Lecture 24: Modulation, part 2 - MIT OpenCourseWare | Free ... · 0 Z 0 0 0 Z 0. Frequency Modulation. Early investigators thought that narrowband FM could have arbitrar ily narrow

6.003: Signals and Systems

Modulation

December 6, 2011 1

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 fiber, E/M

Modulation can improve match based on frequency.

2

Amplitude Modulation

Amplitude modulation can be used to match audio frequencies to

radio frequencies. It allows parallel transmission of multiple channels.

x1(t)

x2(t)

x3(t)

z1(t)

z2(t) z(t)y(t)

z3(t)

cos w1t

cos w2t cos wct

cos w3t

LPF

3

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.

4

Amplitude, Phase, and Frequency Modulation

There are many ways to embed a “message” in a carrier.

Amplitude Modulation (AM) + carrier: y1(t) = x(t) + C cos(ωct)

Phase Modulation (PM): y2(t) = cos(ωct + kx(t)) tFrequency Modulation (FM): y3(t) = cos ωct + k −∞ x(τ )dτ

PM: signal modulates instantaneous phase of the carrier.

y2(t) = cos(ωct + kx(t))

FM: signal modulates instantaneous frequency of carrier. t y3(t) = cos ωct + k x(τ)dτ ' −∞v "

φ(t)d

ωi(t) = ωc + φ(t) = ωc + kx(t)dt

5

Frequency Modulation

sin(ωmt))

Advantages of FM:

• constant power

Compare AM to FM for x(t) = cos(ωmt).

AM: y1(t) = x(t) + C cos(ωct) = (cos(ωmt) + 1.1) cos(ωct)

t

FM: y3(t) = cos ωct + k −∞ x(τ )dτ = cos(ωct +t k ωm

t

• no need to transmit carrier (unless DC important)

• bandwidth? 6

∫


Early investigators thought that narrowband FM could have arbitrar

ily narrow bandwidth, allowing more channels than AM.

t y3(t) = cos ωct + k x(τ)dτ

−∞' v " φ(t)

d ωi(t) = ωc + φ(t) = ωc + kx(t)

dt

Small k → small bandwidth. Right?

7

( ∫ )


Early investigators thought that narrowband FM could have arbitrar

ily narrow bandwidth, allowing more channels than AM. Wrong! 0 t y3(t) = cos ωct + k x(τ)dτ

−∞0 0 t t = cos(ωct) × cos k x(τ)dτ − sin(ωct) × sin k x(τ )dτ

−∞ −∞

If k → 0 then0 t cos k x(τ)dτ → 1

−∞0 t t sin k x(τ )dτ → k x(τ)dτ

−∞ −∞0 t y3(t) ≈ cos(ωct) − sin(ωct) × k x(τ)dτ

−∞

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

(integration does not change the highest frequency in the signal) 8

∫∫ ∫

∫∫ ∫

∫

1

0

−1

1 sin(ωmt)

t

1

0

−1

cos(1 sin(ωmt))

t

Phase/Frequency Modulation

Find the Fourier transform of a PM/FM signal.

y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt))

= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt)))

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

9

2

0

−2

2 sin(ωmt)

t

1

0

−1

cos(2 sin(ωmt))

t






10

3

0

−3

3 sin(ωmt)

t

1

0

−1

cos(3 sin(ωmt))

t






11

4

0

−4

4 sin(ωmt)

t

1

0

−1

cos(4 sin(ωmt))

t






12

5

0

−5

5 sin(ωmt)

t

1

0

−1

cos(5 sin(ωmt))

t






13

6

0

−6

6 sin(ωmt)

t

1

0

−1

cos(6 sin(ωmt))

t






14

7

0

−7

7 sin(ωmt)

t

1

0

−1

cos(7 sin(ωmt))

t






15

8

0

−8

8 sin(ωmt)

t

1

0

−1

cos(8 sin(ωmt))

t






16

9

0

−9

9 sin(ωmt)

t

1

0

−1

cos(9 sin(ωmt))

t






17

10

0

−10

10 sin(ωmt)

t

1

0

−1

cos(10 sin(ωmt))

t






18

20

0

−20

20 sin(ωmt)

t

1

0

−1

cos(20 sin(ωmt))

t






19

50

0

−50

50 sin(ωmt)

t

1

0

−1

cos(50 sin(ωmt))

t






20

m

0

−m

m sin(ωmt)

t

1

0

−1

cos(m sin(ωmt))

t

increasing m






21





x(t) is periodic in T = 2π ωm

, therefore cos(m sin(ωmt)) is periodic in T .

1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 0

22







1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 1

23







1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 2

24







1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 5

25







1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 10

26







1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 20

27







1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 30

28







1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 40

29







1

0

−1

cos(m sin(ωmt))

t

|ak|

k0 10 20 30 40 50 60

m = 50

30

|Ya(jω)|

ωωcωc

100ωm

m = 50


Fourier transform of first part.

x(t) = sin(ωmt)


= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt)))' v " ya(t)

31

m

0

−m

m sin(ωmt)

t

1

0

−1

sin(m sin(ωmt))

t

increasing m

increasing m





x(t) is periodic in T = 2π , therefore sin(m sin(ωmt)) is periodic in T .ωm

32






, therefore sin(m sin(ωmt)) is periodic in T .

1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 0

33







1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 1

34







1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 2

35







1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 5

36







1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 10

37







1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 20

38







1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 30

39







1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 40

40







1

0

−1

sin(m sin(ωmt))

t

|bk|

k0 10 20 30 40 50 60

m = 50

41

' v "ya(t)

|Yb(jω)|

ωωcωc

100ωm

m = 50


Fourier transform of second part.

x(t) = sin(ωmt)


= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt)))' v " yb(t)

42

|Y (jω)|

ωωcωc

100ωm

m = 50


Fourier transform.

x(t) = sin(ωmt)


= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt)))' v " ' v " ya(t) yb(t)

43


Wideband FM is useful because it is robust to noise.

AM: y1(t) = (cos(ωmt) + 1.1) cos(ωct)

t

FM: y3(t) = cos(ωct + m sin(ωmt))

t

FM generates a redundant signal that is resilient to additive noise.

44

Summary

Modulation is useful for matching signals to media.

Examples: commercial radio (AM and FM)

Close with unconventional application of modulation – in microscopy.

45

6.003 Microscopy

Dennis M. Freeman Stanley S. Hong Jekwan Ryu Michael S. Mermelstein Berthold K. P. Horn

46Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.

6.003 Model of a Microscope

microscope

Microscope = low-pass filter

47

Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.

Phase-Modulated Microscopy

microscope

48


49


50


51




many frequencies + many orientations = many images

low resolution high resolution

wx

wy

wx

wy

wx

wy

54


55


56


57


58


59


60


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6.003 Signals and SystemsFall 2011

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Lecture 24: Modulation, part 2 - MIT OpenCourseWare | Free ... · 0 Z 0 0 0 Z 0. Frequency Modulation. Early investigators thought that narrowband FM could have arbitrar ily narrow

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