = varactor diode capacitance

when there is no modulating signal(farad)

ΔC = change in varactor diode capacitance due to modulating signal

Δf = change in frequency

C)L(Cπ21f

(Hz)

Δf = fc-f

FM Reactance Modulator (Direct Method)

VDD

RcR1

Re

C

Cc

R3

Q1 JFET

ModulatingSignalInput

OscillatorTankCircuit Ct

Lt

R4

R

• Ct and Lt forms a tank circuit.

• The JFET acts like a variable reactance load to the LC tank circuit.

• The reactance of the FET is dependent on its transconductance (gm), which is dependent on the gate bias.

• The modulating signal varies the gate bias, and thus the reactance of the JFET. This causes a change in the resonant frequency of the tank circuit.

• Circuit is also direct FM modulator

Output

Indirect FM Modulator (Direct Phase Modulator)

R2R4C2

C3

ModulatingSignalinput

CrystalOscillatorCarrierInput

C1L1

R1

To amplifiersandmultipliers

VD1

• Indirect frequency modulators change the phase of the carrier. (Instantaneous phase of the modulated signal is directly proportional to the modulating signal.)

• L1 and R4, together with VD1 acts as a series resonant circuit to the output frequency of the crystal oscillator.

• The modulating signal changes the capacitance of VD1, which shifts the phase of the carrier signal from the crystal oscillator.

• The frequencies of indirect FM modulators are more stable than those of direct FM modulators.

• Phase deviation is limited to small values because of non-linear characteristics of VD1.

Generation of Frequency ModulationThe following are considerations regarding FM generation:

For Direct FM modulators:• Direct FM modulators based on LC oscillators have the disadvantage of

being not stable enough for communications or broadcast purposes.• Direct FM modulators based on LC oscillators use automatic frequency

control (AFC) to stabilize its frequency.• The advantage of direct FM modulators using LC oscillators is that

relatively high frequency deviations and modulation indices can be achieved because the oscillators are inherently unstable.

For Indirect FM modulators (Direct PM modulators):• Actual oscillator is isolated from modulator and can therefore be extremely

stable source such as a crystal.• Indirect FM modulators (direct PM modulators) using crystals have better

frequency stability than direct FM modulators.• FM modulators using crystals cannot have medium or high modulation index

as is. Frequency multipliers are usually used to achieve high modulation indices.

• There are two ways to perform frequency up conversion: heterodyning and frequency multiplication..

Generation of Frequency ModulationWith the heterodyne method:• A relatively low frequency, angle modulated carrier along with its side

frequencies are applied to one input of a balanced modulator. The second input is a relatively high frequency, unmodulated carrier.

• The two inputs mix nonlinearly and sums and differences of the two inputs are created.

• A bandpass filter is used to eliminate the unwanted frequencies at the output. • The bandpass filter is tuned to the sum frequencies.• Since the side frequencies of the modulated wave are unaffected by the

heterodyning process, frequency deviation is also unaffected and remains as is.• The following are unaffected by the heterodyne process:

– modulating signal frequency– frequency deviation– modulation index– phase deviation– bandwidth

Generation of Frequency ModulationWith frequency multiplication method:• Modulation properties of a modulated signal are increased at the same time

that the carrier frequency is up-converted.• The frequency of the modulated carrier is multiplied by a factor of N.• In addition, the frequency deviation, modulation index, and phase deviation

are also multiplied by the same factor N.• Bandwidth increases as the modulation index is increased.• The modulating signal frequency remains as is.• The separation between side frequencies remains as is (+ or – fm)

• Example: Given a balanced modulator with the following inputs:

a. Frequency modulated signal with the following propertiesf = 4 Khz fm = 8 Khz m = 1 fc = 500 Khz

b. Carrier frequency with fc = 100 Mhz

Generation of Frequency ModulationDetermine the following at the output of the balanced modulator:

a. Carrier frequencyb. Peak frequency deviationc. Modulating signal frequencyd. Modulation index

fc(out) = 100 x 10 6 +500 x 103 = 100.5 x 10 6

f (out) = 4 Khz fm (out) = 8 Khz m (out) = 1 (not changed)

