# Filter Synthesis Formula to implement ZOH

$$h_n = \frac{1}{2\pi}\int_{-\pi}^\pi H_d(\omega) e^{jn\omega}\,\mathrm d\omega$$

I am attempting to design a FIR filter using the formula above for coefficients. Where

$$H_d(\omega) = \frac{\sin \omega}{\omega}.$$

But in doing this I find it is necessary to evaluate the integral

$$\int\frac{e^{j\omega(1+n)}}{\omega}\,\mathrm d\omega.$$

Which I simply cannot do. Is there another way to find the coefficients necessary to perform ZOH sampling while approximating the ideal low pass filter to be the sinc function, or a mistake I am making?

• Have you tried using the discrete formula for the sinc()? Oct 21 '21 at 19:26
• hm, you definitely know the time shape of a zero-order hold filter, right (it's in the name!)? This feels like you're doing it backwards... Oct 21 '21 at 22:20