Time invariant Green's function (inpulse response)

[fisico30]

Hello Forum,

given a input=delta located at time t=0, the system will respond generating a function h(t).

If the delta is instead located at t=t0 (delayed by tau), the system will respond with a function g(t)=h(t-tau), just a shifted version of the response for the delta a t=0...

If this is the case, the system is time invariant and the impulse response is said to be only a function of t-t0...: h(t-t0)

If the system was time variant instead, it will be a function h(t,t-t0), that is, a function of both time t and t-t0...it is as if it was a function of two variables...

I am not clear on this: isn't the function h, the impulse response, a function of time t also in the case of time invariant system?
To be time invariant, does the variable t need to always occur with t0 in a subtraction?

thanks,
Fisico30

 

[kzhu]

My understanding is that once you have proved that the system is time invariant, i.e. h(t) = h(t-t0), you can safely drop t0 in the expression and simply state the impulse response is h(t).