Question

A single-tone real signal \( x[n] \) has its 8-point DFT denoted by \( X(k) \) which has \( X(2) = 2 \). Then, the signal \( x[n] \) will be equal to

• A. \( e^{j \frac{4 \pi n}{2}} \)
• B. \( 2 \cos \left(\frac{\pi n}{2}\right) \)
• C. \( 2 \sin \left(\frac{\pi n}{2}\right) \)
• D. \( 4 \cos \left(\frac{\pi n}{2}\right) \)

Hint:

Recall that the frequency components of DFT are given by \( \frac{2\pi k}{N} \) for \( k = 0, 1, 2, \ldots N-1 \).

Answer 1

The Correct Option is B

- The DFT coefficient \( X(2) = 2 \) implies that the corresponding frequency component is at \( k = 2 \) of an 8-point DFT. - This corresponds to a cosine signal with frequency \( \frac{2\pi k}{N} \), where \( N = 8 \), giving a frequency of \( \frac{\pi}{2} \).
Conclusion: The correct signal is given by option (b).