Angle Modulation

1

Angle Modulation (Phase & Frequency Modulation)EE442 Lecture 7

Spring Semester

Issued 1983

2

Applications for Various Modulation Techniques

We have studied AM, next is FM and PM.

Not a complete list of applications.

3

With few exceptions,Phase Modulation (PM)

is used predominantly in digital communication

Amplitude, Frequency and Phase Modulation

The rate of change in the phase is equal to the frequency.

4

Illustrating AM, PM and FM Signals

Carrier Wave

Modulating Signal m(t)

AM Modulated Signal

PM Modulated Signal

FM Modulated Signal time

Carrier signal

m(t)

AM

PM

FM

AngleModulation

100% modulation

shown

Reference:Lathi & Ding

( )dm t

dt

( )m t

~

~

5

Focus Upon an FM Signal Modulated by a Single-Tone

BA

SEB

AN

D

SIG

NA

L

FM

SI

GN

AL

Single-tone modulating signal( )m t

6

carriercos(Ct)

messagem(t)

amplitudemodulation

phasemodulation

frequencymodulation

Comparing AM, PM and FM for a Ramp m(t)

https://www.princeton.edu/~mvaezi/ece3770/ECE3770_Lecture7.pdf

7

Some Observations on FM and PM Waveforms

1. Both FM and PM waveforms are identical except for a time shift, given that m(t) is a sinusoidal signal.

2. For FM, the maximum frequency deviation occurs when modulatingsignal is at its peak values (i.e., at + mp and – mp).

3. For PM, the maximum frequency deviation takes place at the zerocrossings of the modulating signal m(t).

4. It is usually difficult to know from looking at a waveform whether the modulation is FM or PM.

5. The message resides in the zero-crossings alone, provided the carrier frequency is large.

6. The modulated waveform doesn’t resemble the message waveform.

Reference: Carlson & Crilly, 5th ed., Section 5.1, pages 208 to 212.

8

1. Angle modulation is resistant to propagation-induced selective fadingbecause amplitude variations are not important.

2. Angle modulation is very efficient in rejecting interference (i.e., it minimizes the effect of amplitude noise on the signal transmission).

3. Angle modulation allows for more efficient use of transmitter power.

4. Angle modulation is capable of handling a greater dynamic range in the modulating signal without distortion as would occur in AM.

5. Wideband FM gives significant improvement in the signal-to-noise ratioat the output and is proportional to the square of the modulationindex .

Advantages of Angle Modulation

f

B

=

Bandwidth

Frequency deviation

B

f

9

Phase-Frequency Relationship When Frequency is Constant

0

+ 0

0 is constantCt

( )t

time t

=

= =( )

Slope: ( )i

i

Ct t

d tt

dt

(t) is generalized angle

=( ) cos( ( ))Ct A t

= + 0( ) cos( )C Ct A t

Concept of Instantaneous Frequency

10

(t) is generalized angle

=( ) cos( ( ))Ct A tAngle

Modulation

= + 0( ) cos( )C Ct A t

0

( )t

( )t

time t

=

= ( )

Slope: ( )i

i

Ct t

d tt

dt

ti

+ 0

0 is constantCt

Angle Modulation Gives PM and FM

11

−=

= = ( )

( ) and ( ) ( )t

i i

it t

d tt t d

dt

AngleModulation

PhaseModulation

FrequencyModulation

Frequency modulation and phase modulation are closely related!

12

No. Frequency Modulation (FM) Phase Modulation (PM)

1 Frequency deviation is proportional to modulating signal m(t)

Phase deviation is proportional to modulating signal m(t)

2 Noise immunity is superior to PM (and of course AM)

Noise immunity better than AM, but not FM

3 Signal-to-noise ratio (SNR) is better than PM (and of course AM)

Signal-to-noise ratio (SNR) is not quite as good as with FM

4 FM is widely used for commercialbroadcast radio (88 MHz to 108 MHz)

PM is primarily used for mobile radio services

5 Modulation index is proportional to modulating signal m(t) as well as the modulating frequency fm

Modulation index is proportional to modulating signal m(t)

Comparing Frequency Modulation to Phase Modulation

13

FM has better noise (or RFI)

rejection than AM, as shown in

this dramatic New York

publicity demonstration by

General Electric in 1940. The

radio contained both AM and FM

receivers. With a million-volt arc

as a source of interference

behind it, the AM receiver

produced only a roar of static,

while the FM receiver clearly

reproduced a music program

from Armstrong's experimental

FM transmitter W2XMN in New

Jersey.

FM has superior noise immunity compared to AM

https://en.wikipedia.org/wiki/Frequency_modulation

Note: RFI stands for radio frequency interference.

14

Phase Modulation (PM)

= + + =0 0( ) ( ) ; Usually we set 0,i C pt t k m t

= +( ) cos( ( ))PM C C pt A t k m t

The instantaneous angular frequency (in radians/second) is

= = + = +

( ) ( )( ) '( )ii C p C p

d t dm tt k k m t

dt dt

In phase modulation (PM) the instantaneous angular frequency i

varies linearly with the time derivative of the message signal m(t) [denoted here by m’(t)].

kp is the phase-deviation (sensitivity) constant. Units: radians/volt[Actually it is radians/unit of the parameter m(t).]

Agbo & SadikuSection 4.2; p. 159

15

Frequency Modulation (FM)

= +( ) ( )i C ft k m t

−

= +

( ) cos ( )t

FM C C ft A t k m d

But in frequency modulation the instantaneous angular frequency i varies linearly with the modulating signal m(t),

− −

= + = + ( ) ( ) ( )t t

i C f C ft k m d t k m d

Then

FM and PM are related to each other. In PM the angle is directly proportional to m(t).In FM the angle is directly proportional to the integral ( ) .m t dt

kf is frequency-deviation (sensitivity) constant. Units: radians/volt-sec.

