Convolution integral and fourier transform in linear response theory

[kgz]

Hello,

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.

 

[marcusl]

No, there is only one definition of convolution integral and it extends from -∞ to ∞. Specific times like 0 only have meaning in relation to the specifics of your problem (for instance, the input might be turned on at t=0).

 

[WannabeNewton]

As marcus noted, for your linear time-shift invariant system the convolution integral in generality must be $y(t) = \int_{-\infty}^{\infty}h(t - t')x(t')dt'$. Now if your system happened to be causal and you wanted the response at $t$ then you can write $y(t) = \int_{-\infty}^{t}h(t - t')x(t')dt'$ only because for a causal system, the response at $t$ will only depend on past contributions up to $t$ i.e. $h(t - t')$ vanishes for negative arguments. marcus explained the $t = 0$ part so I'll leave it at that.

 

[kgz]

Thank you guys for your responses.

 

[CellCoree]

Hello,

The first convolution integral is known as the causal convolution, while the second one is known as the non-causal or acausal convolution. The difference between the two is the limits of integration. In the causal convolution, the integration is from 0 to t, which means that the output at any given time t only depends on the input up to that time. This is related to the causality of the system, as the output can only be influenced by the input that came before it. On the other hand, in the non-causal convolution, the integration is from -∞ to t, which means that the output at time t can be influenced by the entire history of the input. This is not related to causality and can result in unrealistic outputs if the system is actually causal.

The choice of which convolution integral to use depends on the initial conditions of the system. If the system is initially at rest (i.e. no input or output), then both integrals will give the same result. However, if the system has some initial conditions, then the causal convolution should be used to take those into account. This is because the non-causal convolution assumes that the system has been operating since -∞ and therefore does not account for any initial conditions.

In terms of using Fourier transform and convolution integral theory to change from frequency domain to time domain, the choice of convolution integral would depend on the type of system you have. If your system is known to be causal, then the causal convolution should be used. However, if the system is non-causal, then the non-causal convolution would be more appropriate. As for not using Laplace transform, it is possible to use the Fourier transform and convolution integral to change from frequency domain to time domain, but it may result in more complex calculations and may not always be possible for all types of systems.

I hope this helps clarify the differences between the two convolution integrals and their importance in linear response theory. Thank you for considering my response.