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6.003: Signals and Systems

Modulation

May 6, 2010

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Communications Systems

Signals are not always well matched to the media through which we

wish to transmit them.

signal applications

audio telephone, radio, phonograph, CD, cell phone, MP3 video television, cinema, HDTV, DVD

internet coax, twisted pair, cable TV, DSL, optical ber, E/M

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Amplitude Modulation

Amplitude modulation can be used to match audio frequencies toradio frequencies. It allows parallel transmission of multiple channels.

x1 (t) z1 (t)

cos 1 t

z2 (t) z(t)

z3 (t)

cos 2 t cos c t

cos 3 t

LPF x2 (t) y(t)

x3 (t)

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Superheterodyne Receiver

Edwin Howard Armstrong invented the superheterodyne receiver,

which made broadcast AM practical.

Edwin Howard

Armstrong

also

invented

and

patented the regenerative (positive feedback)

circuit for amplifying radio signals (while he was

a junior at Columbia University). He also invented wide-band FM.

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Amplitude, Phase, and Frequency Modulation

There are many ways to embed a message in a carrier. Here are

three.

Amplitude Modulation (AM): y1(t ) = x (t ) cos(c t )

Phase Modulation (PM): y2(t ) = cos(c t + kx (t ))

tFrequency Modulation (FM): y3(t ) = cos c t + k x ( )d

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Frequency Modulation

In FM, the signal modulates the instantaneous carrier frequency.

t y3(t ) = cos c t + k x ( )d

(t )

i (t ) = c + d

(t ) = c + kx (t )dt

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Compare AM to FM for x (t ) = cos(m t ).

AM: y1(t ) = (cos(m t ) + 1 .1) cos(c t )

t

FM: y3(t ) = cos(c t + m sin(m t ))

t

Advantages of FM:

constant power

no need to transmit carrier (unless DC important) bandwidth?

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Early investigators thought that narrowband FM could have arbitrar-ily narrow bandwidth, allowing more channels than AM. Wrong!

t y3(t ) = cos c t + k x ( )d

t t

= cos(c t ) cos k x ( )d sin(c t ) sin k x ( )d

If k 0 then t

cos k x ( )d 1

t t sin k x ( )d k x ( )d

t y3(t ) cos(c t ) sin(c t ) k x ( )d

Bandwidth of narrowband FM is the same as that of AM!

(integration does not change bandwidth)

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Phase/Frequency Modulation

Find the Fourier transform of a PM signal.

x (t ) = sin(m t )y(t ) = cos(c t + mx (t )) = cos(c t + m sin(m t ))

= cos(c t ) cos(m sin(m t )) ) sin(c t ) sin( m sin(m t )))2x (t ) is periodic in T = m , therefore cos(m sin(m t )) is periodic in T .

m sin(m t ) m

0

m

t

cos(m sin(m t ))

t increasing m

1

0

1

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cos(m sin(m t ))1

0 t

1 m = 0

|a k |

k0 10 20 30 40 50 60

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cos(m sin(m t ))1

0 t

1 m = 1

|a k |

k0 10 20 30 40 50 60

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cos(m sin(m t ))1

0 t

1 m = 2

|a k |

k0 10 20 30 40 50 60

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cos(m sin(m t ))1

0 t

1 m = 5

|a k |

k0 10 20 30 40 50 60

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cos(m sin(m t ))1

0 t

1 m = 10

|a k |

k0 10 20 30 40 50 60

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cos(m sin(m t ))1

0 t

1 m = 20

|a k |

k0 10 20 30 40 50 60

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cos(m sin(m t ))1

0 t

1 m = 30

|a k |

k0 10 20 30 40 50 60

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cos(m sin(m t ))1

0 t

1 m = 40

|a k |

k0 10 20 30 40 50 60

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cos(m sin(m t ))1

0 t

1 m = 50

|a k |

k0 10 20 30 40 50 60

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Fourier transform of rst part.

x (t ) = sin(m t )

y(t ) = cos(c t + mx (t )) = cos(c t + m sin(m t ))

= cos(c t )cos(m sin(m t ))) sin(c t )sin( m sin(m t )))

ya (t )

|Y a ( j )| m = 50

c c

50m

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= cos(c t )cos(m sin(m t ))) sin(c t ) sin(m sin(m t )) )2x (t ) is periodic in T = m , therefore sin(m sin(m t )) is periodic in T .

m sin(m t ) m

0

m

t

sin(m sin(m t ))1

0

1

t

increasing m

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sin(m sin(m t ))1

0 t

1 m = 0

|bk |

k0 10 20 30 40 50 60

Ph /F M d l i

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sin(m sin(m t ))1

0 t

1 m = 1

|bk |

k0 10 20 30 40 50 60

Ph /F M d l i

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sin(m sin(m t ))1

0 t

1 m = 2

|bk |

k0 10 20 30 40 50 60

Ph /F M d l ti

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sin(m sin(m t ))1

0 t

1 m = 5

|bk |

k0 10 20 30 40 50 60


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sin(m sin(m t ))1

0 t

1 m = 10

|bk |

k0 10 20 30 40 50 60


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0 t

sin(m sin(m t ))1

1 m = 20

|bk |

k0 10 20 30 40 50 60


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sin(m sin(m t ))1

0 t

1 m = 30

|bk |

k0 10 20 30 40 50 60


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sin(m sin(m t ))1

0 t

1 m = 40

|bk |

k0 10 20 30 40 50 60


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sin(m sin(m t ))1

0 t

1 m = 50

|bk |

k0 10 20 30 40 50 60


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Fourier transform of second part.

x (t ) = sin(m t )


= cos(c t )cos(m sin(m t ))) sin(c t ) sin( m sin(m t )))

yb (t )

|Y b( j )| m = 50

c c

50m


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Fourier transform.

x (t ) = sin(m t )


= cos(c t )cos(m sin(m t ))) sin(c t )sin( m sin(m t )))

ya (t ) yb (t ) |Y ( j )| m = 50

c c

50m


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q y

Wideband FM is useful because it is robust to noise.

AM: y1(t ) = (cos(m t ) + 1 .1) cos(c t )

t

FM: y3(t ) = cos(c t + m sin(m t ))

t

FM generates a very redundant signal, which is resilient to additive

noise.

Summary

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y

Modulation is useful for matching signals to media.

Examples: commercial radio (AM and FM)

Close with unconventional application of modulation in microscopy.

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