What's the difference of Frequency Response?

Find Difference Equation From Frequency Response

• Given the frequency response $H (jω) = csc(ω)$ how would I go about finding the difference equation? I know I can find $h[n]$ from here but after that I feel like I'm missing something.

• Answer:

  so, if we're looking around for a discrete-time difference equation, let's modify the notation slightly to make it look like it's a frequency response for a discrete-time system: $$ \begin{align} H \left( e^{j \omega} \right) & = \csc(\omega) \\ & = \frac{1}{\sin(\omega)} \\ & = \frac{2j}{e^{j \omega} - e^{-j \omega}} \\ & = \frac{2j \ e^{-j \omega}}{1 - (e^{-j \omega})^2} \\ \end{align}$$ so i guess that means that $$H(z) = \frac{2j \ z^{-1}}{1 - z^{-2}}$$ so can you get a difference equation from that?