Signals & Systems: Determine System Output

[gamerw00t4]

Homework Statement

Given:
Impulse response:
h(t) = u(t-1) - u(t + 1) (where u(t) is the unit step function)

System input:
x(t) = t*u(t) - t*u(t-2)

Determine System output:
y(10) = ?
y(1) = ?
max y(t) = ?
|y(t)| is max at t = ?

Homework Equations

Convolution integral?
y(t) = _-∞∫^∞ x(τ)h(t-τ) dτ
I think... I just don't know what τ is, and do we really just multiply x(τ)h(t-τ) and integrate with respect to τ?

The Attempt at a Solution

I feel like I could be wrong because every time I think I have to do some integral problem I always try it and get some answer that isn't anything like what we are supposed to get.

How do the "u"'s multiply out?
[t*u(t) - t*u(t-2)][u(t-1) - u(t + 1)]
[t*u(t)*u(t-1)] - [t*u(t) * u(t + 1)] - [t*u(t-2) * u(t-1)] + [t*u(t-2) * u(t + 1)]
I foiled it out, now what?
[t*u(t²-t)] - [t*u(t²+t)] - [t*u(t²-3t+2)] + [t*u(t²-t-2)]
Now I have to integrate this? What about τ, don't we have to integrate with respect to τ? How do I get τ into the equation?

I'm just so confused, what if this isn't even right and I waste time integrating it? According to the equations I used, it seems like I'm being logical, and the next step would be to integrate, but I feel like solving this isn't difficult at all. Is this the convolution integral that we are talking about? or do we have to use a Fourier Series Transform before we integrate to make this easier?

I was watching some videos on it, and my professor showed us this too: they represented x(t) and h(t) graphically, apparently you flip h(t) and you shift it right, but 1.) they never said when you stop shifting. 2) they didnt explain why they were doing so. 3.) they didn't show you how to get the resulting y(t) (they jumped to what you get graphically for y(t) and they didn't even mention the y(t) expression). 4.) I think I'm supposed to do this all mathematically (without graphing) and I haven't seen an example like that
edit: I'm assuming that you flip it because in the equation h(t-τ), its -τ and you shift it t units to the right? See, I feel like this is probably what you do, but they never made this clear, so if I am wrong I could be going down the wrong path and be working on something that is incorrect...
It just seems so difficult to learn, yet its probably not difficult at all...

I tried this in so many different ways, all of which were off.

I just wish there was an example like this in the book so I could follow along... It just seems like the information is scattered everywhere and I have to piece it all together and there are no examples to know if I am doing it right. I want to learn, its just hard to learn when information is in little bits all over the place

Am I even on the right path? Any help would be appreciated.

 

[KataKoniK]

Dear scientist,

Thank you for sharing your struggles with understanding the convolution integral in response to the given forum post. It can be difficult to learn a new concept when the information is scattered and there are not enough examples to follow along. I would like to offer some guidance and clarification to hopefully help you understand the concept better.

Firstly, you are correct in your attempt at solving the problem by using the convolution integral. The convolution integral is used to find the output of a system when the input and impulse response are known. In this case, the impulse response is given as h(t) and the system input is x(t). The output, y(t), can be found by convolving x(t) and h(t), as shown in the equation you mentioned:

y(t) = ∫x(τ)h(t-τ) dτ

Now, to answer your questions about τ - it is a dummy variable used in the integral to represent the time variable. It is similar to how you use x or y in algebra to represent a variable. In this case, you can think of τ as representing the time at which the input and impulse response are being multiplied and integrated.

To solve the problem, you can follow these steps:

1. Rewrite the system input x(t) as a piecewise function using the unit step function u(t):

x(t) = t*u(t) - t*u(t-2)
= t*(u(t) - u(t-2))

2. Rewrite the impulse response h(t) as a piecewise function using the unit step function u(t):

h(t) = u(t-1) - u(t+1)

3. Substitute these expressions into the convolution integral:

y(t) = ∫x(τ)h(t-τ) dτ
= ∫[t*(u(τ) - u(τ-2))][u(t-τ-1) - u(t-τ+1)] dτ

4. Simplify the integrand by using the properties of the unit step function. For example, u(t-τ+1) = 0 for τ > t+1, and u(t-τ+1) = 1 for τ < t+1. Use this to break up the integral into different intervals.

5. Solve the integral for each interval and combine the results to get the final expression for y(t).

6. To find y(10), simply substitute t=10 into