Lecture 24: Modulation, Part 2

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.

z1(t) x1(t)

cos w1t

z2(t) z(t) x2(t) LPF y(t)

cos w2t cos wct

z3(t) x3(t)

cos w3t 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 in vented 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)) t Frequency 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 ω (t) = ωc + φ(t) = ωc + kx(t) i dt 5 Frequency Modulation

Compare AM to FM for x(t) = cos(ωmt).

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

FM: y (t) = cos ω t + k R t x(τ )dτ = cos(ω t + k sin(ω t)) 3 c −∞ c ωm m

Advantages of FM: • constant power • no need to transmit carrier (unless DC important) • bandwidth? 6 Frequency Modulation

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

Z t y3(t) = cos ωct + k x(τ)dτ −∞ ' v " φ(t) d ω (t) = ωc + φ(t) = ωc + kx(t) i dt

Small k → small bandwidth. Right?

7 Frequency Modulation

Early investigators thought that narrowband FM could have arbitrar ily narrow bandwidth, allowing more channels than AM. Wrong! 0 Z t y3(t) = cos ωct + k x(τ)dτ −∞ 0 Z t 0 Z t = cos(ωct) × cos k x(τ)dτ − sin(ωct) × sin k x(τ )dτ −∞ −∞

If k → 0 then 0 Z t cos k x(τ)dτ → 1 −∞ 0 Z t Z t sin k x(τ )dτ → k x(τ)dτ −∞ −∞ 0 Z 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 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(ω t)) is periodic in T . ωm m

1 sin(ωmt) 1

0 t

−1

cos(1 sin(ωmt)) 1

0 t

−1

9 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(ω t)) is periodic in T . ωm m

2 sin(ωmt) 2

0 t

−2

cos(2 sin(ωmt)) 1

0 t

−1

10 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(ω t)) is periodic in T . ωm m

3 sin(ωmt) 3

0 t

−3

cos(3 sin(ωmt)) 1

0 t

−1

11 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(ω t)) is periodic in T . ωm m

4 sin(ωmt) 4

0 t

−4

cos(4 sin(ωmt)) 1

0 t

−1

12 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(ω t)) is periodic in T . ωm m

5 sin(ωmt) 5

0 t

−5

cos(5 sin(ωmt)) 1

0 t

−1

13 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(ω t)) is periodic in T . ωm m

6 sin(ωmt) 6

0 t

−6

cos(6 sin(ωmt)) 1

0 t

−1

14 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(ω t)) is periodic in T . ωm m

7 sin(ωmt) 7

0 t

−7

cos(7 sin(ωmt)) 1

0 t

−1

15 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(ω t)) is periodic in T . ωm m

8 sin(ωmt) 8

0 t

−8

cos(8 sin(ωmt)) 1

0 t

−1

16 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(ω t)) is periodic in T . ωm m

9 sin(ωmt) 9

0 t

−9

cos(9 sin(ωmt)) 1

0 t

−1

17 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(ω t)) is periodic in T . ωm m

10 sin(ωmt) 10

0 t

−10

cos(10 sin(ωmt)) 1

0 t

−1

18 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(ω t)) is periodic in T . ωm m

20 sin(ωmt) 20

0 t

−20

cos(20 sin(ωmt)) 1

0 t

−1

19 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(ω t)) is periodic in T . ωm m

50 sin(ωmt) 50

0 t

−50

cos(50 sin(ωmt)) 1

0 t

−1

20 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(ω t)) is periodic in T . ωm m

m sin(ωmt) m

0 t

−m

cos(m sin(ωmt)) 1

0 t increasing m −1

21 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 0

|ak|

k 0 10 20 30 40 50 60

22 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 1

|ak|

k 0 10 20 30 40 50 60

23 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 2

|ak|

k 0 10 20 30 40 50 60

24 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 5

|ak|

k 0 10 20 30 40 50 60

25 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 10

|ak|

k 0 10 20 30 40 50 60

26 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 20

|ak|

k 0 10 20 30 40 50 60

27 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 30

|ak|

k 0 10 20 30 40 50 60

28 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 40

|ak|

k 0 10 20 30 40 50 60

29 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(ω t)) is periodic in T . ωm m

cos(m sin(ωmt)) 1

0 t

−1 m = 50

|ak|

k 0 10 20 30 40 50 60

30 Phase/Frequency Modulation

Fourier transform of ﬁrst part.

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

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

m = 50 |Ya(jω)|

ω ωc ωc

100ωm

31 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 sin(m sin(ω t)) is periodic in T . ωm m

m sin(ωmt) m

0 t

−m

sin(m sin(ωmt)) 1 increasing m

0 t increasing m −1

32 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 0

|bk|

k 0 10 20 30 40 50 60

33 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 1

|bk|

k 0 10 20 30 40 50 60

34 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 2

|bk|

k 0 10 20 30 40 50 60

35 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 5

|bk|

k 0 10 20 30 40 50 60

36 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 10

|bk|

k 0 10 20 30 40 50 60

37 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 20

|bk|

k 0 10 20 30 40 50 60

38 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 30

|bk|

k 0 10 20 30 40 50 60

39 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 40

|bk|

k 0 10 20 30 40 50 60

40 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 sin(m sin(ω t)) is periodic in T . ωm m

sin(m sin(ωmt)) 1

0 t

−1 m = 50

|bk|

k 0 10 20 30 40 50 60

41 Phase/Frequency Modulation

Fourier transform of second part.

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

= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) ' v " ' v " ya(t) yb(t) m = 50 |Yb(jω)|

ω ωc ωc

100ωm

42 Phase/Frequency Modulation

Fourier transform.

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

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

|Y (jω)| m = 50

ω ωc ωc

100ωm

43 Frequency Modulation

Wideband FM is useful because it is robust to noise.

AM: y1(t) = (cos(ωmt) + 1.1) cos(ωct)

FM: y3(t) = cos(ωct + m sin(ωmt))

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

Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission. 46 6.003 Model of a Microscope

microscope

Microscope = low-pass filter

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

47 Phase-Modulated Microscopy

microscope

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

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

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

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

51 Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission. 52 Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission. 53 many frequencies + many orientations = many images w y

w x w w y y

w w x x

low resolution high resolution

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

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

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

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

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

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

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

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6.003 Signals and Systems Fall 2011

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