Time variant Green function

[fisico30]

Hello forum,

in the case we are in the time domain t, the Green function is a function of time.

In the translation variant case, how do we express and separate the regular time variability of the Green function from the the time variability of its functional form?

1) What is the best notation? G(t, t0) or G(t,tau), G(tau, t0) ? tau=t-t0.

(The Green function is simply the response to a delta impulse. If the delta occurs at t=0, then we get a Green function g(t). If the delta occurs later, delta(t-t0), then the green function is f(t), not equal to g(t-t0).)

2) In taking the Fourier transform of G(t, t0), we can do it with respect to t or t0 (or both).
We would get H(t,w) or H(t0,w) or H(w1, w2)... (transfer function).

what is the different between H(t,w) and H(t0,w) from a physical point of view?
In H(w1,w2) , both w1 and w2 are temporal frequencies. How are they different? Does w1 represent the frequency of the input spectral component and w2 the frequency due to the functional variability of the green function ?

Any comment, clarification, reassurance?

thanks,
fisico30

 

[jvicens]

Dear fisico30,

Thank you for your question regarding the Green function and its time variability. In the translation variant case, the Green function can be expressed as G(t, t0) or G(t, tau) where tau = t - t0. Both notations are commonly used in the scientific community and it ultimately depends on the preference of the researcher. However, it is important to note that G(t, t0) represents the response at time t due to an impulse occurring at time t0, while G(t, tau) represents the response at time t due to an impulse occurring tau seconds ago.

In terms of separating the regular time variability of the Green function from its functional form, this can be achieved by taking the Fourier transform. As you mentioned, we can take the Fourier transform with respect to t, t0, or both. H(t, w) represents the transfer function between the input signal and the output signal at time t, while H(t0, w) represents the transfer function between the input signal and the output signal at time t0. The difference between H(t, w) and H(t0, w) is that they represent the transfer function at different time points, thus capturing the temporal variability of the Green function.

In the case of H(w1, w2), both w1 and w2 represent temporal frequencies. However, they have different interpretations. w1 represents the frequency of the input signal, while w2 represents the frequency due to the functional variability of the Green function. This means that w2 captures the frequency components that are specific to the Green function, while w1 captures the overall frequency components of the input signal.

I hope this clarifies your questions and provides some reassurance. If you have any further questions, please don't hesitate to ask.