Confusion related to PN sequence : Terminology and application in statistical signal processing
If the output $x[\cdot]$ of a discrete-time linear system is related to the input via
$$x[n] = \sum_{i=0}^m h[i]u[n-i] \tag{1}$$
where $u[\cdot]$ is a periodic sequence with period $N \gg m$, then $x[\cdot]$ is also a sequence with period $N$ and
the
periodic cross-correlation function $R_{x,u}[\cdot]$ between $x[\cdot]$
and $u[\cdot]$ is the periodic convolution between the sequence $h[\cdot]$
and the periodic autocorrelation function $R_{u,u}[\cdot]$. That is,
$$R_{x,u} = x \star u = (h * u) \star u = h * (u \star u) = h* R_{u,u}
\tag{2}$$
  If
$u[\cdot]$ is a binary pseudonoise (PN) a.k.a. pseudorandom noise (PRN) sequence that takes on values $\pm 1$, then 
$$R_{u,u}[n] = \begin{cases}N, &amp; n \equiv 0 \bmod N,\\
-1, &amp;\text{otherwise}\end{cases}\tag{3}$$
which is essentially a periodic digital impulse function,
 and by ignoring that $-1$ as inconsequential compared to $N$, we get from $(2)$ that
$$N h[n] = R_{x,u} \implies h[n] = N^{-1} R_{x,u}[n].\tag{4}$$
If we include additive white Gaussian noise $w[n]$ 
of variance $\sigma^2$ on the right side of $(1)$,
then $(2)$ becomes 
$$R_{x+w,u} = (x + w)\star u = (h * u + w) \star u = h * (u \star u) 
+ w \star u= h* R_{u,u} + \hat{w}
\tag{5}$$
where $\hat{w} \sim \mathcal N(0, N\sigma^2)$. Thus, the measured
cross-correlation function is now a Gaussian random variable
with mean $Nh[n]$ and variance $N\sigma^2$.  The estimator for
$h[n]$ is thus just $N^{-1}$ times the measured value of the
cross-correlation function, and its variance is thus
$\displaystyle \frac{\sigma^2}{N}$, This matches up perfectly with
Kay's result
$$\operatorname{var}(\hat{h}[n]) = \frac{\sigma_{w}^2}{Nr_{u,u}[0]}$$
since we have assumed that $u[n] \in \{+1, -1\}$ and hence
$r_{u,u}[0] = 1$: all the $u[n]$, regarded as random instead of
pseudorandom, have variance $1$.