I use a CDMA protocol for a multiuser communication. Let $c_i(t)$ the signature for the $i$-th user and $d_{i,k}$ the $k$-th modulated symbol for the $i$-th user.
At the reception, we have $$r(t)=\sum_{i=1}^{K}\sum_{n=1}^{N}d_{i,n}c_i(t-iT_s)\text.$$
To find the estimated symbol, I use a correlator but I have a problem. In fact, in the litterature, I have these relations:
$$\hat{d}_{i,k}=\int_0^{T_s}c_i^{\ast}(t)r(t)dt$$
and
$$\hat{d}_{i,k}=\int_{(k-1)T_s}^{kT_s}c_i^{\ast}(t)r(t)dt\text.$$
If $k=1$, we have an equality between the formulas but for $k=2$, it's different.
What is the difference between these formulas? Do you have a mathematical proof for the equivalence between the formulas?
