I have read a paper that performs Channel Estimation of Wireless Channel as follows.
A training sequence of length $N$, lets call it $a_N(i)$ for $i\in[0:N-1]$ is repeated twice then sent out over the channel.
Assume the transmit sequence formed of this two sequences is denoted by $s_{CE}(t)$.
The received signal is given by (assuming channel of length $T_{CH}$) is $$r_{CE}(t) = \sum_{t'=0}^{T_{CH}-1}h(t')s_{CE}(t-t')+n(t)$$
The authors claim that if one takes the autocorrelation between a sequence $a_N(i)$ and the received signal then we can mathematically write the autorcorrelation as
$$R(t)= \frac{1}{N}\sum_{d=0}^{N-1}r_{CE}(t+d\underbrace{-N+1}_{????}) a_{N}^*(d) $$
My question is why is there a need for the term I have underbrace $-N+1$. I thought that the autocorrelation is in general expressed as
$$R(t)= \frac{1}{N}\sum_{d=0}^{N-1}r_{CE}(t+d) a_{N}^*(d) $$
Thanks looking forward for your view!
Update: The following is the reference
I particular (25) and (26) are my main concerns.