Agbo & SadikuSection 4.2; p. 159

16

Summary

Angle

Frequency

Phase Modulation Frequency Modulation

= +( ) ( )i C pt t k m t −

= + ( ) ( )t

i C ft t k m d

( )i C p

dm tk

dt = + ( )i C fk m t = +

In phase modulation m(t) drives the time variation of phase i.

In frequency modulation m(t) drives the time variation of frequency fC.

=

( )( ) ii

d tt

dtDefinition: Instantaneous frequency is

Message signal is m(t)

17

A Pictorial View of FM and PM Generation

We require that H(j) be a reversible (or invertible) operationso that m(t) is recoverable.

Phase Modulator

Frequency Modulator

d

dt

( )m t

( )m t

( )dm t

dt

PM ( )t

FM ( )t

−

( )t

m d

Frequency Modulator

Phase Modulator

H(j) = 1/j

H(j) = j

Generation of PM

Generation of FM

cos( )C CA t

cos( )C CA t

Agbo & SadikuFigure 4.1

p. 160

18

Both FM and PM Generation are Nonlinear Processes

( )

( )

( )

1 2

1 1

2 2

1 2

Consider a phase modulated signal:

Let ( ) cos [ ( ) ( )]

If ( ) cos ( ) , and

( ) cos ( )

It then holds that

( ) ( ) ( ) additivity fails

The same a

So PM can't be linear.

C C p

C C p

C C p

s t A t k m t m t

s t A t k m t

s t A t k m t

s t s t s t

= + +

= +

= +

+

rgument holds for FM.

19

Modulation Index for Angle Modulation

Let the peak values of the message signal m(t) and its first derivative m’(t) berepresented by

Peak value of m(t) = mp = ½(mmax – mmin)

Peak value of m’(t) [= dm(t)/dt] = m’p

Frequency Deviation is the maximum deviation of the instantaneous modulated carrier frequency relative to the unmodulated carrier frequency.It is symbolically represented by either or f.

The ratio of the frequency deviation f to the message signal’s bandwidth BIs called the Frequency Deviation Ratio or the Modulation Index, and is denoted by (unitless).

FM: or2

PM: or2

f p

f p

p p

p p

k mk m f

k mk m f

= =

= =

2

f

B B

= =

20

=

=

Carrier signal cos( ) (volts)

Carrier frequency 2 (radians/sec)

Modulating wave ( ) cos( )

Modulating frequency 2 (radians/sec)

Deviation sensitivity (radian

Single-tone modulation

C C

C C

m m

m m

f

A t

f

m t A t

f

k

−

=

= = =

= + = +

= +

s/volt-second)

Frequency deviation (radians/sec)

Modulation Index (unitless)

Instantaneous frequency cos( ) cos( )2

Remember ( ) cos ( )

f m

f m

m m m

mi C f m C m

t

FM C C f

k A

k Af

f

Af f k t f f t

t A t k m d

( )

= +

= +

, generally

Tone modulated wave ( ) cos sin( )

or ( ) cos sin( )

f m

FM C C m

m

FM C C m

k At A t t

t A t t

Equations for FM Wave with Single-Tone Modulation

21

Summary of Mathematical Equations for FM and PM

Type of Modulation

Modulating Signal Angle Modulated Wave

Phase modulation

m(t)

Frequencymodulation

m(t)

Phase modulation

Tone:

m(t) = Am cos(mt)

Frequencymodulation

Tone:

m(t) = Am cos(mt)

( )cos ( )C C pA t k m t +

( )cos ( )t

C C fA t k m d −

+

( )cos cos( )C C p m CA t k A t +

cos sin( )f m

C C C

m

k AA t t

+

f m

m

k A

=

22

( )6 3( ) 10 cos 2 (10 ) 8sin(2 (10 ) )FM t t t = +

( )FM t

Example

f

-

Cf

23

Solution to Example

( )( ) cos 2 sin(2 )FM C C mt A f t f t = +

Start with the basic FM equation:

Compare this to

(a) We see that fC = 1,000,000 Hz & fm = 1000 Hz.(b) The modulation index is = 8.(c) The peak deviation frequency f is

Note: f /fC is 0.008 or 0.8 % deviation frequency to carrierfrequency.

( )6 3( ) 10 cos 2 (10 ) 8sin(2 (10 ) )FM t t t = +

8 1000 8,000 Hzmf f = = =

24

Average Power of a FM or PM Wave

The amplitude AC is constant in a phase modulated or a frequency modulated signal. RF power does not depend upon the frequency or the phase of the waveform.

= +( ) cos ( , ( ))FM or PM C C kt A t g k m t

=2

Average Power (always)2CA

This is a result of FM and PM signals being constant amplitude.

Note: kk becomes kf for FM and kp for PM.

25

Problem:

Consider an angle modulated signal given by

What is the average power of this signal?

Solution:

( )6( ) 6 cos 2 10 2 sin(8000 voltst t t = +

2

2

Average power where 6 volts2

6 36Therefore, 18 watts (assumes 1 ohm resistance)

2 2

Note that the result does not depend upon it being FM or PM.

CC C

C

AP A

P

= = =

= = =

Average Power of a FM or PM Wave

26

# Frequency Modulation (FM) Amplitude Modulation (AM)

1 FM receivers have better noise immunity

AM receivers are very susceptible to noise

2 Noise immunity can be improved by increasing the frequency deviation

The only option in AM is to increase the transmission power

3 Bandwidth requirement is greater and depends upon modulation index

AM bandwidth is less than FM or PM and doesn’t depend upon a modulation index

4 FM (or PM) transmitters and receivers are more complex than for AM

AM transmitters and receivers are less complex than for FM (or PM)

5 All transmitted power is useful so FM is very efficient

Power is wasted in transmitting the carrier and double sidebands in DSB (but DSB-SC & SSB addresses this)

Comparison of FM (or PM) to AM

27

AM, FM and PM Waveforms for Single-Tone m(t)

Review:

Carrier Wave

Modulating Signal m(t)

AM Modulated Signal

PM Modulated Signal

FM Modulated Signal time

Carrier signal

m(t)

AM

PM

FM

AngleModulation

Reference:Lathi & Ding

( )dm t

dt~

( )m t~Refer to slide 16

Focus uponfrequency

28

FM and PM Examples

Sketch FM and PM waveforms for the modulating signal m(t). The constantskf and kp are 2 105 and 10, respectively. Carrier frequency fc = 100 MHz.