Example: Repeat the preceding problem but use a frequency multiplier with a multiplication factor of 15 instead of a balanced modulator.

fc(out) = (10) 500 x 103 = 500 x 104

f (out) = (10) 4 Khz = 40 Khz (changed)fm (out) = (10) 8 Khz = 80 Khz (changed)m (out) = (10) 1 = 10 (changed)fm(out) = 8 Khz (not changed)

Crosby Direct FM Transmitter

Frequency modulator and master oscillatorfc = 5.1 MhzKo

N1X3

N2X2

N3X3

LPF

BPF Mixer

DiscriminatorTuned to 2 Mhz Kd

CrystalReferenceOscillator14.3 Mhz

Buffer and X2Multiplier

N4

PowerAmplifier / couplingnetwork

fin = 2 Mhz

DC correctionvoltage

f = 28.6 Mhz

Automaticfrequency control(AFC) loop

Modulating Signalinput fc

f1 F2=30.6 Mhz

Ft=91.8 MhzVCO


• Modulator could be variable reactance modulator, IC modulator or VCO modulator, which have less stable frequencies compared to FM modulators using crystals.

• For medium and high index FM systems, carrier (master) oscillator cannot use crystals because crystal oscillating frequency could not be significantly varied.

• Automatic frequency control (AFC) is used to stabilize the frequency of the carrier (master) oscillator.– AFC circuit compares frequency of carrier (master) oscillator with that of

a crystal oscillator, and then produces a correction voltage proportional to the difference of the two frequencies.

• The frequency multipliers multiply the following:– center frequency– phase deviations– frequency deviations – modulation index

Crosby Direct FM Transmitter• To achieve maximum frequency deviation at the antenna allowed in US FM

broadcast (75 Khz), the maximum frequency deviation at the modulator must be (assuming the Crosby transmitter in the preceding slides is used):

Δf = (75 khz) / [(N1)(N2)(N3)] = (75 khz) / [(3)(2)(3)] = 4166.7 Hz

The modulation index at the modulator for a maximum modulating signal frequency allowed (15 Khz) is:

m = Δf / fm = 4166.7 Hz / 15 khz = 0.2778

The modulation index at the antenna is:

m=(0.2778) (N1)(N2)(N3) = 0.2778 (3)(2)(3) = 5 (deviation ratio for commercial FM broadcast with 15 Khz modulating signal)

Note that above figures could be different depending on the frequency multipliers used in the transmitter.


Example: Total frequency multiplication = 20, transmit carrier frequency ft = 88.8 Mhz (transmit frequency at antenna). Determine:

a. Master (carrier) oscillator center frequency b. Frequency deviation at the output of the modulator for a frequency

deviation of 75 khz at the antenna.c. Deviation ratio at the output of the modulator for a maximum modulating

signal frequency of fm = 15 khzd. Deviation ratio at the antenna

Solution:a. fc = ft / (N1N2N3) = 88.8 Mhz / 20 = 4.44 Mhzb. Δf = Δf t / (N1N2N3) = 75 khz / 20 = 3750 Hzc. Deviation ratio (DR) at modulator = Δf maximum / fm(maximum) = 3750 / 15,000

