Angle Modulation*

Prepared by Dr Mohamed Hussien

Module outcomeAfter this module, you should be able to:Describe and explain the differences between AM and angle modulation schemes and advantages and disadvantages of each.Describe and explain the differences between frequency and phase modulation and show the relationship between the two.*

IntroductionThree parameters can modulated with information signalAmplitudeFrequencyPhasePhase and frequency changes are related, so we group them together in term angle modulation*

IntroductionIn FM, frequency of modulated signal varies with amplitude of baseband signal.In PM, phase of modulated signal varies with amplitude of baseband signal.In angle modulation amplitude of carrier signal doesnt change with modulation.*

IntroductionFM is more familiar in our daily life:Radio broadcastSound signal in TVMobile radio systemSatellite communicationsCellular telephone systems.PM is less familiar and mostly used in data communication*

Basic DefinitionsLet the unmodulated carrier a sinusoidal waveform,

for simplicity we consider 0 = 0.There are two methods to vary the angle linearly with message signalFrequency modulation (FM)Phase modulation (PM)*

Frequency Modulation (FM)In FM, frequency of modulated signal varies with amplitude of baseband signal.

fsig(t) = signal frequency as a function of time fc = unmodulated carrier frequency kf = frequency sensitivity of modulator in hertz per voltem(t) = modulating signal f = instantaneous frequency deviation*

Frequency Modulation (FM)If the modulating signal is a sine wave

then

The peak frequency deviation (Hz) will be

Thus, frequency deviation is*

Frequency Modulation (FM) Modulation IndexModulation index mf for sine wave is

mf has no theoretical limits and can exceeds one.The frequency deviation formula can be rewritten as

*

Frequency Modulation (FM) Waveform*

Frequency Modulation (FM) ExampleAn FM modulator has kf = 30 kHz/V and operates at a carrier frequency of 175 MHz. Find the output frequency for an instantaneous value of the modulating signal equal to: a) 150 mV b) 2V *

Frequency Modulation (FM) ExampleThe same FM modulator as in previous example is modulated with 3 V sine wave. Calculate the peak frequency deviationSolution: ac voltages are assumed RMS unless otherwise stated*

Frequency Modulation (FM) ExampleAn FM broadcast transmitter operates at its maximum deviation of 75 kHz. Find the modulation index for sinusoidal modulating signal with frequency of : (a) 15 kHz (b) 50 Hz*

Phase Modulation (PM)In PM, (t) is varied linearly with the message signal m(t)

c = 2fct is the angle of the unmodulated carrier.kp is the phase sensitivity of the modulator, expressed in radian per volt.PM signal in the time domain is

*

Phase ModulationIf modulation signal is a sinusoidal one, phase of the modulated carrier is given by

(t) is the phase deviation in radianThe peak phase deviation is defined as phase modulation index, given as*

Phase ModulationExampleA phase modulator has kp = 2 rad/V. What RMS voltage of a sine wave would cause a peak phase deviation of 600?*

Relationships between FM and PMWith FM, the instantaneous frequency is

Since

The FM signal is therefore*

Relationships between FM and PMThe FM signal in time domain is given by

Recalling PM signal in time domain

Thus, FM signal may be regarded as a PM signal in which*

Relationships between FM and PMConversely, a PM signal can be generated by first differentiating em(t) and then using the result as the input to a frequency modulator.

*

Relationships between PM and FMWe may thus deduce all the properties of PM signals from those of FM signals and vice versa. Henceforth, we concentrate our attention on FM signals.

*

Relationship between mf and mpEither FM or PM results in changes in both frequency and phase of modulated waveform.For angle-modulated signal with sine wave, mp or mf represent peak phase deviation from the phase of unmodulated carrier.

*

Relationship between mf and mp ExampleAn FM communications transmitter has maximum frequency deviation of 5 kHz and a range of modulating frequencies from 300 Hz to 3 kHz. What is the maximum phase shift that it produces?*

Relationship between mf and mp ExampleA phase modulator has a sensitivity of kp= 3 rad/V. Find the maximum frequency deviation it produces with a sine-wave input of 2 V peak at frequency of 1 kHz?*

FM SpectrumFM is nonlinear modulation so its spectrum is not related in a simple manner to that of modulating signal.To simplify FM spectral analysis we proceed in the following manner:First, we consider single-tone modulation that produces a narrow band FM signal.Next, we consider single-tone modulation that produces wideband FM signal.*

FM SpectrumIf the modulating signal is a sine wave

then

The peak frequency deviation (Hz) will be

Thus, frequency deviation is*

FM SpectrumWith FM, the instantaneous frequency is

Since

*

FM SpectrumThus, the FM signal is given by

Depending on the value of mf, we may distinguish two cases: Narrowband FM, for which mf is small compared to one ( ).Wideband FM, for which mf is large compared to one. *

FM SpectrumNarrowband FMThe FM signal is given by

By expanding this relation, we get

Assuming mf is small compared to one radian, *

FM SpectrumNarrowband FMBecause

This expression is similar to AM signal.Thus for a narrowband FM signal requires the same transmission bandwidth as the AM signal (2 fm)*

FM SpectrumWideband FMFor mf >> 1, FM produces infinite sidebands even for single tone.Sideband are separated from carrier by multiple of fm.Amplitude of side bands tends to decrease with their distance from carrier.Sidebands with amplitudes less than 1% of total signal voltage can be ignored.*

FM SpectrumWideband FMFM Spectrum in this case can be expressed as a series of sinusoids using Bessel function of first kind.*

FM SpectrumWideband FMBased on Bessel functions s(t) can be rewritten as*

FM SpectrumWideband FM*

FM SpectrumWideband FM*

FM SpectrumWideband FMLet unmodulated carrier have a voltage of A volts RMS across a resistance of RL .Power at carrier frequency is

Similarly, power in each of sidebands arePower in whole signal will be*

FM SpectrumWideband FMLet unmodulated carrier have a peak voltage of A volts across a resistance of RL .Power at carrier frequency is

Similarly, power in each of sidebands arePower in whole signal will be*

FM SpectrumWideband FMAn FM signal has deviation of 3 kHz and modulating frequency of 1 kHz. Its total power PT is 5 W, developed across a 50 load. Carrier frequency is 160 MHz.Calculate the RMS signal voltage VTCalculate RMS voltage at carrier and first three sets of sidebandsCalculate power at carrier and each of first three sidebands*

FM SpectrumWideband FMSignal power is constant with modulation, thus

Modulation index

From Bessel table we have

*

FM SpectrumWideband FM*

FM SpectrumBandwidthFor FM, bandwidth varies directly with fm as well as with mf.For FM, situation is complicated by the fact

Increase in fm will reduce mf and thus number of sidebands.Increase in fm means further apart sidebands in frequency.The above two effects work in opposite directions making FM bandwidth to some extend constant. *

FM SpectrumBandwidth*

FM SpectrumBandwidthCarsons Rule: The bandwidth of FM signal is given by

Example: An FM signal has a deviation of 3 kHz and a modulating frequency of 1 kHz. Using Casons rule calculate the bandwidth.*

FMSpecificationsNo theoretical limits to modulation index or frequency deviation of an FM signal.In general larger values for deviation result in increased S/N. BW of an FM signal is generally limited by government regulations that specify:Maximum frequency deviationMaximum modulating frequency*

FM transmitter *