(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the following input-output relationship:

[tex] y(t) = \int_0^\infty e^{-\sigma}x(t-\sigma) d\sigma [/tex]

A) Is the system time-invariant?

B) Find the output y(t) when the input to the system is [tex] x(t) = \mid t \mid , -\infty < t < \infty [/tex]

2. Relevant equations

These are the equations to check for time invariance

A system with an input-output transformation y(*) = T[x(*)] is time invariant if for any t and [tex] \tau [/tex]

[tex] y(t) = T[x(t)] [/tex]

[tex] z(t) = x(t-\tau)[/tex]

[tex] T[z(t)] = y(t-\tau)

[/tex]

3. The attempt at a solution

Edit: Ok I think it's time to get some sleep. This problem was actually pretty simple and I totally screwed it up earlier.

[tex] z(t) = T[x(t-\tau)] = \int_0^\infty e^{-\sigma}z(t - \tau - \sigma) d\sigma [/tex]

[tex] y(t-\tau) = \int_0^\infty e^{-\sigma}z(t - \tau - \sigma) d\sigma [/tex]

therefore [tex] T[x(t-\tau)] = y(t-\tau) [/tex] , so it is time-invariant

Now how do I atempt part B? I'm not seeing how the given input can be included in the given output function. My main thought is to just substitute [tex] x(t-\sigma) [/tex] with [tex] x(t) [/tex]

My other minor thought is maybe I should use the change of variable rule from calculus to the y(t) fxn.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Checking for time invariance

**Physics Forums | Science Articles, Homework Help, Discussion**