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).

 

[astrokreng]

I can provide clarification on the concept of time invariance in the context of Green's function or impulse response.

Firstly, a time-invariant system is one in which the response to a given input does not change over time. This means that for a given input, the output will always be the same regardless of when the input is applied. This is in contrast to a time-variant system, where the response may change over time.

In the case of Green's function or impulse response, the function h(t) represents the output of the system for a delta input at time t=0. This means that the function h(t) is only a function of time t, as it is solely dependent on the input at t=0.

If the input is delayed by a time tau, the output will be a function g(t)=h(t-tau), which is simply a shifted version of h(t) by tau. This shows that the response of the system is still only dependent on the time difference between the input and the output, and not on the absolute time. Therefore, the impulse response or Green's function is still only a function of t-tau, and not t-tau-t0.

In the case of a time-variant system, the function h(t) would not only depend on the input at t=0, but also on the absolute time t. This means that the response would be a function of both t and t-tau, as you mentioned.

In summary, the time invariance of a system is determined by the dependence of the response on time. For a time-invariant system, the response is only dependent on the time difference between the input and output, while for a time-variant system, the response is dependent on both the absolute time and the time difference.