Question

The radian frequency value(s) for which the discrete-time sinusoidal signal: \[ x[n] = A \cos(\Omega n + \pi/3) \] has a period of 40 is/are \(\_\_\_\_\).

• A. \(0.15\pi\)
• B. \(0.225\pi\)
• C. \(0.3\pi\)
• D. \(0.45\pi\)

Hint:

For discrete-time sinusoidal signals, ensure that the radian frequency satisfies the periodicity condition \( \Omega = 2\pi m / N \) to identify valid periods.

Answer 1

The Correct Option is A

Step 1: Condition for periodicity.
In discrete-time signals, the fundamental period \(N\) satisfies the relationship: \[ \Omega = \frac{2\pi m}{N}, \quad {where } m { and } N { are integers}. \] Given \(N = 40\), solve for the radian frequencies \(\Omega\) that fulfill this condition. Step 2: Validate the options.
Check which values of \(\Omega\) satisfy the periodicity condition with \(N = 40\). The valid frequencies are: \[ \Omega = 0.15\pi \quad {and} \quad \Omega = 0.45\pi. \] Final Answer: \[ \boxed{{(1, 4)}} \]