Fourier Series Revisited AMFM Modulation

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Fourier Series RevisitedAM/FM Modulation
MCS.03
Maj JW Paul
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Nothing is ever a total failure
It can always serve as a bad
example.
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Review
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How to get from here to there
Sky Wave HF
Tropospheric Wave
MF Direct Wave
Surface Wave LF
Reflected Wave
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Review

• What is the difference between Phase
  delay Time delay

• What can you use this for?

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Power Loss

• What does this formula tell us?

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Todays Class

• Fourier Series and Transforms
• AM/FM modulation

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Fourier Series Transforms
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Periodic Signal

• Any signal that repeats itself with a constant
  period
• Assume s(t) is our function
  
  s(tT) s(t) for any value of t
• Period

T
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Fourier Series

• Any periodic signal can be written as the sum of
  harmonics

s(t) a0 ? ancos(2 ? n f t) bnsin(2?n f t)

a0DC or average value.
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Periodic Signals
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Even and Odd functions
sin function odd
cos function even
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sin ( t ? ) cos (t) when ? ?/2 ? ?
phase shift
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Ringing
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Linear systems
vin(t) Acos(?t?) vout(t) Bcos(?t?)
Amplitude A, B Phase ?, ?

• Gives a nice harmonic signal
• Non-linear systems are bad !!!
• May not be a correlation between input and output
  (feedback screech)

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Non-Harmonic Signals

• Express input signal as a sum of harmonics (which
  are know)
• The output is the sum of the responses.

vin(t) A0 A1cos(?1t?1) A2cos(?2t?2)
vout(t) B0 B1cos(?1t?1) Bcos(?2t?2)
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Non-Periodic Signals

• Consider a non-periodic signal as a periodic
  signal with an infinite period
• T ?
• now ? is replaced with ?
• result is a Fourier Transform

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Fourier Transform Pairs
S(?) gives the spectrum of the signal s(t).
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Why Fourier Series/Transforms
s(t) a0 ? ancos(2 ? n f t) bnsin(2?n f t)

• Mathematically, it is much easier to manipulate
  the signal in Fourier space
• filtering, phase shifting, frequency analysis...

t
Aside - needs FFT
f
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Some examples
An interactive overview
http//www.csupomona.edu/apfelzer/demos/toc.html
fourierseries
Intro to Fourier Series (images)
http//cns-alumni.bu.edu/slehar/fourier/fourier.h
tml
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• If a signal is condensed in time, it will be
  expanded in frequency,
• inversely if a signal is expanded in time it will
  be condensed in frequency.
• This means that narrow pulse have large
  bandwidths and wide pulse have narrow bandwidths

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Questions?
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Experience is something you get just after you
needed it...
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Modulation Techniques
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Definition

• Modulation
• process by which some characteristics of a
  carrier (sinusoidal wave) is varied in accordance
  with a modulating wave (data or analog message)
• can change amplitude, phase or frequency of
  carrier or combination according to the message

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Why Modulate a signal?

• An EM signal can easily be generated at voice
  frequencies (speaker, telephone, morse code)
• But to radiate the signal in free space, antenna
  size becomes limiting factor
• For a voice signal with a freq of 3 kHz
• ? c/f 1 x 105 m 100 km

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Amplitude Modulation

• Not superposition
• multiplication
• look at frequency domain...

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If you can read this you have a great LCD
projector
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Frequency Domain Representation of AM Signals
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AM Fourier Transform

• sAM(t) m(t) cos (2?fct) A cos (2?fct)
• SAM(f) 1/2 M(f-fc) M(ffc) A/2 ? (f-fc)
  ? (ffc)

Note - need to add base amplitude
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Other types of AM

• DSB-SC (double side band - suppressed carrier)
• SSB (single side band)

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Why change a good thing?
BW

fc
f
Original Signal
Carrier Signal
Needs tons o power
fc
fc f
fc - f
2 x BW
Needs twice as much bandwidth
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DSB-SC and SSB
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Pros Cons

• DSB-SC
• requires a local carrier with the same phase and
  frequency as the incoming carrier
• needs way less energy (50)
• SSB
• as above - but susceptible to noise
• and uses 1/2 the bandwidth
• recreation of complete signal technically complex

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Frequency Modulation

• Modulation
• process by which some characteristics of a
  carrier (sinusoidal wave) is varied in accordance
  with a modulating wave (data or analog message)
• AM modifies amplitude
• FM modifies frequency

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FM
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FM Bandwidth

• If FM carrier frequency is modulated directly by
  the data signal, then require 2 x BW
• However, can modulate the input signal
• modulation index
• In narrow band FM, can use as little as 30kHz

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Advantages of FM

• Generally better noise immunity
• noise tends to be an amplitude effect
• FM is based on frequency
• Needs bigger bandwidth - but can use it more
  effectively

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AM vs FM (and PM)
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Questions?