FM Broadcasting System

EE 370: Communications Engineering I Chapter 5: Angle Modulations and Demodulations

Chapter 5Angle Modulation and Demodulation

EE 370: Communications Engineering I Chapter 5: Angle Modulations and Demodulations

1. Angle (exponential) Modulation

2. Bandwidth of Angle Modulated Wave

3. Generation of FM Waves

4. Demodulation of FM Signals

Contents

4. Demodulation of FM Signals

6. Superheterodyne analog AM/FM Receivers

7. FM Broadcasting System

1. Angle Modulation

In AM signals, the amplitude of a carrier ismodulated by a signal m(t), and, hence, theinformation content of m(t) is in the amplitudevariations of the carrier. Because a sinusoidalsignal is described by amplitude and anglesignal is described by amplitude and angle(which includes frequency and phase), thereexists a possibility of carrying the sameinformation by varying the angle of thecarrier. This in effect is a nonlinearmodulation technique.

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1. Angle Modulation

The Concept of Instantaneous Frequency

A general sine wave signal can be expressed as

( ) ( )tAt θϕ cos=

( ) θωθ += tt

θ(t) is the generalized angle. For a sine wave with fixed

frequency and phase:

4

( )0θωθ += tt

( ) ( )dt

tdt

θω =

� can be represented as a linear function of time with a slope ω : angular

speed, ω = 2πf.

In general ω is the derivative of the angle. That is

and ( ) ( )∫ ∞−=

t

dt ααωθ

1. Angle Modulation

The Concept of Instantaneous Frequency

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1. Angle Modulation

• Phase Modulation (PM)

( ) ( )tmktt pc ++= 0θωθ

The message signal is modulating the phase of the carrier signal:

θ

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( ) ( )cosPM c p

t A t k m tφ ω = +

( ) ( ) ( )tmkdt

tdt pci

•

+== ωθ

ω

without loosing generalization, we can omit the initial phase θ0 and we get the

following PM signal :

ωi is called the instantaneous frequency of the modulated signal.

1. Angle Modulation

• Frequency Modulation (FM)

The message signal is modulating the frequency of the carrier signal:

( ) ( )tmkt fci += ωω ( ) ( )[ ]∫ ∞−+=

t

fc dmkt ααωθ

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( ) ( )tmkt fci += ωω ( ) ( )[ ]∫ ∞−+= fc dmkt ααωθ

( ) ( )∫ ∞−+=

t

fc dmktt ααωθ ( ) ( )

+= ∫ ∞−

t

fcFM dmktAt ααωϕ cos

1. Angle Modulation

• General Concept of Angle Modulation

( ) ( )[ ]ttAt c ψωϕ += cos ( ) ( ) ( )∫ ∞−−=

t

dthmt αααψ

h(t) = kpδ(t) � phase modulation

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h(t) = kf u(t) � frequency modulation

Further, according to the equations above we can say that the PM and FM are equivalent in certain ways �

• FM with m(t) = a PM with ∫m(t).

• PM with m(t) = a FM with m’(t).

1. Angle Modulation

• General Concept of Angle Modulation

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1. Angle Modulation

Power of Angle Modulation

Regardless the values of kp or kf, the expected

power of the modulated signal is given by: A2/2.

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See example 5.1 and 5.2 of the text book

2. Bandwidth of Angle Modulated

WavesTo determine the bandwidth of an FM wave :

( ) ( )∫ ∞−=

t

dmta αα ( ) ( ) ( )c f fcj t k a t jk a tj t

FMt Ae Ae e

ω ωφ + = =

�

( ) ( )ReFM FMt tφ φ = �

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expanding the factor in power series �

( ) ( ) ( ) ( )2 3

2 3cos sin cos sin ....

2! 3!

f f

FM c f c c c

k kt A t k a t t a t t a t tφ ω ω ω ω

= − − + +

( )fjk a te

2. Bandwidth of Angle Modulated Waves

the FM signal is expressed as an unmodulated carrier plus

spectra of a(t), a2(t), … an(t), … centered at ωc.

Let M(ω) be the spectrum of m(t) with bandwidth B.

The bandwidth of a(t) is also B because the integration is

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equivalent to only a multiplication by 1/jω .

a2(t) has a bandwidth of 2B (M(ω)*M(ω))

a3(t) has a bandwidth of 3B

an(t) has a bandwidth of nB

�


Conclusion: FM signal has infinite bandwidth. (theoretically)

Special cases:

• Narrow-Band Angle Modulation

The angle modulation is not linear in general. However, if

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The angle modulation is not linear in general. However, if |kf a(t)| << 1 � only the 1st two terms are important in the

above equation.

( ) ( )cos sinFM c f c

t A t k a t tφ ω ω ≈ −

This is a linear modulation. It is like an AM wave* with bandwidth = 2B. This is called Narrow Band FM (NBFM).

* However the waveform is entirely different from AM


Narrow-Band Angle Modulation

Similarly the narrow band PM (NBPM) is given by:

( ) ( )cos sinPM c p c

t A t k m t tφ ω ω ≈ −

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The narrow band angle modulation is similar to AM (same

bandwidth, carrier plus spectrum centered on ωc).

