Checking for time invariance

[teknodude]

Homework Statement

Consider the following input-output relationship:
$$y(t) = \int_0^\infty e^{-\sigma}x(t-\sigma) d\sigma$$

A) Is the system time-invariant?
B) Find the output y(t) when the input to the system is $$x(t) = \mid t \mid , -\infty < t < \infty$$

Homework 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 $$\tau$$

$$y(t) = T[x(t)]$$

$$z(t) = x(t-\tau)$$

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

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.

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

$$y(t-\tau) = \int_0^\infty e^{-\sigma}z(t - \tau - \sigma) d\sigma$$

therefore $$T[x(t-\tau)] = y(t-\tau)$$ , 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 $$x(t-\sigma)$$ with $$x(t)$$

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

 

[e(ho0n3]

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 $$x(t-\sigma)$$ with $$x(t)$$

Since x(t) = |t|, then $x(t - \sigma) = |t - \sigma|$. So, all you have to do is evaluate:

$$y(t) = \int_0^\infty e^{-\sigma} |t - \sigma| \, d\sigma$$

 

[piercas]

I would approach this problem by first verifying whether the given system is time-invariant or not. To do this, I would follow the given equations and substitute the given input and output functions into them. From my calculations, it seems that the system is indeed time-invariant, as the output function remains the same when the input function is shifted by a time delay.

For part B, I would approach it by substituting the given input function into the output function and solving for y(t). This can be done by using the properties of integrals and making use of the given limits. After solving for y(t), I would check if the output function satisfies the input-output relationship given in the problem. If it does, then my solution is correct.

In summary, as a scientist, I would approach this problem by following the given equations and using mathematical techniques to verify the time-invariance of the system and to find the output when a specific input is given. I would also double check my solution to make sure it satisfies the given input-output relationship.

 

[piercas]

I would approach this problem by first checking the definition of time-invariance and understanding the equations provided. From the given input-output relationship, it is clear that the system is time-invariant because the output y(t) is dependent on the input x(t) and not on the specific time t. This means that if the input is shifted by a certain amount of time, the output is also shifted by the same amount of time.

For part B, we need to find the output y(t) when the input is x(t) = |t|. To do this, we can substitute x(t) into the given output function and solve for y(t).

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

= \int_0^\infty e^{-\sigma}|t-\sigma| d\sigma

Since the input is an absolute value function, we can split the integral into two parts based on the sign of t-\sigma.

= \int_0^t e^{-\sigma}(t-\sigma) d\sigma + \int_t^\infty e^{-\sigma}(\sigma-t) d\sigma

= \frac{te^{-t}}{2} + \frac{e^{-t}}{2} + \frac{e^{-t}}{2}

= e^{-t}

Therefore, the output y(t) when the input is x(t) = |t| is simply e^{-t}.