= 0.25d. Deviation ratio (DR) at antenna = (0.25)(20) = 5

Crosby Direct FM Transmitter• The Crosby transmitter uses an Automatic Frequency Control (AFC).• Assuming a rock-stable crystal reference oscillator and a perfectly tuned

discriminator, the frequency drift at the output of the second multiplier (N2) without feedback (i.e. open loop) can be computed as:

closed. is loop AFC when theko]k (N1)(N2) [1 offactor aby reduced is multiplier second theofoutput at thedrift frequency theTherefore,

per volt) (hertzfunction transferoscillatormaster ko hz)per (voltsfunction er tor transfdiscriminak

oscillatormaster ofoutput at drift frequency d :where

multiplier second ofoutput at thedrift frequency loop closed kok (N1)(N2) 1

d d

)ko(dk (N1)(N2) - d d:as computed becan drift frequency loop closed The

multiplier second ofoutput at thedrift frequency )(d (N1)(N2) d

d

d

fc

d

fo1fc1

fc1 dfo1fc1

fcfo1

Crosby Direct FM Transmitter• Example: Using the preceding diagram for a Crosby transmitter with the following

parameters: VCO stability = + 200 ppm ko= 10 khz/v kd = 2 v/khzDetermine the reduction in frequency drift at the antenna of the transmitter.

18,360)(3)(2)(3)(5.1x10-91,818,360is antenna at thefrequency in thedrift The

Hz 91,818,360)(3)30,606,120()(N3)30,606,120(ftbecomesdrift with thefrequency transmit antenna The

6,120Hz(3)(2)(5.1)(200)(N2)00ppm)(N1)(5.1Mhz)(2df2 :as computed be alsocan df26,120Hz)(3)(2)(5.1x10- 30,606,120)(N1)(N2)(5.1x10- 30,606,120dfo1df2

is multiplier second theofoutput at thedrift frequency The

30,606,120 01,020)(3)(2)(5,1 (fc) (N1)(N2) f2 is multiplier second theofoutput at thefrequency The

Hz 5,101,020 Mhz) 5.1 x 200ppm( Mhz 5.1 fc isdrift with thefrequency output oscillatormaster theopen, loopfeedback With the

6

66


Hz 208,18 152-18,360 isdrift frequency in reduction The

152Hz)(3)(2)(3)5.1x10(N3))(N1)(N2)(5.1x10( - 91,800,152is closed loopfeedback with theantenna at thedrift frequency The

Hz 91,800,152)(3) 57.050,600,30()(N3) 57.050,600,30(ftbecomesthen frequency transmit antenna The

Hz 57.050,600,3050.57)(3)(2)5.1x10(50.57)(N1)(N2)5.1x10(f2is multiplier second theofoutput at thefrequency The

Hz57.50 /v)Khz Khz)(10 / v(2 (3)(2) 1

6120 kok (N1)(N2) 1

d d

kok (N1)(N2) 1 offactor aby reduced is multiplier second the ofoutput at thedrift frequency theclosed, loopfeedback With the

66

66

d

fo1fc1

d

Phase Lock Loop Direct FM Transmitter

Crystalreferenceoscillatorfo

Phase comparator VCO

Divide byN

Low pass filter

Summer

Modulating signal

FM output to amplifiers

DC correction voltage

Phase lock loop

• The modulating signal varies the output frequency of the VCO based on the voltage of the modulating signal. Thus, FM is achieved.

• Transmitter can be used for wideband (high index) applications.• DC correction voltage is used to achieve crystal frequency stability by adjusting

the VCO center frequency to its proper value.

Phase Lock Loop Direct FM Transmitter

• The VCO output is divided by N and then compared to the output of the crystal oscillator.

• Phase comparator generates a correction voltage proportional to the difference of the two frequencies.

• Correction voltage is added to the modulating signal voltage and applied to the input of the VCO.

• The correction voltage adjusts the VCO center frequency to its proper value.• Low pass filter prevents changes in the VCO output frequency due to modulating

signal from being converted to voltage fed back to the VCO, because it will wipe out the modulation.

