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If I'm given an LTI system. And I'm supposed to find the input $x(t)$, given $y(t)$ and $h(t)$. Am I able to use this relationship:
$$X(F) = \frac{Y(f)}{H(f)}$$
Another question I have is if I'm given $$y(t) = x_1(t) + x_2(t)$$ Does the following hold in the frequency domain:
$$Y(f) = H(f) + X(f)$$
NOTE: $y(t) = x_1(t) + x_2(t)$ Could just be a mistake in the assignment. I think a convolved sign should be there and not addition. I'm not too sure.
The Fourier transform is linear, so a linear combination of multiple signals in the time domain also results into a linear combination in the frequency domain. So the Fourier transform of,
$$ y(t) = a_1\, x_1(t) + a_2\, x_2(t), $$
yields the following relation in the frequency domain,
$$ Y(f) = a_1\,X_1(f) + a_2\,X_2(f), $$
with $G(f)=\mathcal{F}\left\{g(t)\right\}(f)$.
However if instead you mean convolution, instead of addition, then you can use the fact that convolution in the time domain translates to multiplication in the frequency domain (the opposite is also true). So your example then becomes,
$$ y(t) = x_1(t)\, \large{*} \, \normalsize{x_2(t)} \equiv \int_{-\infty}^{\infty} x_1(\tau)\, x_2(t-\tau)\,d\tau, $$
$$ Y(f) = X_1(f)\,X_2(f). $$