( )

( )

8 5

min max

8 5

min

8 5

max

( ) 1 10 1 10 ( );2

1 and 1

10 10 ( 1) 99.9 MHz,

10 10 ( 1) 100.1 MHz

f

i C

i

i

kf f m t m t

m m

f

f

= + = +

= − =

= + − =

= + + =

( )

( )

= + = +

= − =

= + − =

= + + =

8

min max

8

min

8

max

'( ) 1 10 5 '( );2

' 20,000 and ' 20,000

10 5( 20,000) 99.9 MHz,

10 5( 20,000) 100.1 MHz

p

i C

i

i

kf f m t m t

m m

f

f

FM PM

m’(t)m(t)m(t)( )d m t

dtseconds

Fig. 5.4; p. 256 of 4th ed., Lathi & Ding

29

Digital Frequency Shift Keying is Related to FM

Sketch the FM waveform for the modulating signal m(t). The constant kf is2 105. Carrier frequency fc = 100 MHz.

Since m(t) switches from +1 to -1 and vice versa, the FM wavefrequency switches between 99.9 MHz and 100.1 MHz. This is calledFrequency Shift Keying (FSK) and is a digital communication format.

8 5( ) 1 10 1 10 ( )2

f

i C

kf f m t m t

= + = +

FM

m(t)

Fig. 5.5; p. 258 of 4th ed., Lathi & Ding

30

Example – continued

Fig. 5.5; p. 258 of 4th ed., Lathi & Ding

This is carrier PM by a digital signal – it is Phase Shift Keying (PSK)because the digital data is represented by phase of the carrier wave.

= + = +8( ) ( )1

1 102 4

p

i C

k dm t dm tf f

dt dt

PM

Sketch the PM waveform for the modulating signal m(t) from prior slide.The constant kp equals /2. Carrier frequency fc = 100 MHz.

= + = +

= = −

= − =

= − − = − − = −

( ) cos ( ) cos ( )2

( ) sin( ) when ( ) 1

( ) sin( ) when ( ) 1

where jump in ( ) (1) ( 1) 2 or ( 1) (1) 2

PM C C p d C C d

PM C C

PM C C

d

t A t k m t A t m t

t A t m t

t A t m t

m t

m’(t)( )dm t

dt

Evaluate the instantaneous jumps by considering:

( )0

dm t

dt=

Instantaneous jumpsin phase by radians.( ) ~ ( , )p dk m t −

31

Generalized Angle Modulation

Start with equation (4.8) on page 159, which is

( ) cos[ ( )] where ( ) ( ) ( ) ( ) ( )

with ( ) ( ) for PM; ( ) ( ) for FM

Suppose we use the exponential carrier e instead of C

t

A C C

j t

C C

t A t k t t m t h t m h t d

h t t h t u t

A A

−

= + = = −

= =

( ( )) ( ) ( )

( )

cos( ),

then the form for generalized angle modulation becomes

( )

where for PM; for FM

( ) Re ( ) Re ; where ( ) ( ) ( ) ( )

and

(

C C

pC

C

j t k t j t jk t

A C C

p f

jk tj t

PM A C

FM

t

t A e A e e

k k k k

t t A e e t m t t m t

t

+= =

→ →

= = = =

( )

) Re ( ) Re ; where ( ) ( )fC

tjk tj t

A Ct A e e t m d

−

= = =

Agbo & Sadiku; Section 4.2 & 4.3 on pages 158 to 168

32

Generalized Angle Modulation (continued)

0

2 2 3 3

( )( ) Re ( ) Re

!

( ) ( )( ) Re 1 ( )

2! 3!

Now take the real part of the expression above,

( ) cos( ) ( )sin( )

C

C

n n n

fj t

FM A C

n

f fj t

FM C f

FM C C f C

j k tt t A e

n

k t jk tt A e jk t

kt A t k t t

=

= =

= + − + −

= − −

2 2 ( )cos( )

2!

Note: ( ) has a bandwidth = Hz and (t) has a bandwidth = Hz,

but ( ) has a bandwidth = Hz; as , bandwidth

f

C

n

tt

m t B B

t nB n

+

→ →

Consider first Frequency Modulation (FM),

The instantaneous frequency deviations are symmetrical aboutcarrier frequency C, thus FM is double side-banded. The effective FM bandwidth = 2nB Hz.

Agbo & Sadiku; Section 4.2 & 4.3 on pages 158 to 168

33

Generalized Angle Modulation (continued)

( ) cos( ) ( )sin( )

( ) cos( ) ( ) sin( )

FM C C f C

t

FM C C C f C

t A t k t t

t A t A k m d t

−

−

−

Consider the case where kf is small. meaning that |kf (t)| << 1.It is commonly referred to as narrowband FM (NBFM). We takeonly the first two terms in the expansion for FM(t).

Equation (4.15)

By analogy, we can apply same analysis for Phase Modulation (PM).For PM, if kp is small, then |kp (t)| << 1. This is known asnarrowband PM (NBPM).

( ) cos( ) ( ) sin( )PM C C C p Ct A t A k m t t − Equation (4.16)

FM:

PM:

Using these results allows us to generate narrowband FM and PMwith the block diagrams on the next slide (slide #34):

Agbo & Sadiku; Section 4.2 & 4.3 on pages 158 to 168

34

Generation of Narrowband FM and PM

/2

NBFM

ACcos(ct)

m (t)

+

kf

-ACsin(ct)

+

/2

NBPM

ACcos(ct)

m (t)

+

kp

-ACsin(ct)

+

Agbo & Sadiku; Figure 4.5 on page 168

35

Modulation Index Parameter in Angle Modulation

Parameter is the modulation index for angle modulation.

is used to differentiate between narrowband angle modulation andwideband angle modulation.

