Question

Let a discrete time signal \( x(n) \) has Z-transform \( X(z) = \frac{1}{1+2 z^{-1}}, |z|>2 \). If its Fourier transform is denoted as \( X(e^{j\omega}) \), then

• A. \( X(e^{j\omega}) = \frac{1}{1+2 e^{j\omega}} \)
• B. \( X(e^{j\omega}) = \frac{1}{j\omega+2} \)
• C. \( X(e^{j\omega}) = \frac{1}{1+2 e^{-j\omega}} \)
• D. \( X(e^{j\omega}) \) does not exist

Hint:

The Fourier transform of a discrete-time signal can be derived from its Z-transform by setting \( z = e^{j\omega} \).

Answer 1

The Correct Option is C

- The Fourier transform of a discrete-time signal \( x(n) \) is obtained by substituting \( z = e^{j\omega} \) in its Z-transform. 

- Given \( X(z) = \frac{1}{1+2 z^{-1}} \), substituting \( z = e^{j\omega} \) gives \[ X(e^{j\omega}) = \frac{1}{1+2 e^{-j\omega}} \] Conclusion: The correct answer is option (c).