Armstrong Indirect FM Transmitter

Crystal carrier oscillator200 khz

Buffer amplifier

900 Phaseshifter

Combining network

X 72multiplier

Modulating signal input

Poweramplifier

Balancedmodulator

X 72multiplier

Mixer

Buffer amplifier

Crystaloscillator13.15 khz

Vc

V’c

Vusf + Vlsb = Vm(DSBSC)

fm

f1

fo

fo

f2

ft

ft

Phase modulated wave

Armstrong Indirect FM Transmitter• Through the Armstrong transmitter, frequency modulation can be obtained

from phase modulation.• Carrier source is a crystal, which has good frequency stability.• Carrier oscillator frequency is not deviated directly.• Relatively low frequency carrier (200 khz) is phase shifted by 900 and fed to

a balanced modulator. • Output of balanced modulator is AM double sideband suppressed carrier,

which is combined with the original carrier to produce a low index phase modulated wave.

• If an amplitude modulated wave is added to an unmodulated voltage of the same frequency and the two are kept 900 apart, phase modulation will be achieved.

• The effect of mixing (heterodyning) on an FM signal is to change the center frequency only, whereas the effect of frequency multiplication is to multiply the center frequency and frequency deviation by the same amount.

• To achieve sufficient deviation for broadcast purpose, both mixing and multiplication are necessary.

• Original carrier is always 900 out of phase with the sidebands.

Angle Modulation Receivers• Majority of terrestrial FM radio communications systems use conventional

noncoherent demodulation because most of the demodulators use envelope detection.

• Angle modulation receivers are very similar to conventional amplitude modulation (AM) receivers, except for the following basic differences:– Generally much higher operating frequencies for FM– Need for limiting and de-emphasis in FM– Totally different methods of demodulation– Different methods of obtaining AGC

• FM receivers are also superhetrodyne receivers.• RF amplifiers are almost always used in an FM receiver to reduce the noise

figure, and to match the impedance of the receiver to the antenna.– However, RF amplifiers often are not required because of the noise

suppression characteristics of FM.• Local oscillators and mixers take any usual forms such as Colpitts. Tracking is

not normally much of a problem in FM broadcast receivers because tuning frequency range is only 1.25:1, much less than in AM broadcasting.

Angle Modulation Receivers

• IF amplifiers do not differ much from those of AM.– FM receivers generally have much more IF gain than AM receivers.– With FM receivers, it is desirable that the last IF amplifier be saturated so

it could act as a limiter or a passband limiter if the output is filtered. Passband filters are used to preserve the information signals.

– FM receivers using the standard 88 to 108 Mhz broadcast frequency range have an IF which is almost always 10.7 Mhz, with 200 khz bandwidth.

– For broadcast band receivers, two IF sections are typically used. – First IF is at relatively high frequency (such as 10.7 Mhz) to have good

image frequency rejection.– Second IF is at relatively low frequency (such as 455 khz) to achieve high

gain while not being susceptible to oscillations.

Angle Modulation Receivers• An Amplitude limiter must precede the FM demodulator to reduce noise, except

when the FM demodulator itself has limiter capability.– Amplitude Limiter is a circuit that produces constant amplitude output for all

signals above a prescribed minimum input level, which is sometimes called: threshold, quieting, or capture level.

– Amplitude Limiter works on the principle of passing the stronger signal and eliminating the weaker.

– Amplitude variations caused by noise on FM or PM waves can be removed by limiting (clipping) the peaks of the envelope prior to detection (demodulation).

• FM detector’s (demodulator) output voltage is directly proportional to thefrequency deviation, while PM detector’s output voltage is directly proportional to the phase deviation.– FM demodulator is a frequency to amplitude changer.– FM signals can be demodulated by PM receivers, and vice versa but some

signals will be boosted more than it is supposed to be boosted.– AM detector is usually replaced with a limiter, frequency discriminator

(demodulator) , and a deemphasis network in an FM receiver.• Signal to noise ratio could be improved in FM or PM receivers.

