EET260Frequency Modulation

Modulation A sine wave carrier can be modulated by varying

its amplitude, frequency, or phase shift.

In AM, the amplitude of the carrier is modulated by a low-frequency information signal.

sin 2c cv V f t

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Amplitude modulated signal

Frequency modulation In frequency modulation (FM) the instantaneous

frequency of the carrier is caused to deviate by an amount proportional to the modulating signal amplitude.

sin 2c cv V f t

instantaneous frequency changedin accordance with modulating signal

frequency modulated signal

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Phase modulation In phase modulation (PM) the phase of carrier is

caused to deviate by an amount proportional to the modulating signal amplitude.

Both FM and PM are collectively referred to as angle modulation.

sin 2c cv V f t

carrier phase is changed in accordance with modulating signal

phase modulated signal

Frequency modulation Consider the equation below for a frequency

modulated carrier.

We will begin with a simple binary input signal.

FM sin 2 ( )c c d mv V f f v t t

defines the instantaneous frequency

center frequency

frequency deviation

modulating signal

Frequency modulation We will consider a 1-V, 1-kHz square wave as

an input.

Our input signal has 3 levels, and fd = 4-kHz

( ) ( )

no signal (0 V) (0) 10-kHz

"high" signal (+1 V) ( 1) 10 4 14-kHz

"low" signal (-1 V) ( 1) 10 4 6-kHz

i

c d c

c d

c d

v f

f f f f

f f f

f f f

input signal instantaneous frequency of FM signal

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FM sin 2 ( )c c d mv V f f v t t

Frequency modulation

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input signal (1-V, 1-kHz square wave)

FM signal (fc = 10-kHz, fd = 4-kHz)

f = 10-kHzf = 14-kHz

f = 6-kHzf = 14-kHz

f = 6-kHzf = 14-kHz

FM sin 2 ( )c c d mv V f f v t t

Fundamental FM concepts The amount frequency deviation is directly

proportional to the amplitude of the modulating signal.

The frequency deviation rate is determined by the frequency of the modulating signal. The deviation rate is the number of times per second

that carrier deviates above and below its center frequency.

FM sin 2 ( )c c d mv V f f v t t

center frequency

frequency deviation

modulating signal

Fundamental FM concepts

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input signal (1-V, 1-kHz square wave) input signal (2-V, 1-kHz square wave)

Doubling the amplitude of the input doubles the frequency deviation of the carrier.

Fundamental FM concepts

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input signal (1-V, 1-kHz square wave) input signal (1-V, 2-kHz square wave)

Doubling the frequency of the input doubles the frequency deviation rate of the carrier.

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A transmitter operates on a carrier frequency of 915-MHz. A 1-V square wave modulating signal produces 12.5-kHz deviation the carrier. The frequency of the input signal is 2-kHz.

a. Make a rough sketch of the FM signal.

b. If the modulating signal amplitude is doubled, what is the resulting carrier frequency deviation?

c. What is the frequency deviation rate of the carrier?

Example Problem 1

FM with sinusoidal input Consider a sinusoidal modulating input.

input signal (1-V, 500-Hz sine wave)

FM cos 2 sin(2 )

16-kHz

8-kHz

500-Hz

dc c m

m

c

d

i

fv V f t f t

f

f

f

f

500-Hz modulating signal

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Frequency content of an FM signal What does an FM signal look like in the

frequency domain?

We will consider the case of a sinusoidal modulating signal.

FM cos 2 sin(2 )dc c m

m

fv V f t f t

f

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Frequency content of an AM signal

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Amplitude modulated signal vAM

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Carrier signal vc (carrier frequency fc = 5-kHz)

Modulatoror mixer

Information signal vm ( fim= 500-Hz )

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Frequency domain

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Frequency domain

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Frequency domain

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Frequency modulation index The modulation index for FM is defined

Just as in AM it is used to describe the depth of modulation achieved.