Narrowband angle modulation requires << 1 (Typically < 0.3)Wideband angle modulation requires >> 1 (Typically > 5.0)

Equivalently,

Narrowband angle modulation requires f << BWideband angle modulation requires f >> B

Comments:1. Narrowband FM has about the same bandwidth as that of AM.2. Commercial (broadcast) FM is wideband FM (required due to

its superior noise performance).3. Why even consider narrowband FM? Two reasons:

a. NBFM is easier to generate that WBFM.b. It is used as in initial process step in generating WBFM.

36

Narrowband FM with Tone Modulation

Let ( ) cos( ), then ; 2 ; and

Then ( ) cos( ) sin( )

The time-domain NBFM signal is

( ) cos( ) sin( ) sin( )

1( ) cos( ) cos (

2

f p f m

m m p m m

m m

t tf m

f f m m m

m

FM C C C m C

FM C C C

k m k Am t A t m A B

k Ak m d k A d t

t A t A t t

t A t A

− −

= = = = =

= =

−

+

( ) ( )

nd rd

1) cos ( )

2

The 2 term is the upper sideband and the 3 term is the lower sideband.

In comparing to AM:

C m C C mt A t + − −

Equation (4.18)

Equation (4.19)

37

( ) ( )1 1

( ) cos( ) cos ( ) cos ( )2 2

FM C C C C m C C mt A t A t A t + + − −

(C + m)C

AC = 0.2

Narrowband FM (NBFM)

(C - m)

Tone modulation cos(mt)

NBPM requires << 1 radian(generally less than 0.3 radian)

2

CA

2

CA−

( )FM t

Sidebands arein quadrature.

38

Narrowband FM (NBFM)

Cm -m

C rotates faster than m

Phasor lengths adjust to keep constant AC.

39

Review: Phasor Interpretation of AM DSB with Carrier

C

us

ls

cos(Ct)

cos(mt)

DSB AM

C

C + mC - m

m = |us| = |ls|

C rotates faster than m

Spectrum:

uppersideband

lowersideband

40

Narrowband FM Example (Example 4.4)

Exercise: The message signal input to a modulator is m(t) = 4cos(2104t)and the carrier is 10cos(108t). If frequency modulation is performed with kf = 1000, verify that the modulated signal meets the criteria of being narrowband FM. Also, obtain an expression for its spectrum and sketch this spectrum.

Solution: First we calculate the modulation index

Using the equation from slide #37:

4

41000 0.2; 0.3 NBFM

2 10

1 110 thus, (0.2)(10) 1

2 2

mf

m

C C

Ak

A A

= = =

= = =

( ) ( )

( ) ( )

1 1( ) cos( ) cos ( ) cos ( )

2 2

( ) 10 cos( ) cos ( ) cos ( )

FM C C C C m C C m

FM C C m C m

t A t A t A t

t t t t

+ + − −

+ + − −

Agbo & Sadiku; page 170

41

Narrowband FM Example (Example 4.4 continued)

( ) ( )

8

( ) 10 cos( ) cos ( ) cos ( )

The corresponding expression for the specturm becomes

( ) 10 ( ) ( ) ( ) ( )

( ) ( )

where 10 radians/sec and 2 10

FM C C m C m

FM C C C m C m

C m C m

C m

t t t t

+ + − −

= + + − + + + + − −

− + − + − +

= = 4 radians/sec

C

10

-C

(C + m)(C - m)

-(-C + m)(-C - m)

( )FM

Bandwidth = 2m

42

Wideband FM (WBFM)

WBFM requires >> 1 radian (much more complicated)

( )

= +

For wideband FM we have a nonlinear process, with single-tone

modulation:

( ) Re exp sin( )

We need to in order to

analyze ( ). The solut

expand the exponential in a Fourier series

WBFM C C m

WBFM

t A j t j t

t

( )

=−

= +

ion has an expansion in Bessel functions:

( ) ( ) cos 2 ( )

where the coefficients ( ) are Bessel functions.

WBFM C C m

nn

n

t A J f nf t

J

Spectral analysis for tone modulation of WBFM: Agbo & Sadiku, pp. 171-180.We will not cover this section in EE442 but rather focus upon the physicalinterpretation of FM spectrum spread.

m

f f

f B

= =

Modulation Index

43

Digression: Bessel Functions (of the 1st kind)

Bessel functions have many applications: cylindrical waveguides, vibrationalmodes on circular membrane, FM modulation synthesis, acoustic vibrations, etc.

https://www.cefns.nau.edu/~schulz/Bessel/J02.html

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

WBFM (or WBPM) Requires More bandwidth Than AM

44

AM

WBFM

Carrier Signal (frequency fc )

Message Signal (frequency fm)

Amplitude Modulated Signal

Frequency Modulated (FM) Signal

f

f

f

f

fC

fm

A

A

A

A

AC

Am

A

t

t

t

t

A Amplitude Modulated Signal

Frequency Modulated (FM) Signal

Message Signal (frequency fm)

Carrier Signal (frequency fC)

45

= 0.2

= 1.0

= 5

= 10

Number of Sidebands¶

Bandwidth

0.1 2 2 fm

0.3 4 4 fm

0.5 4 4 fm

1.0 6 6 fm

2.0 8 8 fm

5.0 16 16 fm

10.0 28 28 fm

Single-Tone FM Spectra as Function of Modulation Index

= =

m m

f

f

Single-tone Modulation Index

TB or BW

NBFM

WBFM

fm constant

¶Both upper and lowersidebands about fC.

46

Spectra of FM Signals

From A. Bruce Carlson, Communication Systems, An Introduction to Signals and Noise inElectrical Communication, 2nd edition, 1975; Chapter 6, Figure 6.5, Page 229.

