# correlator for cdma

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?

• Hi welcome. Quite obviously, these two formulas just come from different time offsets. The upper formula assumes your symbol started at time 0 (and you're likely not fully giving the same notation as the books that's from, since your $\hat d_{i,k}$ still depends on $k$, but that doesn't appear on the right hand of the equation). The second formula considers the $k$. symbol starts at $(k-1)T_s$. Feb 16 '20 at 17:41