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Consider I have a linear time-invariant (LTI) system, with ##x(t)##, ##y(t)##, and ##h(t)##, as input, output, and impulse response functions, respectively.

I have two choices to write the convolution integral to get ##y(t)##:

$$ 1)\ \ \ y(t) = \int_{0}^{t} h(t-t')x(t')dt' $$

and

$$ 2)\ \ \ y(t) = \int_{-\infty}^{t} h(t-t')x(t')dt' .$$

What are the differences between these two, and are initial conditions important factors in the decision of choosing one of these? Is it related to causality of the system?

Also, suppose I am just given the frequency response of the system,

$$ Y(\omega)=H(\omega)X(\omega) . $$

Using Fourier transform and convolution integral theory, I want to change from frequency domain to time domain. Which one of the convolution integrals above should I pick?

Because I need to work in the freuqency domain, I am trying not to use Laplace transform.

Thank you for sharing your ideas.