Fourier transform of a periodic signal'

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Lecture 18

• Fourier transform of a periodic signal.
• The Sampling Theorem.

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Fourier transform of a periodic signal
This is an impulse train in frequency
0
An impulse in the transform indicates a pure
sinusoid at that frequency. Here, at all
multiples of the fundamental frequency.
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Impulse train in frequency
Periodic function of time
The graph is analogous to the line spectra we
used for Fourier series, replacing lines by
deltas.
Dual property
Periodic function of frequency
Impulse train in time
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(1)
t
0
Periodic impulse train in time
Periodic impulse train in frequency
0
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The Sampling Theorem
2T
kT
T
t
Q Can we recover f(t) from its samples? A In
general, no. No way of knowing what happened
between sample times.
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We narrow it down to a special class of functions.
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Proof consider the following interconnection of
systems

kT
t
The impulse train r(t) depends only on the
samples f(kT).
We will show that the output y(t) reconstructs
the input f(t) This means we have determined
f(t) by its samples.
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Going to the frequency domain
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Filter out the copies
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Formula that interpolates a band-limited function
from its samples
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Aliasing
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Final Exam

• Friday, December 13, 8-11 AM.
  Note Room change4000A MS.
• Closed book. You can bring two (8.5 by 11)
  sheet of formulas (on both sides if you want).
• No calculators allowed (or needed).
• Roughly, 1/3 of the test on pre-midterm material,
  2/3 on post-midterm material.
• Practice Exam posted today on the Web. I will
  post solutions to it by the weekend.
• I will have my regular office hours on Monday,
  not Wednesday. Markus will hold a review session,
  probably on Wednesday. Check the Web in the next
  few days for details.