Double Conversion Superheterodyne FM Receiver

Preselector(Bandpass

Filter)

RFamplifier

1st mixer

Buffer

1st localoscillator

2nd mixer

Buffer

2nd localoscillator

IF amplifier

Limiter Discriminator Deemphasisnetwork

Audioamplifiers

To speaker

Automatic gain control (AGC)

1st IF 2nd IFAudio detector stage

1st IF is at relatively high frequency (such as 10.7 Mhz) to have good image frequency rejection.

2nd IF is at relatively low frequency (such as 455 khz) to achieve high gain while not being susceptible to oscillations.

Double Conversion Superheterodyne FM Receiver

• Preselector, RF amplifier, mixer, local oscillators, IF amplifiers, and audio amplifier are almost identical to those used in AM receivers.

• FM receivers generally have more IF amplification.• RF amplifiers are sometimes not required, due to the noise suppression

characteristics of FM systems.• Limiter, frequency discriminator, and deemphasis network replaces the

detector of AM receiver.• Limiter limits the amplitude of the IF signal prior to demodulating the IF

signal to remove the noise in the envelope of the signal.• Discriminator demodulates the IF signal and extracts the information signal

from the IF signal.• De-emphasis network is a low pass filter which attenuates high frequency

signals to compensate for the emphasis done at the transmitter.

FM Demodulators• FM demodulators produce an output voltage that is directly proportional to the

instantaneous frequency at its input.

Where: Vout = output voltage of demodulator (volts)Δf = difference between the input frequency and the center frequency (hz)K = transfer function of demodulator ( volts per hz)

• Example: An FM demodulator has a transfer function of kd = 0.4 v/khz and an input signal with peak frequency deviation of 75 Khz. Determine the peak output voltage of the discriminator.

• The most common FM demodulators are:– Slope detector (form of tuned-circuit frequency discriminator)– Foster Seeley discriminator (form of tuned-circuit frequency discriminator)– Ratio detector (form of tuned-circuit frequency discriminator)– PLL demodulator– Quadrature demodulator

Vout = Δf K

Vout peak = Δf K = (75 khz)(0.4 v/khz) = 30 volts

FM Demodulators

• Slope detector, Balanced slope detector, Foster Seeley discriminator, and Ratio detector are forms of tuned-circuit frequency discriminator.– Tuned circuit frequency discriminators convert FM to AM and then

demodulate the AM wave using conventional peak detectors.• Most frequency discriminators require a 180 degrees phase inverter, an adder

circuit, and one or more frequency dependent components.

Single Ended Slope Detector

Ci

To audio amplifierD1

RiL

FM in

La Ca

• Similar to AM detector except that IF transformer is off-tuned. (fo is not equal to IF center frequency (fc).

• Uses a tuned circuit (La and Ca) which produces an output voltage proportional to the input frequency.

• Maximum output voltage occurs at the resonant frequency (fo) of tank circuit.• Output voltage increases or decreases proportionately as frequency deviates towards

or away from the resonant frequency (fo).• IF center frequency (fc) is made to fall in the center of the most linear portion of

the voltage-versus-frequency curve. (not equal to fo)• When IF deviates above the IF center frequency (fc) , the output voltage increases.• When IF deviates below the IF center frequency (fc) , the output voltage decreases.• Changes in output voltage results to an AM wave at anode of D1.• D1, Ri, and Ci make up a simple peak detector which demodulates the AM wave.• Detector has poor linearity, difficult to adjust, inefficient, and lacks limiting.

AM wave

frequencyfc

Δf Δf

fo

Volts

Balanced Slope Detector

• Also known as Travis detector, triple tuned discriminator, and amplitude discriminator.

• Uses two slope detectors connected in parallel and fed 1800 out of phase.• La and Ca, Lb and Cb perform the FM to AM conversion.• The balanced peak detectors demodulate the AM signal into information signal.• La and Ca are tuned to a frequency (fa) that is above the IF center frequency (fc) by

approximately 1.33 f.• Lb and Cb are tuned to a frequency (fb) that is below the IF center frequency (fc).• At the IF center frequency (fc), the output voltages from the two tuned circuits are

equal in amplitude but opposite in direction, and output voltage = 0 v.• Detector has poor linearity, difficult to tune (through Ca and Cb), and lacks limiting.

