# Step response given impulse response $h = e^{-|t|}$

I was given the above homework problem and I know how the step response is achieved, but I'm not sure how to set up the integrals in order to evaluate the problem. I've attached my work, but it is wrong because I keep getting the wrong answer. The solution is given as

$$S(t) = u(-t)e^t + u(t)(2-e^{-t})$$

*Note: We have not yet covered laplace transform. In my class, we are expressing the step response as running sums/integrals of the impulse response.

• Given what is mentioned in your question, can you please add a representative plot of the step function and a plot of the given $h$?
– A_A
Feb 13 '18 at 5:34
• Hi A_A, the step function is 1 for all t equal to or less than 0, and the plot is rudimentary. The plot of the step function or impulse response h is not given, as this is a completely analytical problem. Feb 13 '18 at 13:20
• edit: step function is 1 when t is equal to or greater than 0. Feb 13 '18 at 23:22
• No worries. The way the question was phrased initially, I thought it might helped if you simply sketched what it looks like.
– A_A
Feb 13 '18 at 23:25

HINT: In your work you implicitly assume that $t\ge 0$, so your solution is (probably) correct for $t\ge 0$, but you must also consider the case $t<0$.
• @LeeJordan: No, you have to integrate from $-\infty$ to $t$, and if $t <0$ that interval does not include the value zero. Feb 13 '18 at 14:14