# 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.

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?

• Thank you that clears it up. I can't believe that never crossed my mind. – user8044 Feb 25 '14 at 23:44

replace h(z) by the output by input form y(z)/x(z).now cross multiply by the numerator and denominator of the transfer function. now if suppose x(z)*pow(z,-1) appears in the time domain write it as x(n-1). finally bring it into y(n)=(combination of x(n) terms and y(n) terms)..

regards, phani tej