C1To audio amplifier

D1

R1

L

FM inLa

Ca

C2

D2 R2

Lb Cb

T1 +

-

+

-

AM wave

fb

fafc

Vout

+ Δf- Δf

0 volt

Foster Seeley Discriminator

- Also called phase discriminator, center tuned discriminator or phase shift discriminator.- Operation is similar to balanced slope detector.- C1, C2 and Cc act as short circuits at IF frequencies, VL3 = Vp.- At resonant frequency (fo) of secondary (equal to IF center frequency fc), secondary current

and voltage are in phase, Voltage across D1 and C1 is equal in magnitude to voltage across D2 and C2, C1 and C2 charge to equal magnitude with opposite polarities, and Vout= 0 v.

- When IF goes above resonance (XL > XC), secondary tank circuit impedance becomes inductive, secondary current lags the voltage, VD1 + VC1 > VD2 + VC2 , Vout goes positive.

- When the IF goes below resonance (XL < XC), secondary tank circuit impedance becomes capacitive, secondary current leads the voltage, VD1 + VC1 < VD2 + VC2 , Vout goes negative

- Has better response curve (S curve) and easier to tune than balance slope detector, but alsohas no limiter.

C1Vout

(To audio Amplifier)

D1

R1

LpFM in(Vp)

LaCo

C2

D2

R2

Lb

T1

+

-

+

-

-

+

+

-

+ -Cp

Cc

L3

f1

f2fo

Vout

+ Δf- ΔfS curve

0 volt

+

-

VD1 + VC1

VD2 + VC2

VL3 = Vp

VD1 + VC1

VD2 + VC2

VL3 = Vp

VLa

VLb

VLa

VLb

Ratio Detector

- Operation is similar to that of Foster Seeley discriminator except D2 is reversed and Cs is added. Cs charges to the peak voltage across secondary.

- Time constant of Rs and Cs is long enough that rapid changes in amplitude of input signal due to noise are shorted to ground, and have no effect on average voltage across Cs.

- C1 and C2 charge and discharge proportional to the frequency changes in the input signal.- At resonance, output voltage is divided equally between C1 and C2.- When IF deviates from resonance, output voltage across C1 is different to that of C2.- Average voltage of output (Vout) is always positive.- Relatively immune to amplitude variations compared to slope detectors and Foster-Seeley

discriminators.- Has less linear response than discriminators.

C1

Vout

D1

RsLp

FM inLa

Co

C2

D2

Lb

T1

+

-

-

+

Cc

L3

fin<fo

fin > fofo

Vout

+ Δf- Δf

Cs

0 volt

AveragePositiveVoltage

Phase Lock Loop FM Demodulator

• Requires no tuned circuit.• Automatically compensates for changes in carrier frequency due to instability in

transmitter oscillator.• VCO natural frequency is equal to IF center frequency.• Output voltage (Vout) is proportional to the frequency deviation at the FM input,

and is thus the demodulated signal.• IF signal amplitude is usually limited prior to feeding to PLL demodulator for

noise reduction.

Phase Detector

Kd

Low passfilter

AmplifierKa

VCO

DemodulatedAudioOutput

FMinput

fo

VdVout

Vout = f Kd Ka

Quadrature FM Demodulator

Productdetector Cx

FM in Demodulated signal

Ro

Lo

Co

Ci

Rxvi

vo

• Also called coincidence Detector.• Extracts the original information signal from the composite IF signal by

multiplying two quadrature (900 out of phase) signals.• Tank circuit is tuned to the IF center frequency, and produces a phase shift

proportional to the frequency deviation.• Ci produces the 900 phase shift at the IF center frequency.• IF signal (vi) is multiplied by the quadrature signal (vo) in the product detector

which produces an output signal proportional to the frequency deviation.

