Frequency Modulation

ANGLE MODULATION: The intelligence of the modulating signal can be conveyed by varying the frequency or phase of the carrier signal. When this is the case, we have angle modulation, which can be subdivided into two categories:

frequency modulation (FM), and phase modulation (PM).

1

Frequency Modulation

modulating signal.

. The carrier's instantaneous frequency deviation from its unmodulated value varies in proportion to the instantaneous amplitude of the

e FM



A c

sin( 

c t



m f

sin 

m t

)

Phase Modulation

modulating signal; . The carrier's instantaneous phase deviation from its unmodulated value varies as a function of the instantaneous amplitude of the

e PM



A c

sin( 

c t

 

m

sin 

m t

) 2

FIGURE 4-1

The FM and PM waveforms for sine-wave modulation: (a) carrier wave; (b) modulation wave; (c) FM wave; (d) PM wave. (

Note:

The derivative of the modulating sine wave is the cosine wave shown by the dotted lines. The PM wave appears to be frequency modulated by the cosine wave.)

Warren Hioki

Telecommunications,

Fourth Edition 3 Copyright ©2001 by Prentice-Hall, Inc.

Upper Saddle River, New Jersey 07458 All rights reserved.

MODULATION INDEX

• modulation index for an FM signal

m f

 

f m

δ = maximum frequency deviation of the carrier caused by the amplitude of the modulating signal f m = frequency of the modulating signal 4

FREQUENCY ANALYSIS OF THE FM WAVE

e FM



A c J

0 sin 

c t



A c

{

J

1 (

m f

)[sin( 

c

 

m

)

t

 sin( 

c

 

m

)

t

]} 

A c

{

J

2 (

m f

)[sin( 

c

 2 

m

)

t

 sin( 

c

 2 

m

)

t

]} 

A c

{

J

3 (

m f

)[sin( 

c

 3 

m

)

t

 sin( 

c

 3 

m

)

t

]}  ..., etc • Where:e Fm FM wave = the instantaneous amplitude of the modulated Ac = the peak amplitude of the carrier J n = solution to the nth order Bessel function for a modulation index m f .

m f = FM modulation index, Δf/f m 5

FIGURE 4-3

Spectral components of a carrier of frequency, f

c ,

frequency modulated by a sine wave with frequency f

m .

(

Source:

Martin,

Telecommunications and the Computer,

James 2nd ed. [Englewood Cliffs, N.J.: Prentice-Hall, 1976], p. 218. Reprinted with permission from the publisher.)

Warren Hioki

Telecommunications,

Fourth Edition 6 Copyright ©2001 by Prentice-Hall, Inc.

Upper Saddle River, New Jersey 07458 All rights reserved.

Warren Hioki

Telecommunications,

Fourth Edition 7 Copyright ©2001 by Prentice-Hall, Inc.

Upper Saddle River, New Jersey 07458 All rights reserved.

FM signal characters

• The FM wave is comprised of an infinite number of sideband components • As the modulation index increases from m f = 0, the spectral energy shifts from the carrier frequency to an increasing number of significant sidebands.

• J n (m f ) coefficients, decrease in value with increasing order, n.

• negative Jn(mf) coefficients imply a 180' phase inversion.

• The carrier component, J o , and various sidebands, J n , go to zero amplitude at specific values of modulation index, m f .

8

Carrier Frequency Eigenvalues

• in some cases the carrier frequency component, J O , and the various sidebands, J are called

eigenvalues.

n go to zero amplitudes at specific values of m. These values 9

Bandwidth Requirements for FM

• The higher the modulation index, the greater the required system bandwidth

BW

 2 (

n



f m

) where n is the highest number of significant sideband components and f m is the highest modulation frequency .

Carson's Rule

BW

 2 (  

f m

) 10

FIGURE 4-5

Amplitude versus frequency spectrum for various modulation indices (f

m

= 2; (d)

m f

= 5; (e)

m f

= 10.

fixed, & varying): (a)

m f

= 0.25; (b)

m f

= 1; (c)

m f

Warren Hioki

Telecommunications,

Fourth Edition 11 Copyright ©2001 by Prentice-Hall, Inc.

