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Chapter 2 - Signal Properties

Symmetry - Quiz

Question 1

Which of the following continuous signals exhibits even symmetry?

Select all that apply.

Hint: \(f(t) = f(-t),\ \forall\,t\ \Longleftrightarrow\ f(t) \text{ is even symmetric}\)

  • A. $\quad f_{1}(t) = \cos{(3t)}$
  • B. $\quad f_{2}(t) = te^{-t}$
  • C. $\quad f_{3}(t) = e^{-t^{2}}$
  • D. $\quad f_{4}(t) = |t|$
  • E. $\quad f_{5}(t) = \sin{(2t)}$
  • F. $\quad$ None of the above.

Question 2

Which of the following continuous signals exhibits odd symmetry?

Select all that apply.

Hint: \(f(t) = -f(-t),\ \forall\,t\ \Longleftrightarrow\ f(t) \text{ is odd symmetric}\)

  • A. $\quad f_{1}(t) = \cos{(3t)}$
  • B. $\quad f_{2}(t) = te^{-t^{2}}$
  • C. $\quad f_{3}(t) = \sin{(t)}e^{-|t|}$
  • D. $\quad f_{4}(t) = \mathrm{sgn}\left(t\right)$
  • E. $\quad f_{5}(t) = \sin{(2t)}$
  • F. $\quad$ None of the above.

Question 3

For the following signals, determine if they are even, odd, or neither.

a) \(f(t) = t(t^{2} - 1)e^{-t^{2}}\)

  • Even
  • Odd
  • Neither

b) \(f(t) = \cos{(t)} + t^{2}\)

  • Even
  • Odd
  • Neither

c) \(f(t) = \sin{(|t|)}\)

  • Even
  • Odd
  • Neither

d) \(f(t) = u(t)\)

  • Even
  • Odd
  • Neither

e) \(f(t) = \frac{t}{t^{2}+1}\)

  • Even
  • Odd
  • Neither

f) \(f(t) = \begin{cases}\frac{\sin{(t)}}{t},\quad &t\neq 0\\ 1,\quad &t=0\end{cases}\)

  • Even
  • Odd
  • Neither

g) \(f(t) = \begin{cases}\frac{1 - \cos{(t)}}{t},\quad &t\neq 0\\ 0,\quad &t=0\end{cases}\)

  • Even
  • Odd
  • Neither

h) \(f(t) = \begin{cases}\frac{t\cos{t} - \sin{(t)}}{t^{2}},\quad &t\neq 0\\ 0,\quad &t=0\end{cases}\)

  • Even
  • Odd
  • Neither

i) \(f(t) = u(t + a) - u(t - a),\quad a>0\)

  • Even
  • Odd
  • Neither

j) \(f(t) = \mathrm{sgn}(t)t^{2}\)

  • Even
  • Odd
  • Neither

Question 4

Which of the following statements are always true for real valued signals \(x(t)\) and \(y(t)\).

Select all that apply.

Note: You may assume that all signals are differentiable and integrable, where needed.

  • A. If $x(t)$ is even, and $y(t)$ is odd, then $x(t) + y(t)$ is odd.
  • B. If $x(t)$ is odd, and $y(t)$ is odd, then $x(t)y(t)$ is odd.
  • C. If $x(t)$ is even, then its derivative $\tfrac{dx}{dt}$ is odd.
  • D. If $x(t)$ is odd, then $\int_{-a}^{a}x(t)\,\mathrm{d}t = 0,\quad \forall\,a>0$
  • E. The decomposition $x(t) = x_{e}(t) + x_{o}(t)$ is unique, where $x_{e}(t)$ is even and $x_{o}(t)$ is odd.
  • F. None of the above.

Question 6

Which of the following is a valid graphical/visual statement about odd symmetry?

Select all that apply.

  • A. An odd symmetric signal must pass through the origin (0, 0)
  • B. An odd symmetric signal remains unchanged when you flip horizontally.
  • C. An odd symmetric signal remains unchanged when you flip vertically.
  • D. For an odd symmetric signal, if you horizontally flip the signal, it's equivalent to vertically flipping it.
  • E. For an odd symmetric signal, if you rotate it about the origin by 180 degrees, it remains unchanged.
  • F. None of the above.