= 0.2

= 1.0

= 5

= 10

A A

Single-toneModulation Index

= =

m m

f

f

is constant &

is decreasingm

f

f

increasing &

is constantm

f

f

ff

WBFM

47

Broadcast FM Radio covers from 88 MHz to 108 MHz100 stations – 200 kHz spacing between FM stations

Selecting an FM Station

Service Type Frequency Band

Channel Bandwidth

MaximumDeviation

Highest Audio

Commercial FM Radio Broadcast

88.0 to 108.0 MHz

200 kHz 75 kHz 15 kHz

Note: 0 dBu = 0.775 volt into 600 ohms(which is equivalent to 1 mW powerdelivered into the 600 ohm resistor)

48

Measured Spectrum of an FM Radio Signal

noise

Voice modulation

200 kHz

49

Service Type Frequency Band

Channel Bandwidth

MaximumDeviation

Highest Audio

Commercial FM Radio Broadcast

88.0 to 108.0 MHz

200 kHz 75 kHz 15 kHz

Television Sound (analog)

Digital TV has replaced

4.5 MHz above the

picture carrier frequency

100 kHz 25 kHz monaural &50 kHz stereo

15 kHz

Public safety – Police, Fire, Ambulance, Taxi, Forestry, Utilities, & Transportation

50 MHz and 122 MHz to

174 MHz

20 kHz 5 kHz 3 kHz

Amateur, CE class A & Business band Radio

216 MHz to 470 MHz

15 kHz 3 kHz 3 kHz

Specifications for Some Commercial FM Transmissions

50

The 3 Important Parameters for FM and PM

The three important frequencies in FM and PM are

1. Carrier frequency fC (or C)2. Maximum modulation frequency fm (or m), and3. Peak frequency deviation f (or )

Two Definitions of importance:

1. Modulation index

2. Deviation ratio D

Remember: For FM = kf mp & for PM = kp m’p

(can be a very large number)2m m m m

f f

B f B

= = = =

(always much less than unity)C C

fD

f

= =

51

FM Bandwidth and the Modulation Index

A. Narrowband FM (NBFM) – << 1 radian

2 where is the bandwidth of ( )NBFM m mB B B m t

+ = = =

=

+ = +

+

2( 1) , where

is the peak frequency deviation max ( )

2( ) 2( 1)

For PM we have analogous equation,

2( 1

Carson's Ru

)

le

WBFM m

m m m

f

WBFM m m

WBPM m

f fB B

B f

f f k m t

B f B B

B B

B. Wideband FM (WBFM) – >> 1 radian

52

Example: Bandwidth of FM Signal

The message signal input to a modulator is 10cos(2 104t). If frequencymodulation with frequency deviation constant kf = 104 is performed, findthe bandwidth of the resulting FM signal.

Solution:

( )

( ) ( )

4

4

10 1015

2 2 10

2 1 2 5 1 10 kHz 120 kHz,

using to calcuC laarson's rule te bandwidth

f m

m

FM m

FM

k A

f

B f

B

= = =

= + = + =

53

Example: Equal Bandwidth for FM & PM Signals

If phase modulation is performed using the message signal 10cos(2 104t)used in the previous slide, find the phase deviation constant kp giving thePM signal the same bandwidth, namely, 120 kHz.

Solution:

For both the FM and PM signals to have the same bandwidth, and f mustbe the same. For FM, = kf Am ; but for PM, = kp m’p .Expressing the message signal m(t) = Am cos(mt) gives

( )

4

4

'( ) cos( ) sin( ) '

,

10 1

2 10 2

' 1Check: = (10) 5

2

m m m m m p m m

f

f m p m m p

m

p p p m m

p m

m m

dm t A t A t m A

dt

Thus

kk A k A k

k m k Ak A

= = − = −

= − → = = =

= = = =

54

Example: Commercial FM Radio Stations

For commercial FM radio, the audio message signal has a spectral range of30 Hz to 15 kHz, and the FCC allows a frequency deviation of 75 kHz. Estimate the transmission bandwidth for commercial FM using Carson’sRule.

Solution:

( ) ( )

We start by calculating

75 kHz 5

15 kHz

Using Carson's rule gives

2 1 2 5 1 15 kHz 180 kHz

The allowed bandwidth for commercial FM is 200 kHz.

Note that Carson's rul

m

FM m

f

B

B B

= = =

= + = + =

e slightly underestimates the

bandwidth.

55

Why Does FM and PM Take Much More Bandwidth?

Observation: The bandwidth required for AM and NBFM are the same.

However, WBFM (wideband FM) requires much more bandwidth. Why?

A Fourier spectrum of an FM signal shows that to keep the amplitude constant of an FM signal that many components are required to represent the FM waveform. The frequency spectrum of an actual FM signal has components extending infinitely, although their amplitude decreases forsufficiently higher frequencies. Sufficiently higher frequencies appliesto frequencies above the Carson bandwidth rule.

Next we examine the Fourier components this using phasors.

+ = + Carson's Rule2( ) 2( 1)WBFM m mB f B B

Review

Note: Only magnitudes of spectral lines shown.:

FM Tone-Modulated Signal Spectrum

56

For = 2.0

Remember:

57

NBFM Constructed From Phasors in FM Modulation

fCAC

2

CA

2

CA

-fmfm

(t)

AC

2

CA

fC fC + fm

fC - fm

2

CA−

0

NBFM with tone modulation

58

WBFM Phasor Diagram for Arbitrary

AC

carrier

even-order sideband

odd-order sideband

After A. Bruce Carlson and Paul B. Crilly, CommunicationSystems, 5th ed., McGraw-Hill Book Company, New York, NY.

0

59

Sidebands Constructed From Phasors in FM Modulation

Animation showing how phase modulation works in the phasor picture -- phase modulation with a sinusoidal modulation waveform and a modulation depth of π/4 radians. The blue line segments represent the phasors at the carrier and the harmonics of the modulation frequency.

60

Generating FM Signals

There are two basic methods to generate FM:

1. Direct Method (uses voltage-controlled oscillator to vary thethe frequency linearly with the message signal m(t))

Advantage: Can generate large frequency deviation.Disadvantage: Carrier frequency tends to drift and

must be stabilized.

