Question

A continuous-time signal \(x(t) = 2 \cos(8\pi t + \pi/3)\) is sampled at a rate of 15 Hz. The sampled signal \(x_s(t)\) when passed through an LTI system with impulse response: \[ h(t) = \frac{\sin(2\pi t)}{\pi t} \cos(38\pi t - \pi/2), \] produces an output \(x_0(t)\). The expression for \(x_0(t)\) is \(\_\_\_\_\).

• A. \(15 \sin(38\pi t + \pi/3)\)
• B. \(15 \sin(38\pi t - \pi/3)\)
• C. \(15 \cos(38\pi t - \pi/6)\)
• D. \(15 \cos(38\pi t + \pi/6)\)

Hint:

When working with sampled signals, always consider aliasing effects and analyze the system's impulse response to identify the dominant output frequency and corresponding phase.

Answer 1

The Correct Option is C

Step 1: Sampling and aliasing analysis.
The given signal \(x(t) = 2 \cos(8\pi t + \pi/3)\) is sampled at a frequency \(f_s = 15 \, {Hz}\). Sampling results in aliased components at frequencies shifted by multiples of the sampling rate. Among these, the dominant aliased component appears at \(f = 38 \, {Hz}\). Step 2: Determining output frequency and phase.
The impulse response of the LTI system is given by: \[ h(t) = \frac{\sin(2\pi t)}{\pi t} \cos(38\pi t - \pi/2). \] This filters and processes the aliased component. The resulting output signal is: \[ x_0(t) = 15 \cos(38\pi t - \pi/6). \] Final Answer: \[ \boxed{{(3) } 15 \cos(38\pi t - \pi/6)} \]