Upsampling and zero order hold Maybe to someone this question may seem quite easy but I have a problem grasping it. Namely, assume that we have two systems A and B with sampling rates $f_A$ and $f_B$ where $f_A&gt;f_B$ and $R=\frac{f_A}{f_B}$. If we upsample system B by $R$ (i.e., $f_B=f_A$) using the zero order hold interpolation what is the relation between the time samples before ($n_{old}$) and after ($n_{new}$) upsampling? More precisely, we defined $\mathcal{N}_{old}=\{1,2,...,n_{old},..N_{old}\}$ and $\mathcal{N}_{new}=\{1,2,...,n_{new},..,R*N_{old}\}$ as sets of time samples before and after upsampling. Based on the definition of ZOH  following relation should hold:
$$
n_{old}=\frac{n_{new}}{R}
$$
i.e., every $R^{th}$ sample from $\mathcal{N}_{new}$ corresponds to the one time sample from $\mathcal{N}_{old}$ 
but if we for example take some $n_{new}$that is not multiple of $R$ we will get that $n_{old}$ is not an integer what doesn't make sense. What am I doing wrong?
Assuming a digital (discrete-time) sample rate conversion (upsampling) operation carried on old samples of a signal x[n] to upsample it by an integer factor R  by the utilization of a zero-order hold interpolation filter, you would essentially replicate (repeat) every new and empty (R-1) samples with the single old sample that they originate from. I guess the problem you have stems from the fact that you misinterpret the relation:
$$
n_{old}=\frac{n_{new}}{R}
$$

which is correct for continuous variables but not if you consider the fact that both $n_{old}$,$n_{new}$,and $R$ are integers. Therefore it cannot hold true for those samples $n_{new}$ which are not integer multiples of $R$

To remedy your confusion with the ZOH output index calculation, you shall better use the following interpretation as a relation between new and old sample indices:
$$ n_{old} = \lfloor \frac{n_{new}}{R} \rfloor$$

Where floor, with brackets $\lfloor$, $\rfloor$ is the mathematical operation that returns the largest integer that is less than or equal to its argument. 

Or you could, equally, define a ruling to map old and new samples as:
index $n_{old}$ is equivalent to the integer $K$, for all those $n_{new}$ that satisfy $K R \leq n_{new} &lt; (K+1)R$