Find the impulse response from the frequency response

Question

So I'm having a problem here which gives me the frequency response and asks for the impulse response:

$H(\Omega ) = e^{-j\frac{\pi }{2}}$ for $\Omega>0 $ and $H(\Omega ) = e^{j\frac{\pi }{2}}$ for $\Omega<0$

I need to find the impulse response. My first thought was to (obviously) apply Inverse Fourier Transform but I'm thinking that since it's a branch function maybe that's not the correct approach. Also, I see that the two forms look a lot like each other so maybe I can somehow combine them into one and THEN apply the Transform. Any thoughts and guidance?

Answer 1

HINT: (because this is a homework type problem)

Using the sign function the frequency response can be expressed as

$$H(\Omega)=-j\,\textrm{sgn}(\Omega)\tag{1}$$

Direct computation of the inverse Fourier transform is tricky because of convergence issues of the corresponding integral.

If you can use Fourier transform tables, find the inverse Fourier transform of the step function $u(\Omega)$ in the frequency domain. If you can't find it, use the Fourier transform of the step function in the time domain $u(t)$, and figure out a simple transformation to obtain from it the inverse Fourier transform of $u(\Omega)$. Then express $\textrm{sgn}(\Omega)$ in terms of the step function $u(\Omega)$.

As a final hint, that system is called a Hilbert transformer.