2. Armstrong’s Indirect Method (first generate NBFM with the message signal with a small frequency deviation and thenfrequency multiplication is used to increase the frequencyand frequency deviation to desired levels (generates WBFM)

Advantage: More stable carrier frequency.Disadvantage: More complex hardware and cost.

61

Direct Generation of FM Signal Using a VCO

Ceq is capacitance CD plus capacitance of other capacitors.

VCO is “voltage-controlled oscillator”

1

~osc

eqLC

Q

CD

RFC

m(t)

VCO

Varactor diode

( )FM t

+VCC

LC Resonator

62

Direct Generation of FM Signal Using a VCO and PLL

WBFM Output Signal

FrequencyDivider N

Input: m(t)

VCO

LPF

Crystal Oscillatorfosc = fC / N

Mixer used as phase detector

12

1 12 2

cos( ) cos( ) cos( ) cos( )

cos( ) cos(2 ) cos( ) cos(2 )

C C C C C C

C C

t t t t t t

t t

+ = − − + + +

= − + + = + +

Mixer used as phase detector:

X

63

Varactor diode

Narrowband FM Generated by Pulling a Crystal Oscillator

Q1

CD

m(t)

( )FM t

+VCC

Xtal

R4R2

R3

A crystal filter is placed in the feedback loop to stabilize the oscillator.The frequency of oscillation can be pulled slightly from the high-Q crystal resonator’s frequency. The frequency deviates only slightly andis typically only up to about 100 ppm. However, the oscillator is very stable for m(t) = 0.

A crystal is anelectro-mechanical

resonator.

64

Digression: Q-Values for Quartz Crystals in Electronics

A crystal oscillator is an electronic oscillator circuit that uses the mechanical

resonance of a vibrating crystal of piezoelectric material to create an

electrical signal with a precise frequency.

A major reason for the wide use of crystal oscillators is their high Q factor. A

typical Q value for a quartz oscillator ranges from 104 to 106, compared to

perhaps 102 for an LC oscillator.

The maximum Q for a high stability quartz oscillator can be estimated

as Q = 1.6 × 107/f, where f is the resonant frequency in megahertz.

resfQ

f=

Xtal

https://en.wikipedia.org/wiki/Crystal_oscillator

65

Generation of Narrowband Frequency Modulation (NBFM)

−

= +

( ) cos ( )t

FM C C ft A t k m d

NBFM requires << 1 radian

Agbo & SadikuFigure 4.5; page 168

/2

NBFM

ACcos(ct)

m (t)

kf

-ACsin(ct)

DSB-SCmodulator

Carrier

+

+

Oscillator

66

Indirect Generation of FM Using Frequency Multiplication

( )FMNB t ( )FM

WB t

NBFMFrequencyMultiplier

( )m t

In this method, a narrowband frequency-modulatedsignal is first generated and then a frequency multiplier is used to increase the modulation index.The concept is shown below:

A frequency multiplier is used to increase both thecarrier frequency and the modulation index by integer N.

67

Frequency Multipliers

A frequency multiplier is a nonlinear component followed by a bandpassfilter at the multiplied frequency desired.

0

0

( ) cos ( ) , and

( ) cos ( )

t

in C C f

t

out C C f

t A t k m d

t A n t nk m d

= +

= +

We select the nth order nonlinear component of y(t) and pass it throughthe bandpass filter.

Nonlinear Device

Bandpass Filter @ nC

( )out t( )in t ( )y t

Conclusion: Carrier frequency is now nfC and frequency deviation is now nf.Commercial frequency multipliers are generally 2 and 3.

Note: m(t) isnot distortedby multiplier.

Section 4.4; Page 181 ofAgbo & Sadiku

68

Armstrong Indirect FM Transmitter Example

These numbers correspondto an FM broadcast radio station.

( )FMNB t

( )FMWB t

( )m tNBFM

generation

64Multiplier

BPF

Crystal Oscillator

1

1

200 kHz

25HzCf

f

=

=2

2

12.8 MHz

1.6 kHzCf

f

=

=

3

3

1.9 MHz

1.6 kHzCf

f

=

=

48Multiplier

4

4

91.2MHz

76.8 kHzCf

f

=

=

PA

A mixerdoes not

change f

Crystal stabilizedvoltage-controlled

oscillator

69

Why are Two Multiplication Chains Used?

NBFM

generatorMultiplierChain A

MultiplierChain B

Mixer ( )FMWB t

( )FMNB t

Oscillator

70

Many Ways to Perform Frequency Multiplication

In electronics, a frequency multiplier is an electronic circuit that

generates an output signal whose output frequency is a harmonic

(multiple) of its input frequency. Frequency multipliers consist of a

nonlinear circuit that distorts the input signal and consequently

generates harmonics of the input signal.

Most multipliers are doublers or triplers

XOR

71

Frequency Multiplication Using Comb Generation

From our discussion on Fourier series and pulse trains:

Tp

T

Amplitude

Comb frequencies shown

1

pT

2

pT1

T

72

Simple Comb Generator

A step recovery diode (SRD) is a p-n junction diode having the ability to generate

extremely short pulses. It is also called snap-off diode or charge-storage diode,

and has a variety of uses in microwave electronics (e.g., pulse generator or

parametric amplifier).

Comb Generator

Circuit

https://www.edn.com/electronics-blogs/the-emc-blog/4402169/DIY-6-GHz-comb-generator

73

Step Recovery Diode Based Comb Generation

http://www.mwrf.com/analog-semiconductors/designing-step-recovery-diode-based-comb-generator

The key to generating a wide comb of frequencies is togenerate very narrow pulses which step recovery diodesare designed to do.

Time (nanoseconds)

Vo

lts

74

Generation of Narrowband Phase Modulation (NBPM)

= +( ) cos( ( ))PM C C pt A t k m t

/2

NBPM

ACcos(ct)

m (t)kp

-ACsin(ct)

+

+

Agbo & SadikuFigure 4.5; page 168

75

Generation of Narrow Band Phase Modulation

CD

m(t)

+VD

Varactor diode

kp

Carrierfrequency fC

( )PM t

https://www.slideshare.net/sghunio/chapter06-fm-circuits

Limitation 1: Only a small amount of phase shift is generated (low-deviation)Limitation 2: All phase-shift circuits produce amplitude variations.