From the previous example

FM modulation index

where frequency deviation

modulating signal frequency

df

m

d

m

fm

f

f

f

FM cos 2 16,000 sin(2 500 )

8000thus 16

500

c f

df

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v V t m t

fm

f

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Frequency analysis of FM In order to determine the frequency content

of

we can use the Fourier series expansion given

where Jn(mf) is the Bessel function of the first kind of order n and argument mf.

( ) ( )cos( )

where 2 carrier frequency

2 modulating signal frequency

FM n f c mn

c c

m m

v t J m n t

f

f

FM cos 2 sin(2 )c c f mv V f t m f t

Expanding the series,

we see that a single-frequency modulating signal produces an infinite number of sets of side frequencies.

Frequency analysis of FM

Each sideband pair includes an upper and lower side frequency

The magnitudes of the side frequencies are given by coefficients Jn(m).

Although there are an infinite number of side frequencies, not all are significant.

Frequency analysis of FM

For the case for mf = 2.0, refer to the table in Figure 5-2 to determine significant sidebands.

FM spectrum for mf = 2.0

Bessel functions Jn(mf)

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1J0

J1J2 J3 J4 J5 J6 J7 J8 J9 J10

if mf = 2.0, then the side frequencies we need to consider are J0, J1, J2, J3, J4

For the case for mf = 2.0, the series can be rewritten

Substituting the values for J0(2), J1(2),…, J4(2)

FM spectrum for mf = 2.0

From the equation, the spectrum can be plotted.

What is the bandwidth of this signal?

FM spectrum for mf = 2.0

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Am

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0.030.13

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fc

fc + fm

fc + 2fm fc + 4fm

fc + 3fmfc - 3fm fc - fm

fc - 2fmfc - 4fm

The bandwidth of the previous signal is

More generally, the bandwidth is given

where N is the number of significant sidebands.

FM spectrum for mf = 2.0

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( 4 ) ( 4 )

8c m c m

m

BW f f f f

f

fc + 4fmfc - 4fm

2 mBW f N

A signal vm(t) = sin (21000t) is frequency modulates a carrier vc(t) = sin (2500,000t). The frequency deviation of the carrier is fd = 1000 Hz.

a. Determine the modulation index.

b. The number of sets of significant side frequencies.

c. Draw the frequency spectrum of the FM signal.

Example Problem 2

Example Problem 2

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FM Spectrum with mf=1.0

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FM bandwidth increases with modulation index.

FM bandwidth as function of mf

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FM spectrum with mf=1.00

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FM spectrum with mf=0.25

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FM spectrum with mf=2.00

0.030.13

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0.130.03

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FM spectrum with mf=4.00

0.020.050.13

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0.430.36

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0.360.43

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0.130.050.02

Note the case mf = 0.25

In this special case, FM produces only a single pair of significant sidebands, occupying no more bandwidth than an AM signal.

This is called narrowband FM.

FM bandwidth

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FM spectrum with mf=0.25

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FM systems with mf < /2 are defined as narrowband. This is true despite the fact that only values of mf in

the range of 0.2 to 0.25 have a single pair of sidebands.

The purpose of NBFM is conserve spectrum and they are widely used in mobile radios.

Narrowband FM

An approximation for FM bandwidth is given by Carson’s rule:

The bandwidth given by Carson’s rule includes ~98% of the total power.

(max) (max)BW 2 d mf f

Carson’s Rule

What is the maximum bandwidth of an FM signal with a deviation of 30 kHz and maximum modulating signal of 5 kHz as determined the following two ways:

a. Using the table of Bessel functions.

b. Using Carson’s rule.

Example Problem 3

Example Problem 3

950 960 970 980 990 1000 1010 1020 1030 1040 10500

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FM spectrum with mf=6.00

0.02 0.060.13

0.25

0.36 0.36

0.11

0.24 0.28

0.15

0.28 0.24

0.11

0.36 0.36

0.25

0.130.06 0.02

BW = 90 kHz (Bessel functions)

BW = 70 kHz (Carson’s rule)