EGR 277 Digital Logic

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Lecture 24 EGR 261 Signals and Systems
Read Ch. 17 in Electric Circuits, 8th Edition
by Nilsson Ch. 7 in Linear Systems
and Signals, 2nd Edition by Lathi
Parsevals Theorem Parsevals theorem relates the
energy associated with a time-domain function of
finite energy (i.e., an energy signal) to the
Fourier transform of the function.
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Lecture 24 EGR 261 Signals and Systems
Parsevals Theorem
Energy calculated in the time domain
Energy calculated in the frequency domain
Example Calculate total energy for the function
f(t) e-atu(t) a) In the time domain b) In the
frequency domain
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Lecture 24 EGR 261 Signals and Systems

• Notes on Parsevals theorem
• For real functions, f(t), F(w) is even and
  F(w) F(-w), so
• Recall that our definition for energy is not
  dimensionally correct. However, if f(t) v(t)
  or i(t) then the energy could be viewed as energy
  for a 1O load as shown below
• Parsevals theorem gives a physical
  interpretation of F(w)2
• F(w)2 is an energy density in joules per
  hertz (to a 1O load)
• Energy of the frequency range (w1, w2) can be
  calculated as follows

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Lecture 24 EGR 261 Signals and Systems
Bandwidth of a filter Recall in our earlier
discussions of band pass filters, that we defined
the bandwidth B of a filter as the width of the
passband, typically defined by the 3dB points, as
illustrated below.
Bandwidth of a signal Suppose that now we
define the bandwidth of a signal as positive
range of frequency over which F(w) extends, as
illustrated below on the left. For any practical
signal, F(w) extends to ?, so a practical
estimate of bandwidth might be the range where
most (perhaps 95?) of the signal energy lies.
This is sometimes called the essential bandwidth
of the signal.
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Lecture 24 EGR 261 Signals and Systems

• Example If f(t) i(t) 50e-10tu(t) mA is the
  current delivered to a 1 O resistor
• a) Find total energy in the time domain
• Find total energy in the frequency domain
• Find the percentage of energy for w lt 10 rad/s
• Find the bandwidth, B, of the signal (i.e., the
  value of w such that 95 of the energy is
  delivered from 0 to w)
• Sketch F(w)2 and the results from parts c and d.

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Lecture 24 EGR 261 Signals and Systems
Bandwidth of sinc(x) Recall that the function
rect(t/?) is an important function and has the
following Fourier transform
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Lecture 24 EGR 261 Signals and Systems
Bandwidth of sinc(x) Using the result that the
essential bandwidth is approximately 1/? for a
gated pulse of width ?
The relationships above are illustrated on the
following page.
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Lecture 24 EGR 261 Signals and Systems