# Use convolution integral to find step response of a system

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1. Apr 30, 2015

### TEEE

1. The problem statement, all variables and given/known data
An electrical network has the unit-impulse response :h(t)=3t⋅e-4t .If a unit voltage step is applied to the network, use the convolution integral to work out the value of the output after 0.25 seconds.

2. Relevant equations
Convolution integral: y(t)=f(t)*h(t)
Unit step function u(t) = 1 when t>=0, 0 when t<0

3. The attempt at a solution
This is a problem on the problem sheet of my circuit analysis course. The lecturer also gives a solution which is not complete:
v(τ)=∫0 1⋅3(τ-t)e(-4(τ-t)) dt
v(0.25) = 4.95×10-2V
I'm not sure how to proceed from the above equation to get the final answer. I have tried solving using convolution:y(t) =∫0h(τ)u(t-τ)dτ or y(t)=∫0u(τ)h(t-τ)dτ,but get something like 0 or -0.192. I guess I dealt with the u(τ) or u(t-τ) in the integral wrong as I'm not sure how to deal with it in an integral. I know that u(t) might affect the limit of the integral but I'm not sure how.(p.s.the∫ part above means integrate from ∞ to 0, if that causes any confusion)
Any help or advise would be really appreciated as I've been bothered by this for quite some time.Thanks!

2. May 2, 2015

### Zondrina

Let $h(t) = 3t e^{-4t}$ and $u(t)$ be the unit step function, then:

$$(h * u)(t) = \int_0^{\infty} h(x) u(t-x) \space dx = 3 \int_0^{0.25} x e^{-4x} \space dx$$

Integration by parts will yield the answer.

3. May 2, 2015

### TEEE

Thank you for your reply, it helped me find out where I went wrong in my approach to the solution. I changed the limit to infinity to 0.25 instead of 0.25 to 0 when evaluating (h*u)(t). Thanks again much appreciated.

4. May 6, 2015

### rude man

Actually, the complete convolution integral limits are -∞ to +∞. But the lower limit changes to zero because your h(τ) = 0, τ < 0, and the upper limit changes to t since U(t-τ) = 0 for τ > t. So then v(t) = 3∫0t τ e-4τ
and you can let t = 0.25 for your particular problem after performing the definite integral.

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