Find Difference Equation From Frequency Response
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:

$$  \begin{align}
H \left( e^{j \omega} \right) &amp; = \csc(\omega) \\
&amp; = \frac{1}{\sin(\omega)} \\
&amp; = \frac{2j}{e^{j \omega} - e^{-j \omega}} \\
&amp; = \frac{2j \ e^{-j \omega}}{1 - (e^{-j \omega})^2} \\
\end{align}$$

so i guess that means that
$$H(z) = \frac{2j \ z^{-1}}{1 - z^{-2}}$$

so can you get a difference equation from that?