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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?

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    $\begingroup$ 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 $(k-1)T_s$. $\endgroup$
    – Marcus Müller
    Feb 16 '20 at 17:41
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The formula is how you estimate the symbol at the receiver. It is simply the dot product of the spreading sequence (conjugate) and the received sequence for the duration of the spread symbol. The k is used to define subsequent symbols at the receiver.

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