Autocorrelation function between received and training signal used for channel estimation

Question

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

Link

I particular (25) and (26) are my main concerns.

Answer 1

I think it's just a trick which makes $R(t)=0, \mathrm{when}\quad t<0$.

First, $-N+1$ is just a time offset and not affect the essence of function.

$r_{CE}(t)$ is bounded to $[0, 2N+T_{CH}-2]$, and $a_N(t)$ to $[0, N-1]$. In the form you gave, $R(t)$ is bounded to $[-N+1, 2N+T_{CH}-2]$. May be the author want the non-zero value of $R(t)$ also starts at 0, so he just move it right with $N-1$ length, which causes the $-N+1$ term in the equation.

--------- Why $R(t)$ is bounded as that -------------

This is just a explanation to give some intuition, not a formal proof. Consider two function $f(t)$ and $g(t)$, which $f(t)=0, t \notin [a, b]$ and $g(t)=0, t\notin [c, d]$. Now we want to calculate the sum of products, i.e., $S=\sum_t f(t)g(t)$, $[a,b]$ and $[c,d]$ must be overlapped, right? Thus we get $a \le d $ and $c\le b$.

Get back to the $R(t)=\sum _d r_{CE}(t+d)a^*(d)$(note here the argument is $d$), $a^*(d)$ is bounded to $[0, N-1]$ and $r_{CE}(t+d)$ to $[-t, L-1-t]$, in which $L=2N+T_{CH}-1$ is the length of $r_{CE}(d)$. Here we get $a=0, b=N-1, c=-t, d=L-1-t$. Using $a \le d $ and $c \le b$, we get $-(N-1) \le t \le L-1$.

As to why $r_{CE}$ is bounded as that, you can make a same process to understand it.