Question

Let an input \( x(t) = 2\sin(10\pi t) + 5\cos(15\pi t) + 7\sin(42\pi t) + 4\cos(45\pi t) \) be passed through an LTI system having an impulse response, \[ h(t) = 2\left(\frac{\sin(10\pi t)}{\pi t}\right) \cos(40\pi t). \] The output of the system is

• A. ( 2\sin(10\pi t) + 5\cos(15\pi t) \)
• B. ( 5\cos(15\pi t) + 7\sin(42\pi t) \)
• C. ( 7\sin(42\pi t) + 4\cos(45\pi t) \)
• D. ( 2\sin(10\pi t) + 4\cos(45\pi t) \)

Hint:

In LTI systems, convolution in the time domain corresponds to multiplication in the frequency domain, and the output depends on the frequency response of the system.

Answer 1

The Correct Option is C

Step 1: Understanding the Convolution Operation.
For an LTI system, the output \( y(t) \) is the convolution of the input \( x(t) \) and the impulse response \( h(t) \). In the frequency domain, the output can be obtained by multiplying the Fourier transforms of the input and the impulse response. Step 2: Frequency Analysis.
Given that the impulse response contains a term \( \frac{\sin(10\pi t)}{\pi t} \), which corresponds to a low-pass filter, it will filter out higher frequencies. Therefore, the components of \( x(t) \) with frequencies close to \( 10\pi t \) will be filtered out, leaving the components at \( 42\pi t \) and \( 45\pi t \). Step 3: Conclusion.
The output of the system contains the terms \( 7\sin(42\pi t) \) and \( 4\cos(45\pi t) \), so the correct answer is (C).