# Signals and Systems - LTI - Transforms - Impulse Response

I have $$x(t)-> LTI -> y(t)$$ where $$LTI=h(t),H(jw)$$.

and

$$H(jw)=ab/((a+jw)(b+jw))$$ where a and B are real numbers.

I am wanting to find the impulse response $$h(t)$$ as well as the input/output differential equation to the system.

If my understanding is correct I would want to take the Fourier transform of the frequency response/input and I am not sure where to go from there.

• Please write a better title. The topic of this whole site is signals and systems. Your title must be more specific to your problem. – Marcus Müller Oct 18 '18 at 18:55
• Okay. I'll update. First time posting. – Orthanc Oct 18 '18 at 19:23

## 1 Answer

$$H(jw) = \frac{output}{input} = \frac{Y(jw)}{X(jw)}$$

$$\frac{Y(s)}{X(s)} = \frac{ab}{(a+s)(b+s)}$$

$$s=jw$$

Then you can use some of the notions from this example to solve for your impulse response / differential equation. Hope that helps!

• I assume I could also take the inverse Laplace in order to find h(t) correct? – Orthanc Oct 18 '18 at 19:02
• Correct. the inverse laplace of H(s) will give you h(t). – spet Oct 18 '18 at 19:57