Tank ciruit

Amplitude Limiters and FM Thresholding

• Amplitude Limiter is a circuit that produces constant amplitude output for all signals above a prescribed minimum input level, which is sometimes called: threshold, quieting, or capture level.

• Some FM demodulators have limiting capability while others don’t have it.• An amplitude limiter can be an IF amplifier which is overdriven to cutoff and

saturation. • One limiter stage, Two limiter stages (double limiting), three limiter stages

(triple limiting), and others, can be used.• A class C IF amplifier can also be used as a limiter but it will require more

filtering at the output.• Amplitude Limiter works on the principle of passing the stronger signal and

eliminating the weaker.• Noise is more prevalent at the peaks of the FM waveform and relatively

insignificant during the zero crossings.• With amplitude limiters, at the signal to noise ratio output of the demodulator

can be improved compared to the signal to noise ratio at the input of the limiter.

• The improvement in the signal to noise ratio due to limiting is called FM Thresholding, FM quieting, or FM Capture Effect.


• The three criteria needed before FM thresholding can occur are:• Predetection signal to noise ratio must be 10 db or greater.• IF signal must be sufficiently amplified to overdrive the limiter.• Signal must have modulation index equal to or greater than unity.

• The output voltage from an FM detector is proportional to m2.• Doubling m (modulation index) increases the signal to noise ratio by a

factor of 4 (6 db).• Due to limiting, capture effect can be experienced when a mobile receiver

passes from one FM transmitter area to another.• Capture effect is the inherent ability of FM to diminish the effects of

interfering signals.• Weaker of two FM signals is eliminated until it is about half of the stronger

signal.• Capture ratio of an FM receiver is the minimum db difference in signal

strength between two received signals necessary for capture effect to suppress the weaker signal.


• Example: An FM receiver has a bandwidth of 200 Khz, power noise figure = 8 db, input noise temperature = 90 K. Determine the minimum receiver carrier power necessary to achieve a postdetection signal to noise ratio of 37 db.

Stereophonic FM SystemsLeft

Channel in

RightChannel in

Sum (L+R)

Difference (L-R)

19 khzSubcarriergenerator 19 khz

Frequencydoubler

BalancedModulator

SCAGenerator

Audio in

FrequencyModulator

Adder

50Hz-15 khz

23-53 khz (DSBSC)

19 khz

59.5-74.5 khz

38 khz

FMOut

Matrix

SCA-Subsidiary Communications Authorization. It is an optional low quality information signal, which may be used for background music in buildingsDSBSC – Double sideband suppressed carrier

Stereophonic FM Systems• Stereophonic FM transmission is a modulation system in which sufficient

information is sent to the receiver to enable it to reproduce original stereo material.

• Stereophonic FM radio systems are compatible with non-stereophonic systems.• In stereo FM, it is not possible to transmit a left and a right channel

independently because a monaural system will not receive acceptably all the information.

• The sum (L+R) modulates the carrier in the same manner as in monaural FM.• (L-R) is shifted in frequency to 23-53 khz through the balanced modulator, to

allow frequency division multiplexing with the (L+R) signal, which occupies the 50 hz-15 khz range.

• The 19 khz subcarrier, (L+R), and (L-R) signals are used to modulate the carrier signal at the FM modulator.

• The 19 khz subcarrier is doubled at the receiver, and the result is used as the pilot carrier for the difference signal. It is transmitted at 19 khz to make the extraction of the pilot carrier easier at the receiver.

• SCA can be used as a second medium quality information signal which is optional. It uses a 67 khz subcarrier, modulated to a depth if + o - 7.5 khz.

• After demodulation in a stereo receiver, (L-R) will be added to (L+R) to produce the left channel, while the difference between the two will produce the right channel.

Documents

Electronic Communications Presentation Materials Part 3