Problem designing a specific filter I have the next problem.



$H_{c1}(j\omega )$ is the ideal antialising filter and $H_{c2}(j\omega )$ is a real one. I'm asked to design $H(e^{j\Omega })$ so that $y[n]$ in the second diagram (the one on the right) is exactly the same as $x[n]$ in the first diagram (the one on the left).

I did something but I'm pretty sure that it isn't right. I found the equation for $H_{c2}(j\omega )$ as a function of $\omega$. Then I normalized the $\omega$-axis due to the A/D converter and divided the magnitude by $T$. If I didn't make any mistakes while doing the algebra, we get that

$$H_{c2}(e^{j\Omega })=\frac{9}{10}\cdot \frac{1}{\pi - \omega _{p}T}\cdot \Omega + \frac{9}{10}\cdot \frac{1}{\pi - \omega _{p}T}\cdot \pi + \frac{1}{10}\hspace{0.5cm} for\hspace{0.5cm} -\pi &lt; \Omega &lt; -\omega _{p}T$$

After that, I just thought of finding the inverse function of $H_{c2}(e^{j\Omega })$ for $-\pi &lt; \Omega &lt; -\omega _{p}T$... And that was it. Let $B(\Omega)=\frac{1}{\frac{9}{10}\cdot \frac{1}{\pi - \omega _{p}T}\cdot \Omega + \frac{9}{10}\cdot \frac{1}{\pi - \omega _{p}T}\cdot \pi + \frac{1}{10}}$. Then,

$$H(e^{j\Omega }) =
\left\{
 \begin{array}{l1}
  B(\Omega )  &amp; \mbox{if } -\pi &lt;\Omega &lt;-\omega _{p}T \\
  1 &amp; \mbox{if } -\omega _{p}T &lt; \Omega&lt;\omega _{p}T \\
B(-\Omega) &amp; \mbox{if } \omega _{p}T &lt; \Omega&lt;\pi
 \end{array}
\right.$$

The problem with this is that I'm pretty sure that the filter I "designed" has no antitransform. I think that there is a smarter way of approaching this exercise but I just can't see it.
so that $y[n]$ in the second diagram (the one on the right) is exactly the same as $x[n]$ in the first diagram (the one on the left).


Considering that $H_{c1}$ is per your very own definition $\ne H_{c2}$ means that without restricting your system's input $r_c$, there's nothing you can do. You will have to implement a filter with transfer function $H_{c1}$ if you need a system with transfer function  $H_{c1}$; there's no way around that.

Now,  $H_{c1}$ being impossible with finitely long filters (as it is infinitely sharp in frequency domain), this is impossible.