ELEC 361/W: Midterm exam Solution: Fall 2005 Professor: A. Amer TA: M. Ghazal

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ELEC 361/W Midterm exam Solution Fall
2005Professor A. AmerTA M. Ghazal

• Q1
• True According to the Shifting property of the
  FT
• False Causality is not determined based on the
  input signal x(t)
• True Using shifting and linearity properties of
  the FS
• True If (x(t) is bounded and since cos(1/t)
  is bounded by 1
• False The fundamental period of this signal is
  24 which is the least common multiple of 3 (the
  fundamental period of the first term) and 8 (the
  fundamental period of the second term)

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Q2 Fourier Transform
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Q2 Solution

• We have
• Taking the inverse transform, we get

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Q2 Solution

• We have
• Taking the inverse transform, we get

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Q3 Convolution
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Q3 Solution Step 1

• (a) Write down the x(t) and h(t) functionally and
  graphically
• Note that h(t) is a scaled version of x(t)

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Q3 Solution Step 2

• Sketch h(-t) and h(t-t)
• h(-t)
• Rreflection around y-axis
• Chage t to t
• h(t-t) h(-tt)
• Add t to all axis points
• Move the graph away to the left

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Q3 Solution Step 3

• Slide h(t-t) to the right and collect the overlap
• As you go, find
• Limits for y(t)
• Limits for integration

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Q3 Solution Step 3
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Q3 Solution Step 3
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Q3 Solution Step 3
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Q3 Solution Step 4

• Put the limits together to make y(t)

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Q3 Solution Step 5

• (b) find the first derivative of y with respect
  to t both functionally and graphically
• This function has 4 discontinuities, only when a
  1, it has 3 discontinuities (two
  discontinuities become one)
• Note that we know 0lt a ?1

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Q4 Fourier Series
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Q4 Solution

• (1) Graph both xn and xn-1 to get gn
• From the graph, we can write gn as
• Note that gn is periodic with N 10
• (2) From the expression for gn, the FS
  coefficients are

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Q4 Solution

• (3) Since gn xn xn-1, the FS
  coefficients ak and bk are related by
• Therefore,