Let an input x(t) = 2 sin(10πt)+ 5 cos(15πt) +7 sin(42πt) + 4cos(45πt) is passed through an LTI system having an impulse response,

h(t)=2(sin(10πt)/πt) cos(40πt)

The output of the system is

The correct answer is: 7 sin(42πt) + 4 cos(45πt)

To find the output of the Linear Time-Invariant (LTI) system, we can analyze both the input signal and the system's impulse response in the frequency domain.

Step 1: Identify the frequencies present in the input signal
The given input signal is:

$x(t)\; =\; 2\; sin(10\pi t)\; +\; 5\; cos(15\pi t)\; +\; 7\; sin(42\pi t)\; +\; 4\; cos(45\pi t)$

The angular frequencies ($\omega$) of the individual components of $x(t)$ are:
1. For $2\; sin(10\pi t)$: $\omega \_1\; =\; 10\pi$ rad/s
2. For $5\; cos(15\pi t)$: $\omega \_2\; =\; 15\pi$ rad/s
3. For $7\; sin(42\pi t)$: $\omega \_3\; =\; 42\pi$ rad/s
4. For $4\; cos(45\pi t)$: $\omega \_4\; =\; 45\pi$ rad/s

Step 2: Find the frequency response of the system
The impulse response of the system is given by:

$h(t)\; =\; 2\; (sin(10\pi t)\; /\; (\pi t))\; cos(40\pi t)$

Let us define a basic signal $g(t)\; =\; sin(10\pi t)\; /\; (\pi t)$.
The Fourier transform of $g(t)$ is a rectangular function in the frequency domain:

$G(\omega )\; =\; 1$ for $|\omega |\; \le \; 10\pi$, and $0$ otherwise.

Using the modulation property of the Fourier transform:

$F\{g(t)\; cos(\omega \_0\; t)\}\; =\; 1/2\; [G(\omega \; -\; \omega \_0)\; +\; G(\omega \; +\; \omega \_0)]$

Here, $\omega \_0\; =\; 40\pi$. Therefore, the frequency response $H(\omega )$ of the system is:

$H(\omega )\; =\; 2\; *\; 1/2\; [G(\omega \; -\; 40\pi )\; +\; G(\omega \; +\; 40\pi )]\; =\; G(\omega \; -\; 40\pi )\; +\; G(\omega \; +\; 40\pi )$

This indicates that the system is a bandpass filter. The passbands are:
- A positive frequency band centered at $40\pi$ spanning from $40\pi \; -\; 10\pi \; =\; 30\pi$ to $40\pi \; +\; 10\pi \; =\; 50\pi$.
- A negative frequency band centered at $-40\pi$ spanning from $-50\pi$ to $-30\pi$.
Thus, the system allows frequencies in the range $30\pi \; \le \; |\omega |\; \le \; 50\pi$ to pass with a gain of $1$, and completely blocks all other frequencies.

Step 3: Determine the system output
Now we check which input frequencies fall within the filter's passband ($30\pi \; \le \; |\omega |\; \le \; 50\pi$):
- For $\omega \_1\; =\; 10\pi$: Outside the passband, so this component is filtered out (output = 0).
- For $\omega \_2\; =\; 15\pi$: Outside the passband, so this component is filtered out (output = 0).
- For $\omega \_3\; =\; 42\pi$: Inside the passband, so this component passes through with a gain of 1.
- For $\omega \_4\; =\; 45\pi$: Inside the passband, so this component passes through with a gain of 1.

Combining the components that pass through the filter, the output $y(t)$ is:

$y(t)\; =\; 7\; sin(42\pi t)\; +\; 4\; cos(45\pi t)$