Given a discrete signal $ r(nT_c)$ specified at a rate $T_c$, assume we want to resample with a 3/2 rate change to match the sampling rate of some other signal.
We would like to upsample it by 3, then filter it by filter of length $K$ and then downsample by 2.
Also assume we have the following relationship exists $$ \frac{3}{2}T_s = T_c$$
Step 1: Upsample then we have
$r'(n\frac{Ts}{2}) = \left\{ \begin{array}{ll} r(n\frac{Ts}{3}) & \mbox{if $n = 0,3,6$};\\ 0 & \mbox{if OW}.\end{array} \right.$
Step 2: Filter
$\hat{r}(n\frac{Ts}{2})= \sum_{k=0}^{K-1}r(\frac{(n-k)T_s}{2})h_{filter}(k)\,\,\,\,\,\,\,n=0,1....$
Step 3:
$r_o(nT_s)= \hat{r}(n\frac{Ts}{2}-\underbrace{\frac{K-1}{2}\frac{T_s}{2}}{???}) \,\,\,\,\,n=0,1,$
I don't understand two of the above steps
1) why do we need use a filter to resample and
2) why isnt it the part underbrace $$\hat{r}(n\frac{Ts}{2}) \,\,\,\,\,n=0,1,....$$
Thank you very much.