The difference: in angle modulation the sideband spectrum is

π/2 phase shifted with respect to the carrier. The waveform is

completely different.


Narrow-Band FM and PM wave generation

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2. Bandwidth of Angle Modulated

Waves• Wide-Band Angle Modulation

This is the situation where we cannot ignore the higher terms because (|kf a(t)| << 1) is not satisfied. (can be due

to high kf ).

In this case the bandwidth of the FM signal is found to be

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f

In this case the bandwidth of the FM signal is found to be

given by the following approximation:

( )

+=+∆= B

mkBfB

pf

FMπ2

22 Carson’s rule

For truly wideband case, ∆ f >> B � BFM ≈ 2∆ fCarson’s rule can be in terms of the deviation ratio βas


• Phase Modulation

( )12 += βBBFMB

f∆=β

Deviation ratio in FM plays the role of modulation index in AM

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All the analysis developed for the FM can be applied to the PM by replacing mp by mp’ and kf by kp. That is

Examples 5.3 – 5.5 (pg. 217-220)

( )2 22

p p

PM

k mB f B B

π

= ∆ + = +

�

3. Generation of FM Wave

• Indirect Method of Armstrong

We start with the generation of a NBFM as

described previously. Then we use a frequencymultiplier ( x N ) to obtain a WBFM. After filtering

using a bandpass filter centered at Nωc, we get an

FM signal with N∆f.

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FM signal with N∆f.

Sometimes the frequency increase of the carrier is

not needed.

Solution: after the multiplier we insert a mixer to

down convert the carrier to the wanted one.


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Example 5.7


• Direct Generation

Using a VCO

Design of VCO:

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1) OPAMP + hysteric circuit (i.e. Smith trigger)

2) Variation of L or C of a tank of a resonant circuit

: reverse biased semiconductor (i.e. diode) can be

used as a variable capacitor.


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22


• Direct Generation

Also we can use variable inductor. It can be

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Also we can use variable inductor. It can be

achieved by winding two inductors in the same

core. Then controlling the inductance of the inner

inductor by injecting a current in the outer one.

4. Demodulation of FM Wave

In an FM signal the information resides in the

instantaneous frequency :

�a network with a response linear to ω would be

( )i c fk m tω ω= +

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�a network with a response linear to ω would be

able to detect the message signal.e

�

example : |H(ω)| = aω + b centered around the

carrier frequency in the FM band.


|H(ω)| = aω + b

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Direct

Differentiation

Method


( ) ( )

+= ∫ ∞−

t

fcFM dmktAt ααωϕ cos

Let the FM signal:

The differentiator output will be:

2626

( ) ( ){ } ( )

+== ∫ ∞−

t

fcFMFM dmktAdt

dt

dt

dt ααωϕϕ cos�

( ) ( )[ ] ( )

++= ∫ ∞−

t

fcfcFM dmkttmkAt ααωωϕ sin�


� the above expression shows that the output is

FM and AM modulated. See figure 5.12b. Since

ωc > kfm(t) all the time, an envelop detector can

be used to extract m(t) as shown in the previous

figure.

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figure.

� Problem: A must be a constant. If not (due to

channel noise, fading, …), it must be fixed before

demodulation.

Solution: Bandpass limiter

4. Demodulation of FM WaveBandpass limiter

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4. Demodulation of FM Wave� Input: distorted FM signal

� Output of the hard limiter: A(t) >=0

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� Output of the bandpass filter:


Practical Frequency Demodulators

Differentiation: OPAMP differentiator can be used to

convert frequency variation to amplitude variationthat can be detected using a simple envelop detector.

Slope detection: Any tuned circuit which has a linear

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Slope detection: Any tuned circuit which has a linearsegment of positive slope in the frequency responseunder or above the resonance can be used instead ofthe OPAMP differentiator.

Example: high pass RC filter.

Limitation: narrow bandwidth.


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Ratio detector: balanced demodulator. Not verysensitive to the amplitude variation of the FM signal.widely used in the past.

Zero-crossing detectors: frequency counter that

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Zero-crossing detectors: frequency counter thatmeasures the instantaneous frequency by counting therate of zero-crossing.

Phase-Locked Loop (PLL): because of its low cost andgood performance, it is widely used in FM receivers.

Phase-Locked Loop (PLL)

• PLL is used to track the phase and frequency of the carrier

component of an incoming signal.

• Useful for synchronous demodulation of AM signals with

suppressed carrier (no pilot).

• Can be used for demodulation of angle-modulated signals

especially under low SNR conditions.

• Also has applications in clock recovery clock recovery clock recovery clock recovery systems in

digital receivers.