Advantages of frequency modulation

1. Resilient to noise: The main advantage of frequency modulation is a reduction in noise. As most noise is amplitude based, this can be removed by running the received signal through a limiter so that only frequency variations remain.

2. Resilient to signal strength variations: In the same way that amplitude noise can be removed, so too can signal variations due to channel degradation because it does not suffer from amplitude variations as the signal level varies. This makes FM ideal for use in mobile applications where signal levels constantly vary.

3. Does not require linear amplifiers in the transmitter: As only frequency changes contain the information carried, amplifiers in the transmitter need not be linear.

4. Enables greater efficiency : The use of non-linear amplifiers (e.g., class C and class D/E amplifiers) means that transmitter efficiency levels can be higher. This results from linear amplifiers being inherently inefficient.

76

Advantages of FM

Disadvantages of frequency modulation

1. Requires a more complicated demodulator: One of the disadvantages is that the demodulator is a more complicated, and hence more expensive than the very simple diode detectors used in AM.

2. Sidebands extend to infinity: The sidebands for an FM transmission theoretically extend out to infinity. To limit the bandwidth of the transmission, filters are used, and these introduce some distortion of the signal.

77

Disadvantages of FM

78

Ideal FM Differentiator Demodulator

The ideal FM detector converts the FM signal‘s instantaneous frequency i

to an amplitude that is proportional to i.

( )

( )

−

−

−

= + = +

= +

= − + +

Input: ( ) cos ( ) cos ( )

( )Output: cos ( )

( )( ) sin ( )

t

FM C C C C f

t

FMC C f

t

FMC C f C f

t A t t A t k m d

d t dA t k m d

dt dt

d tA k m t t k m d

dt

Differentiation performs FM to AM conversion

EnvelopeDetector

d

dt

Limiter Differentiator

( )FM t

( )FMd t

dt ( )( )C C C fA A k m t +

After DC removal

Both AM and FM included

AM allows theenvelope detector

to be used

79

Bandpass Limiter at the Receiver

For an envelope detector to work well the FM signal’s amplitudeshould be constant or flat. We can accomplish with a “hard limiter.”Factors such as channel noise, interference and channel fading resultin amplitude variations in an FM signal’s amplitude at the receiver.

Band-passFilter@ C

Limiter

( )FM t

t →

Output ofLimiter

( )FM t

Input

t→

VL

ConstantAmplitude Output

( )4

( ) ( ) cos ( )t

FM C fx t t t k t m d −

= = +

80

Practical FM Differentiator Demodulator

( )( )

= = =+ +

=

3

3

3

3

Multiplica

s

tion by in

/ 1( ) ; where

1 1 /

1F

s l

or << ; t

hen ( )

The high-pass filter

the frequency domain i equiva ent

to differentiation in t

a

ime domai acts n!

dB

dB

dB

dB

jj RCH j

j RC j RC

H j j C

j

RRC

+

a differentiator for an FM signal. Therefore,

y(t) = ( )C C C C fA RC A RCk m t

RC+

_x(t) Envelope

Detector

+

_y(t)Differentiator

at low frequencies

Envelope detector extracts m(t)

81

Bode Plot of CR High-Pass Filter

Log

amp

litu

de

82

Practical Frequency Demodulators

Frequency discriminators can be built in various ways:

• Time-delay demodulator

• FM slope detector

• Balanced discriminator

• Quadrature demodulators

• Phase locked loops (a superior technique)

• Zero crossing detector

83

Time Delay Demodulator

This is an implementation of discrete time approximation to differentiation.

( )

( )0 0

1( ) ( ) ( )

( ) 1lim ( ) lim ( ) ( )

FM FM

FMFM FM

y t t t

d ty t t t

dt

→ →

= − −

= = − −

It can be shown that an adequate value for is less than T/4, where T is theperiod of the unmodulated carrier for the FM signal. Again, this relies uponFM to AM conversion after which the envelope detector recovers m(t).

Amplifier(Gain = 1/)

EnvelopeDetector

yd(t)( )FM t

Time Delay Time-Delay Differentiator

+ y(t)

84

An FM Slope Detector Performs FM to AM Conversion

( )FM t

Slope sets frequency toamplitude conversion scale

EnvelopeDetector Comment: The

differentiationoperation is performed by anycircuit acting as a frequency-to-amplitude converter.

Slopeapproximation

x(t) y(t)

Operates on theskirt of the LCresonance curve

85

Balanced Discriminator (Foster-Seeley Discriminator) – 1936

( )FM t

Centered around fc

•••

Transfer Characteristic

f

Two tuned circuits

Envelopedetectors

Another example of the use of symmetry in design.

km(t)

0U Cf f

0L Cf f

0Uf

0Lf

86

Quadrature Demodulator – Block Diagram

FM signal is converted into PM signal

PM signal is used to recover the message signal m(t)

Phase Shifting Circuit

Low-Pass Filter

Phase Comparator

Circuit

FM ( )t

( )m t

Phase DetectorSignal delay 0 timescarrier frequency fC

= 90 degrees (or /2).

