Inverting a sampled system I'm doing some self-study for an upcoming exam and came across the following question: 
My first idea to solve it was using the bilinear transform to get some approximation of $H(Z)$ (or just using the plain sampled version of the $H(s)$) then $V(Z)$ would be $H^{-1}(Z)$ to invert the effect of $H$.
Is this the correct approach? If yes, is there any other approach to this type of problem, like not using inverses?
Thanks for the help!
Can't be done.  Next question?
First, the author does not define $x[n]$.  Let's assume, then, that $x[n] = x(nT_s)$, where $T_s$ is the sampling interval; i.e. $f_s = 1/T_s$.
So assume that $x_1(t) = x_1[n]\ \delta \left(t - \left(n + \epsilon\right) T_s \right)$, where $\epsilon$ is any number where $0 &lt; |\epsilon| &lt; 1$, and $x_2(t) = x_2[n] \left( u \left(t - T_s \left(n - \frac{1}{2}\right) \right) - u \left(t - T_s \left(n + \frac{1}{2}\right) \right)\right)$.
Basically, $x_1(t)$ is a string of Dirac delta functionals that do not land on $t = n T_s$, where $x_2(t)$ is a string of rectangular pulses (or a &quot;stair-step wave&quot;) where the changes don't land on $t = n T_s$.  I contend without proof* that for any given signal $b[n]$, both an $x_1[n]$ and an $x_2[n]$ could be found that would cause the system to generate $b[n]$.  These $x_1[n]$ and $x_2[n]$ would have the following properties:

In general, $x_1[n] \ne x_2[n]$.
In the case where $x(t) = x_1(t)$, $x[n] = x(nT_s)$ would be zero for all $n$
In the case where $x(t) = x_2(t)$ (if I've dotted my 'i's and crossed my 't's right), $x[n] = x(nT_s) = x_2[n]$ would be true for all $n$.

Since you can have more than one version of $x(t)$ that leads to the same $b[n]$, there is no inverse.  You can define some constraints on the form of $x(t)$ that depend on some $x[n]$, as I did above, and if you do it properly you can then find an inverse system from $b[n]$ to $x[n]$ -- but because the sampling process loses information, you can never reconstruct some generic $x(t)$.
It could be that the author was just implicitly thinking (or stated explicitly, and you missed it), that $x(t)$ was constrained to be bandlimited to $f_s/2$ (i.e., the Nyquist criteria).  If that's the case, and if the relationship between $x(t)$ and $x[n]$ were defined then you could invert the system.
* Proof is left an an exercise for the reader, heh heh heh.