# Calculate Z transform of a sum

I'm new on the forum and hence not familiar with formal writing of a post so before going in on this post I wanted to make this clear.

I'm an second year engineering student with a small knowledge of stability, discrete systems, continuous systems, laplace-transform, fourier-transform etcetera. This year I'm taking control system classes and we have given a system of the following discrete signal: $$u_{0}, u_{1}+\frac{u_{0}}{2}, u_{2}+\frac{u_{1}}{2},+\frac{u_{0}}{4},...,u_{k}+\frac{u_{k-1}}{2}+...+\frac{u_{0}}{2^k}$$

It is asked to design a filter system to reconstruct the initial signal. Therefore, we need to find the z-transform of the given signal. That's at least my plan. From the z-transform can other properties of the system be found: stability, poles, zeros, block-scheme, impulse response, ...

But there is my problem: I have rewritten the given signal like this: $$u[k]=\sum_{i=0}^{k}\frac{u_{i}}{2^{k-i}}=\frac{1}{2^{k}}\sum_{i=0}^{k}{u_{i}2^{i}}$$. But now I don't have a clue on how to derive the z-transformation. I've tried some other things like rewriting the formula, however it did not work in finding the z-transform.

What can I do?

Extra question: If the z-transform is found of this equation, can this be considered as the transfer function of the filter system?

EDIT: Due to a good suggestion in the comment section, I have written out a small block diagram and derived the transfer function, but I still have trouble finding a way to design a filter from the given question because the coefficients of $$u$$ are not further defined. Here I give my new insights:

• Note that you are not obliged to use the $z$ transform to get the inverse filter equation... Homework constraint? Commented Mar 5, 2022 at 11:51
• No, it is not a homework constraint. It was just intuitively in my mind better to use a z transform. Due to your suggestion I have written out a block diagram (I updated my question with my new notations), but I have no idea how to derive a filter from this diagram because the given u-constants are not further defined. Commented Mar 5, 2022 at 12:26

Let us define the $$Z$$ transform by

$$U(z) = \sum_{n=0}^{\infty}u_n z^{-n}$$

$$V(z) = \sum_{n=0}^{\infty}v_n z^{-n}$$

Here we implicitly assume that $$u_n=0, v_n=0$$ for $$n<0$$ (causal signals).

From the relation $$u_{0}, u_{1}+\frac{u_{0}}{2}, u_{2}+\frac{u_{1}}{2},+\frac{u_{0}}{4},...,u_{k}+\frac{u_{k-1}}{2}+...+\frac{u_{0}}{2^k}$$, we can directly derive

$$v_k=u_k+ \frac{1}{2}v_{k-1}$$

$$u_k = v_k - \frac{1}{2}v_{k-1}$$

and then that

$$U(z)=\left(1-\frac{z^{-1}}{2}\right)V(z)$$

$$V(z)=\left(\frac{1}{1-\frac{z^{-1}}{2}}\right)U(z)$$

• I guess the there is a typo in the final expressions. That should be $z^{-1}$ instead of $z$, no?
– KBS
Commented Mar 5, 2022 at 16:02
• @KBS Corrected. Thank you. Commented Mar 5, 2022 at 16:55
• No worries. Here to help.
– KBS
Commented Mar 5, 2022 at 16:58