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Z-Transforms

This quiz will test your understanding of Z-Transforms, a mathematical tool used to analyze discrete-time signals and systems.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = a^n), where (a) is a constant?

  1. \(X(z) = \frac{z}{z - a}\)
  2. \(X(z) = \frac{z}{z + a}\)
  3. \(X(z) = \frac{1}{z - a}\)
  4. \(X(z) = \frac{1}{z + a}\)
Question 2 Multiple Choice (Single Answer)

What is the Z-Transform of the unit step sequence (u[n])?

  1. \(U(z) = \frac{1}{1 - z^{-1}}\)
  2. \(U(z) = \frac{z}{z - 1}\)
  3. \(U(z) = \frac{1}{z - 1}\)
  4. \(U(z) = \frac{z}{1 - z^{-1}}\)
Question 3 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = n)?

  1. \(X(z) = \frac{z}{(z - 1)^2}\)
  2. \(X(z) = \frac{z^2}{(z - 1)^2}\)
  3. \(X(z) = \frac{1}{(z - 1)^2}\)
  4. \(X(z) = \frac{z}{(z + 1)^2}\)
Question 4 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = \sin(\omega_0 n))?

  1. \(X(z) = \frac{z \sin(\omega_0)}{z^2 - 2z \cos(\omega_0) + 1}\)
  2. \(X(z) = \frac{z \cos(\omega_0)}{z^2 - 2z \sin(\omega_0) + 1}\)
  3. \(X(z) = \frac{z}{z^2 - 2z \cos(\omega_0) + 1}\)
  4. \(X(z) = \frac{z}{z^2 - 2z \sin(\omega_0) + 1}\)
Question 5 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = \cos(\omega_0 n))?

  1. \(X(z) = \frac{z^2 - 1}{z^2 - 2z \cos(\omega_0) + 1}\)
  2. \(X(z) = \frac{z^2 + 1}{z^2 - 2z \cos(\omega_0) + 1}\)
  3. \(X(z) = \frac{z}{z^2 - 2z \cos(\omega_0) + 1}\)
  4. \(X(z) = \frac{1}{z^2 - 2z \cos(\omega_0) + 1}\)
Question 6 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = e^{\alpha n}), where (\alpha) is a constant?

  1. \(X(z) = \frac{z}{z - e^{\alpha}}\)
  2. \(X(z) = \frac{1}{z - e^{\alpha}}\)
  3. \(X(z) = \frac{z}{z + e^{\alpha}}\)
  4. \(X(z) = \frac{1}{z + e^{\alpha}}\)
Question 7 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = \delta[n]), where (\delta[n]) is the unit impulse function?

  1. \(X(z) = 1\)
  2. \(X(z) = z\)
  3. \(X(z) = \frac{1}{z}\)
  4. \(X(z) = 0\)
Question 8 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = n^2)?

  1. \(X(z) = \frac{z}{(z - 1)^3}\)
  2. \(X(z) = \frac{z^2}{(z - 1)^3}\)
  3. \(X(z) = \frac{1}{(z - 1)^3}\)
  4. \(X(z) = \frac{z}{(z + 1)^3}\)
Question 9 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = \cos(\omega_0 n) + j \sin(\omega_0 n))?

  1. \(X(z) = \frac{z(z - \cos(\omega_0))}{z^2 - 2z \cos(\omega_0) + 1}\)
  2. \(X(z) = \frac{z(z + \cos(\omega_0))}{z^2 - 2z \cos(\omega_0) + 1}\)
  3. \(X(z) = \frac{z}{z^2 - 2z \cos(\omega_0) + 1}\)
  4. \(X(z) = \frac{1}{z^2 - 2z \cos(\omega_0) + 1}\)
Question 10 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = \left{\begin{array}{ll} 1, & n = 0\ 2, & n = 1\ 3, & n = 2\ 0, & \text{otherwise} \end{array}\right.)?

  1. \(X(z) = \frac{z^2 + 2z + 3}{z^3}\)
  2. \(X(z) = \frac{z^2 - 2z + 3}{z^3}\)
  3. \(X(z) = \frac{z^2 + 2z - 3}{z^3}\)
  4. \(X(z) = \frac{z^2 - 2z - 3}{z^3}\)
Question 11 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = \left{\begin{array}{ll} 1, & n \text{ is even}\ 0, & n \text{ is odd} \end{array}\right.)?

  1. \(X(z) = \frac{1}{1 - z^{-2}}\)
  2. \(X(z) = \frac{z}{1 - z^{-2}}\)
  3. \(X(z) = \frac{1}{1 + z^{-2}}\)
  4. \(X(z) = \frac{z}{1 + z^{-2}}\)
Question 12 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = n \cos(\omega_0 n))?

  1. \(X(z) = \frac{z(z - \cos(\omega_0))}{(z - \cos(\omega_0))^2 + \sin^2(\omega_0)}\)
  2. \(X(z) = \frac{z(z + \cos(\omega_0))}{(z + \cos(\omega_0))^2 + \sin^2(\omega_0)}\)
  3. \(X(z) = \frac{z}{(z - \cos(\omega_0))^2 + \sin^2(\omega_0)}\)
  4. \(X(z) = \frac{1}{(z - \cos(\omega_0))^2 + \sin^2(\omega_0)}\)
Question 13 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = \left{\begin{array}{ll} 1, & n = 0\ -1, & n \text{ is odd}\ 0, & n \text{ is even and } n \ne 0 \end{array}\right.)?

  1. \(X(z) = \frac{1 - z^{-1}}{1 + z^{-1}}\)
  2. \(X(z) = \frac{1 + z^{-1}}{1 - z^{-1}}\)
  3. \(X(z) = \frac{1}{1 + z^{-1}}\)
  4. \(X(z) = \frac{1}{1 - z^{-1}}\)
Question 14 Multiple Choice (Single Answer)

What is the Z-Transform of the sequence (x[n] = \left{\begin{array}{ll} 1, & n = 0\ 2, & n = 1\ 3, & n = 2\ 4, & n = 3\ 0, & \text{otherwise} \end{array}\right.)?

  1. \(X(z) = \frac{1 + 2z^{-1} + 3z^{-2} + 4z^{-3}}{1 - z^{-1}}\)
  2. \(X(z) = \frac{1 + 2z^{-1} + 3z^{-2} + 4z^{-3}}{1 + z^{-1}}\)
  3. \(X(z) = \frac{1 - 2z^{-1} + 3z^{-2} - 4z^{-3}}{1 - z^{-1}}\)
  4. \(X(z) = \frac{1 - 2z^{-1} + 3z^{-2} - 4z^{-3}}{1 + z^{-1}}\)
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