# 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!

• some systems are not invertible. if $H(s)$ has poles with non-negative real parts, then there is no stable inverse. now, if $V(z)$ has a sufficiently long impulse response and if a known and constant delay $\tau>0$ is no problem, then to within some error you can approximate $b[n] \approx x(n - \tau)$ – robert bristow-johnson Oct 9 '20 at 1:18

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 $$0 < |\epsilon| < 1$$, 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 "stair-step wave") 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.