Question

The Z-transform of the sequence \{1, 0, 1, 0, 1\ is:}

• A. \(1 + Z^{2} + Z^{4}\)
• B. \(1 + Z + Z^{2}\)
• C. \(Z + Z^{3} + Z^{5}\)
• D. \(Z + Z^{2} + Z^{3}\)

Hint:

When calculating the Z-transform of a finite sequence, only the non-zero terms matter. Multiply each non-zero term by the corresponding power of \(Z\).

Answer 1

The Correct Option is A

Step 1: Recall the definition of Z-transform. For a sequence \(x[n]\), the Z-transform is defined as \[ X(Z) = \sum_{n=0}^{\infty} x[n] Z^n. \] Step 2: Write the given sequence. The sequence is \(\{1, 0, 1, 0, 1\}\), i.e.: \[ x[0] = 1, \quad x[1] = 0, \quad x[2] = 1, \quad x[3] = 0, \quad x[4] = 1. \] Step 3: Apply the Z-transform formula. \[ X(Z) = x[0]Z^0 + x[1]Z^1 + x[2]Z^2 + x[3]Z^3 + x[4]Z^4 \] \[ = (1)(1) + (0)Z + (1)Z^2 + (0)Z^3 + (1)Z^4 \] \[ = 1 + Z^2 + Z^4. \] \[ \boxed{1 + Z^2 + Z^4} \]