87

Using a XOR Gate for Phase Frequency Detector

XOR

A B Output

0 0 0

0 1 1

1 0 1

1 1 0

Exclusive OR gate

A

B

Logic Table

88

Quadrature Demodulator – Implementation

The signal is split into two components. One passes through a network

providing a basic 90° phase shift in addition to the phase shift from the

signal’s frequency deviation. The mixer output is dependent upon the phase difference between the two signals; that is, it acts as a phase detector producing a voltage output proportional to the phase difference and thus the frequency deviation on the FM signal.

http://www.radio-electronics.com/info/rf-technology-design/fm-reception/fm-quadrature-detector-demodulator.php

C1

C2

R L

Low-passFilter

MixerIncomingFM signal

Demodaudio

89

Phase-Locked Loops (Using Feedback)

A PLL consists of three basic components:❑ Phase detector❑ Loop filter❑ Voltage-controlled oscillator (VCO)

PLL Diagram:

+cos ( )C C iA t t

+2 cos ( )VCO C oA t t( )oe t

( )H s

Phase Detector

Output signalis phase

Low-PassFilter

Oscillator(VCO)

Bias Generator

90

Zero-Crossing Detectors

91

Zero-Crossing Detector Illustration

https://www.slideshare.net/avocado1111/angle-modulation-35636989

Averagingcircuit

m(t)( )FM tMulti-

vibrator

Zero-crossingcircuit

Hard Limiter

More frequentZC’s gives higherinstantaneousfrequency whichcauses greateraverage signal.

92

Noise in Frequency Modulation

In FM systems noise has a greater effect on the higher modulating frequencies. It is common practice to boost the signal level of the higher modulating frequencies to improve the signal-to-noise ratio of the overall transmitted FM signal.

This artificial boosting at the transmitter is called “pre-emphasis” and the removal of the boost at the receiver is called “de-emphasis.”

The result is an improvement in the discernible quality of received FM signals.

Power Spectral Density (PSD) of output noise in an FM receiver.(Increases because noise is differentiated in FM receiver)

( )2

22

2( ) for

o

TO

N

C

Bf

N fS f

A

=

fB-B

0

( )oNS f

-f - 2B f +2B

Fig. 10.9; p. 578 of 4th ed., Lathi & Ding

93

Pre-Emphasis and De-Emphasis in FMIn

terf

eren

ce

Frequency f

PM

FM

FM with Pre- and De-emphasis filters

Channel noise acts as interference inFM and is uniform over the entire BW.Voice and music have more energy atlower frequencies, so we need to “emphasize “their upper frequencies by filtering. However, the HF emphasis must be removed at the receiver usinga “de-emphasis” filter.

Pre-emphasis Filter

De-emphasis Filter

FM Transmitter

FM Receiver

AWG Noise

Channel

m(t)

R1

R2C

R1 C

Filtering improves SNR in FM transmission.

(Widely used commercially in the recording industry)

94

Typical Pre-Emphasis and De-Emphasis Filters

Transmitter Receiver

Pre-emphasis Filter De-emphasis Filter

1

1( )

1out

in

VH

V j R C

= =

+( )1

1 2

1( )

1out

in

j R CVH

V j R R C

+= =

+

1

1

R C

( ) ( )H dB

log( )

-6 dB/octave

1

1

R C ( )1 2

1

R R C

( ) ( )H dB

log( )

+6 dB/octave

2.1 kHz 2.1 kHz33 kHz

R1

R2C

R1 C

95

Analog and Digital FM Cellular Telephones

1G analog cellular telephone (1983) – AMPS (Advanced Mobile Phone Service)First use of cellular concept . . . Used 30 kHz channel spacing (but voice BW was B = 3 kHz)

Peak frequency deviation f = 12 kHz, and BT = 2(f + B) = 2(12 kHz + 3 kHz) = 30 kHz

Two channels (30 kHz each); one for uplink and one for downlinkUsed FM for voice and FSK (next slide) for data communicationNo protection from eavesdroppers!

Successor to AMPS was GSM (Global System for Mobile) in early 1990sGSM is 2G cellular telephone Still used by nearly 50% of world’s population (as of 2017)GSM was a digital communication system

Modulating signal is a bit stream representing voice signalUses Gaussian Minimum Shift Keying (GMSK)Channel bandwidth is 200 kHz (simultaneously shared by 32 users

This is 4.8 times improvement over AMPS

More to come on cellular . . .

Digital Carrier Modulation – ASK, FSK and PSK

96

AmplitudeShift Keying

FrequencyShift Keying

PhaseShift Keying

Dig

ital

Sig

nal

s

https://slideplayer.com/slide/12711804/

( ) ~ ( , )p dk m t −

m(t)

97

Digital Phase Shift Modulation

Binary Phase Shift Keying (BPSK)

The wave shape is ‘symmetrical’ at each phase transition because the bit rate is a sub-multiple of the carrier frequency ωC/(2π). In addition, the message transitions are timed to occur at the zero-crossings of the carrier.

( ) ~ ( , )p dk m t −

98

Questions?

https://www.tutorialspoint.com/principles_of_communication/principles_of_communication_modulation.htm

99

Additional slides

100-1 -1

Switch

Integrator

m(t)

Schmitt Trigger

Inverter

FM(t)

time

Triangular-Wave FM Generation

vin

vout

vinvout

Able to generateFM up to 30 MHz

After A. Bruce Carlson and Paul B. Crilly, CommunicationSystems, 5th ed., McGraw-Hill Book Company, New York, NY.

Schmitt Trigger characteristic

A

101

Switching-Circuit Phase Modulator

m(t) Sawtooth wave

Comparator Output

Flip-Flop Output

BPF+ Flip-

Flop

Comparatorm(t)

Sawtooth Generator

Carrier Oscillator

PM(t)

time

And filtering removes harmonics. After A. Bruce Carlson and Paul B. Crilly, Communication

Systems, 5th ed., McGraw-Hill Book Company, New York, NY.

102

FM System Improvement in SNR

The signal-to-noise ratio (SNR) improvement in an FM system is afunction of modulation index ,

Example: For FM transmission bandwidth BT of 200 kHz and a messagebandwidth Bm of 15 kHz ( = 5.67), the improvement in the SNR at theoutput of an FM receiver to have an FM gain of 27 dB above the CNR.

This is essentially a tradeoff between message signal quality (SNR) andFM transmission bandwidth. Thus, greater transmission bandwidth is the key to FM’s superior performance.

33 ( 1) , where is carrier-to-noise ratio

32

FM

TFM

m

SNR CNR CNR

BSNR CNR

B

= +

=