Upper Saddle River, New Jersey 07458 All rights reserved.

Warren Hioki

Telecommunications,

Fourth Edition 12 Copyright ©2001 by Prentice-Hall, Inc.

Upper Saddle River, New Jersey 07458 All rights reserved.

Broadcast FM

• extends from 88 to 108 MHz • is divided into 100 channels • Channels range from 88.1 MHz, where N = 201, to 107.9 MHz, where N = 300 • N=5(f-47.9) where N = the FM broadcast channel number f = the frequency in MHz Guard Band: A range of frequences separating transmitted channels in which no signals should be transmitted.

13

FIGURE 4-6

Commercial FM broadcast band.

Warren Hioki

Telecommunications,

Fourth Edition 14 Copyright ©2001 by Prentice-Hall, Inc.

Upper Saddle River, New Jersey 07458 All rights reserved.

Commercial FM broadcast band

• The maximum permissible carrier deviation, δ, is ±75 kHz • Modulating frequencies is ranging from 50 Hz to 15 kHz • The modulation index can range from as low as 5 for f m = 15 kHz (75 kHz/15 kHz) to as high as 1500 for f m = 50 Hz (75 kHz/50 Hz).

• The ±75-kHz carrier deviation results in an FM bandwidth requirement of 150 kHz for the receiver.

• A 25-kHz

guard band

above and below the upper and lower FM sidebands.

• Total bandwidth of one channel is 200Hz.

15

Narrowband FM

• NBFM uses low modulation index values, with a much smaller range of modulation index across all values of the modulating signal.

• An NBFM system restricts the modulating signal to the minimum acceptable value, which is 300 Hz to 3 KHz for intelligible voice.

• 10 to 15 kHz of spectrum.

• Used in police, fire, and Taxi radios, GSM, amateur radio, etc.

16

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POWER IN THE FM WAVE

• power of the unmodulated carrier

P T

2 

V crms R

• For a modulated carrier

P T



P J

0 

P J

1 

P J

2 

P J

3  ...



P J n



V J

2 0

R

 2

V J

2 1

R

 2

V J

2 2

R

 2

V J

2 3

R

 ...

 2

V J

2

n R

19

FM NOISE

• increased bandwidth of an FM -- enhance the signal-to noise ratio (SNR). Advantages of FM over AM.

• To take this advvantage, large m f is necessary– high order sidebands are important – wider bandwidth is required.

• Phasor Analysis of FM Noise   sin  1

V N V c

where α = the maximum

phase deviation

caused by the noise of the carrier frequency V N = noise voltage V c = carrier voltage 20

FIGURE 4-7

Phasor addition of noise on an FM signal’s carrier frequency causes a phase shift, whose maximum value is  .

Warren Hioki

Telecommunications,

Fourth Edition 21 Copyright ©2001 by Prentice-Hall, Inc.

Upper Saddle River, New Jersey 07458 All rights reserved.

The ratio of carrier voltage to noise voltage, (voltage)

V c V n

is the SNR

SNR



V V N c

  sin  1 1

SNR

α represents the equivalent modulation index produced by the noise.

 

N

 

f m SNR FM

  

N

22

• The effect of noise on an FM carrier signal is directly proportional to the modulation frequency f • Increasing f m , degrades the

SNR

  

N

   

f m

m .

Voice, data, and music contain many frequencies, which are distributed throughout the given modulation passband. Therefore, the SNR is

not

uniform throughout. To maintain a flat SNR, following techniques are employed .

23

Pre-emphasis and De-emphasis

• Pre-emphasis: boost the signal levels of the higher modulating frequencies before the modulation process to maintain a uniform SNR.—high-pass filter • De-emphasis: FM receiver brings back pre emphasized signals to their original amplitudes Low-pass filter.

• Direct FM: the modulation signal is used to directly change the carrier signal's frequency or phase • Indirect FM: the modulation signal is used to change the phase of the carrier signal, which indirectly changes its frequency. 24