Phase-Locked Loop (PLL)Three basic components:

1. Voltage Controlled Oscillator (VCO)

2. Multiplier-works as the phase detector(PD) or phase comparator

3. Loop filter

( ) where is a constantVCO c o

ce t cω ω= +


ωc is the free running frequency of VCO

Instantaneous frequency of the VCO :

Further, considering the output of VCO

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( )VCO c o tω ω θ= + �

( ) ( )Therefore (5.24)o o

t ce tθ =�

Further, considering the output of VCO as Bcos[ωct+θo(t)]:

c and B are constants of the PLL.


Although in Figure (a), incoming frequency and the VCO

Output frequency are equal (ωc), the analysis is also valid

when they are different as shown below.

We assume the incoming signal (input to the PLL) be

Asin[ωct+θi(t)]. Asin[ωct+θi(t)].

If the incoming signal happens to be

Asin[ωot+ѱ(t)],

it can still be expressed as Asin[ωct+θi(t)],

where θi(t)=(ωo-ωc)t+ ѱ(t).

Thus the analysis is not restricted to equal frequencies

of the incoming signal and the free-running VCO signal.


The multiplier output is ABsin[ωct + θi(t)]cos[ωct + θo(t)]

=(1/2)AB (sin[θi(t) - θo(t)]+sin[2ωct + θi(t) + θo(t)]).

The sum frequency term is suppressed by the loop filter.

Hence the effective input to the loop filter is (1/2)AB sin[θi(t) - θo(t)].

37These equations suggest the model in part (b) of the Figure

If h(t) is the unit impulse response of the loop filter,

eo(t)=h(t)*(1/2) AB sin[θi(t) - θo(t)]

Since , we have

where K=cB/2 and θe(t) is the phase error, defined by

θe(t) = θi(t) - θo(t).

( ) ( ) ( )0

1sin

2

t

i oAB h t x x x dxθ θ = − − ∫

( ) ( )o ot ce tθ =� ( ) ( ) ( )0

sin

t

o et AK h t x x dxθ θ = − ∫�

Demodulation of FM, PM Waves with PLL

When the incoming FM signal is A sin[ωct + θi(t)],

Hence,

and assuming a small error θe(t),

( ) ( ) .

t

i ft k m dθ α α−∞

= ∫

( ) ( ) ( )t

o f et k m d tθ α α θ−∞

= −∫

k

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and assuming a small error θe(t),

Thus, the PLL acts as an FM demodulator. If the incoming

signal is a PM wave, and

In this case the output of the PLL should be integrated to obtain the message signal m(t).

( ) ( ) ( ) ( )i.e., f

o f o

kt k m t e t m t

cθ =� �

( ) ( )i pt k m tθ = ( ) ( )1

.o pe t k m tc

= �

6. Superheterodyne FM/AM Receiver

[A+m(t)]cos(ωIF

t)

Or

Acos(wIF

t+ΨΨΨΨ(t))

[A+m(t)]cos(ωct)

Or

Acos(ωct+ΨΨΨΨ(t))

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AM superheterodyne receiver: Intermediate frequency = 455kHz and enelope

detection is used.

monophonic FM receiver: Identical to the superheterodyne AM receiver

except that the intermediate frequency is 10.7MHz and envelope detector is

replaced by a PLL or a frequency discriminator.

Superheterodyne FM/AM Receiver

� All of the frequency (channel) selectivity is realized in the

IF section; the RF section plays a negligible role.� The main function of the RF section is image frequency suppression.� The mixer (converter output consists of components of the difference

between the incoming (fc) and the local-oscillator (fLO) frequencies.i.e., fIF=| fLO-fc |

e.g., if the incoming carrier frequency fc=1000kHz, then fLO=fc+fIF=1000+455=1455 kHz. But another carrier , with frequency fLO=fc+fIF=1000+455=1455 kHz. But another carrier , with frequency f’c=1455+455=1910kHz will also be picked up because the differencef’c – fLO is also 455kHz. The station at 1910kHz is said to be theimage of the station of 1000kHz. Stations that are 2fIF=910 kHz apart

are called image stations and would both appear simultaneously at theIF output. The RF filter at receiver input prevents this from happening.The RF filter may provide poor selectivity against adjacent stationsseparated by 10kH, but can provide good selectivity against stations separated by 910kHz. Thus when we wish to tune in a station at 1000kHZ, the RF filter tuned to 1000kHz suppresses the image station at 1910kHz.


FCC assigned the following frequency bands for FM

broadcasting.

Frequency range: 88 to 108MHz

Separation: 200 kHz

Max. frequency deviation: 75 kHz

Old FM receivers are monophonic. (one signal m(t))

New FM receivers are stereophonic. (left and right

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New FM receivers are stereophonic. (left and rightaudio signals, i.e. two different microphones). �

more natural effect.

FCC ruled that:

� monophonic receivers must be able to receive

stereo FM signals.

� Total transmission band for the stereo FM signal =

200kHz with ∆f = 75KHz.


19kHz

Composite

Baseband

Signal

Left

Signal

(Base band)

Right

Signal

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Diagram for stereo FM transmitter

Signal

(Baseband)


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Spectrum of a baseband stereo FM signal


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FM stereo receiver

Documents

FM Broadcasting System