I'm trying to solve a problem about filter design.
We want to design a equalizer filter which have the following frequency response :
How can i get the difference equation from the impulse responce which is defined by:
$$ \frac{2 f_1}{F_s} \left ( {\rm sinc}(2 \pi f_1 n T_s) (g_1-g_2) \right )+ \frac{2 f_2}{F_s} \left ( {\rm sinc}(2 \pi f_2 n T_s) (g_2-g_3) \right )+ \frac{2}{F_s} \left ( f_3 g_3 {\rm sinc}(2 \pi f_3 n T_s)-f_0\cdot g_1 {\rm sinc}(2 \pi f_0 n T_s)\right ) $$
How can i get the difference equation from the impulse response
That's easy enough to do
$$ y[k] = \sum_{n=-\infty}^\infty x[k-n] \cdot \left[ \frac{2 f_1}{F_s} \left ( {\rm sinc}(2 \pi f_1 n T_s) (g_1-g_2) \right )+ \frac{2 f_2}{F_s} \left ( {\rm sinc}(2 \pi f_2 n T_s) (g_2-g_3) \right )+ \frac{2}{F_s} \left ( f_3 g_3 {\rm sinc}(2 \pi f_3 n T_s)-f_0\cdot g_1 {\rm sinc}(2 \pi f_0 n T_s)\right) \right] $$
However, that doesn't help you much since it's an infinite sum that's also non-causal, so you can't implement it.
If you want something practical, you need to deploy any of the standard filter design algorithm for this type of problem (least square IIR or FIR, inverse DFT, windowing, etc.) to approximate the target response. The best method depends a lot on your specific requirements and what type of error you can tolerate.
